Level I Math
Black Line Masters
(Part 1)
NSSAL
(Draft)
C. David Pilmer
2012
(Last Updated: May, 2014)
This resource is the intellectual property of the Adult Education Division of the Nova Scotia
Department of Labour and Advanced Education.
The following are permitted to use and reproduce this resource for classroom purposes.
• Nova Scotia instructors delivering the Nova Scotia Adult Learning Program
• Canadian public school teachers delivering public school curriculum
• Canadian non-profit tuition-free adult basic education programs
The following are not permitted to use or reproduce this resource without the written
authorization of the Adult Education Division of the Nova Scotia Department of Labour and
Advanced Education.
• Upgrading programs at post-secondary institutions (exception: NSCC's ACC program)
• Core programs at post-secondary institutions (exception: NSCC)
• Public or private schools outside of Canada
• Basic adult education programs outside of Canada
Individuals, not including teachers or instructors, are permitted to use this resource for their own
learning. They are not permitted to make multiple copies of the resource for distribution. Nor
are they permitted to use this resource under the direction of a teacher or instructor at a learning
institution.
Acknowledgments
The Adult Education Division would like to thank the following ALP instructors for piloting this
resource and offering suggestions during its development.
Andre Davey (Metroworks)
Shannon Davis (YCLA)
Andrea Fitzgerald (CLANS)
Elizabeth Grzesik (EHALA)
Cheryl Mycroft (GALA)
Joyce Power (Metroworks)
David Sweeny (YCLA)
Kirsteen Thomson (Can-U)
Table of Contents
Difficulty
Introduction …………………………………………………………
iv
Number Magnitude (Whole Numbers)………..……..………………
Comparing Quantities …………………………………………..
Expanded Form (A and B) ………………………………………
Write the Number (A to C) ………………………………………
Write the Number (D and E) ……………………………………
Place Value ………………………………………………………
Before, After, or Between (A and B) ……………………………
Closer To, and Odd or Even (A and B) …………………………
Find the Odd or Even Numbers (A and B) ……………………..
Whole Numbers and Number Lines (A to C) …………………..
Order the Numbers ………………………………………………
Give an Example ………………………………………………..
Closer To (A and B) …………………..…………………………
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Operations with Whole Numbers ………………………………….…
Connect Four Addition Game (A to C) ………………..………..
What Does the Star Represent? (Addition) ………………………
Adding Multi-Digit Numbers ……………………………………
Connect Four Subtraction Game (A to D) ……….……………..
Subtraction Search ………………………..……………………...
Subtracting Multi-Digit Numbers ………………………………..
Multiplication Models ……………………………………………
Multiplication Array Game ………………………………………
Connect Four Multiplication Game (A to E) …………………….
Connect Four Multiples Game (A to C) …………………………
Capture the Flag Multiplication Game ………………………….
Multiples Puzzle …………………………………………………
Factors ……………………………………………………………
Factor Flowers ……………………………………………………
Random Multiplication Facts Quizzes …………………………..
What Does the Star Represent? (Multiplication) ………………..
Multiplying on Your Hands …………………...…………………
Put the Number in the Right Box …………………………….....
Investigation: Multiplying by Multiples of 10, 100, and 1000 ….
Multiplying by Multiples of 10, 100, and 1000 ………………….
Multiplying Two Digit Numbers, Part 1 (Expanded Form) ……..
Multiplying Two Digit Numbers, Part 2 (Lattice Method) ………
Multiplying Multi-Digit Numbers ………………………………..
Connect Four Division Game ………………………..…………..
Division Search …………………………………………………..
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or
Puzzle
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90
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107
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Multiple Operations (Whole Numbers) …………………..………….
Express the Number in Multiple Ways …………………………..
Find the Center …………………………………………………..
Name the Preceding or Next …………………………..…………
One of these Things is Not Like the Others ……………………..
Fact Family Puzzle ………………………………....…………….
Provide the Other Members of the Fact Family ………………….
What Number Does the Star Represent? …………………………
Pathways ……………………………………………..…………..
Two of These Boxes Just Don't Belong (A and B) ……………….
Equivalent …………………………………………………………
Greater Than, Less Than or Equal To; Whole Number Operations..
Find the Digit Based on the Reasonable Estimate ………………
Venn Diagrams and Whole Numbers ……………………………
Whole Number Crossword Puzzle (A to D) …………………….
KenKen Puzzles (A and B) ………………………………………
KenKen Puzzles (C and D) ………………………………………
KenKen Puzzles (E and F) ……………………………………….
Find the Two Numbers …………………………………………..
Which Combination of Numbers Works? ……………………….
Magic Squares ……………………………………………….…..
Addition Pyramids ………………………………………………
Row Factors and Column Factors ……………………………….
Letter and Number Sentences ……………………………………
Math Logic Puzzles ………………………………………………
Number Sentences (A) …………………………………………..
Number Sentences (B) ………………………………………….
Order of Operations (A) …………………………………………
Order of Operations (B) …………………………………………
Order of Operations (C) …………………………………………
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Patterns ……………………………………………………………….
What's the Pattern? (A) …………………………………………..
What's the Pattern? (B) …………………………………………..
Toothpick Patterns ……………………………………………….
Create the Pattern (A and B) ……………………………………..
Number Patterns (A and B) ………………………………………
Row, Column, and Diagonal Pattern …………………………….
What's the Relationship? …………………………………………
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168
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Divisibility Chart …………………………………………………
More Divisibility (A and B) ……………………………………..
Divisibility or Prime Connect Four Game ………………………
Division with Remainders ……………………………………….
Long Division (Partial Quotient Method) ……………………….
Prime Factorization ………………………………………………
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Input Output (A to D) ……………………………………………
Filling or Draining ……………………………………………….
Travelling Towards or Away From Home ………………………
Weight of the Water ……………………………………………..
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Word Problems ………………………………………………………
Describing the Relationships with Words ……………………….
List the Numbers Based on the Written Description ……………
Addition and Subtraction Crossword ……………………………
Multiplication and Division Crossword …………………………
Operations Crossword ……………………………………………
Word Sentence to Number Sentence to Answer (A to C) ………
What are the Possibilities? (A) ………………………………….
What are the Possibilities? (B) ………………………………….
Describing Relationships with Words and Numbers ……………
More Describing Relationships with Words and Numbers ……..
Recognizing the Important Information …………………………
Does It Make Sense? …………………………………………….
Insert Your Own Numbers and Words ………………………….
Complete the Statement: Addition and Subtraction ……………..
Complete the Statement: Multiplication and Division …………..
Complete the Statement (A and B) ………………………………
Complete the Statement (C) ……………………………………..
Complete the Statement (D) ……………………………………..
Not Enough Information is Provided ……………………………
Word Problems with Too Much Information ……………………
Create Your Own Math Statement ………………………………
Word Problems (A and B) ………………………………………
Same Numbers, Similar Context, Different Math (A) ………….
Same Numbers, Similar Context, Different Math (B) ………….
More than One Question …………………………………………
Food Chart (A and B) ……………………………………………
Keeping Track of the New Stock (A and B) ……………………
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Answers ………………………………………………………………
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Introduction
The concepts covered in Level I Math fit into one of the following five categories.
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Number and Operations (limited to whole numbers)
Patterns and Relations
Statistics and Probability
Shape, Space, and Measurement
Consumer Math
The specific outcomes aligned with each of these categories can be found in the ALP Level I
Math Curriculum Guide.
Over the years, Community-Learning Organizations have collected, and the Adult Education
Division has supplied, a variety of print resources used in the delivery of Level I Math. Many of
those resources can still be used with this new curriculum but we emphasize that there is a much
greater emphasis on mathematical understanding and multiple representations of concepts in this
new program. Although we want our learners to develop a level of automaticity as it pertains to
operations with whole numbers, we do not want this math course, or any other ALP math course,
to focus primarily on the mastery of skills. Unfortunately many of the "traditional" textbooks
used in adult basic education programs do have this as their primary focus. For this reason, the
Adult Education division would like all instructors to use the following ALP resources in the
delivery of Level I Math, and to supplement that material with the more traditional resources
they have collected over the years.
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Level I Math Black Line Masters Part 1 and Part 2
Mental Math
Customized Practice
Number Sense
These resources include activities, exercises, investigations, and games that encourage
understanding and thinking, rather than solely focussing on the mastery of algorithms. Learners
are ultimately better served when mathematical concepts are examined and taught in this matter.
We do not expect all Level I learners to complete all the worksheets or activities in the resources
above, rather instructors will use their professional judgement to choose the items that are most
appropriate for their individual learners. By supplying these materials the LAE is providing a
greater variety of education tools for ALP instructors; the instructors have to decide what tools
are best suited for their learners, at what times, and in what sequence. For example, let's
consider multiplication of two multi-digit numbers. Most instructors are familiar with the
traditional algorithm for such multiplication, but some instructors are unfamiliar with
multiplication of multi-digit numbers using the expanded forms of the numbers and/or lattice
multiplication. These latter two techniques are found in this resource. Does that mean that all
learners need to know all three methods? Definitely not; chose the technique that works best for
your learner.
Please do not view these specific resources as textbooks. Although within sections, the activity
sheets are generally arranged from easiest to hardest, a seamless flow from one activity to the
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next was not created. These booklets are merely a collection of black line masters to be used as
the instructor sees fit.
All of these materials are available at the
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NSSAL site (http://gonssal.ca/documents/NSSALdevelopedresources.pdf)
NSSAL Practitioners Website (http://instructors.gonssal.ca/login)
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Number Magnitude
(Whole Numbers)
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Comparing Quantities
Insert the appropriate word or phrase. You can choose from "more," "fewer" or "the same
number of."
1. Most families have:
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___________________________ cars
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___________________________ windows
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___________________________ pillows
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___________________________ toothbrushes
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___________________________ forks
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___________________________ running shoes
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___________________________ dogs
as compared to hamsters
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___________________________ fingers
as compared to toes
as compared to bicycles
as compared to doors
as compared to beds
as compared to hair dryers
as compared to spoons
as compared to socks
2. Most cars have:
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___________________________ headlights as compared to bumper stickers
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___________________________ headrests as compared to seatbelts
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___________________________ steering wheels as compared to windows
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___________________________ tires as compared to rear view mirrors
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___________________________ floor mats as compared to horns
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___________________________ gas caps as compared to tail lights
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Expanded Form (A)
1. Write the multi-digit number in its expanded form. Two examples have been done for you.
Number
Expanded Form
Number
Expanded Form
e.g.
396
300 + 90 + 6
e.g.
2056
2000 + 50 + 6
(a)
42
(b)
694
(c)
3985
(d)
569
(e)
78
(f)
4281
(g)
867
(h)
31
(i)
6497
(j)
528
(k)
826
(l)
5923
(m)
59
(n)
3045
(o)
4808
(p)
703
(q)
6420
(r)
5099
(s)
810
(t)
9603
2. Given the expanded form, write the multi-digit number. The last eight have had their
expanded forms scrambled.
Expanded Form
Number
Expanded Form
(a)
50 + 8
(b)
600 + 20 + 9
(c)
5000 + 800 + 70 + 4
(d)
200 + 80 + 6
(e)
8000 + 300 + 80 + 7
(f)
30 + 5
(g)
400 + 90 + 3
(h)
700 + 20 + 1
(i)
600 + 10 + 9
(j)
7000 + 80 + 2
(k)
800 + 50
(l)
500 + 3
(m)
1000 + 700 + 8
(n)
6000 + 80 + 9
(o)
5000 + 300 + 40
(p)
8000 + 600 + 50 + 7
(q)
8 + 50
(r)
50 + 300 + 2
(s)
200 + 50 + 1000 + 9
(t)
9 + 600 + 40
(u)
8000 + 7 + 300 + 60
(v)
500 + 80 + 7000
(w)
4 + 6000 + 80
(x)
8000 + 5 + 30
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Expanded Form (B)
1. Write the multi-digit number in its expanded form. An example has been done for you.
Number
Expanded Form
e.g.
132 794
100 000 + 30 000 + 2000 + 700 + 90 + 4
(a)
54 982
(b)
5685
(c)
746 173
(d)
27 959
(e)
306 781
(f)
43 908
(g)
372 080
(h)
50 736
(i)
270 480
(j)
908 704
2. Given the expanded form, write the multi-digit number. The last four have had their
expanded forms scrambled.
Expanded Form
Number
(a)
60 000 + 7000 + 500 + 90 + 1
(b)
500 000 + 60 000 + 2000 + 700 + 10 + 3
(c)
40 000 + 1000 + 200 + 70 + 8
(d)
800 000 + 50 000 + 300 + 70 + 4
(e)
600 000 + 2000 + 800 + 90 + 2
(f)
90 000 + 5000 + 40 + 3
(g)
500 000 + 30 000 + 900 + 5
(h)
600 000 + 80 000 + 4000 + 700 + 10
(i)
6 + 700 + 20 000 + 50 + 9000
(j)
30 000 + 5000 + 700 000 + 60 + 200 + 9
(k)
400 + 200 000 + 8 + 90 000 + 60
(l)
900 000 + 5 + 7000 + 30 + 100
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Write the Number (A)
1. Write the number that has been described using words.
e.g.
Word Description
fifty-seven
(a)
Number
57
e.g.
Word Description
nineteen
eighty-two
(b)
forty-nine
(c)
sixteen
(d)
ten
(e)
forty-six
(f)
twenty-seven
(g)
seventy-four
(h)
thirteen
(i)
fifty-six
(j)
eleven
(k)
thirty-eight
(l)
twenty-three
(m)
eight
(n)
ninety-nine
(o)
seventeen
(p)
eighty-three
(q)
twelve
(r)
fifty-two
(s)
ninety
(t)
thirty-one
(u)
seven
(v)
sixty-three
(w)
fifteen
(x)
sixty-eight
Number
19
2. Write out the number using words.
Number
e.g.
72
(a)
59
(b)
42
(c)
18
(d)
37
(e)
61
(f)
95
(g)
21
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Word Description
seventy-two
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Write the Number (B)
1. Write the number that has been described using words.
e.g.
Word Description
three hundred forty-nine
(a)
Number
349
e.g.
Word Description
six hundred eight
nine hundred thirty-two
(b)
two hundred forty-six
(c)
seven hundred twelve
(d)
three hundred sixty
(e)
seventy-nine
(f)
six hundred twenty-one
(g)
one hundred seven
(h)
five hundred eighty-nine
(i)
four hundred ninety
(j)
forty-two
(k)
eight hundred eleven
(l)
two hundred seventy-six
(m)
seven hundred two
(n)
three hundred nineteen
(o)
twelve
(p)
five hundred thirty-one
(q)
six hundred seventy
(r)
nine
(s)
one hundred eighty-six
(t)
nine hundred
(u)
two hundred sixteen
(v)
sixty-five
(w)
seven hundred twenty
(x)
four hundred sixty-two
Number
608
2. Write out the number using words.
Number
e.g.
452
(a)
578
(b)
352
(c)
79
(d)
217
(e)
906
(f)
740
(g)
541
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Word Description
four hundred fifty-two
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Write the Number (C)
1. Write the number that has been described using words.
Word Description
e.g. three thousand, two hundred ninety-one
e.g. six thousand, fourteen
Number
3 291
6 014
(a)
eight thousand, three hundred twenty-two
(b)
four thousand, six hundred eighty-three
(c)
seven thousand, five hundred thirteen
(d)
four hundred eleven
(e)
nine thousand, five hundred twelve
(f)
three thousand, four hundred twenty-nine
(g)
two thousand, nine hundred fifty
(h)
one thousand, seventy-eight
(i)
two hundred seven
(j)
five thousand, nine hundred eighty-three
(k)
six thousand, eight hundred seven
(l)
nine thousand, forty-six
(m) five hundred ninety-seven
(n)
eight thousand, two hundred seventy-four
(o)
fifteen
(p)
four thousand, nine hundred twenty-eight
(q)
one thousand, three hundred eighteen
(r)
thirty-eight
(s)
three thousand, seventy-six
(t)
six thousand, nine hundred
(u)
seven thousand, three hundred eight
(v)
seven hundred sixteen
(w)
nine thousand, five hundred seventy
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2. Write out the number using words.
Number
e.g.
3 612
(a)
4 567
(b)
2 351
(c)
547
(d)
9 189
(e)
6 911
(f)
63
(g)
8 063
(h)
1 904
(i)
708
(j)
7 850
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Word Description
three thousand, six hundred twelve
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Write the Number (D)
1. Write the number that has been described using words.
Word Description
e.g. twenty-three thousand, eight hundred two
Number
23 802
e.g. one hundred fifty thousand, six hundred twenty-three
150 623
(a)
fifty-six thousand, seven hundred forty-six
(b)
two hundred thirty-nine thousand, one hundred fifteen
(c)
forty thousand, three hundred seventy-one
(d)
three thousand six hundred five
(e)
five hundred twenty-three thousand, ninety
(f)
sixty thousand, two hundred eight
(g)
ninety-three
(h)
three hundred five thousand, sixty-eight
(i)
nine hundred one
(j)
thirteen thousand, seven hundred fifteen
(k)
five hundred thirty-six thousand
(l)
four hundred seven thousand, fifty-two
(m) nine thousand, four hundred sixty
(n)
fifty thousand, six hundred nine
(o)
seven hundred thirteen thousand, three hundred ninety-one
(p)
twelve thousand, ninety-six
(q)
six hundred thirty
(r)
two hundred thousand, five hundred sixteen
(s)
eighty thousand, five hundred seventy
(t)
ten thousand, four hundred
(u)
three hundred six thousand, one hundred eleven
(v)
nine hundred fifteen
(w)
eight hundred seven thousand, two
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2. Write out the number using words.
Number
Word Description
e.g.
254 703
two hundred fifty-four thousand, seven hundred three
(a)
34 781
(b)
245 359
(c)
780
(d)
12 692
(e)
304 562
(f)
7 023
(g)
70 650
(h)
634 904
(i)
53 011
(j)
940 060
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Write the Number (E)
1. Write the number that has been described using words.
Word Description
e.g. six million, fifty-three thousand, eight hundred seven
Number
6 053 807
e.g. twenty-four million, one hundred thousand, fifty
24 100 050
(a)
ten million, ninety-six thousand, eight hundred two
(b)
one million, two hundred five thousand, sixteen
(c)
seven hundred thirty-four million
(d)
eighty million, five hundred twenty-nine thousand, seventy
(e)
four hundred twelve million, six hundred seventy thousand
(f)
eighty-five million, fifteen thousand, nine hundred
(g)
ninety-seven thousand, eight hundred twelve
(h)
six hundred twenty-seven million, seven hundred fifty
(i)
forty million, sixty-five thousand, ninety
(j)
five hundred six million, seventy thousand, nine hundred
(k)
three hundred two thousand, twenty-eight
(l)
eleven million, three thousand, forty-seven
(m) nine million, three hundred thirteen thousand, four
(n)
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2. Write out the number using words.
Number
Word Description
e.g.
8 290 043
Eight million, two hundred ninety thousand, forty-three
(a)
7 305 411
(b)
23 078 600
(c)
328 109 000
(d)
13 436 500
(e)
6 009 740
(f)
498 315
(g)
540 679 020
(h)
95 811 002
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Place Value
Complete each of the following.
1.
3.
5.
7.
9.
11.
For 2 345, what number is in the:
2.
For 65 721, what number is in the:
(a) tens' place?
_____
(a) ten thousands' place?
_____
(b) thousands' place?
_____
(b) hundreds' place?
_____
(c) ones' place?
_____
(c) thousands' place?
_____
For 7 890, what number is in the:
4.
For 48 156, what number is in the:
(a) ones' place?
_____
(a) hundreds' place?
_____
(b) thousands' place?
_____
(b) ten thousands' place?
_____
(c) hundreds' place?
_____
(c) tens' place?
_____
For 423 719, what number is in the:
6.
For 120 693, what number is in the:
(a) ten thousands' place?
_____
(a) tens' place?
_____
(b) hundred thousands' place?
_____
(b) thousands' place?
_____
(c) ones' place?
_____
(c) hundred thousands' place? _____
For 65 807, what number is in the:
8.
For 835 270, what number is in the:
(a) thousands' place?
_____
(a) ten thousands' place?
_____
(b) hundreds' place?
_____
(b) ones' place?
_____
(c) tens' place?
_____
(c) hundred thousands' place? _____
For 8 143 625, what number is in the:
10.
For 7 015 428, what number is in the:
(a) hundreds' place?
_____
(a) tens' place?
_____
(b) hundred thousands' place?
_____
(b) millions' place?
_____
(c) millions' place?
_____
(c) ten thousands' place?
_____
For 6 324 857, what number is in the:
12.
For 5 391 207, what number is in the:
(a) tens' place?
_____
(a) millions' place?
_____
(b) ten thousands' place?
_____
(b) hundreds' place?
_____
(c) hundred thousands' place?
_____
(c) thousands' place?
_____
NSSAL
©2012
13
Draft
C. D. Pilmer
Before, After, or Between (A)
Number
Your Answers:
Word Description
e.g. What number is before 8?
7
seven
e.g. What number is between 12 and 14?
13
thirteen
e.g. What number is after 29?
30
thirty
1.
What number is after 6?
2.
What number is before 11?
3.
What number is between 18 and 20?
4.
What number is after 15?
5.
What number is between 23 and 25?
6.
What number is before 27?
7.
What number is between 28 and 30?
8.
What number is after 34?
9.
What number is before 37?
10.
What number is after 11?
11.
What number is between 46 and 48?
12.
What number is after 59?
13.
What number is before 70?
14.
What number is between 81 and 83?
15.
What number is after 42?
16.
What number is between 90 and 92?
17.
What number is before 77?
18.
What number is after 99?
19.
What number is between 73 and 74?
20.
What number is before 80?
NSSAL
©2012
14
Draft
C. D. Pilmer
Before, After, or Between (B)
Number
Your Answers:
Word Description
e.g. What number is before 120?
119
one hundred nineteen
e.g. What number is between 456 and 458?
457
four hundred fifty-seven
e.g. What number is after 599?
600
six hundred
1.
What number is after 325?
2.
What number is before 421?
3.
What number is between 188 and 190?
4.
What number is after 239?
5.
What number is between 356 and 358?
6.
What number is before 650?
7.
What number is between 286 and 288?
8.
What number is before 700?
9.
What number is before 998?
10.
What number is after 437?
11.
What number is between 638 and 640?
12.
What number is after 399?
13.
What number is before 900?
14.
What number is between 513 and 515?
15.
What number is after 661?
16.
What number is before 712?
17.
What number is between 600 and 602?
18.
What number is after 807?
19.
What number is after 999?
20.
What number is between 499 and 501?
NSSAL
©2012
15
Draft
C. D. Pilmer
Closer to, and Odd or Even (A)
For each of the following, write the number based on the description, indicate whether it is closer
to 0 or 100, and state whether it is an odd or even number.
e.g.
Word Description
fifty-nine
e.g.
thirty-six
(a)
eighteen
(b)
seventy-six
(c)
eighty-three
(d)
forty
(e)
twenty-four
(f)
thirty-nine
(g)
ninety-three
(h)
sixty-five
(i)
forty-six
(j)
seventy-one
(k)
eleven
(l)
thirty-eight
(m)
forty-nine
(n)
seventy-two
(o)
ninety-six
(p)
twenty-seven
(q)
sixteen
(r)
twelve
(s)
forty-four
(t)
eighty-five
(u)
fifty-eight
(v)
thirty-seven
NSSAL
©2012
Number
59
Closer to 0 or 100
100
Odd or Even
odd
36
0
even
16
Draft
C. D. Pilmer
Closer to, and Odd or Even (B)
For each of the following, write the number based on the description, indicate whether it is closer
to 0 or 100, and state whether it is an odd or even number.
e.g.
Word Description
three hundred seventy-two
e.g.
eight hundred eleven
(a)
two hundred forty-nine
(b)
six hundred twenty-three
(c)
seven hundred ninety-four
(d)
one hundred eighty-six
(e)
five hundred twelve
(f)
three hundred seven
(g)
four hundred sixty
(h)
ninety-nine
(i)
two hundred seventy-six
(j)
five hundred forty-seven
(k)
one hundred eighty-four
(l)
nine hundred eight
(m)
fifty-three
(n)
three hundred fifty
(o)
nine hundred ninety-six
(p)
six hundred forty-five
(q)
one thousand, thirty-one
(r)
two hundred seventy-seven
(s)
three thousand, ten
(t)
seven hundred sixty-nine
(u)
two thousand, three hundred
(v)
eight hundred fifteen
NSSAL
©2012
Number
372
Closer to 0 or 1000
0
Odd or Even
even
811
1000
odd
17
Draft
C. D. Pilmer
Find the Odd or Even Numbers (A)
(a) State all the even numbers between 6 and 13.
________________________________
(b) State all the odd numbers between 5 and 16.
________________________________
(c) State all the even numbers between 11 and 18.
________________________________
(d) State all the odd numbers between 9 and 19.
________________________________
(e) State all the even numbers between 18 and 26.
________________________________
(f) State all the odd numbers between 14 and 23.
________________________________
(g) State all the even numbers between 36 and 47.
________________________________
(h) State all the odd numbers between 52 and 62.
________________________________
(i) State all the even numbers between 27 and 39.
________________________________
(j) State all the odd numbers between 88 and 97.
________________________________
(k) State all the even numbers between 38 and 51.
________________________________
(l) State all the odd numbers between 46 and 56.
________________________________
(m) State all the even numbers between 72 and 81.
________________________________
(n) State all the odd numbers between 66 and 81.
________________________________
(o) State all the even numbers between 54 and 68.
________________________________
(p) State all the odd numbers between 86 and 100.
________________________________
(q) State all the even numbers between 93 and 100.
________________________________
How can you tell if a number is even?
How can you tell if a number is odd?
NSSAL
©2012
18
Draft
C. D. Pilmer
Find the Odd or Even Numbers (B)
(a) State all the even numbers between 128 and 137.
________________________________
(b) State all the odd numbers between 92 and 105.
________________________________
(c) State all the even numbers between 215 and 225.
________________________________
(d) State all the odd numbers between 67 and 83.
________________________________
(e) State all the even numbers between 459 and 471.
________________________________
(f) State all the odd numbers between 324 and 334.
________________________________
(g) State all the even numbers between 583 and 591.
________________________________
(h) State all the odd numbers between 796 and 804.
________________________________
(i) State all the even numbers between 697 and 708.
________________________________
(j) State all the odd numbers between 197 and 211.
________________________________
(k) State all the even numbers between 996 and 1005. ________________________________
(l) State all the odd numbers between 2395 and 2403. ________________________________
(m) State all the even numbers between 6544 and 6552. ________________________________
(n) State all the odd numbers between 4992 and 5002. ________________________________
(o) State all the even numbers between 1683 and 1692. ________________________________
(p) State all the odd numbers between 3919 and 3930. ________________________________
(q) State all the even numbers between 5989 and 5997. ________________________________
(r) State all the odd numbers between 7688 and 7696. ________________________________
(s) State all the even numbers between 8028 and 8035. ________________________________
NSSAL
©2012
19
Draft
C. D. Pilmer
Whole Numbers and Number Lines (A)
1. Place each number at its approximate location on the number line.
7
18
5
21
12
0
20
2. Place each number at its approximate location on the number line.
62
30
24
11
56
0
60
3. Place each number at its approximate location on the number line.
14
18
36
9
22
10
30
4. Place each number at its approximate location on the number line.
32
27
37
21
17
20
40
5. Place each number at its approximate location on the number line.
62
20
40
95
48
0
NSSAL
©2012
100
20
Draft
C. D. Pilmer
Whole Numbers and Number Lines (B)
1. Place each number at its approximate location on the number line.
72
183
5
148
96
0
200
2. Place each number at its approximate location on the number line.
99
270
304
213
52
0
300
3. Place each number at its approximate location on the number line.
215
240
369
285
196
200
400
4. Place each number at its approximate location on the number line.
321
362
329
348
357
300
360
5. Place each number at its approximate location on the number line.
192
31
905
699
560
0
NSSAL
©2012
1000
21
Draft
C. D. Pilmer
Order the Numbers
1. In each case, order the numbers from smallest to largest. No work needs to be shown.
(a) 17, 32, 9, 2, 39
_____, _____, _____, _____, _____
(b) 87, 73, 29, 32, 7
_____, _____, _____, _____, _____
(c) 54, 19, 56, 49, 91
_____, _____, _____, _____, _____
(d) 28, 70, 37, 8, 74, 35
_____, _____, _____, _____, _____, _____
(e) 61, 12, 85, 65, 4, 39
_____, _____, _____, _____, _____, _____
(f) 43, 96, 3, 25, 47, 28
_____, _____, _____, _____, _____, _____
(g) 86, 27, 15, 37, 80, 8
_____, _____, _____, _____, _____, _____
(h) 19, 54, 16, 49, 67, 9
_____, _____, _____, _____, _____, _____
(i) 34, 26, 21, 12, 49, 6, 41
_____, _____, _____, _____, _____, _____, _____
(j) 73, 52, 18, 61, 23, 78, 30
_____, _____, _____, _____, _____, _____, _____
(k) 46, 37, 93, 51, 33, 29, 72
_____, _____, _____, _____, _____, _____, _____
(l) 86, 67, 27, 91, 25, 62, 45
_____, _____, _____, _____, _____, _____, _____
(m) 65, 78, 13, 35, 82, 32, 58
_____, _____, _____, _____, _____, _____, _____
(n) 12, 95, 46, 6, 60, 42, 98
_____, _____, _____, _____, _____, _____, _____
2. Put in numbers in order from smallest to largest. These numbers will form a sequence where
the terms (i.e. numbers) have a common difference. Find the next term (i.e. number) in the
sequence.
(a) 27, 12, 7, 17, 22
_____, _____, _____, _____, _____, _____
(b) 17, 20, 14, 26, 23
_____, _____, _____, _____, _____, _____
(c) 47, 31, 39, 39, 27, 43
_____, _____, _____, _____, _____, _____, _____
(d) 52, 56, 46, 58, 54, 48
_____, _____, _____, _____, _____, _____, _____
3. Create your own sequence where the terms have a common difference of 3.
NSSAL
©2012
22
Draft
C. D. Pilmer
Give an Example
e.g. Give an example where the number 365 is used to represent
something.
Answer:
There are 365 days in one year.
1. Give an example where the number 10 is used to represent something.
2. Give an example where the number 12 is used to represent something.
3. Give an example where the number 18 is used to represent something.
4. Give an example where the number 24 is used to represent something.
5. Give an example where the number 25 is used to represent something.
6. Give an example where the number 30 is used to represent something.
7. Give an example where the number 50 is used to represent something.
8. Give an example where the number 60 is used to represent something.
9. Give an example where the number 100 is used to represent something.
10. Give an example where the number 1000 is used to represent something.
NSSAL
©2012
23
Draft
C. D. Pilmer
Closer To (A)
Three whole numbers are provided. You are asked to determine whether the first number is
closer to the second number or closer to the third number. Circle the correct answer. Two
examples have been provided.
e.g.
Is 8 closer to 5 or 10?
e.g.
Answer: 10
Is 43 closer to 40 or 50?
Answer: 40
1.
Is 6 closer to 5 or 8?
2.
Is 3 closer to 1 or 7?
3.
Is 4 closer to 0 or 6?
4.
Is 7 closer to 4 or 9?
5.
Is 5 closer to 1 or 8?
6.
Is 8 closer to 4 or 10?
7.
Is 6 closer to 3 or 10?
8.
Is 2 closer to 0 or 3?
9.
Is 9 closer to 7 or 12?
10.
Is 8 closer to 3 or 12?
11.
Is 6 closer to 0 or 11?
12.
Is 7 closer to 5 or 11?
13.
Is 9 closer to 6 or 13?
14.
Is 10 closer to 7 or 15?
15.
Is 11 closer to 10 or 14?
16.
Is 10 closer to 6 or 12?
17.
Is 15 closer to 13 or 19?
18.
Is 13 closer to 10 or 15?
19.
Is 16 closer to 14 or 20?
20.
Is 12 closer to 8 or 15?
21.
Is 19 closer to 15 or 21?
22.
Is 18 closer to 16 or 22?
23.
Is 24 closer to 20 or 30?
24.
Is 27 closer to 20 or 30?
25.
Is 39 closer to 30 or 40?
26.
Is 46 closer to 40 or 50?
27.
Is 73 closer to 70 or 80?
28.
Is 94 closer to 90 or 100?
29.
Is 28 closer to 20 or 30?
30.
Is 60 closer to 0 or 100?
31.
Is 40 closer to 0 or 100?
32.
Is 70 closer to 0 or 100?
33.
Is 30 closer to 20 or 50?
34.
Is 70 closer to 50 or 100?
NSSAL
©2012
24
Draft
C. D. Pilmer
Closer To (B)
Three whole numbers are provided. You are asked to determine whether the first number is
closer to the second number or closer to the third number. Circle the correct answer. Two
examples have been provided.
e.g.
Is 34 closer to 30 or 40?
e.g.
Answer: 30
Is 459 closer to 400 or 500?
Answer: 500
1.
Is 36 closer to 30 or 40?
2.
Is 44 closer to 40 or 50?
3.
Is 67 closer to 60 or 70?
4.
Is 50 closer to 20 or 60?
5.
Is 60 closer to 10 or 90?
6.
Is 80 closer to 70 or 100?
7.
Is 100 closer to 80 or 150?
8.
Is 150 closer to 130 or 160?
9.
Is 270 closer to 250 or 300?
10.
Is 420 closer to 400 or 500?
11.
Is 200 closer to 0 or 300?
12.
Is 500 closer to 400 or 800?
13.
Is 700 closer to 600 or 750?
14.
Is 600 closer to 550 or 700?
15.
Is 640 closer to 600 or 700?
16.
Is 870 closer to 800 or 900?
17.
Is 81 closer to 20 or 100?
18.
Is 67 closer to 30 or 80?
19.
Is 58 closer to 0 or 90?
20.
Is 37 closer to 0 or 100?
21.
Is 99 closer to 0 or 200?
22.
Is 230 closer to 100 or 300?
23.
Is 341 closer to 300 or 350?
24.
Is 789 closer to 750 or 800?
25.
Is 699 closer to 600 or 750?
26.
Is 219 closer to 100 or 250?
27.
Is 224 closer to 200 or 300?
28.
Is 547 closer to 500 or 550?
29.
Is 839 closer to 800 or 900?
30.
Is 658 closer to 650 or 700?
31.
Is 2399 closer to 2000 or 3000?
32.
Is 1837 closer to 1000 or 2000?
33.
Is 5643 closer to 5000 or 6000?
34.
Is 2845 closer to 2000 or 4000?
NSSAL
©2012
25
Draft
C. D. Pilmer
Operations with Whole
Numbers
NSSAL
©2012
26
Draft
C. D. Pilmer
Connect Four Addition Game (A)
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place two paper clips on two numbers on the Addend Strip whose sum is that desired square.
Once they have chosen the two numbers, they can capture one square with that appropriate
product. They either mark the square with an X or place a colored counter on the square.
There may be other squares with that same product but only one square can be captured at a
time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips on the Addend Strip. They then mark the square with that product using a O or a
different colored marker. If a player cannot move a single paperclip to capture a square, a
paperclip must still be moved on the addend strip in order to ensure that the game can
continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
4
0
8
2
3
5
2
6
7
6
0
3
1
4
5
4
8
2
7
3
6
0
4
5
6
4
1
7
3
6
2
5
8
3
5
1
Addend Strip:
0
NSSAL
©2012
1
2
3
4
27
Draft
C. D. Pilmer
Connect Four Addition Game (B)
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place two paper clips on two numbers on the Addend Strip whose sum is that desired square.
Once they have chosen the two numbers, they can capture one square with that appropriate
product. They either mark the square with an X or place a colored counter on the square.
There may be other squares with that same product but only one square can be captured at a
time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips on the Addend Strip. They then mark the square with that product using a O or a
different colored marker. If a player cannot move a single paperclip to capture a square, a
paperclip must still be moved on the addend strip in order to ensure that the game can
continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
10
12
7
13
8
11
9
11
10
12
6
13
6
14
9
11
10
9
8
11
12
7
14
11
13
10
8
6
9
10
9
7
14
10
12
8
6
7
Addend Strip:
3
NSSAL
©2012
4
5
28
Draft
C. D. Pilmer
Connect Four Addition Game (C)
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place two paper clips on two numbers on the Addend Strip whose sum is that desired square.
Once they have chosen the two numbers, they can capture one square with that appropriate
product. They either mark the square with an X or place a colored counter on the square.
There may be other squares with that same product but only one square can be captured at a
time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips on the Addend Strip. They then mark the square with that product using a O or a
different colored marker. If a player cannot move a single paperclip to capture a square, a
paperclip must still be moved on the addend strip in order to ensure that the game can
continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
12
11
18
14
16
14
15
14
13
16
12
13
16
12
15
10
14
17
15
17
14
18
15
13
13
10
13
16
11
18
15
12
11
17
14
10
Addend Strip:
5
NSSAL
©2012
6
7
8
9
29
Draft
C. D. Pilmer
What Number Does the Star Represent? (Addition)
Example
For each question, determine what number is represented by the star?
(a) 7 +  = 10
(b)  +  = 18
(c)  + 3 = 8
(d)  +  +  = 12
Answers:
(a) What number when added to 7 gives you 10? The answer is 3. ( = 3)
(b) What number when added to itself gives you 18? The answer is 9. ( = 9)
(c) What number when added to 3 gives you 8? The answer is 5. ( = 5)
(d) What number when added to together three times gives you 12? The answer is 4. ( = 4)
Questions
In each case determine the number that is represented by the star? No work needs to be shown.
(a)  + 6 = 10
 = _____
(b)  +  = 8
 = _____
(c)  +  +  = 6
 = _____
(d) 11 +  = 18
 = _____
(e) 2 + 8 = 
 = _____
(f) 3 +  = 10
 = _____
(g)  + 5 = 14
 = _____
(h)  +  = 16
 = _____
(i) 7 +  = 15
 = _____
(j)  +  +  = 21
 = _____
(k) 3 +  + 2 = 10
 = _____
(l) 6 + 7 = 
 = _____
(m)  + 5 + 4 = 11
 = _____
(n)  +  +  = 15
 = _____
(o) 2 +  = 14
 = _____
(p)  +  = 6
 = _____
(q)  +  + 2 = 4
 = _____
(r)  + 7 = 18
 = _____
(s) 6 +  + 1 = 10
 = _____
(t)  + 11 = 19
 = _____
(u)  +  +  = 30
 = _____
(v) 5 +  +  = 11
 = _____
(w)  +  = 14
 = _____
(x)  = 9 + 4
 = _____
(y) 9 +  = 10
 = _____
(z)  +  +  = 9
 = _____
NSSAL
©2012
30
Draft
C. D. Pilmer
Adding Multi-Digit Numbers
To add multi-digit whole numbers, start by stacking the numbers vertically such that
corresponding place values line up (e.g. units with units, tens with tens) and add from right to
left. If the sum in any corresponding place value is 10 or greater, we regroup (i.e. carry the
excess to the next larger place value).
e.g. 324 + 45
Answer:
Stack the numbers vertically (i.e. one on top of another) such corresponding place values
line up.
Add the Units
Add the Tens
↓
↓
3 2 4
+
4 5
3 2 4
+
4 5
9
6 9
4 units plus 5 units is 9
units.
Add the Hundreds
2 tens plus 4 tens is 6
tens.
↓
+
3 2 4
4 5
3 6 9
3 hundreds plus 0
hundreds is 3 hundreds.
Does this answer of 369 look reasonable? The easiest way to check is to round the original
numbers to values that we can mentally add. We can round 324 to 320, and round 45 to 50.
When 320 is added to 50, we obtain 370, which is very close to the original answer of 369.
Our answer looks reasonable.
e.g. 158 + 265
Answer:
Add the Units
Add the Hundreds
↓
↓
1 5 8
+ 2 6 5
1 5 8
+ 2 6 5
1 5 8
+ 2 6 5
3
2 3
4 2 3
1
8 units plus 5 units is 13
units. Regroup the 13 to
1 ten and 3 units. Write
the 3 in the units place
and carry the 1 to the
next place value (tens).
NSSAL
©2012
Add the Tens
↓
1
1
1
1
1 ten plus 5 tens plus 6
1 hundred plus 1
tens is 12 tens. Regroup hundred plus 2 hundreds
the 12 to 1 hundred and is 4 hundreds
2 tens. Write the 2 in
the tens place and carry
the 1 to the next place
value (hundreds).
31
Draft
C. D. Pilmer
e.g. 451 + 75 + 192
Answer:
Add the Units
Add the Tens
↓
4 5 1
7 5
+ 1 9 2
↓
4 5 1
7 5
+ 1 9 2
Add the Hundreds
↓
2
8
2
4 5 1
7 5
+ 1 9 2
1 8
7 1 8
1 unit plus 5 units plus 2 5 tens plus 7 tens plus 9
units is 8 units.
tens is 21 tens. Regroup
the 21 to 2 hundreds and
1 ten. Write the 1 in the
tens place and carry the
2 to the next place value
(hundreds).
2 hundreds plus 4
hundreds plus 0
hundreds plus 1 hundred
is 7 hundreds.
Does this answer of 718 look reasonable? The easiest way to check is to round the original
numbers to values that we can mentally add. We can round 451 to 450, round 75 to 100,
and round 192 to 200. When we add 450, 100, and 200, we obtain 750. This estimate is
higher than the original answer of 718, but this is to be expected because we rounded two of
the numbers up significantly. Regardless of this, the answer of 718 seems reasonable.
e.g. 926 + 437
Answer:
1
9 2 6
+ 4 3 7
1 3 6 3
Questions
1. The following addition questions are partially completed. Fill in the missing portions. The
regrouping (or carrying) has been shown for each question.
(a)
+
NSSAL
©2012
(b)
1
4
2
7
8
6
(c)
5
6
+
32
4
1
5
1
+
1
3
8
2
7
9
Draft
C. D. Pilmer
(d)
(e)
+
(g)
(a)
(d)
2
5
6
7
5
7
1
8
+
1
(h)
1
+
1
2.
3
2
2
8
0
0
6 2
+ 3 5
1 4 6
(e)
+ 5 2 3
3.
1
6
7
4
3
7
(f)
9
3
5
7
3
7
1
+
1
(i)
2
+
(b)
1
8
6
9
2 8
+ 4 7
6 2 4
+
1
(c)
(f)
+ 3 5 7
7
9
1
5
7
1
2
5
4
3
3
6
0
5
5 0 7
+ 8 8 3
5 1 3
2 5 2
+
4 6
(h)
1 5 2
4 7
+ 6 7 4
(i)
2 6 1
7 8 5
+ 5 9 1
(a)
68 + 95
(b)
72 + 56
(c)
313 + 925
(d)
679 + 493
(e)
367 + 45 + 243
(f)
68 + 371 + 964
33
9
7
4
7 5
+ 6 1
(g)
NSSAL
©2012
4
2
Draft
C. D. Pilmer
4. With the following addition questions, we are provided with the final answer, but need to find
the missing parts of the original question. Fill in those missing parts. The regrouping (or
carrying) has been shown in each case.
(a)
(b)
4
0
9
6
+
1
(d)
7
6
(e)
3
7
(g)
7
1
(h)
7
+
2
9
1
(f)
2
8
9
+
2
2
2
6
3
1
1
3
9
+
1
8
(i)
1
5
0
7
+
1
8
5
+
1
8
8
1
7
2
6
1
9
+
+
(c)
1
6
4
3
+
1
6
5
4
2
1
3
6
7
5
4
0
8
5
6
3
1
5. Open-ended Questions (i.e. More than one acceptable answer.)
(a) Provide two two-digit numbers that add up to 82.
(b) Provide two three-digit numbers that add up to 419.
(c) Provide a two-digit number and three-digit number that when added together give 374.
NSSAL
©2012
34
Draft
C. D. Pilmer
Connect Four Subtraction Game (A)
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place two paper clips on two numbers; one from Value 1 and one from Value 2. Once they
have chosen the two numbers, they can capture one square with that appropriate difference
(i.e. Value 1 subtract Value 2). They either mark the square with an X or place a colored
counter on the square. There may be other squares with that same difference but only one
square can be captured at a time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips. They then mark the square with that difference using a O or a different colored
marker. If a player cannot move a single paperclip to capture a square, a paperclip must still
be moved in order to ensure that the game can continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
7
8
5
7
6
9
4
9
7
3
10
5
5
6
8
4
6
7
10
8
5
3
7
8
6
4
7
9
5
6
9
8
3
10
6
4
Value 1:
10
NSSAL
©2012
Value 2:
9
8
7
6
0
35
1
2
3
Draft
C. D. Pilmer
Connect Four Subtraction Game (B)
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place two paper clips on two numbers; one from Value 1 and one from Value 2. Once they
have chosen the two numbers, they can capture one square with that appropriate difference
(i.e. Value 1 subtract Value 2). They either mark the square with an X or place a colored
counter on the square. There may be other squares with that same difference but only one
square can be captured at a time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips. They then mark the square with that difference using a O or a different colored
marker. If a player cannot move a single paperclip to capture a square, a paperclip must still
be moved in order to ensure that the game can continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
5
4
2
4
6
3
3
6
7
3
0
5
0
4
1
5
2
1
4
2
6
3
7
4
3
5
4
0
3
2
2
7
2
1
5
6
Value 1:
10
NSSAL
©2012
Value 2:
9
8
7
6
3
36
4
5
6
Draft
C. D. Pilmer
Connect Four Subtraction Game (C)
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place two paper clips on two numbers; one from Value 1 and one from Value 2. Once they
have chosen the two numbers, they can capture one square with that appropriate difference
(i.e. Value 1 subtract Value 2). They either mark the square with an X or place a colored
counter on the square. There may be other squares with that same difference but only one
square can be captured at a time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips. They then mark the square with that difference using a O or a different colored
marker. If a player cannot move a single paperclip to capture a square, a paperclip must still
be moved in order to ensure that the game can continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
3
2
5
4
6
4
7
4
6
2
3
5
6
0
1
0
5
6
3
5
3
7
2
4
4
3
2
4
1
0
1
7
0
5
3
6
Value 1:
13
NSSAL
©2012
Value 2:
12
11
10
9
6
37
7
8
9
Draft
C. D. Pilmer
Connect Four Subtraction Game (D)
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place two paper clips on two numbers; one from Value 1 and one from Value 2. Once they
have chosen the two numbers, they can capture one square with that appropriate difference
(i.e. Value 1 subtract Value 2). They either mark the square with an X or place a colored
counter on the square. There may be other squares with that same difference but only one
square can be captured at a time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips. They then mark the square with that difference using a O or a different colored
marker. If a player cannot move a single paperclip to capture a square, a paperclip must still
be moved in order to ensure that the game can continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
3
5
4
5
6
5
6
8
7
9
10
7
9
7
6
3
5
9
7
10
5
7
8
4
8
6
9
4
9
3
4
7
8
10
5
6
Value 1:
15
NSSAL
©2012
Value 2:
14
13
12
5
38
6
7
8
9
Draft
C. D. Pilmer
Subtraction Search
There are twenty subtraction facts (e.g. 12 - 8 = 4) hidden in this grid. Check for three adjoining
numbers that produce this fact. These numbers could be oriented horizontally, vertically, or
diagonally. Circle the three adjoin numbers and record the fact below. Some of the facts cross
over each other (e.g. vertical facts intersect with horizontal facts).
14
0
20
9
11
5
12
1
18
28
6
6
2
8
1
9
5
4
10
3
8
30
4
9
16
22
7
0
8
14
5
12
4
2
9
15
19
7
1
6
15
4
11
3
3
7
0
19
2
8
3
4
5
11
6
1
5
10
0
16
6
8
10
13
2
5
10
3
20
9
17
11
6
15
4
20
1
8
13
7
3
0
7
9
20
8
12
7
2
8
28
2
17
6
1
9
9
0
5
23
Subtraction Facts:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
NSSAL
©2012
39
Draft
C. D. Pilmer
Subtracting Multi-Digit Numbers
To subtract multi-digit whole numbers, start by stacking the numbers vertically such that
corresponding place values line up (e.g. units with units, tens with tens) and subtract from right
to left. If the digit being subtracted is larger than the digit from which it is being subtracted,
regroup (i.e. borrow) one from the digit in the next larger place value.
e.g. 597 - 62
Answer:
Stack the numbers vertically (i.e. one on top of another) such corresponding place values
line up.
Subtract the Units
Subtract the Tens
↓
Subtract the Hundreds
↓
↓
5 9 7
−
6 2
5 9 7
−
6 2
5 9 7
−
6 2
5
3 5
5 3 5
7 units minus 2 units is
5 units.
9 tens minus 6 tens is 3
tens.
5 hundreds minus 0
hundreds is 5 hundreds.
To determine if our answer is reasonable, we can round 597 to 600, round 62 to 60, and
take the difference. Since 600 − 60 =
540 , it appears that our answer of 535 is reasonable.
e.g. 392 - 145
Answer:
Subtract the Units
↓
↓
8
12
8
12
3
9
2
3
9
2
− 1
4
5
− 1
4
5
4
7
7
We cannot take 5 units
from 2 units. Therefore
we regroup (i.e. borrow)
1 from the tens, which
leaves us with 8 tens
and 12 units. 12 units
minus 5 units is 7 units.
NSSAL
©2012
Subtract the Tens
↓
8 tens minus 4 tens is 4
tens.
40
Subtract the Hundreds
8
12
9
4
2
5
2 4
7
3
− 1
3 hundreds minus 1
hundred is 2 hundreds.
Draft
C. D. Pilmer
e.g. 647 - 391
Answer:
Subtract the Units
Subtract the Tens
↓
↓
5
14
6
4
7
3
9
1
2
5
6
↓
5
14
6 4 7
6
4
7
3
9
1
5
6
− 3 9 1
−
6
7 units minus 1 unit is 6
units.
Subtract the Hundreds
−
We cannot take 9 tens
3 hundreds minus 1
from 4 tens. Therefore
hundred is 2 hundreds.
we regroup (i.e. borrow)
1 from the hundreds,
which leaves us with 5
hundreds and 14 tens.
14 tens minus 9 tens is 5
tens.
e.g. 934 - 268
Answer:
Subtract the Units
Subtract the Tens
↓
↓
9
− 4
12
12
2
14
8
2
14
8
2
14
3
6
4
8
9
3
4
9
3
4
4
6
8
4
6
8
6
6
4
6
6
6
We cannot take 8 units
from 4 units. Therefore
we regroup (i.e. borrow)
1 from the tens, which
leaves us with 2 tens
and 14 units. 14 units
minus 8 units is 6 units.
NSSAL
©2012
Subtract the Hundreds
↓
−
−
We cannot take 6 tens
8 hundreds minus 4
from 2 tens. Therefore
hundreds is 4 hundreds.
we regroup (i.e. borrow)
1 from the hundreds,
which leaves us with 8
hundreds and 12 tens.
12 tens minus 6 tens is 6
tens.
41
Draft
C. D. Pilmer
e.g. 803 - 288
Answer:
Subtract the Units
Subtract the Tens
Subtract the Hundreds
↓
−
↓
9
13
7
10
13
0
3
8
0
3
8
8
2
8
8
5
1
5
9
↓
7
10
13
7
10
8
0
3
8
2
8
8
2
9
−
5
−
1 5
We cannot take 8 units
from 3 units. Therefore
we regroup (i.e.
borrow). However, we
have a zero in the tens
place. We need to
borrow 1 hundred from
the hundreds place, then
borrow 10 from the tens
place. That leaves us
with 7 hundreds, 9 tens,
and 13 units. 13 units
minus 8 units is 5 units.
9 tens minus 8 tens is 1
ten.
7 hundreds minus 2
hundreds is 5 hundreds.
Questions
1. The following subtraction questions are partially completed. Fill in the missing portions.
The regrouping (or borrowing) has been shown for each question.
(a)
(b)
-
4
1
3
8
2
-
(d)
(e)
-
NSSAL
©2012
3
2
9
5
4
7
1
-
42
5
14
6
3
4
9
5
7
15
8
2
5
5
7
(c)
-
8
11
9
2
1
5
(f)
5
3
-
9
5
7
12
8
5
2
4
Draft
C. D. Pilmer
(g)
-
2.
(h)
13
5
3
10
6
4
4
7
0
8
2
(i)
14
-
7
4
12
8
3
4
5
8
2
5
9
-
(a)
8 7
− 5 2
(b)
5 8
− 2 4
(c)
8 9
− 3 1
(d)
7 3
(e)
8 3
(f)
9 6
− 5 8
− 5 8
(g)
− 1 7
10
11
9
7
0
5
1
6
6 7 8
− 3 1 3
(h)
(j)
7 4 5
− 5 8 2
(k)
9 5 3
− 3 4 8
(l)
8 3 5
−
7 9
(m)
9 4 1
− 5 8 3
(n)
6 0 3
− 4 5 5
(0)
5 0 5
− 1 4 6
NSSAL
©2012
5 6 7
(i)
8
− 2 4 7
43
4 1 8
−
6 7
Draft
C. D. Pilmer
3. (a) 83 - 57
(d) 951 - 827
(b) 75 - 28
(c) 649 - 263
(e) 342 - 186
(f) 407 - 239
4. With the following subtraction questions, we are provided with the final answer, but need to
find the missing parts of the original question. Fill in those missing parts. The regrouping (or
borrowing) has been shown in each case.
(a)
(b)
4
7
4
2
7
-
(e)
9
1
6
(g)
1
3
2
6
2
NSSAL
©2012
9
8
12
6
(h)
13
5
(f)
7
5
3
4
2
3
(i)
3
9
9
12
2
10
14
0
6
15
4
2
5
7
3
-
-
44
2
4
9
6
9
4
-
13
7
-
3
9
3
-
11
1
9
16
4
6
5
(d)
-
(c)
12
1
1
4
9
Draft
C. D. Pilmer
5. Open-ended Questions (i.e. More than one acceptable answer.)
(a) Provide two two-digit numbers that differ by 23 where no regrouping (i.e. borrowing) is
required to complete the question.
(b) Provide two two-digit numbers that differ by 23 where regrouping (i.e. borrowing) is
required to complete the question.
NSSAL
©2012
45
Draft
C. D. Pilmer
Multiplication Models
There are three common models used to represent the operation of multiplication. These are the
area model, set model and number line model. An example of each has been provided below for
the product of 3 and 4.
Example:
Mathematic
al Sentence
Area Model
Set Model
Number Line Model
3 × 4 = 12
A rectangle
measuring 3 units
by 4 units.
12
Three sets of four
For each of the following, complete the mathematical statement and draw the three models.
(a) 2 × 7 = _______
Area Model:
Set Model:
Number Line Model:
(b) 6 × 3 = _______
Area Model:
Set Model:
Number Line Model:
NSSAL
©2012
46
Draft
C. D. Pilmer
(c) 5 × 5 = _______
Area Model:
Set Model:
Number Line Model:
(d) 7 × 4 = _______
Area Model:
Set Model:
Number Line Model:
(e) 3 × 6 = _______
Area Model:
Set Model:
Number Line Model:
NSSAL
©2012
47
Draft
C. D. Pilmer
Multiplication Array Game
Roll two dice. Multiply the two numbers that are rolled and write down the mathematical
sentence (e.g. 3 × 4 = 12) in the space provided. In the circular array recording sheet that has
been provided, create an array using the rolled numbers. (e.g. 3 rows of 4, or 3 columns of 4).
Once this is done, roll the dice again and repeat the procedure. The only thing is that the new
array cannot overlap other array. Continue to roll the dice, write the sentences and draw the
arrays until you are unable to draw new arrays (i.e. cannot be drawn without overlapping existing
arrays). See how many arrays you can draw on the recording. It is a combination of luck and
skill.
Mathematical Sentence:
Circular Array Recording Sheet:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
NSSAL
©2012
Number of Arrays Drawn: _______
48
Draft
C. D. Pilmer
Connect Four Multiplication Game A
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place two paper clips on two numbers on the Factor Strip whose product is that desired
square. Once they have chosen the two numbers, they can capture one square with that
appropriate product. They either mark the square with an X or place a colored counter on the
square. There may be other squares with that same product but only one square can be
captured at a time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips on the Factor Strip. They then mark the square with that product using a O or a
different colored marker. If a player cannot move a single paperclip to capture a square, a
paperclip must still be moved on the fraction strip in order to ensure that the game can
continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
9
2
5
45
15
0
3
0
15
6
18
45
0
18
9
27
10
5
18
27
2
10
9
3
3
10
0
15
4
6
45
2
6
3
10
27
Factor Strip:
0
NSSAL
©2012
1
2
3
5
49
9
Draft
C. D. Pilmer
Connect Four Multiplication Game B
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place two paper clips on two numbers on the Factor Strip whose product is that desired
square. Once they have chosen the two numbers, they can capture one square with that
appropriate product. They either mark the square with an X or place a colored counter on the
square. There may be other squares with that same product but only one square can be
captured at a time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips on the Factor Strip. They then mark the square with that product using a O or a
different colored marker. If a player cannot move a single paperclip to capture a square, a
paperclip must still be moved on the fraction strip in order to ensure that the game can
continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
6
45
27
5
45
8
10
0
36
18
20
15
36
8
12
4
0
36
2
18
45
27
6
12
20
4
15
0
10
9
27
12
3
6
36
20
Factor Strip:
0
NSSAL
©2012
1
2
3
4
50
5
9
Draft
C. D. Pilmer
Connect Four Multiplication Game C
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place two paper clips on two numbers on the Factor Strip whose product is that desired
square. Once they have chosen the two numbers, they can capture one square with that
appropriate product. They either mark the square with an X or place a colored counter on the
square. There may be other squares with that same product but only one square can be
captured at a time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips on the Factor Strip. They then mark the square with that product using a O or a
different colored marker. If a player cannot move a single paperclip to capture a square, a
paperclip must still be moved on the fraction strip in order to ensure that the game can
continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
18
2
30
8
12
24
9
54
12
18
10
6
24
5
8
6
54
20
10
30
18
5
24
3
24
4
20
12
2
18
12
54
9
30
5
8
Factor Strip:
1
NSSAL
©2012
2
3
4
5
51
6
9
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C. D. Pilmer
Connect Four Multiplication Game D
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place two paper clips on two numbers on the Factor Strip whose product is that desired
square. Once they have chosen the two numbers, they can capture one square with that
appropriate product. They either mark the square with an X or place a colored counter on the
square. There may be other squares with that same product but only one square can be
captured at a time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips on the Factor Strip. They then mark the square with that product using a O or a
different colored marker. If a player cannot move a single paperclip to capture a square, a
paperclip must still be moved on the fraction strip in order to ensure that the game can
continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
14
63
6
28
15
30
42
12
30
63
14
10
8
21
54
18
54
21
35
15
8
28
42
12
18
54
14
63
6
35
10
28
42
12
21
18
Factor Strip:
2
NSSAL
©2012
3
4
5
6
52
7
9
Draft
C. D. Pilmer
Connect Four Multiplication Game E
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place two paper clips on two numbers on the Factor Strip whose product is that desired
square. Once they have chosen the two numbers, they can capture one square with that
appropriate product. They either mark the square with an X or place a colored counter on the
square. There may be other squares with that same product but only one square can be
captured at a time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips on the Factor Strip. They then mark the square with that product using a O or a
different colored marker. If a player cannot move a single paperclip to capture a square, a
paperclip must still be moved on the fraction strip in order to ensure that the game can
continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
42
12
16
8
24
48
6
72
45
54
15
18
56
24
21
16
56
20
14
30
10
40
6
27
54
18
36
12
42
21
15
72
27
14
35
10
5
6
Factor Strip:
2
NSSAL
©2012
3
4
53
7
8
9
Draft
C. D. Pilmer
Connect Four Multiples Game (A)
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place one paper clip on the appropriate "Multiple of" Strip and one paper clip on the
appropriate "Range for Multiple" Strip. For example to capture a 32, the first paper clip
could be on the "2" or "4" (because 32 is a multiple of 2 or 4), while the second paper clip
must be on the "30 to 39" range (because 32 is within this range). They either mark the
square with an X or place a colored counter on the square. Only one square can be captured
at a time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips. They then mark the square with that product using an O or a different colored
marker. If a player cannot move a single paperclip to capture a square, a paperclip must still
be moved in order to ensure that the game can continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
27
12
38
28
15
22
26
30
14
36
34
10
35
24
33
20
16
25
18
10
12
32
24
32
38
21
35
25
12
18
15
32
27
16
21
36
"Multiple of" Strip:
2
NSSAL
©2012
3
4
"Range for Multiple" Strip:
5
6
10 to 19
54
20 to 29
30 to 39
Draft
C. D. Pilmer
Connect Four Multiples Game (B)
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place one paper clip on the appropriate "Multiple of" Strip and one paper clip on the
appropriate "Range for Multiple" Strip. For example to capture a 45, the first paper clip
could be on the "5" or "9" (because 45 is a multiple of 5 or 9), while the second paper clip
must be on the "40 to 49" range (because 45 is within this range). They either mark the
square with an X or place a colored counter on the square. Only one square can be captured
at a time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips. They then mark the square with that product using an O or a different colored
marker. If a player cannot move a single paperclip to capture a square, a paperclip must still
be moved in order to ensure that the game can continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
28
36
45
44
25
36
45
24
35
21
30
27
30
49
32
40
48
42
48
42
20
49
36
24
36
27
45
35
21
32
20
35
24
42
30
28
"Multiple of" Strip:
4
NSSAL
©2012
5
6
"Range for Multiple" Strip:
7
9
20 to 29
55
30 to 39
40 to 49
Draft
C. D. Pilmer
Connect Four Multiples Game (C)
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place one paper clip on the appropriate "Multiple of" Strip and one paper clip on the
appropriate "Range for Multiple" Strip. For example to capture a 56, the first paper clip
could be on the "7" or "8" (because 56 is a multiple of 7 or 8), while the second paper clip
must be on the "50 to 59" range (because 56 is within this range). They either mark the
square with an X or place a colored counter on the square. Only one square can be captured
at a time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips. They then mark the square with that product using an O or a different colored
marker. If a player cannot move a single paperclip to capture a square, a paperclip must still
be moved in order to ensure that the game can continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
36
49
64
42
36
45
66
32
45
56
35
63
40
54
63
30
66
48
60
64
42
54
36
56
48
30
35
63
40
42
36
56
54
32
60
35
"Multiple of" Strip:
6
NSSAL
©2012
7
8
"Range for Multiple" Strip:
9
30 to 39
56
40 to 49
50 to 59
60 to 69
Draft
C. D. Pilmer
Capture the Flag Multiplication Game
Mission: The winner is the first player to capture the opposing player's flag on the opposite
side of the board.
Rules:
• With this game each player starts with six markers on opposing sides of the board at the
designated spots. Coins can be used as markers. For example, six pennies for one player,
and six nickels for the other player.
• Each round a player can only move one marker one square either forwards, backwards,
diagonally, or sideways. However, the player must know the multiplication fact to make
that move. For example if the marker is first on a "2" square and wishes to move to a "5"
square, then they must tell the other player that "2 times 5 is 10." If they do not know the
fact, then they must choose another marker to move.
• Markers can move two spaces if they are jumping an opponent's marker and thus
eliminating that marker. Again the player must know the multiplication fact associated
with their starting square and their landing square.
• Each player is required to move a marker each round.
• Two markers cannot share the same square.
• There are a few squares that have "blockers" that limit movement and the opportunities to
jump and eliminate markers.
• You are not permitted to guard your flag by placing one of your markers on your own
flag.
NSSAL
©2012
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C. D. Pilmer
Player 2
Start
Player 2
Start
Player 2
Start
Player 2
Start
Player 2
Start
Player 2
Start
2
6
7
7
6
2
3
8
4
5
9
5
4
8
3
0
10
1
6
4
6
1
10
0
9
3
2
7
2
3
9
3
4
5
6
7
6
5
4
3
8
7
10
4
4
10
7
8
3
4
5
6
7
6
5
4
3
9
3
2
7
2
3
9
0
10
1
6
4
6
1
10
0
3
8
4
5
9
5
4
8
3
2
6
7
7
6
2
Player 1
Start
Player 1
Start
Player 1
Start
Player 1
Start
Player 1
Start
Player 1
Start
NSSAL
©2012
10
10
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C. D. Pilmer
Multiples Puzzle
Start at each of the bolded numbers and move to the left, right, up or down to the next adjoining
multiple of that bolded number. You will create different continuous pathways for each bolded
number. If your bolded number is 9, then your pathway would contain numbers like 18, 27, 36,
45,… (i.e. multiples of 9). No two pathways will cross, meaning a number will not be shared by
multiple pathways. However, every number in the grid will be used. You are not permitted to
cross obstructions (i.e. the thick walls). We recommend that you use different colored pens or
pencils for each of the separate pathways.
(a)
(c)
(e)
4
70
49
56
21
14
50
24
36
22
10
6
63
5
12
20
16
18
5
14
35
14
9
30
32
30
2
25
8
28
8
20
27
15
28
21
15
35
7
42
12
18
3
24
18
8
3
9
40
10
45
35
21
6
22
14
2
48
40
6
36
24
54
14
7
10
8
9
12
16
24
64
81
63
30
63
60
12
16
27
6
3
33
48
99
18
12
56
28
30
3
18
15
20
18
8
15
54
27
18
24
54
10
35
21
12
6
9
36
4
45
6
45
15
25
5
24
44
8
28
16
40
9
63
99
14
9
33
21
4
55
88
121
77
44
81
77
42
24
55
3
16
33
3
18
27
66
45
63
18
27
30
40
36
11
20
32
12
7
18
49
28
7
11
25
12
28
8
21
33
49
72
9
22
55
44
40
48
24
77
63
21
56
77
110
88
5
15
45
32
16
72
8
35
14
27
72
45
15
10
35
9
56
6
2
25
14
63
16
8
42
35
7
49
21
4
NSSAL
©2012
(b)
(d)
(f)
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C. D. Pilmer
Factors
Factors are numbers you can multiply together to get another number.
e.g. What are the factors of 12?
Answer:
• 3 and 4 are factors of 12 because 3 × 4 =
12 .
• 6 and 2 are factors of 12 because 6 × 2 =
12 .
• 1 and 12 are factors of 12 because 1×12 =
12 .
• Therefore the factors of 12 are 1, 2, 3, 4, 6, and 12.
(One can also say that 12 is divisible by 1, 2, 3, 4, 6, and 12.)
1. List all the factors of each of these numbers.
(a) 6
(b) 14
(c) 20
(d) 18
(e) 24
(f) 30
(g) 36
(h) 40
(i) 35
(j) 63
2. What factor(s) are shared by each of these pairs of numbers?
(a) 16 and 28
(b) 50 and 20
(c) 8 and 12
(d) 18 and 30
3. Open-ended Questions (i.e. More than one correct answer.)
__________
(a) Provide a number that has the factors 2, 3, and 4.
__________
(b) Provide a number that has the factors 1, 3, and 5.
__________
(c) Provide a number that has the factors 2, 4, and 5.
__________
(d) Provide two numbers that have an odd number of factors.
NSSAL
©2012
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C. D. Pilmer
Factor Flowers
In this puzzle, we are going to find all the factor flowers in the flowing grid. For the purpose of
this activity, a factor flower is described as a whole number surrounded on all four sides by four
of its factors, although the whole number may have additional factors that are not shown. Three
examples of factor flowers are shown below.
6
2
12
2
3
18
18
10
3
6
30
2
1
9
5
1, 2, 3 and 6
are four factors
of 12.
2, 3, 9 and 18
are four factors
of 18.
2, 5, 6 and 10
are four factors
of 30.
Find the15 factor flowers in the following grid. Shade them in lightly with a pencil.
5
7
2
1
9
9
12
20
7
35
35
7
72
19
8
24
3
11
8
5
6
25
8
21
1
6
25
9
4
30
2
16
2
13
15
6
18
2
9
32
9
3
1
3
10
8
15
4
12
2
42
4
6
6
18
15
8
40
20
7
16
13
3
1
5
22
4
30
13
11
5
9
3
2
6
18
2
8
23
1
20
10
24
15
30
36
9
14
7
3
5
16
8
11
27
3
1
20
5
17
24
11
22
8
48
6
8
3
2
13
8
5
6
8
7
9
17
4
36
12
11
24
30
6
4
4
12
10
23
8
1
81
5
30
6
24
9
22
4
15
19
5
17
16
9
6
17
8
32
4
7
21
14
2
2
14
9
1
21
7
17
12
3
3
6
18
10
13
2
6
NSSAL
©2012
6
22
1
3
22
61
8
2
10
10
19
7
56
20
5
11
2
2
42
21
3
4
6
2
6
Draft
C. D. Pilmer
Random Multiplication Facts Quizzes
Please note that there are three different types of quizzes (A, B & C). These three types
correspond to the suggested order that the multiplication facts are taught.
Quiz A1
×
Name: ________________
2
7
5
8
Quiz A2
×
4
1
7
5
0
0
8
9
6
2
3
Quiz A3
×
Name: ________________
9
6
3
4
×
7
6
2
8
1
1
5
3
0
7
×
Name: ________________
8
3
2
7
×
6
8
3
2
4
4
1
7
9
6
62
2
0
1
9
Name: ________________
1
Quiz B2
5
NSSAL
©2012
5
Quiz A4
9
Quiz B1
Name: ________________
5
9
2
0
Name: ________________
4
9
0
3
5
Draft
C. D. Pilmer
Quiz B3
×
Name: ________________
8
6
7
1
Quiz B4
×
3
9
7
2
6
4
2
5
4
3
8
Quiz C1
×
Name: ________________
8
7
5
6
×
3
4
4
8
8
6
2
9
7
7
×
Name: ________________
1
6
9
7
×
3
3
8
7
4
4
7
8
6
9
63
5
1
4
9
Name: ________________
6
Quiz C4
5
NSSAL
©2012
3
Quiz C2
6
Quiz C3
Name: ________________
9
2
8
7
Name: ________________
2
7
6
9
3
Draft
C. D. Pilmer
What Number Does the Star Represent? (Multiplication)
Example
For each question, determine what number is represented by the star?
(a) 7 ×  = 28
(b)  ×  = 49
(c) 5 ×  × 2 = 30
(d)  ×  ×  = 125
(e)  ×  × 3 = 12
Answers:
(a) What number when multiplied by 7 gives you 28? The answer is 4. ( = 4)
(b) What number when multiplied by itself gives you 49? The answer is 7. ( = 7)
(c) What number when multiplied by 5 and 2 gives you 30? The answer is 3. ( = 3)
(d) What number when multiplied together three times gives you 125? The answer is 5.
( = 5)
(e) What number when multiplied by itself and 3 gives you 12? The answer is 2. ( = 2)
Questions
In each case determine the number that is represented by the star? No work needs to be shown.
(a)  × 3 = 12
 = _____
(b)  ×  = 9
 = _____
(c) 5 × 8 = 
 = _____
(d) 6 ×  = 42
 = _____
(e)  ×  ×  = 8
 = _____
(f) 11 ×  = 11
 = _____
(g)  × 8 = 24
 = _____
(h)  ×  = 16
 = _____
(i) 9 ×  = 45
 = _____
(j)  ×  = 81
 = _____
(k) 3 ×  × 1 = 18
 = _____
(l) 8 × 4 = 
 = _____
(m)  × 5 × 4 = 20
 = _____
(n)  × 9 = 27
 = _____
(o) 4 × 2 ×  = 40
 = _____
(p)  = 7 × 8
 = _____
(q) 6 ×  = 48
 = _____
(r)  ×  = 25
 = _____
(s)  ×  × 2 = 18
 = _____
(t)  × 5 = 30
 = _____
(u) 6 ×  × 2 = 24
 = _____
(v) 9 ×  = 63
 = _____
(w)  × 10 = 80
 = _____
(x)  ×  ×  = 27
 = _____
(y) 5 ×  ×  = 20
 = _____
(z) 2 × 3 ×  = 36
 = _____
NSSAL
©2012
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C. D. Pilmer
Multiplying on Your Hands
Most people are OK with the multiplication facts with the numbers 0, 1, 2, 3, 4, and 5.
Examples:
1× 8 = 8
2×3 = 6
3 × 7 = 21
4 × 6 = 24
5 × 9 = 45
However, many people struggle remembering the facts for larger numbers (6, 7, 8, and 9).
Examples:
6 × 7 = 42
7 × 7 = 49
8 × 6 = 48
8 × 8 = 64
9 × 7 = 63
There is a neat way to get these multiplication facts using your hands. First we need to identify
the numbers associated with the fingers on your hands.
Example 1
Complete the following operation. 8 × 8 = ?
Step 1
Touch the appropriate fingers to find the multiplication fact. For example, if you wanted to
work out 8 × 8 , take finger eight from the left hand and touch it to finger eight from the right
hand.
NSSAL
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C. D. Pilmer
Step 2
Count the number of fingers above the touching fingers on the left hand. In this case there
are 2. Count the number of fingers above the touching fingers on the right hand. In this case
there are 2. Now multiply these two numbers together ( 2 × 2 = 4 )
Step 3
Count the two touching fingers and all the fingers dangling below the touching fingers. In
this case we have 6 fingers. Now multiply this number by 10 ( 6 × 10 = 60 )
Step 4
Add the two numbers that were generated in steps 2 and 3.
4 + 60 = 64
Therefore: 8 × 8 = 64
NSSAL
©2012
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C. D. Pilmer
Example 2
Complete the following operation. 7 × 9 = ?
Step 1
Touch the correct fingers.
Step 2
Count the fingers above the touching fingers left hand and count the fingers above
the touching fingers on the right hand. Multiply these two numbers.
3 ×1 = 3
Step 3
Count the two touching fingers and all the fingers dangling below the touching
fingers. Now multiply this number by 10.
6 × 10 = 60
Step 4
NSSAL
©2012
Add the numbers from steps 2 and 3 ( 3 + 60 = 63 ).
Therefore: 7 × 9 = 63
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C. D. Pilmer
Example 3
Complete the following operation. 6 × 7 = ?
Step 1
Step 2
4 × 3 = 12
Step 3
3 × 10 = 30
Step 4
Since 12 + 30 = 42 , then 6 × 7 = 42
"Hand" clipart by scarlett was downloaded on April 5, 2011 from http://www.clker.com/cliparthand-11.html
NSSAL
©2012
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C. D. Pilmer
Put the Number in the Right Box
With each question you have been provided with a list of numbers. Place those numbers in the
appropriate box found on the right. A sample question has been provided.
e.g. List of Numbers
16, 12, 9,
5, 20, 6,
24, 17, 8,
27, 22, 36
1.
2.
3.
4
5.
List of Numbers
15, 4, 11,
6, 19, 18,
10, 14, 7,
12, 9, 21
List of Numbers
19, 15, 10,
9, 16, 30,
22, 25, 18,
40, 45, 21
List of Numbers
24, 20, 8,
15, 30, 9,
28, 25, 27,
40, 45, 26
List of Numbers
16, 24, 21,
30, 9, 12,
8, 18, 6,
28, 15, 36
List of Numbers
27, 25, 20,
12, 10, 40,
12, 19, 32,
80, 35, 28
NSSAL
©2012
Multiple of 4
Not a Multiple of 4
Multiple of 3
12, 24, 36
9, 6, 27
Not a Multiple of 3
16, 20, 8
5, 17, 22
Multiple of 2
Not a Multiple of 2
Multiple of 2
Not a Multiple of 2
Multiple of 3
Not a Multiple of 3
Multiple of 4
Not a Multiple of 4
Multiple of 5
Not a Multiple of 5
Multiple of 3
Not a Multiple of 3
Multiple of 5
Not a Multiple of 5
Multiple of 5
Not a Multiple of 5
Multiple of 6
Not a Multiple of 6
Multiple of 4
Not a Multiple of 4
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C. D. Pilmer
Investigation: Multiplying by Multiples of 10, 100, and 1000
Numbers are multiples of 10 if they can be expressed as a whole number multiplied by 10
e.g. 70 = 7 × 10
e.g. 340 = 34 × 10
e.g. 30 = 3 × 10
Numbers are multiples of 100 if they can be expressed as a whole number multiplied by 100.
e.g. 300 = 3 × 100 e.g. 700 = 7 × 100
e.g. 3400 = 34 × 100
Numbers are multiples of 1000 if they can be expressed as a whole number multiplied by 1000.
e.g. 3000 = 3 × 1000 e.g. 7000 = 7 × 1000 e.g. 34000 = 34 × 1000
In this section we will discover how we can multiply numbers that are multiples of 10, 100 and
1000. (e.g. 60 × 400 ). Use a calculator to work out the answers to the following two sets of
questions. Look for a pattern. (Hint: Look at the number of zeros in each question.)
First Set of Questions
4× 2 =
40 × 2 =
4 × 20 =
40 × 20 =
400 × 2 =
4 × 200 =
400 × 20 =
40 × 200 =
4000 × 2 =
4 × 2000 =
Second Set of Questions
5× 7 =
50 × 7 =
5 × 70 =
50 × 70 =
500 × 7 =
5 × 700 =
500 × 70 =
50 × 700 =
5000 × 7 =
5 × 7000 =
Questions
1. Based on the work that you have done above, what do you think the answers to each of these
is. (Only use a calculator to check your answers.)
(a) 4000 × 200 =
(b) 40 × 2000 =
(c) 5000 × 7000 =
(d) 500 × 7000 =
(e) 60 × 3 =
(f) 60 × 300 =
(g) 900 × 8 =
(h) 90 × 8000 =
2. Explain the rule that you have discovered. It may be easiest to do by looking at a specific
question or questions (e.g. 40 × 70 = 2800 , 400 × 7000 = 2800000 , …)
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Multiplying by Multiples of 10, 100, and 1000
In the previous investigation, you discovered how to multiply numbers that are multiples of 10,
100, and 1000 (e.g. 60 × 400 = 24000 ). Here are two examples that cover the material that you
learned in that investigation.
Example 1
Evaluate 400 × 70 .
Answer:
To work out 400 × 70 , it is a three step process
(i) Omitting the zeros, multiply the two numbers ( 4 × 7 = 28 )
(ii) Next count the number of zeros in the original question (There are 3; two from the
number 400 and one from the 70)
(iii) Take the product from step (a) and tack on the number of zeros from step (b).
Therefore: 400 × 70 = 28 000
Example 2
Evaluate 800 × 6000 .
Answer:
To work out 800 × 6000 , it is a three step process.
(i) Omitting the zeros, multiply the two numbers ( 8 × 6 = 48 )
(ii) Next count the number of zeros in the original question (There are 5; two from the
number 800 and three from the 6000)
(iii) Take the product from step (a) and tack on the number of zeros from step (b).
Therefore: 800 × 6000 = 4 800 000
Questions:
1. Complete the operation and express the answer in written form. Two sample questions has
been completed for you.
Answer in Written Form
e.g.
90 × 6000 = 540 000
five hundred forty thousand
e.g.
70 × 30 = 2 100
two thousand one hundred
(a)
2000 × 600 =
(b)
9 × 4000 =
(c)
300 × 500 =
(d)
80 × 400 =
(e)
60 × 60 =
(f)
800 × 5 =
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2. Complete the following operations. Do not use a calculator.
(a) 20 × 70 =
(b) 9 × 60 =
(c) 400 × 30 =
(d) 5 × 5000 =
(e) 300 × 8 =
(f) 1000 × 400 =
(g) 90 × 7000 =
(h) 2000 × 6000 =
(i) 40 × 8 =
(j) 700 × 700 =
(k) 9 × 80 =
(l) 500 × 60 =
(m) 3000 × 90 =
(n) 7000 × 4 =
(o) 2000 × 900 =
(p) 80 × 500 =
(q) 800 × 6 =
(r) 1000 ×1000 =
(s) 2 × 4000 =
(t) 90 × 90 =
(u) 700 × 8000 =
(v) 80 × 800 =
(w) 5000 × 2000 =
(x) 90 × 8 =
(y) 100 × 900 =
(z) 7 × 5000 =
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Multiplying Two Digit Numbers, Part 1 (Expanded Form)
In order to complete this activity sheet, you should already know:
1. How to express a number in its expanded form
e.g. 47 = 40 + 7
e.g. 392 = 300 + 90 + 2
2. How to multiply numbers that are multiples of 10, 100, or 1000.
e.g. To work out 400 × 70 :
(a) Omitting the zeros, multiply the two numbers ( 4 × 7 = 28 )
(b) Next count the number of zeros in the original question (There are 3; two from the
number 400 and one from the 70)
(c) Take the product from step (a) and tack on the number of zeros from step (b).
Therefore: 400 × 70 = 28 000
e.g. To work out 800 × 6000 :
(a) Omitting the zeros, multiply the two numbers ( 8 × 6 = 48 )
(b) Next count the number of zeros in the original question (There are 5; two from the
number 800 and three from the 6000)
(c) Take the product from step (a) and tack on the number of zeros from step (b).
Therefore: 800 × 6000 = 4 800 000
Let's look at multiplying two digit numbers. The easiest way to do this is work through a sample
problem. You may wish to view the following video that corresponds to the examples below.
• http://www.youtube.com/watch?v=gEc_7V_UpB4
(or Google Search: YouTube Multiplying Two Digit Numbers Part 1 (Expanded Form))
Example 1
Complete the following operation. 73× 24
Answer:
We first need to express the two numbers in their expanded forms.
73 = 70 + 3
24 = 20 + 4
Next we set the numbers up so that we can do the multiplication. Note that the 20 and the 4
must both be multiplied by the 70 and the 3. That means we have to do four sets of
multiplication.
70 + 3
× 20 + 4
1
2 8
6
1 4 0
2
0
0
0
4 × 3 ; first set of multiplication
4 × 70 ; second set of multiplication
20 × 3 ; third set of multiplication
20 × 70 ; fourth set of multiplication
1 7 5 2
Therefore: 73 × 24 = 1752
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Example 2
Complete the following operation. 67 × 49
Answer:
We first need to express the two numbers in their expanded forms.
67 = 60 + 7
49 = 40 + 9
Next we set the numbers up so that we can do the multiplication.
60 + 7
× 40 + 9
6 3
5 4 0
2 8 0
2 4 0 0
The 9 must be multiplied by both the 60 and the 7.
The 40 must be multiplied by both the 60 and the 7.
3 2 8 3
Therefore: 67 × 49 = 3283
Example 3
Complete the following operation. 84 × 57
Answer:
80 + 4
× 50 + 7
2
5 6
2 0
4 0 0
8
0
0
0
The 7 must be multiplied by both the 80 and the 4.
The 50 must be multiplied by both the 80 and the 4.
4 7 8 8
Example 4
Complete the following operation. 91× 35
Answer:
90 + 1
× 30 + 5
5
4 5 0
3 0
2 7 0 0
The 5 must be multiplied by both the 90 and the 1.
The 30 must be multiplied by both the 90 and the 1.
3 1 8 5
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Questions
1. The answer to 63× 95 is partially completed. Fill in the four missing components.
60 + 3
× 90 + 5
The 5 must be multiplied by both the 60 and the 3.
The 90 must be multiplied by both the 60 and the 3.
5 9 8 5
2. The answer to 74 × 38 is partially completed. Fill in the five missing components.
70 + 4
× 30 + 8
The 8 must be multiplied by both the 70 and the 4.
The 30 must be multiplied by both the 70 and the 4.
3
Complete each of the operations. Only use a calculator to check your answers.
(a) 26 × 43
(b) 57 × 35
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(c) 29 × 71
(d) 53× 83
(e) 79 × 36
(f) 83× 41
(g) 48 × 61
(h) 72 × 79
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4. In this section we have focused on multiplying two digit numbers using a specific technique.
This technique, however, can be used to multiply even larger numbers. An example has been
provided below. Look at this example and then multiply the numbers 436 and 72 on your
own.
Example: 651× 37 = ?
600 + 50 + 1
30 + 7
×
3 5
4 2 0
3
1 5 0
1 8 0 0
7
0
0
0
0
0
The two numbers are written in their expanded forms.
The 7 must be multiplied by the 600, 50, and 1.
The 30 must be multiplied by the 600, 50, and 1.
2 4 0 8 7
Now work out 436 × 72 in the space provided.
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Multiplying Two Digit Numbers, Part 2 (Lattice Method)
Up to this point if we wanted to multiply two digit numbers, we had to write both numbers in
their expanded form and then do the four sets of multiplication, and add the four numbers. For
example, if we wanted to multiply 82 and 53, we would have to do the following.
82 = 80 + 2
53 = 50 + 3
80 + 2
× 50 + 3
Expanded Forms
Remember that the 3 must be multiplied by both the 80 and 2.
The same is true with the 50; it must also be multiplied by the
80 and 2.
6
2 4 0
1 0 0
4 0 0 0
4 3 4 6
Therefore: 82 × 53 = 4346
There is an alternate method that is directly tied to the technique but it is much easier and faster.
It involves using a two column two row chart where each cell is divided in two by a diagonal.
You can view explanations that correspond to examples 1 and 2 by going to the following
website.
http://www.youtube.com/watch?v=Yt2atjULffY
(or Google Search: YouTube Multiplying Two Digit Numbers (Lattice Method))
Example 1
Complete the following operation. 82× 53
We can use the following chart to multiply two digit numbers. We put the 82 along the top
of the chart, and the 53 along the right side of the chart
8
2
5
3
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Multiply the 5 by the 8, and place the two digits of the product (40) in the two spaces in the
upper left hand corner of the chart.
8
2
4
5
0
3
Now multiply the 5 by the 2, and place the two digits of the product (10) in the two spaces in
the upper right hand corner of the chart.
8
2
4
1
0
5
0
3
Repeat this procedure by multiplying the 3 by the 8 (product: 24), the 3 by 2 (product: 6),
and fill in the appropriate spaces in the chart.
8
2
4
1
0
0
2
0
4
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5
3
6
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Now ignore the 82 and 53 along the outside of our chart. Starting at the bottom, add the
numbers along each diagonal placing the answer along the outer edge of the chart. If a sum
exceeds 9, carry the tens digit up to the next diagonal (This does not occur in this example.).
4
4
1
0
0
3
2
Start here and work your way up to the
next diagonal.
0
6
4
4
6
These numbers along the outside, starting at the upper left, represent the digits of your
product. Therefore: 82 × 53 = 4346
Example 2
Complete the following operation. 34 × 26
3
4
2
6
3
0
6
1
carry 1
4
0
8
2
4
8
2
0
6
8
0
0
6
1
8
2
8
4
8
4
34 × 26 = 884
Example 3
Complete the following operation. 71× 65
7
1
6
5
7
4
2
3
5
carry 1
1
0
6
0
5
6
4
5
6
0
4
2
3
0
5
1
6
5
5
71× 65 = 4615
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Example 4
Complete the following operation. 68 × 95
carry 1
6
8
9
5
6
5
4
3
0
8
7
2
4
0
6
5
4
5
9
7
4
3
4
0
6
2
0
0
68 × 95 = 6460
Why Does this Work?
Let's take the last example and do it another way where we express 68 as 60 + 8, and 95 as
90 + 5. When we multiply these expanded forms of the numbers, make sure the 5 is multiplied
by both the 60 and 8, and similarly the 90 must be multiplied by both the 60 and 8.
60 + 8
× 90 + 5
4 0
3 0 0
7 2 0
5 4 0 0
6
4
5
4
3
2
4
0
6
6 4 6 0
7
0
0
68 × 95 = 6460
Notice how the columns in the first technique match up to the diagonals in the second technique.
The nice thing about the second technique is that it takes care of the place values (units, tens,
hundreds, and thousands) for us.
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Questions
1. Multiply the numbers using the chart provided.
(a) 32 × 75
(b) 41× 36
(c) 92 × 81
(d) 74 × 24
(e) 67 × 18
(f) 58 × 36
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2. Work out 57 × 38 using both techniques (The chart has not been provided; you must draw
your own.). Which technique do you prefer?
3. Complete the following operations.
(a) 42 × 46
(c) 39 × 72
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(b) 52 × 71
(d) 84 × 37
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Multiplying Multi-Digit Numbers
On previous activity sheets, you learned how to multiple two digit numbers using two
techniques. Those same techniques can be used with numbers having three or more digits, and
as before, the second technique (i.e. chart method) is far easier and faster.
You may wish to view the following video.
• http://www.youtube.com/watch?v=VZOHBbTvS-0
(or Google Search: YouTube Multiplying Multi-Digit Numbers (Lattice Method))
Example 1
Complete the operation. 382 × 57
Answer: First Technique (Expanded Form with Repeated Multiplication)
300 + 80 + 2
×
50 + 7
The two numbers are written in their expanded forms.
1 4
The 7 must be multiplied by the 300, 80, and 2.
5 6 0
2 1 0 0
1 0 0
4 0 0 0
1 5 0 0 0
The 50 must be multiplied by the 300, 80, and 2.
Therefore: 382 × 57 = 21 774
2 1 7 7 4
Answer: Second Technique (Chart Method)
Our chart will have three columns because one number is a three digit number. The chart
will have two rows because the other number is a two digit number. Remember that each
cell is divided in two by a diagonal.
3
8
2
5
7
We could have set the chart up so that it had two columns and three rows. If that was the
case, then 57 would be along the top and 382 would be down along the right side.
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We will now multiply the numbers.
3
8
2
1
4
1
5
2
0
5
0
1
1
6
4
5
7
Now ignore the 382 and 57 along the outside of our chart. Starting at the bottom, add the
numbers along each diagonal placing the answer along the outer edge of the chart. If a sum
exceeds 9, carry the tens digit up to the next diagonal.
Carry 1
2
1
1
4
1
5
2
0
5
0
1
1
6
7
4
7
4
Therefore: 382 × 57 = 21 774
Example 2
Complete the following operation using the chart method. 725 × 613
Answer:
Set up the chart with the three digit numbers along the sides and do the multiplication.
7
2
4
1
2
0
3
2
0
7
2
0
0
2
0
1
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5
1
6
5
6
1
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Ignore the numbers along the sides, and add the numbers along the diagonals.
Carry 1
4
1
2
4
4
4
Carry 1
3
0
2
0
0
0
7
2
2
5
0
1
1
4
6
5
2
5
Therefore: 725 × 613 = 444 425
Example 3
Complete the following operation using the chart method. 9271× 46
Answer:
9
2
0
3
6
7
2
5
8
1
4
1
0
8
4
4
4
0
2
2
6
6
Carry 1 Carry 1 Carry 1
4
3
0
6
5
2
8
1
4
2
6
8
4
4
0
2
4
0
2
6
6
6
Therefore: 9271× 46 = 426 466
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Questions:
1. Complete the following operations using the chart provided.
(a) 453× 62
(b) 571× 245
(c) 4372 × 59
2. Complete the operations.
(a) 45 × 623
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(b) 341× 908
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Connect Four Division Game
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place two paper clips on two numbers; one from Value 1 and one from Value 2. Once they
have chosen the two numbers, they can capture one square with that appropriate quotient (i.e.
Value 1 divided by Value 2). They either mark the square with an X or place a colored
counter on the square. There may be other squares with that same quotient but only one
square can be captured at a time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips. They then mark the square with that quotient using a O or a different colored
marker. If a player cannot move a single paperclip to capture a square, a paperclip must still
be moved in order to ensure that the game can continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
6
24
12
15
12
2
8
3
6
30
4
15
18
12
10
9
8
12
6
8
2
24
6
9
30
4
15
12
4
3
6
18
9
2
10
18
Value 1:
30
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Value 2:
24
18
12
6
1
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Division Search
There are twenty division facts (e.g. 12 ÷ 6 = 2) hidden in this grid. Check for three adjoining
numbers that produce this fact. These numbers could be oriented horizontally, vertically, or
diagonally. Circle the three adjoin numbers and record the fact below. Some of the facts cross
over each other (e.g. vertical facts intersect with horizontal facts).
30
2
48
6
8
0
18
14
5
9
5
28
15
11
2
33
9
13
3
9
32
4
8
5
9
17
2
27
6
1
11
7
1
5
3
10
5
2
31
0
24
64
45
3
45
4
19
1
42
23
21
8
9
25
7
16
4
4
6
7
1
8
3
0
5
12
43
8
7
36
20
4
5
27
1
5
4
1
5
6
4
56
8
7
22
9
8
0
12
6
3
0
18
5
36
8
18
2
9
25
Division Facts:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
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Divisibility Chart
You need three colored pencils (red, yellow, and blue) to complete this activity.
Each number between 1 and 60 has three blocks below it. Using a calculator, determine whether
the number is divisible by 2, 3, and/or 5. If the number is divisible by 2, shade the first block
below the number red. If the number is divisible by 3, shade the second block below the number
yellow. If the number is divisible by 5, shade the third block below the number blue. For
example, the number 15 is both divisible by 3 and 5, therefore its second block is shaded yellow,
and its third block is shaded blue.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Now that the chart is complete, look for patterns.
Look at all the numbers that are divisible by 2. How can you determine if a number is divisible
by 2 by just looking at the number (versus using a calculator)?
Look at all the numbers that are divisible by 3. How can you determine if a number is divisible
by 3 by just looking at the number (versus using a calculator)? (Hint: Consider adding the
digits.)
Look at all the numbers that are divisible by 5. How can you determine if a number is divisible
by 5 by just looking at the number (versus using a calculator)?
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More Divisibility (A)
You have already learned how to identify numbers that are divisible by 2, 3, and 5.
• Numbers that are divisible by 2 are even numbers.
e.g. 6, 34, 798, 2050, and 3942 are all divisible by 2
•
If the sum of the digits of a multi-digit number produces a number that is divisible by
three, then the original multi-digit number is divisible by 3.
e.g. 561 is divisible by 3 because 5 + 6 + 1 = 12 and 12 is divisible by 3.
•
If the ones digit is a 0 or a 5, then the number is divisible by 5.
e.g. 75, 120, 375, and 2960 are all divisible by 5.
Questions:
1. Beside each number you will find a box corresponding to a number that might divide evenly
into the original number. Check off the appropriate boxes to identify whether the original
number is divisible by 2, 3, and/or 5. Do not use a calculator.
e.g.
78
(a)
2
3


5
2
e.g.
345
64
(b)
35
(c)
81
(d)
90
(e)
40
(f)
42
(g)
105
(h)
307
(i)
208
(j)
635
(k)
915
(l)
410
(m)
720
(n)
816
(o)
1245
(p)
2036
(q)
4109
(r)
7281
(s)
9130
(t)
3075
(u)
8374
(v)
7320
3
5


2. (a) Create 4 three-digit numbers that are divisible by 2 and 5 (but not 3). Do not use
numbers encountered in question 1.
(b) Create 4 three-digit numbers that are divisible by 2 and 3 (but not 5). Do not use
numbers encountered in question 1.
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More Divisibility (B)
You have already learned how to identify numbers that are divisible by 2, 3, and 5. Numbers
that are divisible by 2 are even numbers. If the sum of the digits of a multi-digit number
produces a number that is divisible by three, then the original multi-digit number is divisible by
3. If the ones digit is a 0 or a 5, then the number is divisible by 5.
• Did you know that if a number is divisible by both 2 and 3, then that number is also divisible
by 6?
• Did you know that if a number is divisible by both 2 and 5, then that number is also divisible
by 10?
• Did you know that if a number is divisible by both 3 and 5, then that number is also divisible
by 15?
Questions:
1. Beside each number you will find a box corresponding to a number that might divide evenly
into the original number. Check off the appropriate boxes to identify whether the original
number is divisible by 2, 3, 5, 6, 10, and/or 15. Do not use a calculator.
2
e.g.
220
e.g.
315
(a)
12
(b)
45
(c)
30
(d)
14
(e)
36
(f)
19
(g)
430
(h)
114
(i)
207
(j)
96
(k)
225
(l)
600
(m)
704
(n)
425
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5




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
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2
(o)
408
(p)
570
(q)
615
(r)
1078
(s)
2310
(t)
7131
(u)
8706
(v)
5603
(w)
4700
(x)
6125
(y)
3210
3
5
6
10
15
2. (a) Create 4 three-digit numbers that are divisible by 2, 3, and 6 (but not 5, 10, or 15). Do
not use numbers encountered in question 1.
(b) Create 4 three-digit numbers that are divisible by 3, 5, and 15 (but not 2, 6, or 10). Do
not use numbers encountered in question 1.
(c) Create 4 four-digit numbers that are divisible by 3 (but not 2, 5, 6, 10, or 15). Do not use
numbers encountered in question 1.
3. Fill in the blanks.
(a) If a number is divisible by both 2 and 7, then that number is also divisible by _____.
(b) If a number is divisible by both 3 and 11, then that number is also divisible by _____.
(c) If a number is divisible by both 5 and 6, then that number is also divisible by _____.
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Divisibility or Prime Connect Four Game
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place one paper clip on the Tens strip and one paperclip on the Ones strip. They have now
generated a two digit number. That two digit number is either divisible by a single digit
whole number greater than 1 (i.e. 2, 3, 4, 5, 6, 7, 8, 9), or the number is a prime. The player
captures a single square that describes the number. For example if the two digit number is
14, it is divisible by 2 or 7 (of the choices we are given), then the player can capture either a
square with a 2 on it, or a square with a 7 on it. If the number is prime, then a square marked
P can be captured.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips on either the Tens or Ones strip. They then mark the square that describes that
number using a O or a different colored marker. If a player cannot move a single paperclip
to capture a square, a paperclip must still be moved in order to ensure that the game can
continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
6
4
7
2
6
3
P
9
6
8
P
2
5
3
P
5
4
9
4
8
9
7
3
2
7
2
4
6
8
P
6
P
9
3
2
5
Tens Strip:
1
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2
3
1
2
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Division with Remainders
Up to this point, we have only worked with division questions that worked out evenly (i.e. no
remainders). Examples of this are shown below.
14 ÷ 7 =
2
21 ÷ 3 =
7
45 ÷ 5 =
9
64 ÷ 8 =
8
70 ÷ 10 =
7
But what happens when division questions do not work out evenly.
19 ÷ 7 =
?
22 ÷ 3 =
?
44 ÷ 5 =
?
67 ÷ 8 =
?
78 ÷ 10 =
?
With these types of questions, we have to talk about remainders. Consider the examples that
follow.
Example 1
Complete the operation: 9 ÷ 4
Answer:
We are going to show you three ways to solve this question. You are only required to learn
the third method (i.e. most efficient method). We have shown the first two methods so that
you understand why the third method works.
Method 1: Sharing Items
Suppose you had 9 apples that you had to share evenly between 4 people. How many apples
would each person get? Are there any apples left over?
Each person gets 2 apples, and 1 apple is left over. This
1 apple is what remains.
Therefore we can conclude that:
9÷4 =
2 with a remainder of 1.
This can also be written as 9 ÷ 4 =
2, R:1
Method 2: Using Cuisenaire Rods
Take the Cuisenaire rod for 9 (color: blue) and figure out how many Cuisenaire rods for 4
(color: purple) fit into it. Since your exercise sheet is photocopied in black and white, we
have used two different textures so that you can distinguish the two types of rods
Rod Representing 9:
Rod Representing 4:
We can place two purple rods (i.e. rods representing 4) on top of the blue rod (i.e. rod
representing 9), but they don't cover the whole thing. A small portion, which is 1 unit long,
is sticking out.
4
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So 2 sets of 4 fit into 9, with 1 left over. That means that 9 ÷ 4 is equal to 2 with a remainder
of 1. This is written as 9 ÷ 4 =
2, R:1 (Same answer that we got when we used the sharing
items method.)
Method 3: Using the Rule
Four does not divide evenly into 9, so start by finding the closest number to 9 that is smaller
than 9 which 4 does divide evenly into. That number is 8 ( 8 ÷ 4 =
2 ) . Now find the
difference between the 8 and 9. The difference is 1 ( 9 − 8 =
1) . This difference is our
remainder.
Therefore: 9 ÷ 4 =
2, R:1
Example 2
Complete the operation: 10 ÷ 3
Answer:
Method 1: Sharing Items
Suppose you had 10 apples that you had to share evenly between 3 people. How many
apples would each person get? Are there any apples left over?
Each person gets 3 apples, and 1 apple is left
over. This 1 apple is what remains.
Therefore we can conclude that:
10 ÷ 3 =
3 with a remainder of 1.
This can also be written as 10 ÷ 3 =
3, R:1
Method 2: Using Cuisenaire Rods
Take the Cuisenaire rod for 10 (color: orange) and figure out how many Cuisenaire rods for 3
(color: lime green) fit into it. Since your exercise sheet is photocopied in black and white,
we have used two different textures so that you can distinguish the two types of rods
Rod Representing 10:
Rod Representing 3:
We can place three lime green rods (i.e. rods representing 3) on top of the orange rod (i.e. rod
representing 10), but they don't cover the whole thing. A small portion, which is 1 unit long,
is sticking out.
3
3
3
So 3 sets of 3 fit into 10, with 1 left over. That means that 10 ÷ 3 is equal to 3 with a
remainder of 1. This is written as 10 ÷ 3 =
3, R:1 (Same answer that we got when we used
the sharing items method.)
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Method 3: Using the Rule
Three does not divide evenly into 10, so start by finding the closest number to 10 that is
smaller than 10 which 3 does divide evenly into. That number is 9 ( 9 ÷ 3 =
3) . Now find the
1) . This difference is our
difference between the 9 and 10. The difference is 1 (10 − 9 =
remainder.
Therefore: 10 ÷ 3 =
3, R:1
For the remaining examples, we will use the "Rule" to obtain our answer. This is the method we
would like you to use when you complete the exercise questions.
Example 3
Complete the following division questions.
(a) 19 ÷ 7
(b) 22 ÷ 3
(c) 44 ÷ 5
(d) 67 ÷ 8
Answers:
(a) Start by finding the closest number to 19 that is smaller than 19 which 7 does divide
2 ) . Now find the difference between the 19 and
evenly into. That number is 14 (14 ÷ 7 =
14. The difference is 5 (19 − 14 =
5 ) . This difference is our remainder.
Therefore: 19 ÷ 7 =
2, R:5
(b) If 21 ÷ 3 =
7 , and 22 − 21 =
1 , then 22 ÷ 3 =
7, R:1
(c) If 40 ÷ 5 =
8 , and 44 − 40 =
4 , then 44 ÷ 5 =
8, R:4
(d) If 64 ÷ 8 =
8 , and 67 − 64 =
3 , then 67 ÷ 8 =
8, R:3
Questions:
1. Complete the following division. Most, but not all, of these questions involve remainders.
(a) 13 ÷ 5
(b) 7 ÷ 2
(c) 14 ÷ 3
(d) 18 ÷ 3
(e) 33 ÷ 7
(f) 43 ÷ 8
(g) 25 ÷ 6
(h) 19 ÷ 2
(i) 29 ÷ 3
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(j) 40 ÷ 6
(k) 42 ÷ 7
(l) 22 ÷ 5
(m) 60 ÷ 9
(n) 31 ÷ 8
(o) 13 ÷ 4
(p) 38 ÷ 5
(q) 48 ÷ 9
(r) 32 ÷ 4
(s) 18 ÷ 4
(t) 19 ÷ 6
(u) 54 ÷ 5
(v) 35 ÷ 8
(w) 50 ÷ 7
(x) 69 ÷ 9
2. There are twenty apples to share equally between six people. How many apples does each
person get? How many apples are left over?
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Long Division (Partial Quotient Method)
Division can be thought as "splitting a number into equal parts." Another way of thinking about
division is to say "how many times can one number be subtracted from another number" (i.e.
repeated subtraction).
e.g. 12 ÷ 4 = ?
"Splitting a Number into Equal Parts"
We were able to split the
number 12 into four equal
parts of three; therefore
12 ÷ 4 = 3
"Repeated Subtraction"
12 - 4 - 4 - 4 = 0
We were able to subtract 4 from 12, three
times. Therefore 12 ÷ 4 = 3
Taking these approaches with larger numbers is far too difficult and time-consuming.
e.g. 52 ÷ 4 = ?
"Splitting a Number into Equal Parts"
We were able to split the
number 52 into four equal
parts of thirteen; therefore
52 ÷ 4 = 13
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"Repeated Subtraction"
52 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 = 0
We were able to subtract 4 from 52, thirteen
times. Therefore 52 ÷ 4 = 13
Now there is the traditional algorithm for long division that some of you may have learned in the
past, but we are going to use a different approach called the Partial Quotient Method. The
biggest problem with the traditional method is that on it is very easy to make a mistake, there is
only one way to complete the question, and that learners often do not understand why the
procedure works. The Partial Quotient Method tends to make more sense because it relies on
repeated subtraction, and allows learners to use multiple ways to obtain the correct answer. To
use this technique, you must have a good grasp of your multiplication facts, be able to multiply
by multiples of 10, 100, and 1000, and be able to subtract multi-digit numbers. These are all
things that we have done in previous lessons.
Example 1
Solve 2856 ÷ 8
Answer:
8 2856
1600
8 2856
1600
1256
8 2856
1600
1256
800
8 2856
1600
1256
800
456
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200
How many times does 8 go into 1600? This learner goes with
200 because he/she knows that 8 × 200 = 1600 . (The learner
could have gone with a larger number like 300, but it does not
matter with this method.)
Now he/she subtracted 1600 from 2856.
200
200
How many times does 8 go into 1256. This learner goes with
100 because he/she knows that 8 × 100 = 800 .
100
Now he/she subtracted 800 from 1256.
200
100
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8 2856
1600
1256
200
800
456
100
400
50
8 2856
1600
1256
800
456
400
56
8 2856
1600
1256
800
456
400
How many times does 8 go into 456. This learner goes with 50
because he/she knows that 8 × 50 = 400 .
Now he/she subtracted 400 from 456.
200
100
50
200
How many times does 8 go into 56. This learner goes with 7
because he/she knows that 8 × 7 = 56 . After we filled in the 56,
we did the subtraction and found that we had a remainder of 0.
That means 8 divides evenly into 2856.
100
50
56
56
0
357
8 2856
1600
1256
800
456
400
56
56
0
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In our last step the learner simply adds 200, 100, 50, and 7. That
means that the quotient is 357.
200
2856 ÷ 8 = 357
100
50
7
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View the following YouTube video that walks you through Examples 2 and 3.
• http://www.youtube.com/watch?v=yx4vRU233zU
(or Google Search: YouTube Long Division (Partial Quotient Method) nsccalpmath)
Example 2
Solve 4785 ÷ 7
Answer:
We have shown you three solutions to this question. The first student was the most efficient
because he/she did the question in the fewest number of steps, however, all of the students
have correct answers. That is the great thing about the partial quotient method; there is more
than one way to do it right. In this case, 7 does not go evenly into 4785; we have a
remainder of 4 when we complete the division.
First Learner:
683 R: 4
7 4785
600
4200
585
560
80
25
3
21
4
Second Learner:
683 R: 4
7 4785
400
2800
1985
1400
200
585
80
560
25
3
21
4
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Third Learner:
683 R: 4
7 4785
600
4200
585
350
50
235
30
210
25
3
21
4
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Example 3
Solve 31 654 ÷ 9
Answer:
We have again supplied multiple solutions; all of them are correct.
First Learner
3517 R: 1
9 31654
27000
4654
4500
154
90
64
63
1
3000
500
10
7
Second Learner:
3517 R: 1
9 31654
2000
18000
13654
9000
1000
4654
400
3600
1054
900
154
90
64
63
1
100
10
Third Learner:
3517 R: 1
9 31654
3000
27000
4654
4500
500
154
10
90
64
5
45
19
2
18
1
7
Questions
Complete each of the division questions. Show all of your work.
1.
2.
5 3185
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3.
4.
3 2256
5.
6 1524
6.
4 1489
7.
5 4642
8.
9 3479
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9.
10.
8 4664
11.
6 5236
12.
4 3833
13.
3 2903
14.
3 12651
5 13705
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15.
16.
4 22945
17.
6 28581
18.
7 25144
19.
9 58712
20.
8 41738
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Prime Factorization
Prime numbers are numbers that are only divisible by one or itself. The first eight prime
numbers are 2, 3, 5, 7, 11, 13, 17, and 19. Factors are numbers that multiply together to get
another number. For example, the numbers 3 and 4 are factors of 12 because 3 × 4 =
12 . The
numbers 2 and 6 are also factors of 12 because 2 × 6 =
12 . Prime factorization is the process of
finding the prime numbers that multiply together to make another number.
Example
Determine the prime factors of each of the following.
(a) 12
(b) 40
(c) 150
Answers:
We these questions, there are often multiple ways of arriving at the final answer.
(a) Method 1:
In this case the learner starts by expressing 12 as the product of 2 and 6.
12= 2 × 6
The learner then realizes that 6 is not a prime number, so proceeds to factor the 6.
12= 2 × 6
12 =2 × ( 2 × 3)
12 = 2 × 2 × 3
← Final Answer
Method 2:
In this case the learner starts by expressing 12 as the product of 3 and 4.
12= 3 × 4
The learner then realizes that 4 is not a prime number, so proceeds to factor the 4.
12= 3 × 4
12 =3 × ( 2 × 2 )
12 = 2 × 2 × 3 ← Final Answer
The two learners got the same answer even though they started out on slightly different
paths.
(b) 40= 8 × 5
40 = ( 2 × 4 ) × 5
← 8 is not prime so it's expressed as 2 × 4
40 = 2 × 4 × 5
40 =2 × ( 2 × 2 ) × 5 ← 4 is not prime so it's expressed as 2 × 2
40 = 2 × 2 × 2 × 5
← Final Answer
(c) 150= 15 ×10
150 = ( 3 × 5 ) × ( 2 × 5 )
150 = 2 × 3 × 5 × 5
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← Neither 15 nor 10 is a prime number
← Final Answer
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Questions
1. Determine the prime factors for each of the following.
(a) 6
(b) 21
(c) 10
(d) 35
(e) 49
(f) 26
(g) 33
(h) 20
(i) 44
(j) 42
(k) 45
(l) 66
(m) 30
(n) 70
(o) 27
(p) 63
(q) 110
(r) 18
(s) 16
(t) 100
(u) 36
(v) 250
(w) 81
(x) 24
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Multiple Operations
(Whole Numbers)
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Express the Number in Multiple Ways
For each number, express it as:
• At least two number sentences involving addition,
• At least two number sentences involving subtraction,
• At least two number sentences involving multiplication,
• At least two number sentences involving division, and
• At least three written sentences.
(Please note that answers will vary from learner to learner.)
Example: Number 10
• Two number sentences involving addition: 6 + 4 = 10 , 5 + 5 = 10
• Two number sentences involving subtraction: 12 − 2 = 10 , 29 − 19 = 10
• Two number sentences involving multiplication: 1 × 10 = 10 , 2 × 5 = 10
• Two number sentences involving division: 30 ÷ 3 = 10 , 80 ÷ 8 = 10
• Three written sentences: The number is three more than seven
The number is half of twenty.
The number is five times bigger than two.
Questions
(a) Number 4
(b) Number 6
(c) Number 8
(d) Number 9
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Find the Center
With these puzzles, you must determine the missing number in the center of the cross. You are
provided with two numbers in each branch of the cross. Those two numbers are used to generate
the number in the center but you are not told what operation (addition, subtraction,
multiplication, or division) applies to each branch. However, you do know that each operation is
represented by a single branch.
Example
Find the missing number in the center of the puzzle.
12
Answer:
The center number must be 4 because:
9−5 =
4
12 ÷ 3 =
4
1+ 3 =
4
2× 2 =
4
3
9
5
1
3
2
2
Questions
Find the missing number in the center of each puzzle.
(a)
(b)
6
1
4
2
(d)
18
3
4
2
3
10
5
5
6
3
1
(e)
2
(f)
24
6
3
7
5
14
2
20 10
4
40
1
3
4
1
(i)
21
2
28
2
2
40
2
30
10
4
12
8
4
7
10
3
2
2
111
5
50
7
14
6
2
9
(h)
30
8
2
5
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24
12
7
1
12
6
(g)
4
2
15
18
(c)
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Name the Preceding or Next
Fill in the blank with the missing number. Two examples have been completed to assist you.
e.g. Name the next even number to each of the following.
(a) 36 : _____
(b) 74 : _____
Answers:
(a) 36 : 38
(b) 74 : 76
(c) 88 : 90
e.g. Name the preceding multiple of five.
(a) 45 : _____
(b) 30 : _____
Answers:
(a) 45 : 40
1.
(c) 55 : _____
(b) 30 : _____
(c) 90 : _____
(b) 15 : ____
(c) 27 : _____
(b) 28 : _____
(c) 16 : _____
(b) 9 : _____
(c) 27 : _____
Name the next multiple of four to each of the following.
(a) 28 : _____
9.
(b) 20 : _____
Name the preceding multiple of three to each of the following.
(a) 15 : _____
8.
(c) 50 : _____
Name the preceding multiple of four to each of the following.
(a) 36 : _____
7.
(b) 58 : _____
Name the next multiple of three to each of the following.
(a) 21 : _____
6.
(c) 29 : _____
Name the next multiple of ten to each of the following.
(a) 60 : _____
5.
(b) 63 : _____
Name the preceding multiple of five to each of the following.
(a) 35 : _____
4.
(c) 100 : 95
Name the preceding even number to each of the following.
(a) 36 : _____
3.
(b) 30 : 25
(c) 100 : _____
Name the next odd number to each of the following
(a) 47 : _____
2.
(c) 88 : _____
(b) 20 : _____
(c) 4 : _____
Name the next multiple of six to each of the following.
(a) 24 : _____
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One of these Things is Not Like the Others
With each of these questions, you must identify which one of the four does not belong. You
must also explain why that one does not belong and why the remaining three belong together.
(Hint: Think about odd and even, prime and composite, divisibility, patterns, perfect squares…)
e.g.
13
15
2
9
The 2 does not belong because it is an even number and the remaining numbers, 13,
15, and 9, are all odd numbers.
1.
10
7
12
4
2.
5
12
6
9
3.
15
6
8
11
4.
3, 5, 7, 9, 11…
1, 2, 4, 8, 16,…
2, 7, 12, 17, 22,…
1, 4, 7, 10, 13,…
5.
25
9
16
6
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6.
22
15
40
55
7.
17
7
9
19
8.
31, 34, 37, 40,…
18, 21, 24, 27…
66, 63, 60, 57,…
52, 55, 58, 61,…
9.
dozen
4+4+4
2× 6
fourteen
10.
one hundred five
twenty-six
eighty-seven
fifty-two
11.
8+7
10 - 5
5×3
17 - 2
12.
ninety-nine
90 +9
9×9
one less than 100
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Fact Family Puzzle (Multiplications and Division)
Print the following onto rigid paper, cut each fact family into four puzzle pieces, shuffle all the
family facts together, and ask the learners to sort them into their appropriate families.
3× 4
4× 3
2× 7
7× 2
12÷ 3
12÷ 4
14÷ 2
14÷ 7
5× 6
6× 5
8×1
1×8
30÷ 5
30÷ 6
8÷ 8
8÷1
2× 9
9× 2
4× 6
6× 4
18÷ 2
18÷ 9
24÷ 4
24÷ 6
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3× 5
5× 3
6× 9
9× 6
15÷ 3
15÷ 5
54÷ 6
54÷ 9
8× 4
4×8
3× 7
7× 3
32÷ 8
32÷ 4
21÷ 3
21÷ 7
8× 6
6×8
6× 7
7× 6
48÷ 8
48÷ 6
42÷ 6
42÷ 7
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Provide the Other Members of the Fact Family
Example
In each case, one member of a four member fact family has been provided. Provide the other
three members of that family.
(b) 5600 ÷ 800 =
(c) 80 + 90 =
(a) 9 − 6 =
3
7
170
Answers:
(a) 9 − 3 =
6
3+ 6 =
9
6+3=
9
(b) 5600 ÷ 7 =
800
7 × 800 =
5600
800 × 7 =
5600
(c) 90 + 80 =
170
170 − 80 =
90
170 − 90 =
80
Questions
1. For each question, provide the three missing members of the fact families.
(a) 35 ÷ 7 =
5
(b) 6 × 7 =
42
(c) 8 + 5 =
13
(d) 16 − 9 =
7
(e) 60 + 50 =
110
(f) 240 ÷ 3 =
80
(g) 800 − 500 =
300
(h) 40 × 7 =
280
(i) 1500 ÷ 50 =
30
2. Complete each question and then provide the three missing members of the fact family.
(a) 9 × 4 =
_______
(b) 140 − 60 =
_______
(c) 2100 ÷ 700 =
_______
(d) 400 + 900 =
_______
(e) 90 × 60 =
_______
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What Number Does the Star Represent?
Example
For each question, determine what number is represented by the star?
(a) 4 +  = 10
(b)  +  = 14
(d)  ×  = 16
(c)  × 8 = 40
(e) 12 -  = 7
(f)  ÷ 2 = 6
Answers:
(a) What number when added to 4 gives you 10? The answer is 6. ( = 6)
(b) What number when added to itself gives you 14? The answer is 7. ( = 7)
(c) What number when multiplied by 8 gives you 40? The answer is 5. ( = 5)
(d) What number when multiplied by itself gives you 16? The answer is 4. ( = 4)
(e) What number subtracted from 12 gives you 7? The answer is 5. ( = 5)
(f) What number divided by 2 gives you 6? The answer is 12. ( = 12)
Questions
In each case determine the number that is represented by the star? No work needs to be shown.
(a)  + 2 = 10
 = _____
(b)  +  = 18
 = _____
(c) 9 -  = 6
 = _____
(d)  × 7 = 28
 = _____
(e)  +  = 10
 = _____
(f)  ÷ 3 = 5
 = _____
(g) 30 ÷  = 6
 = _____
(h)  ×  = 9
 = _____
(i)  + 7 = 13
 = _____
(j)  +  = 6
 = _____
(k)  ÷ 5 = 9
 = _____
(l) 6 ×  = 24
 = _____
(m)  ×  = 25
 = _____
(n)  - 8 = 3
 = _____
(o)  +  = 16
 = _____
(p) 6 +  = 14
 = _____
(q) 36 ÷  = 4
 = _____
(r)  ×  = 49
 = _____
(s) 7 ×  = 21
 = _____
(t) 15 -  = 6
 = _____
(u)  ×  = 16
 = _____
(v)  +  = 20
 = _____
(w)  ÷ 9 = 8
 = _____
(x) 8 ×  = 48
 = _____
(y) 9 +  = 10
 = _____
(z)  ×  = 81
 = _____
NSSAL
©2012
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Pathways
Find the pathway from the upper left hand corner to the lower right hand corner of each grid by
moving to equivalent and adjacent squares (i.e. squares to the left, right, top, or bottom).
Start
Start
2× 4
7×3
9 ÷1
9−7
24 ÷ 8
12 ÷ 2
9−3
3× 2
24 ÷ 3
7 +1
16 ÷ 2
6+2
6−4
3× 6
11 + 1
5+3
4×3
10 − 4
4+4
8×8
7−2
12 − 4
8+3
19 − 7
18 + 2
6 ×1
18 ÷ 3
5 +1
7×7
9−2
9+2
4× 2
3+5
12 ÷ 4
3−2
0+6
18 − 6
4×5
8 ÷1
7+5
7×6
16 ÷ 4
32 ÷ 4
8 ×1
9 −1
15 − 9
3+3
30 ÷ 5
8−2
4+2
Finish
Finish
Start
Start
15 − 3
8× 2
9−6
15 + 3
25 ÷ 5
9× 2
21 − 3
18 × 1
17 − 6
2×8
6+6
10 − 2
7+9
8× 6
15 ÷ 3
1 + 19
3× 3
9+9
24 − 6
6×6
12 × 1
7×8
8+4
12 ÷ 1
2×6
32 ÷ 4
0×6
36 ÷ 9
13 + 5
27 ÷ 3
18 − 6
3× 4
20 − 8
7×4
0 + 12
28 − 9
35 ÷ 7
4 × 10
6×3
19 − 4
7+6
21 ÷ 7
8+9
18 − 4
9+3
10 × 8
12 + 7
17 ÷ 1
7 + 11
18 + 0
Finish
NSSAL
©2012
Finish
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C. D. Pilmer
Two of These Boxes Just Don't Belong (A)
Three boxes in each row have the same answers; the remaining two boxes just don't belong.
Circle the three boxes in each row that have the same answers.
1.
5×1
10 ÷ 2
2×3
6+1
9-4
4+3
5-2
21 ÷ 3
8+1
11 - 4
8-5
10 - 4
24 ÷ 4
3× 2
9+6
20 - 2
17 - 7
3×3
40 ÷ 4
2×5
4×4
5+3
4×2
12 - 2
40 ÷ 5
15 - 4
8+6
33 ÷ 3
11 × 1
3×4
13 + 4
30 ÷ 2
7×3
20 - 5
5×3
6×2
12 - 2
4×3
7+8
9+3
19 - 6
5×4
14 + 6
40 ÷ 2
22 ÷ 2
21 - 3
8 ÷2
6+7
3×6
9+9
2.
3.
4.
5.
6.
7.
8.
9.
10.
NSSAL
©2012
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C. D. Pilmer
Two of These Boxes Just Don't Belong (B)
Three boxes in each row have the same answers; the remaining two boxes just don't belong.
Circle the three boxes in each row that have the same answers.
1.
3×3
8×3
36 ÷ 4
12 - 5
4+5
30 + 20
4 × 10
15 × 2
5×8
80 ÷ 2
8× 2
20 - 4
24 ÷ 4
4× 4
15 - 1
9+5
12 - 2
28 ÷ 2
2×7
14 × 0
4×7
90 ÷ 3
26+4
40 - 20
5×6
25 + 25
60 - 20
40 + 20
100 ÷ 2
5 × 10
100 - 20
90 - 30
20 × 3
120 ÷ 2
70 + 10
20 - 4
40 × 4
240 ÷ 3
10 × 8
50 + 30
0+0
50 - 50
9×1
7÷7
0×8
50 + 60
2 + 45
3 × 30
450 ÷ 5
100 - 10
2.
3.
4.
5.
6.
7.
8.
9.
10.
NSSAL
©2012
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C. D. Pilmer
Equivalent
Determine the missing number.
Example 1:
5× 6 = 3×
Example 2:
6+8=
×2
The answer is 7 because 6 + 8 and 7 × 2
both equal 14.
The answer is 10 because 5 × 6 and 3 × 10
both equal 30.
Part 1
(a) 4 + 6 = 2 +
(b)
9+3=
5+ 4 = 3+
(e)
Part 2
(a) 12 − 2 = 14 −
24 − 3 = 29 −
(c)
8 + 8 = 12 +
6+8= 7+
(f)
5+8 =
+ 11
(b)
14 − 7 = 11 −
(c)
9−3=
−1
(e)
18 − 3 = 20 −
(f)
17 − 6 =
Part 3
(a) 2 × 6 = 3 ×
(b)
5× 4 =
×2
(c)
8× 2 =
5×8 = 4×
(e)
6× 4 =
×8
(f)
4×9 = 6×
(b)
30 ÷ 5 =
(c)
14 ÷ 2 = 28 ÷
(e)
27 ÷ 9 = 15 ÷
(f)
70 ÷ 7 =
(b)
16 ÷ 4 = 9 −
(c)
5 × 7 = 39 −
(d)
(d)
(d)
Part 4
(a) 24 ÷ 3 = 16 ÷
(d)
42 ÷ 7 =
÷5
Part 5
(a) 4 × 7 = 20 +
+7
÷2
−4
×4
÷4
(d)
72 ÷ 9 = 11 −
(e)
6 × 6 = 32 +
(f)
4 + 5 = 54 ÷
(g)
15 − 10 = 30 ÷
(h)
9 × 7 = 64 −
(i)
1× 6 =
(j)
19 − 3 = 8 ×
(k)
39 + 3 =
×6
(l)
15 − 7 = 32 ÷
(m)
7 × 3 = 26 −
(n)
56 ÷ 8 =
+3
(o)
18 + 3 = 3 ×
(p)
8× 0 = 9 −
(q)
7×5 =
(r)
32 ÷ 8 = 10 −
(s)
45 ÷ 9 =
(t)
7×7 = 4 +
(u)
23 + 7 = 6 ×
(v)
9 − 1 = 16 ÷
(w)
42 ÷ 7 =
(x)
9+3=
NSSAL
©2012
−3
+4
×6
122
÷3
−5
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C. D. Pilmer
Greater Than, Less Than or Equal To; Whole Number Operations
Greater Than: >
Less Than: <
Equal To: =
1. Place the appropriate sign (<, >, or =) between the two operations.
(a) 3+7
5+5
(b) 4 × 5
3× 6
(c) 12 ÷ 4
10 ÷ 2
(d) 12 − 7
9−5
(e) 3 × 4
2× 6
(f) 6 + 9
11 + 5
(g) 24 ÷ 3
12 − 7
(h) 11 + 11
3× 7
(i) 5 + 2
63 ÷ 7
(j) 4 × 7
30 − 2
(k) 50 + 40
8 ×10
(l) 80 − 50
120 ÷ 3
(m) 30 + 90
30 × 40
(n) 120 − 50
9× 6
(o) 160 ÷ 8
13 + 6
(p) 180 ÷ 90
17 − 12
(q) 3 × 70
70 + 70 + 70
(r) 20 + 30
5× 2× 5
(s) 60 − 20
2× 4×3
(t) 280 ÷ 7
3× 3× 5
(u) 130 − 40
20 + 50 + 30
(v) 6 × 2 ×1
240 ÷ 80
2. Using the numbers 2, 3, 4, 5, or 6, fill in the blank such that the statement is correct. In some
cases, there may be more than one acceptable answer.
(a) 6 × 5=
×10
(c) 35 ÷ 5 <
(e)
(b) 24 ÷
+3
(d)
+ 6 > 15 − 5
(g) 45 ÷
>9−2
× 2 = 48 ÷ 6
(f) 5 ×
=
16 − 7
(h)
< 28 − 7
× 7 = 10 + 30
3. Fill in the blanks such that the statement is correct. You can use any numbers you wish.
Naturally there are an infinite number of correct answers.
(a)
NSSAL
©2012
÷
<
×
(b)
123
+
>
×
Draft
C. D. Pilmer
Find the Digit Based on the Reasonable Estimate
Find the missing digit based on the estimate provided.
1. If __845 ÷6 is about 300, then what number should be filled in to replace
the missing digit?
______
2. If __2 ×81 is close to 3400, then what number should be filled in to replace
the missing digit?
______
3. If __49 - 754 is about 200, then what number should be filled in to replace
the missing digit?
______
4. If 3945 + __078 is close to 6000, then what number should be filled in to
replace the missing digit?
______
5. If __95 ÷5 is about 80, then what number should be filled in to replace
the missing digit?
______
6. If __93 + 389 is almost 900, then what number should be filled in to
replace the missing digit?
______
7. If 71 × __9 is close to 6300, then what number should be filled in to replace
the missing digit?
______
8. If 4238 ÷ __1 is about 60, then what number should be filled in to replace
the missing digit?
______
9. If __93 + 211 is about 700, then what number should be filled in to
replace the missing digit?
______
10. If 62 × __9 is close to 2400, then what number should be filled in to replace
the missing digit?
______
11. If __51 - 349 is about 600, then what number should be filled in to replace
the missing digit?
______
12. If __041 ÷62 is about 50, then what number should be filled in to replace
the missing digit?
______
13. If __10 ×39 is close to 28 000, then what number should be filled in to replace
the missing digit?
______
14. If __92 + 704 is almost 1600, then what number should be filled in to
replace the missing digit?
NSSAL
©2012
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C. D. Pilmer
Venn Diagrams and Whole Numbers
A Venn diagram, which is normally comprised of overlapping circles, is used to show
relationships between different things. For this activity we are going to use them to illustrate
relationships between different whole
numbers. Suppose a learner is given the
11
5
numbers 3, 5, 6, 9, 12, 11, 14, 16, 20, 21, and
23. They are asked to take those numbers and
16
3
identify those that are divisible by 2, and
6
those that are divisible by 3. Now some of
Divisible by 2
these numbers (e.g. 5, 11, 23) are not divisible
by either 2 or 3. Some numbers (e.g. 14, 16,
20) are only divisible by 2, others (e.g. 3, 9,
21) are divisible by only 3, and still others
(e.g. 6, 12) are divisible by both 2 and 3. A
Venn diagram, like the shown on the right,
can be used to illustrate this.
14
23
Divisible by 3
9
12
20
21
With the questions below, you have been given an incomplete Venn diagram and a list of
numbers. Your mission is to place the numbers correctly in the Venn diagram.
1. List:
4, 5, 8, 9, 10, 14, 15, 20, 21, 22, 25, 30
Divisible by 2
Divisible by 5
Divisible by 3
Divisible by 5
2. List:
6, 9, 10, 11, 12, 15, 16, 18, 20, 25, 27, 30
NSSAL
©2012
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C. D. Pilmer
3. List:
2, 3, 4, 7, 9, 10, 13, 15, 18, 20, 21, 23, 25
Note: A prime number is a number that
can be divided evenly only by 1 or itself.
(e.g. The number 11 is a prime number
because it is only divisible by 1 and 11.)
Odd
Prime
4. List:
3, 4, 7, 9, 10, 15, 16, 18, 25, 27, 28, 30, 36
Note: For our purposes, a perfect square is
a number that can be expressed as a whole
number squared. (e.g. 81 is a perfect
square because 81 = 92)
Perfect
Square
Even
5. List:
7, 9, 10, 13, 15, 18, 30, 36, 45, 48, 55, 90
Multiple of 9
NSSAL
©2012
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Multiple of 5
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C. D. Pilmer
Whole Number Crossword Puzzle (A)
A
B
C
F
H
D
E
G
I
J
K
L
M
O
R
N
P
S
Q
T
V
U
W
Across:
Down:
A. Next even number after 384
B. 8 × 10
C. 22 + 10 + 10
D. 2000 + 100 + 30 + 9
G. one thousand, four hundred twenty
E.
337 - 10
I. 5 more than 228
F.
8× 90
J. Double 25
H. 7 less than 470
L. The product of 4 and 8
K. Next number in the following sequence.
70, 74, 78, 82, ____
O. 196 + 231
Q. 143 - 87
R. 5 times 7
T. The number of minutes in 1 hour and 34
minutes
M. 3 sets of 9
N. increase 734 by 20
P.
11 + 5 + 3 + 9
S.
Next number in the following sequence.
63, 60, 57, 54, ____
V. 42
W. A number between 10 and 20 that is
divisible by both 5 and 3
NSSAL
©2012
U. The number of cents in 2 quarters, 1 dime,
and 1 nickel
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Whole Number Crossword Puzzle (B)
A
B
C
F
H
D
E
G
I
J
K
L
M
O
R
N
P
S
Q
T
V
U
W
Across:
Down:
A. 50 × 9
B. The next odd number after 51
C. 5 more than 81
D. six thousand, four hundred thirty-nine
G. 4000 + 800 + 10 + 5
E.
213 rounded to the nearest tens
I. increase 153 by 30
F.
Next number in the following sequence
394, 399, 404, 409, ____
J. 63 - 29
L. ____ ÷ 7 = 4
O. The number of minutes in 6 hours and 4
minutes
Q. A number between 10 and 20 that is
divisible by 2, 3, 6, and 9
R. decrease 70 by 7
T. The product of 2 and 7
V. 5 + 10 + 2 + 30
H. 156 + 316
K. 72
M. Double 12
N. 1542 ÷ 3
P.
6 sets of 11
S.
6 less than double 20
U. Next number in the following sequence
60, 54, 48, 42, ____
W. The even number before 88
NSSAL
©2012
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C. D. Pilmer
Whole Number Crossword Puzzle (C)
A
B
C
F
H
D
E
G
I
J
K
L
M
O
R
N
P
S
Q
T
V
U
W
Across:
Down:
A. 70 × 8
B. 6 + 20 + 2 + 40
C. Triple 6 plus 1
D. nine thousand, seven hundred twelve
G. 7558 rounded to the nearest hundreds
E.
Next number in the following sequence
886, 890, 894, 898, 902, ____
F.
15× 23
I. increase 361 by 40
J. Number of cents in 3 quarters and 2 dimes
L. 9 times 6
O. 800 + 70 + 4
Q. 37 + 56
R. 581 ÷ 7
T. Double 13
H. The next multiple of 5 that follows 130
K. 8 sets of 4
M. 82
N. Number of minutes in 3 hours and 16
minutes
P.
161 - 87
V. Next number in the following sequence
50, 47, 44, 41, ____
S.
39 decreased by 6
W. ____ ÷ 6 = 8
U. A number between 20 and 30 that is
divisible by 2, 4, 7, and 14
NSSAL
©2012
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Whole Number Crossword Puzzle (D)
A
B
C
F
H
D
E
G
I
J
K
L
M
O
R
N
P
S
Q
T
V
U
W
Across:
Down:
A. The next odd number after 769
B. 87 decreased by 9
C. 6 sets of 3
D. 8000 + 600 + 20 + 9
G. six thousand, three hundred seven
E.
746 increased by 60
I. Next number in the following sequence
338, 344, 350, 356, ____
F.
50 less than 784
J. 444 ÷ 6
H. Number of minutes in 2 hours and 37
minutes
L. 25+47
K. triple 8 plus 4
O. 1 + 30 + 4 + 100 + 50
M. Next number in the following sequence
107, 104, 101, 98, ____
Q. 6 times 7
R. ____ ÷ 9 = 4
T. Number of cents in 2 quarters and 3 dimes
V. A number between 40 and 50 that is a
multiple of 3, 5, 9, and 15
N. 24 × 35
P.
92
S.
Double 32
U. 119 - 37
W. The product of 8 and 9
NSSAL
©2012
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C. D. Pilmer
KenKen Puzzles (A)
KenKen puzzles were invented in 2004 by Japanese math teacher Tetsuya Miyamoto. The goal
is to fill a grid with numbers such that no number is repeated in the same row or column. For a 3
by 3 KenKen grid, one can only use the numbers 1, 2, and 3. For a 4 by 4 KenKen grid, one can
only use the digits 1, 2, 3, and 4.
In addition to this, grids are divided into heavily outlined groups of cells, called cages. The
numbers in these cages have to produce the target number using the indicated operation. For
example if two cells are within a cage and 5+ is shown in an upper corner, then one find two
numbers that add to 5. If working with a 3 by 3 grid where we are restricted to the numbers 1, 2,
and 3, the only possible numbers are 2 and 3.
All of this will make more sense after viewing the following online video.
http://www.youtube.com/watch?v=eik2syOmwSM
(or Google Search: YouTube KenKen Will Shortz Introduces New Puzzle)
Questions:
Complete the following KenKen Puzzles. Since these are 3 by 3 puzzles, you can only use the
numbers 1, 2, and 3. Work in pencil and make sure you have a good eraser.
(a)
5+
4+
1
(b) 3+
5+
2
3+
(c)
3+
(d) 6+
4+
3+
5+
3
NSSAL
©2012
5+
4+
5+
3+
4+
4+
131
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C. D. Pilmer
(e)
5+
3+
(f) 4+
4+
(g)
2
4+
1
5+
5+
3
8+
3+
(h) 3+
4+
4+
4+
7+
(j) 5+
5+
3+
(i)
4+
6+
5+
7+
3+
NSSAL
©2012
1
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C. D. Pilmer
KenKen Puzzles (B)
Insert the numbers 1, 2, and 3 into the grid such that:
• no number is repeated in the same row or column, and
• the numbers in the cages produce the target number using the indicated operation. In
these questions, we have limited ourselves to the operations of addition and
multiplication.
(a)
2×
4+
5+
(b) 9 ×
6×
3+
3+
6×
(c)
6×
4+
(d) 12 ×
18 ×
3×
4+
3+
(e)
6×
4+
(f) 6 ×
3+
2
NSSAL
©2012
3+
5+
3×
3×
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KenKen Puzzles (C)
Normally with 3 by 3 KenKen Puzzles, we only use the numbers 1, 2, and 3 in the grid. We are
going to change this. For the puzzles below, we are going to use the numbers 3, 4, and 5.
New Instructions: Insert the numbers 3, 4, and 5 into the grid such that:
• no number is repeated in the same row or column, and
• the numbers in the cages produce the target number using the indicated operation. In
these questions, we have limited ourselves to the operations of addition and
multiplication.
(a)
12 ×
(b) 15 ×
8+
4
12 ×
9+
15 ×
(c)
5
15 ×
7+
4
(d) 8+
12+
7+
12+
12 ×
8+
20 ×
(e)
20 ×
9+
(f) 12 ×
11+
15 ×
8+
14+
12 ×
5
NSSAL
©2012
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C. D. Pilmer
KenKen Puzzles (D)
Normally with 3 by 3 KenKen Puzzles, we only use the numbers 1, 2, and 3 in the grid. We are
going to change this. For the puzzles below, we are going to use the numbers 5, 6, and 7.
New Instructions: Insert the numbers 5, 6, and 7 into the grid such that:
• no number is repeated in the same row or column, and
• the numbers in the cages produce the target number using the indicated operation.
(a)
35 ×
(b) 42 ×
13+
5
11+
12+
42 ×
(c)
30 ×
11+
30 ×
19+
(d) 17+
7
35 ×
11+
18+
13+
(e)
7
42 ×
11+
5
13+
13+
35 ×
NSSAL
©2012
42 ×
(f) 17+
30 ×
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KenKen Puzzles (E)
Insert the numbers 1, 2, 3, and 4 into the grid such that:
• no number is repeated in the same row or column, and
• the numbers in the cages produce the target number using the indicated operation. In
these questions, we have limited ourselves to the operations of addition and
multiplication.
(a)
8×
12 ×
4+
1
(b) 3 ×
6+
6+
4
6×
6×
12 ×
(d) 6+
1
2
NSSAL
©2012
3×
7+
8+
4
3+
24 ×
3×
9+
4+
8×
1
12 ×
4×
7+
6×
6+
(c)
2
1
2
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C. D. Pilmer
KenKen Puzzles (F)
Insert the indicated numbers into the grid such that:
• no number is repeated in the same row or column, and
• the numbers in the cages produce the target number using the indicated operation (e.g.
8 × : find two numbers when multiplied give you 8).
(a) 1, 2, 3, 4 Puzzle
3×
(b) 1, 2, 3, 4 Puzzle
8×
6×
6×
12 ×
5+
5+
12 ×
4×
3+
8+
12 ×
15 ×
5+
9+
2×
12+
6+
6+
10 ×
8×
6+
NSSAL
©2012
3
(d) 2, 3, 4, 5 Puzzle
20 ×
15 ×
3+
5+
(c) 2, 3, 4, 5 Puzzle
8×
5+
6×
8+
5
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(e) 3, 4, 5, 6 Puzzle
24 ×
(f) 3, 4, 5, 6 Puzzle
7+
8+
11+
30 ×
18 ×
10+
15 ×
15 ×
(g) 4, 5, 6, 7 Puzzle
11+
30×
42×
4
28×
4
17+
20×
20×
15+
13+
35×
11+
(i) 5, 6, 7, 8 Puzzle
48 ×
NSSAL
©2012
42×
4
(j) 6, 7, 8, 9 Puzzle
15+
30 ×
13+
56 ×
72 ×
15+
14+
35 ×
12+
10+
(h) 4, 5, 6, 7 Puzzle
11+
12+
11+
8+
30 ×
11+
12 ×
11+
56 ×
42 ×
17+
63 ×
14+
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C. D. Pilmer
Find the Two Numbers
Example:
Find the two numbers that multiply to give 12, and add to give 7.
Answer:
The numbers are 3 and 4 because 3 × 4 = 12 and 3 + 4 = 7 .
Questions:
Find two numbers that:
(a)
multiply to give 15,
and add to give 8.
(b)
Answer: ____ & ____
(d)
multiply to give 12,
and add to give 13.
multiply to give 16,
and add to give 8.
(e)
multiply to give 12,
and add to give 8.
(h)
multiply to give 22,
and add to give 13.
(k)
multiply to give 40,
and add to give 14.
(n)
multiply to give 35,
and add to give 12.
(q)
multiply to give 8, and
add to give 9.
Answer: ____ & ____
NSSAL
©2012
multiply to give 10,
and add to give 11.
multiply to give 42,
and add to give 13.
multiply to give 63,
and add to give 16.
(t)
multiply to give 50,
and add to give 27.
(l)
multiply to give 60,
and add to give 23.
Answer: ____ & ____
139
multiply to give 25,
and add to give 10.
Answer: ____ & ____
(o)
multiply to give 24,
and add to give 14.
Answer: ____ & ____
(r)
multiply to give 32,
and add to give 12.
Answer: ____ & ____
(u)
Answer: ____ & ____
(w)
multiply to give 36,
and add to give 13.
Answer: ____ & ____
Answer: ____ & ____
Answer: ____ & ____
(v)
(i)
Answer: ____ & ____
Answer: ____ & ____
(s)
multiply to give 30,
and add to give 17.
multiply to give 40,
and add to give 13.
Answer: ____ & ____
Answer: ____ & ____
Answer: ____ & ____
(p)
(f)
Answer: ____ & ____
Answer: ____ & ____
(m)
multiply to give 18,
and add to give 9.
multiply to give 28,
and add to give 11.
Answer: ____ & ____
Answer: ____ & ____
Answer: ____ & ____
(j)
(c)
Answer: ____ & ____
Answer: ____ & ____
(g)
multiply to give 20,
and add to give 12.
multiply to give 24,
and add to give 10.
Answer: ____ & ____
(x)
multiply to give 100,
and add to give 20.
Answer: ____ & ____
Draft
C. D. Pilmer
Which Combination of Numbers Works?
In each case you have been given three numbers and an incomplete calculation. Insert the
numbers in the appropriate positions to make the calculation complete.
(a)
1
4
5
×
-
=
19
(b)
2
5
7
×
+
=
17
(c)
3
6
8
+
-
=
5
(d)
4
5
6
×
+
=
34
(e)
3
6
7
×
-
=
15
(f)
4
5
20
÷
+
=
10
(g)
3
7
9
+
-
=
13
(h)
5
7
8
×
-
=
27
(i)
4
8
12
÷
+
=
14
(j)
2
5
20
÷
-
=
2
(k)
6
7
10
+
-
=
9
(l)
3
6
7
×
+
=
45
(m)
3
5
15
÷
+
=
6
(n)
8
16
24
÷
+
=
26
(o)
5
6
9
×
-
=
49
NSSAL
©2012
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C. D. Pilmer
Magic Squares
In a magic square, the numbers in each column, row, and diagonal all
add up to the same number. For example, with the magic square on the
right, the numbers in each column, row, and diagonal all add up to 30.
7
14
9
12
10
8
11
6
13
Complete each of the magic squares below.
(a)
(b)
3
5
(c)
6
3
7
2
7
6
7
(d)
5
(e)
5
(f)
4
4
10
8
6
1
8
(g)
5
(i)
4
12
7
NSSAL
©2012
7
8
7
(h)
2
3
12
5
9
4
10
3
141
9
8
6
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C. D. Pilmer
Addition Pyramids
With addition pyramids, the two numbers in adjoining boxes add to give the number in the box
immediately above.
8
18
3
5
7
34
11
5
2
14
9
5
4
20
9
11
1
8
3
Insert the missing numbers in each of the following addition pyramids.
1.
2.
4
9
6
13
5
3.
4.
8
3.
10
14
2
6.
9
9
7.
8
8.
9.
9
7
2
10.
10
18
9
3
6
11.
2
12.
4
19
11
3
8
13.
14.
7
21
12
3
NSSAL
©2012
10
1
7
15.
9
2
14
4
142
6
8
Draft
C. D. Pilmer
16.
17.
18.
30
16
10
7
6
0
5
8
19.
2
7
1
5
20.
13
21.
12
7
12
3
8
9
22.
2
23.
9
11
143
6
22
12
6
2
40
20
10
NSSAL
©2012
3
24.
17
3
7
10
5
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C. D. Pilmer
Row Factors and Column Factors
In each question you have been provided with a chart that is missing four numbers. These
numbers are the factors of the numbers found to the right of each row, and factors of the numbers
found at the bottom of each column. Find the missing numbers.
Example:
35
Answer:
15
5
3
15
28
7
4
28
35
12
12
Questions:
(a)
10
(b)
18
12
8
6
27
(e)
60
24
18
(h)
(k)
NSSAL
©2012
30
35
(l)
14
32
28
(n)
28
56
40
72
7
8
20
30
54
40
42
16
(m)
21
(i)
54
18
27
3
4
18
15
24
28
45
10
(j)
30
(f)
20
20
36
8
8
9
(g)
12
6
7
21
20
2
15
(d)
(c)
16
(o)
35
6
21
16
144
27
15
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C. D. Pilmer
Letter and Number Sentences
1.
A+ B = 9
2.
C−D=5
(a)
If A is 5, how much is B?
(a)
If C is 12, how much is D?
(b)
If B is 7, how much is A?
(b)
If D is 4, how much is C?
(c)
If A is 3, how much is B?
(c)
If C is 20, how much is D?
(d)
If B is 1, how much is A?
(d)
If D is 8, how much is C?
3.
E × F = 24
4.
G÷H =6
(a)
If E is 6, how much is F?
(a)
If G is 30, how much is H?
(b)
If F is 3, how much is E?
(b)
If H is 3, how much is G?
(c)
If E is 12, how much is F?
(c)
If G is 42, how much is H?
(d)
If F is 1, how much is E?
(d)
If H is 10, how much is G?
5.
I−J =7
6.
K × L = 30
(a)
If I is 16, how much is J?
(a)
If K is 10, how much is L?
(b)
If J is 5, how much is I?
(b)
If L is 5, how much is K?
(c)
If I is 11, how much is J?
(c)
If K is 2, how much is L?
(d)
If J is 10, how much is I?
(d)
If L is 30, how much is K?
7.
M ÷N =4
8.
P + Q = 13
(a)
If M is 36, how much is N?
(a)
If P is 9, how much is Q?
(b)
If N is 3, how much is M?
(b)
If Q is 2, how much is P?
(c)
If M is 28, how much is N?
(c)
If P is 6, how much is Q?
(d)
If N is 5, how much is M?
(d)
If Q is 8, how much is P?
NSSAL
©2012
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C. D. Pilmer
Math Logic Puzzles
For each, find the numbers represented by the symbols , �, and . Hint: For each of the
puzzles, one of the equations, not necessarily the first equation, allows you to solve for a symbol
very quickly.
Puzzle 1:
-�=2
�+1=6
+=8
Puzzle 4:
�-=3
 + � +  = 10
 ×4 = 8
Puzzle 7:
�-3=
� +  = 2 ×
14 ÷ � = 2
Answers (in no particular order)
 = 4, � = 6,  = 2
 = 14, � = 6,  = 4
 = 2, � = 5,  = 3
NSSAL
©2012
Puzzle 2:
Puzzle 3:
+=8
+�=6
+�=7
Puzzle 5:
+�=8
3 × = 6
 + � = 10
Puzzle 6:
-2=7
 + � +  = 17
� ×  = 18
Puzzle 8:
-�=5
�+�+�=9
+�=
Puzzle 9:
+�+2=
 ÷ 3=4
-=6
24 ÷ � = 
++1=9
 +  = 3 ×�
 = 6, � = 4,  = 12
 = 11, � = 3,  = 8
 = 4, � = 2,  = 5
 = 6, � = 2,  = 9
 = 7, � = 5,  = 1
 = 1, � = 7,  = 4
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C. D. Pilmer
Number Sentences (A)
In each case create four number sentences using the three numbers provided.
Example 1:
2
18
Example 2:
9
Answer:
2 × 9 = 18
9 × 2 = 18
8
5
Answer:
8 + 5 = 13
5 + 8 = 13
18 ÷ 9 = 2
18 ÷ 2 = 9
13
13 − 5 = 8
13 − 8 = 5
Questions:
1.
12
4
8
2.
10
15
5
3.
8
16
2
4.
4
5
9
5.
28
7
4
6.
9
45
5
7.
12
3
9
8.
3
15
5
9.
20
10
30
10.
30
180
6
NSSAL
©2012
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C. D. Pilmer
Number Sentences (B)
Create as many number sentences as possible using three numbers from the chart at a time and
limiting yourself to the operations of addition, subtraction, multiplication, and division.
Example:
2
28
3
You can create 16 number sentences using these numbers.
4
6
7
Answer:
2×3 = 6
3× 2 = 6
6÷2=3
6÷3 = 2
2+4=6
4+2=6
6−2= 4
6−4= 2
4 × 7 = 28
7 × 4 = 28
28 ÷ 4 = 7
28 ÷ 7 = 4
3+ 4 = 7
4+3=7
7−3= 4
7−4=3
1. You can create 12 number sentences using these numbers.
4
27
20
5
3
9
2. You can create 20 number sentences using these numbers.
2
12
24
3
4
8
3. You can create 24 number sentences using these numbers.
3
9
6
12
18
2
NSSAL
©2012
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C. D. Pilmer
Order of Operations (A)
Below, the same question was done by four different learners. The problem was that everyone
ended up with different answers.
Dave's Answer:
Nashi's Answer
Rana's Answer
Montez's Answer
3× 8 − 4 ÷ 2
3× 8 − 4 ÷ 2
24 − 4 ÷ 2
3× 4 ÷ 2
3× 8 − 4 ÷ 2
24 − 4 ÷ 2
3× 8 − 4 ÷ 2
3× 8 − 2
20 ÷ 2
12 ÷ 2
10
6
24 − 2
22
3× 6
18
Dave worked from
left to right. He did
the multiplication
first, followed by the
subtraction, and then
did the division.
Nashi started with
the subtraction,
followed by
multiplication, and
then did the division.
Rana started with the
multiplication,
followed by the
division, and then did
the subtraction.
Montez worked from
right to left. He
started with the
division, followed by
the subtraction, and
then did the
multiplication.
All of these learners started with the same question, but ended up with very different answers
based on the order they chose to do the operations. Only one of the learners is correct. Do you
know which one? The correct answer is 22. Rana did the question correctly because she knew
the order of operations, the rules used to clarify which mathematical operations are done first in a
mathematical expression. Most people remember the proper order by using the acronym
BEDMAS.
B
E
DM
AS
- brackets first
- then exponents (e.g. squaring, cubing)
- followed by division and multiplication in the order they appear (i.e. from left to right)
- followed by addition and subtraction in the order they appear (i.e. from left to right)
For this activity sheet we are only going to look at questions involving division, multiplication,
addition, and subtraction (i.e. only the "DMAS" portion of "BEDMAS")
Example 1
Evaluate each of the following.
(a) 5 + 7 × 3
(b) 15 ÷ 3 − 2
(c) 10 − 3 + 8 ÷ 4
(d) 5 × 4 + 6 ÷ 3
Answers:
(a) The mathematical expression 5 + 7 × 3 only involves the operations of addition and
multiplication. According to BEDMAS, we do multiplication before addition.
5 + 7×3
5 + 21
26
NSSAL
©2012
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C. D. Pilmer
(b) The mathematical expression 15 ÷ 3 − 2 only involves the operations of division and
subtraction. According to BEDMAS, we do division before subtraction.
15 ÷ 3 − 2
5−2
3
(c) The mathematical expression 10 − 3 + 8 ÷ 4 involves the operations of subtraction,
addition and multiplication. According to BEDMAS, multiplication is done before
subtraction or addition. Once this is done we have to decide between the subtraction and
addition. When these operations occur in the same question, we always work from left to
right. That means we will do the subtraction before the addition.
10 − 3 + 8 ÷ 4
10 − 3 + 2
7+2
9
(d) The mathematical expression 5 × 4 + 6 ÷ 3 involves the operations of multiplication,
addition, and division. According to BEDMAS we do division and multiplication before
addition. However, do we do the division before the multiplication, or vice versa? When
these two operations occur in the same question, we always work from left to right.
5× 4 + 6 ÷ 3
20 + 6 ÷ 3
20 + 2
22
Questions
Evaluate each of the mathematical expressions. Show your work and do not use a calculator.
(a) 6 × 2 − 1
(b) 10 − 3 × 2
(c) 18 + 10 ÷ 2
(d) 25 − 10 ÷ 5
(e) 5 × 8 + 2
(f) 9 + 2 × 3
NSSAL
©2012
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C. D. Pilmer
(g) 12 − 6 + 5
(h) 18 ÷ 2 × 3
(i) 18 ÷ 2 + 4
(j) 6 + 5 × 4
(k) 3 + 2 × 6 − 1
(l) 10 − 3 × 3 + 1
(m) 8 + 12 ÷ 4 − 3
(n) 4 × 2 + 3 × 5
(o) 3 × 6 − 2 × 5
(p) 12 ÷ 4 + 3 × 9
(q) 28 ÷ 4 − 2 × 3
(r) 10 − 6 + 3 × 5
(s) 10 + 15 ÷ 5 × 2
(t) 12 − 5 × 4 ÷ 10
(u) 7 × 3 − 16 ÷ 2
(v) 45 ÷ 5 − 6 + 4
(w) 3 × 4 + 5 × 7 − 2
(x) 6 + 16 ÷ 4 + 3 × 8
NSSAL
©2012
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Order of Operations (B)
Below, the same question was done by four different learners. The problem was that everyone
ended up with different answers.
Kendrick's Answer:
Helen's Answer
Jun's Answer
Nasrin's Answer
6 + 42 ÷ 2
6 + 42 ÷ 2
6 + 42 ÷ 2
6 + 42 ÷ 2
102 ÷ 2
100 ÷ 2
50
6 + 22
6+4
10
6 + 16 ÷ 2
22 ÷ 2
6 + 16 ÷ 2
6+8
11
14
Kendrick worked
from left to right. He
did the addition first,
followed by the
squaring, and then
did the division.
Helen started with
the division,
followed by the
squaring, and then
did the addition.
Jun started with the
squaring, followed
by the addition, and
then did the division.
Nasrin started with
the squaring,
followed by the
division, and then did
the addition.
All of these learners started with the same question, but ended up with very different answers
based on the order they chose to do the operations. Only one of the learners is correct. Do you
know which one? The correct answer is 14. Nasrin did the question correctly because he knew
the order of operations, the rules used to clarify which mathematical operations are done first in a
mathematical expression. The proper order can be remembered using the acronym BEDMAS.
B
E
DM
AS
- brackets first
- then exponents (e.g. squaring, cubing)
- followed by division and multiplication in the order they appear (i.e. from left to right)
- followed by addition and subtraction in the order they appear (i.e. from left to right)
For this activity sheet we are only going to look at questions involving exponents, division,
multiplication, addition, and subtraction (i.e. only the "EDMAS" portion of "BEDMAS")
Example 1
Evaluate each of the following.
(a) 5 + 32 × 2
(b) 4 × 5 − 23
(c) 62 ÷ 9 + 2 × 5
(d) 29 − 33 + 24 ÷ 6
Answers:
(a) The mathematical expression 5 + 32 × 2 involves the operations of addition, squaring, and
multiplication. According to BEDMAS, we would do the squaring (i.e. exponents) first,
followed by multiplication, and finally the addition.
5 + 32 × 2
5 + 9× 2
5 + 18
23
NSSAL
©2012
Remember: 32 means 3 × 3
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(b) The mathematical expression 4 × 5 − 23 involves the operations of multiplication,
subtraction, and cubing. According to BEDMAS, we would do the cubing (i.e.
exponents) first, followed by multiplication, and finally the subtraction.
4 × 5 − 23
4×5 − 8
20 − 8
12
Remember: 23 means 2 × 2 × 2
(c) The mathematical expression 62 ÷ 9 + 2 × 5 involves the operations of squaring, division,
addition, and multiplication. According to BEDMAS, we would do the squaring (i.e.
exponents) first. Next we have to decide between the division and multiplication. When
these operations occur in the same question, we always work from left to right. That
means we will do the division before the multiplication. The last operation we will
complete is the addition.
62 ÷ 9 + 2 × 5
36 ÷ 9 + 2 × 5
4 + 2×5
4 + 10
14
(d) The mathematical expression 29 − 33 + 24 ÷ 6 involves the operations of subtraction,
cubing, addition, and division. According to BEDMAS, we would do the cubing (i.e.
exponents) first. Next we would do the division. Once this is done we have to decide
between the subtraction and addition. When these operations occur in the same question,
we always work from left to right. That means we will do the subtraction before the
addition.
29 − 33 + 24 ÷ 6
29 − 27 + 24 ÷ 6
29 − 27 + 4
2+4
6
Questions
Evaluate each of the mathematical expressions. Show your work and do not use a calculator.
(a) 24 + 42 ÷ 8
NSSAL
©2012
(b) 32 + 5 × 6
(c) 30 − 23 × 3
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C. D. Pilmer
(d) 4 × 3 + 52
(e) 6 − 5 + 33
(f) 42 − 2 × 3
(g) 5 × 32 − 2
(h) 20 − 42 ÷ 2
(i) 18 − 23 + 6
(j) 2 × 32 + 5 × 2
(k) 23 × 5 − 30 ÷ 6
(l) 30 − 22 × 6 + 1
(m) 10 + 4 × 32 − 2
(n) 9 − 42 ÷ 2 + 1
(o) 52 − 3 × 23
(p) 42 + 32 × 2
(q) 5 × 23 − 4 × 32
(r) 62 ÷ 4 + 2 × 52
NSSAL
©2012
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C. D. Pilmer
Order of Operations (C)
Below, the same question was done by four different learners. The problem was that everyone
ended up with different answers.
Kimi's Answer:
Ryan's Answer
Paulette's Answer
Ajay's Answer
4 × (5 − 2)
4 × (5 − 2)
4 × (5 − 2)
4 × (5 − 2)
2
2
2
2
20 − 22
20 − 4
16
4 × 32
4×9
36
4×5 − 4
4 ×1
4
122
Kimi started with the
multiplication,
followed by the
squaring, and then
did the subtraction.
Ryan started with the
operation in the
brackets, followed by
the squaring, and
then did the
multiplication.
Paulette started with
the squaring,
followed by the
subtraction, and then
did the
multiplication.
Ajay started with the
operation in the
brackets, followed by
the multiplication,
and then did the
squaring.
4 × 32
144
All of these learners started with the same question, but ended up with very different answers
based on the order they chose to do the operations. Only one of the learners is correct. Do you
know which one? The correct answer is 36. Ryan did the question correctly because he knew
the order of operations, the rules used to clarify which mathematical operations are done first in a
mathematical expression. The proper order can be remembered using the acronym BEDMAS.
B
E
DM
AS
- brackets first
- then exponents (e.g. squaring, cubing)
- followed by division and multiplication in the order they appear (i.e. from left to right)
- followed by addition and subtraction in the order they appear (i.e. from left to right)
Example 1
Evaluate each of the following.
2
3
(a) 3 × ( 6 − 1)
(b) ( 7 − 5 ) + 3 × 7
(c) 9 + 20 × (1 + 1) − 33
2
Answers:
2
(a) With the mathematical expression 3 × ( 6 − 1) we have multiplication, subtraction
embedded within a set of brackets, and squaring. According to BEDMAS, we start with
the operations in the brackets, followed by the squaring (i.e. exponents), and then finish
off with the multiplication.
3 × ( 6 − 1)
2
3 × 52
3 × 25
75
NSSAL
©2012
Remember: 52 means 5 × 5
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(b) With the mathematical expression ( 7 − 5 ) + 3 × 7 we have subtraction embedded within a
set of brackets, cubing, addition, and multiplication. According to BEDMAS, we start
with the operations in the brackets. This will be followed by the cubing (i.e. exponents).
We will then do the multiplication, and then finish up with the addition.
3
( 7 − 5)
3
+ 3× 7
23 + 3 × 7
8 + 3× 7
8 + 21
Remember: 23 means 2 × 2 × 2
29
(c) With the mathematical expression 9 + 20 × (1 + 1) − 33 we will start with the addition that
is embedded within a set of brackets. Next we will work with the exponents (i.e. the
squaring and the cubing). Following this we do the multiplication. That leaves us with
the addition and multiplication. When these operations occur in the same question, we
always work from left to right. That means we will do the addition before the
subtraction.
2
9 + 20 × (1 + 1) − 33
2
9 + 20 × 22 − 33
9 + 20 × 4 − 27
9 + 80 − 27
89 − 27
62
Questions
Evaluate each of the mathematical expressions. Show your work and do not use a calculator.
(a) 10 + ( 5 − 1)
2
(b) ( 8 − 6 ) × 4
(c) 36 ÷ (1 + 2 )
(d) 7 2 − ( 4 + 6 )
(e) ( 3 + 6 ) × 23
(f) 2 × ( 8 − 3)
NSSAL
©2012
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(g) ( 3 + 7 ) × ( 6 − 2 )
(h) (10 + 25 ) ÷ ( 7 − 2 )
(i) 3 × 4 × ( 6 − 4 )
(j) 32 × (11 − 2 )
(k) ( 7 + 20 ) ÷ 32
(l) 32 ÷ ( 5 − 3)
(m) 42 ÷ (1 + 7 ) × 3
(n) 3 + ( 7 − 5 ) × 5
(o) 4 × ( 2 + 3) − 6
2
(p) (1 + 1) × ( 8 − 5 ) − 6
2
NSSAL
©2012
(q) 5 + ( 7 − 4 ) × (1 + 1)
157
3
2
3
(r) 4 + ( 5 − 2 ) ÷ ( 7 + 2 )
3
Draft
C. D. Pilmer
Patterns
NSSAL
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C. D. Pilmer
What's the Pattern? (A)
Look for a pattern and then fill in the three missing symbols.
(a)
, , , , , , , , , , , , , , ___, ___, ___
(b)
, , , , , , , , , , , , , , ___, ___, ___
(c)
, , , , , , , , , , , , , , , ___, ___, ___
(d)
, , , , , , , , , , , ___, ___, ___
(e)
, , , , , , , , , , ___, ___, ___
(f)
A, , B, , C, , D, , E, , F, , ___, ___, ___
(g)
, , , , , , , , , , , ___, ___, ___
(h)
, , , , , , , , , , , , , ___, ___, ___
(i)
, , , , , , , , , , , ___, ___, ___
(j)
, R, , P, , R, , P, , R, , P, , R, , ___, ___, ___
(k)
, , , , , , , , , , , , , , , , ___, ___, ___
(l)
, , , , , , , , , , , , , , , ___, ___, ___
(m) ,
, , , , , , , , , , , , , ___, ___, ___
(n)
Z, z, Y, y, X, x, W, w, V, v, U, u, ___, ___, ___
(o)
, , , , , , , , , , , , , , , , ___, ___, ___
(p)
, , , , , , , , , , , , , , ,, ___, ___, ___
NSSAL
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What's the Pattern? (B)
Look for a pattern and then fill in the three missing symbols. Hint: Sometimes more than one
trait is changing.
e.g. Every figure is rotated slightly counter clockwise (45o) and every fifth figure is not bolded.
, , , , , , , , , , , , , , 
(a)
, , , , , , , , , , , , , , ___, ___, ___
(b) ,
, , , , , , , , , , , , , ___, ___, ___
(c)
, , , , , , , , , , , ___, ___, ___
(d)
, , , , , , , , , , , , , , ___, ___, ___
(e)
, , , , , , , , , , , , ,, ___, ___, ___
(f)
, , , , , , , , , , ___, ___, ___
(g)
, , , , , , , , , ___, ___, ___
(h)
, , , , , , , , , , , , , , ___, ___, ___
(i)
, , , , , , , , , , , , , ___, ___, ___
(j)
, , , , , , , , , , , , , , ___, ___, ___
(k)
, , , , , , , , , , , , , , , ___, ___, ___
(l)
, , , , , , , , , , , ,, , ___, ___, ___
(m) ,
, A, , , B, , , C, ,, D, , ___, ___, ___
(n)
, , , , , , , , , , , , , , , , ___, ___, ___
(o)
, , , , , , , , , , , , , ___, ___, ___
NSSAL
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Toothpick Patterns
You have been supplied with a sequence of shapes where each shape is created using toothpicks.
For example a triangle is made up of 3 toothpicks. In these questions, you are going to be
looking for a pattern in terms of the number of toothpicks as you move from one figure to the
next. There are three specific parts to each question.
(a) Draw the next figure in the sequence.
(b) Describe the sequence in terms of numbers (i.e. numbers of toothpicks in each figure).
(c) In words, describe what is happening to the numbers as you move from figure to figure in the
sequence.
Example:
Answer:
(a) Next Figure:
(b) Sequence Using Numbers: 4, 7, 10, 13, 16
(c) Describe Sequence Using Works: Start at 4 and keep adding 3.
1.
2.
3.
NSSAL
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4.
5.
6.
7.
8.
NSSAL
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C. D. Pilmer
Create the Pattern (A)
Using the instructions, create the first five numbers in the sequence.
e.g. Start at 7 and go up by 3 each time.
Answer: 7, 10, 13, 16, 19
(a) Start at 9 and go up by 2 each time.
______, ______, ______, ______, ______
(b) Start at 24 and go down by 1 each time.
______, ______, ______, ______, ______
(c) Start at 8 and go up by 3 each time.
______, ______, ______, ______, ______
(d) Start at 4 and go up by 5 each time.
______, ______, ______, ______, ______
(e) Start at 33 and go down by 3 each time.
______, ______, ______, ______, ______
(f) Start at 29 and go down by 2 each time.
______, ______, ______, ______, ______
(g) Start at 11 and go up by 4 each time.
______, ______, ______, ______, ______
(h) Start at 30 and go down by 2 each time.
______, ______, ______, ______, ______
(i) Start at 3 and go up by 6 each time.
______, ______, ______, ______, ______
(j) Start at 2 and go up by 4 each time.
______, ______, ______, ______, ______
(k) Start at 11 and go up by 5 each time.
______, ______, ______, ______, ______
(l) Start at 38 and go down by 3 each time.
______, ______, ______, ______, ______
(m) Start at 36 and go down by 4 each time.
______, ______, ______, ______, ______
(n) Start at 23 and go up by 6 each time.
______, ______, ______, ______, ______
(o) Start at 40 and go down by 5 each time.
______, ______, ______, ______, ______
(p) Start at 13 and go up by 10 each time.
______, ______, ______, ______, ______
(q) Start at 72 and go down by 2 each time.
______, ______, ______, ______, ______
(r) Start at 99 and go down by 10 each time.
______, ______, ______, ______, ______
(s) Start at 41 and go up by 4 each time.
______, ______, ______, ______, ______
NSSAL
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Create the Pattern (B)
Using the instructions, create the first four numbers in the sequence.
e.g. Start at 125 and go up by 6 each time.
Answer: 125, 131, 137, 143
(a) Start at 63 and go up by 7 each time.
________, ________, ________, ________
(b) Start at 234 and go down by 2 each time.
________, ________, ________, ________
(c) Start at 81 and go up by 3 each time.
________, ________, ________, ________
(d) Start at 79 and go down by 3 each time.
________, ________, ________, ________
(e) Start at 126 and go up by 5 each time.
________, ________, ________, ________
(f) Start at 540 and go down by 10 each time.
________, ________, ________, ________
(g) Start at 352 and go up by 20 each time.
________, ________, ________, ________
(h) Start at 47 and go up by 4 each time.
________, ________, ________, ________
(i) Start at 68 and go down by 4 each time.
________, ________, ________, ________
(j) Start at 275 and go up by 25 each time.
________, ________, ________, ________
(k) Start at 134 and go up by 6 each time.
________, ________, ________, ________
(l) Start at 456 and go down by 100 each time.
________, ________, ________, ________
(m) Start at 99 and go down by 11 each time.
________, ________, ________, ________
(n) Start at 347 and go up by 3 each time.
________, ________, ________, ________
(o) Start at 605 and go down by 5 each time.
________, ________, ________, ________
(p) Start at 710 and go up by 30 each time.
________, ________, ________, ________
(q) Start at 670 and go down by 20 each time.
________, ________, ________, ________
(r) Start at 412 and go up by 6 each time.
________, ________, ________, ________
(s) Start at 364 and go down by 3 each time.
________, ________, ________, ________
NSSAL
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Number Patterns (A)
Look at the pattern and fill in the missing numbers.
(a) 6, 8, 10, 12, 14, 16, ______, ______, ______
(b) 23, 22, 21, 20, 19, 18, 17, 16, 15, ______, ______, ______
(c) 5, 7, 9, 11, 13, 15, 17, 19, 21, ______, ______, ______
?
(d) 30, 28, 26, 24, 22, 20, 18, ______, ______, ______
(e) 0, 3, 6, 9, 12, 15, ______, ______, ______
(f) 10, 15, 20, 25, 30, 35, ______, ______, ______
(g) 40, 36, 32, 28, 24, 20, ______, ______, ______
(h) 31, 28, 25, 22, 19, 16, 13, ______, ______, ______
(i) 7, 12, 17, 22, 27, 32, 37, 42, ______, ______, ______
(j) 45, 47, 49, 51, 53, 55, 57, ______, ______, ______
(k) 19, 22, 25, 28, 31, 34, 37, ______, ______, ______
(l) 51, 47, 43, 39, 35, 31, 27, 23, ______, ______, ______
(m) 67, 62, 57, 52, 47, 42, 37, ______, ______, ______
(n) 44, 43, ______, 41, ______, 39, 38, ______, 36
(o) 23, ______, 27, 29, 31, ______, ______, 37, 39
(p) 30, ______, ______, 21, 18, 15, 12, ______, 6
(q) 17, 21, ______, ______, 33, ______, 41, 45, 49, 53
(r) ______, 14, 19, 24, ______, ______, 39, 44, 49
(s) ______, 33, ______, 39, 42, ______, 48, 51, 54, 57
(t) ______, ______, 41, 36, 31, ______, 21, 16, 11, 6
(u) 91, ______, 93, ______, 95, ______, 97
NSSAL
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Number Patterns (B)
Look at the pattern and fill in the missing numbers.
(a) 2, 4, 6, 8, 10, 12, ______, ______, ______
(b) 4, 7, 10, 13, 16, ______, ______, ______
(c) 29, 27, 25, 23, 21, ______, ______, ______
?
(d) 55, 50, 45, 40, 35, ______, ______, ______
(e) 3, 10, 17, 24, 31, 38, ______, ______, ______
(f) 64, 56, 48, 40, 32, 24, ______, ______, ______
(g) 0, 6, 12, 18, 24, 30, 36, ______, ______, ______
(h) 1, 12, 23, 34, 45, 56, ______, ______, ______
(i) 44, 40, 36, 32, 28, 24, 20, ______, ______, ______
(j) 100, 104, 108, 112, 116, 120, ______, ______, ______
(k) 675, 680, 685, 690, 695, 700, 705, 710, ______, ______, ______
(l) 190, 210, 230, 250, 270, 290, 310, 330, ______, ______, ______
(m) 326, 324, 322, 320, 318, 316, ______, ______, ______
(n) 6, 10, 14, ______, 22, 26, ______, ______, 38
(o) 40, ______, 34, 31, ______, 25, 22, ______, 16
(p) 56, 61, _______, 71, 76, 81, ______, ______, 96, 101, 106
(q) ______, 25, 29, ______, 37, 41, 45, ______, 53, 57
(r) ______, 245, ______, 233, 227, 221, 215, ______, 203
(s) 16, ______, ______, 40, 48, ______, 64, 72, 80
(t) ______, ______, 292, ______, 284, 280, 276, 272, 268
(u) 420, ______, 460, ______, 500, ______, 540
NSSAL
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Row, Column, and Diagonal Patterns
Describe the pattern between numbers found in columns, rows and diagonals. (Note: Rows go
from left to right. Columns go from top to bottom. Diagonals go from the upper left to the lower
right.)
e.g.
4
6
8
4
6
6
8
4
6
8
10 12
14
10
12 14
10 12
14
10 12
14
16 18
20
16
18 20
16 18
20
16 18
20
8
4
Row Pattern:
Add 2
1.
4.
5
8
11
14 17
23 26
2.
Column Pattern:
Add 6
25 23
21
20
19 17
29
13 11
3.
Diagonal Pattern:
Add 8
1
11
21
15
31 41
51
9
61 71
81
Row Pattern:
Row Pattern:
Row Pattern:
Column Pattern:
Column Pattern:
Column Pattern:
Diagonal Pattern:
Diagonal Pattern:
Diagonal Pattern:
42 38
34
30 26
18 14
5.
29 31
33
22
23 25
10
17 19
6.
17 12
7
27
32 27
22
21
47 42
37
Row Pattern:
Row Pattern:
Row Pattern:
Column Pattern:
Column Pattern:
Column Pattern:
Diagonal Pattern:
Diagonal Pattern:
Diagonal Pattern:
NSSAL
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What's the Relationship?
A chart containing numbers between and equal to 1 and 50 is provided. Some of the numbers
are enlarged and bolded. What is the relationship amongst those enlarged and bolded numbers?
Write the answer in as many ways as you can think of.
Example 1:
Example 2
6
7
8
9
10
1
2
15
16
17
18
19
20
11
12
25
26
27
28
29
30
21
22
31
41
32
33
34
42
43
44
1
2
3
4
5
11
12
13
14
21
22
23
24
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Answers:
Multiples of 6
Divisible by 6
Divisible by both 2 and 3
Start at 6 and keep adding 6
Start at 48 and keep subtracting 6
33
4
14
24
34
35
6
16
26
36
31
41
42
43
44
45
1
2
3
4
11
12
13
14
21
22
23
24
31
32
33
34
41
42
43
44
21
5
6
8
9
10
16
7
17
14
15
18
20
26
27
28
19
29
24
25
35
36
39
40
46
37
47
38
45
48
49
50
30
(b)
2
12
22
32
11
4
Answers:
All are prime numbers
Only divisible by 1 and themself
(a)
1
3
13
23
3
13
23
37
8
18
28
38
46
47
5
6
15
25
35
45
5
15
25
39
10
20
30
40
48
49
50
7
8
9
10
1
2
3
4
5
6
7
8
9
10
16
17
18
19
11
12
13
14
15
16
17
18
19
20
26
27
28
29
21
22
23
24
25
26
27
28
29
30
36
37
38
39
31
32
33
34
35
36
37
38
39
40
46
47
48
49
20
30
40
50
41
42
43
44
45
46
47
48
49
50
7
17
27
9
19
29
(c)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
(d)
NSSAL
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(e)
(f)
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
41
42
43
44
45
46
47
48
49
50
1
2
3
4
5
6
7
8
9
10
2
4
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
42
9
19
29
39
49
10
13
7
17
27
37
47
8
12
5
15
25
35
45
6
11
3
13
23
33
43
1
2
3
4
5
6
7
8
9
10
1
2
3
9
11
12
13
14
15
16
17
18
19
20
11
22
23
24
25
26
27
28
29
30
21
12
22
13
21
31
32
33
34
35
36
37
38
39
40
31
33
41
42
43
44
45
46
47
48
49
50
41
32
42
(g)
(h)
(i)
1
11
21
31
41
12
22
32
14
24
34
44
16
26
36
46
18
28
38
48
20
30
40
50
(j)
(k)
4
14
24
15
25
6
16
26
43
34
44
35
45
36
46
23
5
7
17
27
37
47
8
18
28
19
29
10
20
30
38
48
39
49
40
50
(l)
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
41
42
43
44
45
46
47
48
49
50
NSSAL
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Input Output (A)
A number is put in (the input number) and a different number is spit out (the output number). In
each case, determine the mathematical rule that changes the input number to the output number.
Three examples have been provided to help you understand what needs to be done.
Example I:
Input
2
5
10
Example II:



Output
6
Input
10
15
7
30
21
Rule:
Input × 3 = Output
Example III:



Output
4
Input
16
1
20
15
6
Rule:
Input - 6 = Output
(a)
5
2



Input
3
9
7
6
4



Output
6
Input
27
14
12
8
18
Rule:
(d)
16
5



Input
6
11
2
0
9



Output
30
Input
18
10
9
45
7
Rule:
(g)
32
8
Rule:
NSSAL
©2012



6



Output
15
6
4
Rule:
(h)
Input
20
4
(f)
Output
9
Rule:



Output
9
Rule:
(e)
Input
14
3
(c)
Output
11
Rule:
10
Rule:
Input ÷ 2 = Output
(b)
Input
7



Output
8
(i)
Output
5
Input
4
8
9
2
2



Rule:
Output
10
Input
9
15
3
8
4



Output
36
12
16
Rule:
170
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C. D. Pilmer
Input Output (B)
A number is put in (the input number) and a different number is spit out (the output number). In
each case, determine the mathematical rule that changes the input number to the output number.
Three examples have been provided to help you understand what needs to be done.
Example I:
Input
4
7
6
Example II:



Output
28
Input
15
49
20
42
10
Rule:
Input × 7 = Output
Example III:



Output
4
Input
60
11
18
1
36
Rule:
Input - 9 = Output
(a)
9
7



Input
13
81
19
63
11



Output
6
Input
4
12
15
4
9
Rule:
(d)
21
56



Input
7
3
4
8
9



Output
42
Input
48
24
8
54
32
Rule:
(g)
9
12
Rule:
NSSAL
©2012



18



Output
6
1
4
Rule:
(h)
Input
16
24
(f)
Output
5
Rule:



Output
13
Rule:
(e)
Input
35
6
(c)
Output
36
Rule:
3
Rule:
Input ÷ 6 = Output
(b)
Input
4



Output
10
(i)
Output
23
Input
25
16
36
19
19



Rule:
Output
15
Input
3
26
5
9
9



Output
24
40
72
Rule:
171
Draft
C. D. Pilmer
Input Output (C)
A number is put in (the input number) and a different number is spit out (the output number). In
each case, determine the mathematical rule that changes the input number to the output number.
Three examples have been provided to help you understand what needs to be done.
Example I:
Input
900
30
7
Example II:



Output
54000
Input
150
1800
70
420
210
Rule:
Input × 60 = Output
(a)
Example III:



Output
130
Input
120
50
600
190
3000
Rule:
Input - 20 = Output
(b)
Input
17
120
73



Input
7
150
20
103
9



Output
350
Input
190
1000
72
450
350
2800
36



Input
500
700
986
9
1700



Output
300
Input
60
786
3
1500
90
Rule:
(g)
1000
97
Rule:
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©2012



290



Output
24000
1200
36000
Rule:
(h)
Input
150
12
(f)
Output
40
Rule:



Output
130
Rule:
(e)
Input
160
1000
Rule:
Input ÷ 3 = Output
Rule:
(d)
200
(c)
Output
47
Rule:



Output
40
(i)
Output
80
Input
2670
930
800
27
156



Rule:
Output
2970
Input
1600
1100
60000
456
480



Output
800
30000
240
Rule:
172
Draft
C. D. Pilmer
Input Output (D)
In each case, determine the mathematical rule that changes the input number to the output
number. For these questions, the input number is multiplied by 2, 3, 4, or 5 and then has a
number added to or subtracted from it.
Example I:
Input
4
2
9
Example II:



Output
7
Input
1
3
6
17
3
Rule:
(Input × 2 ) - 1 = Output
(a)
Example III:



Output
7
Input
8
27
10
15
3
Rule:
(Input × 4 ) + 3 = Output
(b)
Input
6
8
3



Input
10
21
1
11
7



Output
31
Input
2
4
8
22
9
2
7



Input
3
6
0
26
6



Output
14
Input
7
5
4
23
1
Rule:
(g)
4
2
Rule:
NSSAL
©2012



15



Output
36
21
6
Rule:
(h)
Input
9
13
(f)
Output
14
Rule:



Output
1
Rule:
(e)
Input
4
11
Rule:
(Input × 5 ) - 4 = Output
Rule:
(d)
46
(c)
Output
17
Rule:



Output
36
(i)
Output
37
Input
3
17
8
9
1



Rule:
Output
13
Input
6
38
2
3
5



Output
16
4
13
Rule:
173
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C. D. Pilmer
Filling or Draining
A large container, which can hold 20 litres of water when filled to the brim, is either being filled
or drained at a constant rate. You will be able to tell based on the sequence of diagrams of the
container that have been supplied. Your mission is to complete the next diagram in the
sequence, then describe each situation using words, using a table, and using an equation, and
finally predict when the container will be full or empty.
e.g.
10 L
time = 0 minutes
10 L
10 L
time = 1 minute
time = 2 minutes
10 L
time = 3 minutes
Answer:
In the last diagram, we should show the water level at the 8 L mark.
Written Description:
The container initially had 14 L of water in it
and it is being drained at a rate of 2 litres per
minute.
Table of Values:
Time
Litres
0
14
1
12
2
10
3
8
10 L
Equation: Litres = 14 - 2 × Time
or
L = 14 - 2T
If the container initially held 14 L of water and it's losing 2 L per minute, then in 7 minutes
the container will be empty (i.e. hold 0 L of water)
NSSAL
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C. D. Pilmer
1.
10 L
time = 0 minutes
10 L
10 L
time = 1 minute
time = 2 minutes
10 L
time = 3 minutes
Written Description:
Table of Values:
Time
Litres
Equation:
Empty:
2.
10 L
time = 0 minutes
10 L
10 L
time = 1 minute
time = 2 minutes
10 L
time = 3 minutes
Written Description:
Table of Values:
Time
Litres
Equation:
Full:
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C. D. Pilmer
3.
10 L
time = 0 minutes
10 L
10 L
time = 1 minute
time = 2 minutes
10 L
time = 3 minutes
Written Description:
Table of Values:
Time
Litres
Equation:
Full:
4.
10 L
time = 0 minutes
10 L
10 L
time = 1 minute
time = 2 minutes
10 L
time = 3 minutes
Written Description:
Table of Values:
Time
Litres
Equation:
Empty:
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C. D. Pilmer
5.
10 L
time = 0 minutes
10 L
10 L
time = 1 minute
time = 2 minutes
10 L
time = 3 minutes
Written Description:
Table of Values:
Time
Litres
Equation:
Empty:
6.
10 L
time = 0 minutes
10 L
10 L
time = 1 minute
time = 2 minutes
10 L
time = 3 minutes
Written Description:
Table of Values:
Time
Litres
Equation:
Full:
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Draft
C. D. Pilmer
7.
10 L
time = 0 minutes
10 L
10 L
time = 1 minute
time = 2 minutes
10 L
time = 3 minutes
Written Description:
Table of Values:
Time
Litres
Equation:
Empty:
8.
10 L
time = 0 minutes
10 L
10 L
time = 1 minute
time = 2 minutes
10 L
time = 3 minutes
Written Description:
Table of Values:
Time
Litres
Equation:
Full:
NSSAL
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C. D. Pilmer
Travelling Towards or Away From Home
Montez is either travelling towards or away from his home at a constant speed. You will be able
to tell by looking at the sequence of diagrams that have been provided. Your mission is to
describe each situation using words, using a table of values (where the times go from 0 seconds
to 4 seconds) and using an equation, and finally predict how far Montez is from home at t = 7
seconds.
e.g.
t = 0 seconds
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
t = 1 second
t = 2 seconds
Answer:
Written Description:
Table of Values:
Time
Distance
from Home
0
7
1
11
2
15
3
19
4
23
Montez is initially 7 metres from home and runs away from the home
at a rate of 4 metres per second.
Equation: distance = 7 + 4 × time or d = 7 + 4t
At t = 7 seconds, Montez will be 35 metres from home. The easiest way to figure this out is
to substitute 7 in for t in the equation d = 7 + 4t, and then solve for d. You could also take
the table of values and expand it until you reach a time of 7 seconds or jump along the
number line on the diagram four spaces for every second until you reach the desired time.
NSSAL
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C. D. Pilmer
1.
t = 0 seconds
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
t = 1 second
t = 2 seconds
Written Description:
Table of Values:
Time
Distance
from Home
Equation:
At t = 7 seconds
2.
t = 0 seconds
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
t = 1 second
t = 2 seconds
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C. D. Pilmer
Written Description:
Table of Values:
Time
Distance
from Home
Equation:
At t = 7 seconds
3.
t = 0 seconds
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
t = 1 second
t = 2 seconds
Written Description:
Table of Values:
Time
Distance
from Home
Equation:
At t = 7 seconds
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C. D. Pilmer
4.
t = 0 seconds
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
t = 1 second
t = 2 seconds
Written Description:
Table of Values:
Time
Distance
from Home
Equation:
At t = 7 seconds
5.
t = 0 seconds
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
t = 1 second
t = 2 seconds
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C. D. Pilmer
Written Description:
Table of Values:
Time
Distance
from Home
Equation:
At t = 7 seconds
6.
t = 0 seconds
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
t = 1 second
t = 2 seconds
Written Description:
Table of Values:
Time
Distance
from Home
Equation:
At t = 7 seconds
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Draft
C. D. Pilmer
7.
t = 0 seconds
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
t = 1 second
t = 2 seconds
Written Description:
Table of Values:
Time
Distance
from Home
Equation:
At t = 7 seconds
8.
t = 0 seconds
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
t = 1 second
t = 2 seconds
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C. D. Pilmer
Written Description:
Table of Values:
Time
Distance
from Home
Equation:
At t = 7 seconds
9.
t = 0 seconds
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
0m
5m
10 m
15 m
20 m
25 m
30 m
35 m
t = 1 second
t = 2 seconds
Written Description:
Table of Values:
Time
Distance
from Home
Equation:
At t = 7 seconds
NSSAL
©2012
185
Draft
C. D. Pilmer
Weight of the Water
A spring balance is used to measure the weight of an object that is suspended
below it. That weight will be measured in newtons (N). In this situation we
have a container suspended below our spring balance that is either being
filled or drained of water at a constant rate. Your mission is to describe each
situation using words, using a table of values (where the times go from 0
seconds to 4 seconds) and using an equation, and finally predict the weight of
the water at t = 6 seconds. The scale was adjusted so that the weight of the
empty container is not included.
e.g.
time = 0 seconds
Newtons
time = 1 second
Newtons
time = 2 seconds
Newtons
0
0
0
5
5
5
10
10
10
15
15
15
20
20
20
25
25
25
30
30
30
Answer:
Written Description: The container initially contained water weighting 5 newtons and then
water was added such that the weight increased by 2 newtons per
second.
Table of Values:
Time
Weight
0
5
1
7
2
9
3
11
4
13
Equation: weight = 5 + 2 × time or w = 5 + 2t
At t = 6 seconds, the weight of the container and its contents should be 17 newtons.
NSSAL
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C. D. Pilmer
1.
time = 0 seconds
Newtons
time = 1 second
Newtons
time = 2 seconds
Newtons
0
0
0
5
5
5
10
10
10
15
15
15
20
20
20
25
25
25
30
30
30
Written Description:
Table of Values:
Time
Weight
Equation:
At t = 6 seconds
2.
time = 0 seconds
Newtons
NSSAL
©2012
time = 1 second
Newtons
time = 2 seconds
Newtons
0
0
0
5
5
5
10
10
10
15
15
15
20
20
20
25
25
25
30
30
30
187
Draft
C. D. Pilmer
Written Description:
Table of Values:
Time
Weight
Equation:
At t = 6 seconds
3.
time = 0 seconds
Newtons
time = 1 second
Newtons
time = 2 seconds
Newtons
0
0
0
5
5
5
10
10
10
15
15
15
20
20
20
25
25
25
30
30
30
Written Description:
Table of Values:
Time
Weight
Equation:
At t = 6 seconds
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Draft
C. D. Pilmer
4.
time = 0 seconds
Newtons
time = 1 second
Newtons
time = 2 seconds
Newtons
0
0
0
5
5
5
10
10
10
15
15
15
20
20
20
25
25
25
30
30
30
Written Description:
Table of Values:
Time
Weight
Equation:
At t = 6 seconds
5.
time = 0 seconds
Newtons
NSSAL
©2012
time = 2 seconds
Newtons
time = 4 seconds
Newtons
0
0
0
5
5
5
10
10
10
15
15
15
20
20
20
25
25
25
30
30
30
189
Draft
C. D. Pilmer
Written Description:
Table of Values:
Time
Weight
Equation:
At t = 6 seconds
6.
time = 0 seconds
Newtons
time = 2 seconds
Newtons
time = 4 seconds
Newtons
0
0
0
5
5
5
10
10
10
15
15
15
20
20
20
25
25
25
30
30
30
Written Description:
Table of Values:
Time
Weight
Equation:
At t = 6 seconds
NSSAL
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190
Draft
C. D. Pilmer
Word Problems
NSSAL
©2012
191
Draft
C. D. Pilmer
Describing the Relationships with Words
Example:
Given the diagram on the right, describe as many
mathematical relationships using the different shapes.
Answer:
There are 3 arrows.
There are 5 lightning bolts.
There are 6 hearts.
There are a total of 14 shapes.
There is 1 more heart than lightning bolts.
There is 1 less lightning bolt than hearts.
There are 2 more lightning bolts than arrows.
There are 2 less arrows than lightning bolts.
There are 3 more hearts than arrows.
There are 3 less arrows than hearts.
There are twice as many hearts as arrows.
There are half as many arrows as hearts.
Now do the same with this diagram that has moons, suns and hearts.
__________________________________________________
__________________________________________________
__________________________________________________
__________________________________________________
__________________________________________________
__________________________________________________
__________________________________________________
__________________________________________________
__________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
NSSAL
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192
Draft
C. D. Pilmer
List the Numbers Based on the Written Description
1. List all the whole numbers between 8 and 15.
2. List all the even numbers between and equal to 10 and 22.
3. List all the multiples of 10 between and equal to 1 and 100.
4. List all the numbers that are divisible by 5 between 13 and 41.
5. List all the two digit numbers whose tens digit and one digit are the same.
6. List all the numbers that are divisible by 3 between 16 and 32.
7. List all prime numbers between and equal to 5 and 18.
8. List all the odd numbers that are divisible by 3 between 2 and 26.
9. List all the even numbers that are divisible by 5 between 3 and 47.
10. List all the two digit numbers whose digits add to 7.
11. List all the two digit numbers greater than 30 that are multiples of 9.
12. List all the numbers less than 35 that are divisible by 8.
13. List all composite numbers (i.e. not prime) between and equal to 9 and 21.
14. List all the numbers that are divisible by 2 and 3 between 5 and 35.
15. List all the odd numbers that are divisible by 5 between 7 and 56.
NSSAL
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193
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C. D. Pilmer
Addition and Subtraction Crossword
Complete the following crossword using words to express your answer (e.g. seven). Do not use
a calculator.
1.
3.
4.
2.
5.
8.
6.
7.
9.
10.
11.
12.
13.
14.
17.
15.
18.
19.
21.
16.
20.
22.
23.
24.
25.
27.
26.
28.
29.
30.
31.
33.
32.
34.
35.
36.
37.
Across:
1. nine subtract seven
2. four plus eight
3. forty plus ten
5. eight subtract four
6. three add four
8. six increased by seven
10. nineteen decreased by three
13. five plus six
15. eleven minus two
NSSAL
©2012
Down:
1. six plus four
2. five minus two
3. sixty subtract twenty
4. six add six
5. decrease sixteen by one
7. increase nine by two
9. seventeen subtract eight
10. thirty plus thirty
11. six add two
194
Draft
C. D. Pilmer
Across:
17. twenty subtract eighteen
19. sixteen decreased by fifteen
21. subtract seventy from eighty
23. two increased by ten
25. fourteen subtract eleven
27. one plus seven
29. add eight to twelve
31. sixty decreased by fifty
33. eighteen minus twelve
34. decrease thirteen by eight
35. seventeen subtract ten
36. increase ten by thirty
37. thirteen minus four
NSSAL
©2012
Down:
12. seventy decreased by seventy
14. add five to five
16. seventeen minus nine
18. decrease twenty by nineteen
20. increase four by seven
22. fifteen subtract seven
24. zero plus five
26. subtract ten from twenty-one
28. fifty subtract twenty
30. increase eight by three
31. fifteen decreased by twelve
32. add zero to nine
195
Draft
C. D. Pilmer
Multiplication and Division Crossword
Complete the following crossword using words to express your answer (e.g. twenty-seven).
Include hyphens (-) where appropriate. Do not use a calculator.
1.
2.
3.
4.
5.
7.
6.
8.
9.
10.
11.
12.
13.
14.
15.
16.
19.
17.
18.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
37.
35.
36.
38.
39.
40.
41.
42.
44.
45.
Across
2. four multiplied by twenty
5. five times six
NSSAL
©2012
43.
Down
1. eight times four
3. four multiplied by four
196
Draft
C. D. Pilmer
Across
7. eighteen divided by nine
8. nine times four
9. eight multiplied by three
12. seven times five
15. twenty-seven divided by three
17. three multiplied by ten
19. three times five
20. six multiplied by seven
23. twenty-five divided by five
24. twenty-four divided by two
25. seven divided by seven
26. sixty divided by six
28. thirty-five divided by five
32. three times three
33. two multiplied by eight
34. eighty divided by eight
36. forty-eight divided by six
37. fifteen divided by three
38. seven times ten
40. six divided by one
41. sixty divided by thirty
42. seven multiplied by two
44. one hundred divided by ten
45. six times nine
NSSAL
©2012
Down
4. four times seven
5. two times six
6. forty divided by ten
10. forty divided by eight
11. eighteen divided by three
13. nine times nine
14. ten multiplied by five
16. four times five
18. twenty-one divided by seven
21. fifty-six divided by eight
22. eleven times one
27. ten times nine
29. sixty-three divided by seven
30. eight times seven
31. eight multiplied by eight
34. two times six
35. thirty divided by six
37. twenty-four divided by six
38. fifty-four divided by nine
39. twelve divided by three
40. forty-nine divided by seven
41. sixty divided by twenty
43. one hundred divided by fifty
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Operations Crossword
Complete the following crossword using words to express your answers (e.g. thirty-five).
Include hyphens (-) where appropriate. Do not use a calculator.
1.
2.
4.
5.
3.
6.
7.
9.
8.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
43.
Across
2. nine times six
4. seven multiplied by three
7. thirty-two divided by eight
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42.
44.
45.
Down
1. sixteen minus six
2. thirty-five divided by seven
3. forty-three plus three
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Across
9. one hundred subtract ten
11. twelve minus nine
12. forty-five divided by nine
14. seven increased by three
15. nine times nine
16. fifty-six divided by eight
18. twenty-one decreased by two
21. subtract thirty from thirty
22. seven multiplied by six
23. ten times seven
26. seventeen decreased by six
29. forty-nine divided by seven
31. eight divided by eight
36. six multiplied by six
38. three times four
41. one hundred subtract ninety-nine
42. eighty divided by forty
43. fourteen decreased by five
44. twenty multiplied by three
45. forty-four divided by four
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Down
4. eighteen subtract five
5. sixteen divided by eight
6. four add five
8. five multiplied by two
10. three times eight
13. eight multiplied by nine
17. nine increased by four
19. twenty-seven divided by nine
20. subtract fifteen from nineteen
23. eighteen divided by three
24. sixty-three divided by seven
25. twelve times zero
27. nine multiplied by nine
28. eighty minus seventy
30. two times forty
32. thirty divided by two
33. subtract six from eleven
34. thirty divided by five
35. four multiplied by two
36. one hundred divided by five
37. zero multiplied by six
39. seventeen minus eight
40. sixty divided by ten
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Word Sentence to Number Sentence to Answer (A)
In each case take the word sentence, make a number sentence out of it, and then provide the
answer. A few examples have been provided.
Word Sentence
Number Sentence
Answer
e.g. What is the product of seven and three?
7×3
21
e.g. What is five increased by six?
5+6
11
e.g. What is a third of twelve?
12 ÷ 3
4
(a)
What is the sum of three and eight?
(b)
What is six multiplied by five?
(c)
What do you get when you double three?
(d)
What is nine decreased by four?
(e)
What is half of ten?
(f)
Given seven and five, what is their total?
(g)
What is eight times three?
(h)
What is twelve divided by two?
(i)
What is seven increased by six?
(j)
What do you get when you triple six?
(k)
What do you get when eight is taken away from ten?
(l)
What is seven less four?
(m) What is three plus eleven?
(n)
What do we get when ten is broken into five equal parts?
(o)
How much more is nine compared to two?
(p)
What is a quarter of eight?
(q)
What is the product of eight and two?
(r)
What is six combined with eight?
(s)
What is six taken from thirteen?
(t)
What is ten increased by six?
(u)
How many threes fit into fifteen?
(v)
What do you get when nine is removed from ten?
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Word Sentence to Number Sentence to Answer (B)
In each case take the word sentence, make a number sentence out of it, and then provide the
answer. A few examples have been provided.
Word Sentence
Number Sentence
Answer
e.g. The sum of seven and nine
7+9
16
e.g. What is twenty-two decreased by nine
22 − 9
13
e.g. What is a quarter of twenty
20 ÷ 4
5
(a)
What do you get when you double eleven?
(b)
What is thirty-five divided by seven?
(c)
What is seven times six?
(d)
What is seventeen increased by eight?
(e)
What do you get when nine is taken away from fifteen?
(f)
What is the sum of twelve and seven?
(g)
What do we get when ten is broken into two equal parts?
(h)
What is sixteen plus ten?
(i)
What is the product of nine and seven?
(j)
What is half of twenty-four?
(k)
What is thirty-seven less five?
(l)
What is six multiplied by eight?
(m) What is three taken from forty-nine?
(n)
What is twenty-six increased by eleven?
(o)
What do you get when six is removed from twenty-three?
(p)
How many nines fit into eighty-one?
(q)
What is forty-five decreased by six?
(r)
What is eighteen combined with nine?
(s)
What do you get when you triple twelve?
(t)
Given seven and seventeen, what is their total?
(u)
How much more is thirty-six compared to three?
(v)
What is a third of twenty-seven?
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Word Sentence to Number Sentence to Answer (C)
In each case take the word sentence, make a number sentence out of it, and then provide the
answer. A few examples have been provided.
Word Sentence
Number Sentence
Answer
e.g. What is the product of nine hundred and eight
900 × 8
7200
e.g. How many eights are in three hundred twenty?
320 ÷ 8
40
e.g. What do we get when sixty is increased by seventy
60 + 70
130
(a)
Given five hundred and two hundred, what is their total?
(b)
What is fifty multiplied by three?
(c)
What is three hundred increased by four hundred twenty?
(d)
What is forty removed from one hundred?
(e)
What is a third of nine thousand?
(f)
How many sevens fit into three hundred fifty?
(g)
What is seventy times eighty?
(h)
What is the sum of eleven and eighty?
(i)
How much more is ninety compared to thirty?
(j)
What is five thousand plus eight thousand?
(k)
What is two hundred sixty decreased by twenty?
(l)
What do you get when you double four thousand?
(m) What is six hundred increased by two hundred thirty?
(n)
What do you get when you triple forty?
(o)
What is one thousand nine hundred less eight hundred?
(p)
What is half of sixteen thousand?
(q)
What is two hundred eighty divided by seven?
(r)
What do you get when ten is taken away from ninety?
(s)
What is sixty combined with eighty?
(t)
What do we get when six is broken into six equal parts?
(u)
What is seventy taken from ninety-six?
(v)
What is the product of six and seven thousand?
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What are the Possibilities? (A)
The following questions have more than one answer. Find all the possible answers in each case.
Example 1
The product of two whole numbers is greater than 8 and less than 13. What are the possibilities
for the two numbers?
Answer:
The word "product" means that we are dealing with the operation of multiplication. We need
to find all the pairs of whole numbers (e.g. 0, 1, 2, 3,…) that multiply to give us 9, 10, 11, or
12 (i.e. greater than 8 and less than 13).
1× 9 =
9
3× 3 =
9
1×10 =
10
2×5 =
10
1×11 =
11
1×12 =
12
2× 6 =
12
3× 4 =
12
1 and 10
2 and 6
2 and 5
3 and 4
Therefore the possibilities are:
1 and 9
3 and 3
1 and 11
1 and 12
Example 2
The difference of two single digit numbers is 3. What are the possibilities for the two numbers?
Answer:
The word "difference" means that we are dealing with the operation of subtraction. Notice
that we are told to consider only single digit numbers (i.e. number 1 through 9), rather than
multi-digit numbers (e.g. 13, 25, 159).
3−0 =
3
4 −1 =
3
5−2 =
3
7−4=
3
8−5 =
3
9−6 =
3
Therefore the possibilities are:
3 and 0
4 and 1
7 and 4
8 and 5
5 and 2
9 and 6
6−3 =
3
6 and 3
Example 3
The sum of two even numbers is 8. What are the possibilities for the two numbers?
Answer:
The word "sum" means that we are dealing with the operation of addition. We are told to
work only with even numbers (e.g. 2, 4, 6, 8, 10,… ) whose sum is 8.
8+0 =
8
2+6 =
8
4+4 =
8
Therefore the possibilities are:
8 and 0
2 and 6
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Questions
1. The sum of two single digit numbers is 5. What are the possibilities for the two numbers?
2. The product of two whole numbers is 18 or 20. What are the possibilities for the two
numbers?
3. When dividing two single digit numbers, the quotient is 2. What are the possibilities for the
two numbers?
4. The difference of two single digit numbers is 5. What are the possibilities for the two
numbers?
5. The product of two whole numbers is a whole number that is between, or equal to, 4 and 6.
What are the possibilities for the two numbers?
6. The sum of two single digit numbers is greater than 14 and less than 19. What are the
possibilities for the two numbers?
7. When dividing a two digit number that is 24 or less by a single digit number, the quotient is
4. What are the possibilities for the two numbers?
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8. The difference of two whole numbers, which are both less than or equal to 6, is 2 or 3. What
are the possibilities for the two numbers?
9. The product of an even and odd number is equal a whole number that is 10 or less. What are
the possibilities for the two numbers?
10. The sum of two odd numbers is 8 or 10. What are the possibilities for the two numbers?
11. When dividing a two digit number by a single digit odd number, the quotient is 2. What are
the possibilities for the two numbers?
12. The difference of two even numbers, which are both less than or equal to 10, is 4. What are
the possibilities for the two numbers?
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What are the Possibilities? (B)
The following questions have more than one answer. Find all the possible answers in each case.
e.g. The product of two single digit odd numbers is greater than 8 and less than 40. What are the
possibilities for the two numbers?
Answer:
Start by listing all single digit odd numbers.
1, 3, 5, 7, 9
The word product tells us that we are multiplying. We need to look at all the possible
ways of multiplying two of those odd numbers. These are listed below.
1×1 = 1
1× 3 = 3
1× 5 = 5
1× 7 = 7
1× 9 = 9
3 × 7 = 21
3× 3 = 9
3 × 5 = 15
3 × 9 = 27
5 × 9 = 45
5 × 5 = 25
5 × 7 = 35
7 × 7 = 49
7 × 9 = 63
9 × 9 = 81
Now we only want those numbers whose products are greater than 8 and less than 40.
That means we are limited to the following.
1× 9 = 9
3 × 7 = 21
3 × 5 = 15
3× 3 = 9
5 × 5 = 25
3 × 9 = 27
5 × 7 = 35
So there are seven combinations of numbers that work.
1 and 9
3 and 3
3 and 5
3 and 7
3 and 9
5 and 5
5 and 7
1. The product of two single digit even numbers is 16 or greater. What are the possibilities for
the two numbers?
2. The product of two different single digit odd numbers is less than 30. What are the
possibilities for the two numbers?
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3. The sum of two odd numbers is 14, and their product is greater than 20. What are the
possibilities for the two numbers?
4. Two single digit numbers differ by 3, and their product is less than 45 and greater than 5.
What are the possibilities for the two numbers?
5. The product of two whole numbers is 24, and their sum is 14 or less. What are the
possibilities for the two numbers?
6. The product of two even numbers is 24 or less, and their sum is greater than 8. What are the
possibilities for the two numbers?
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Describing Relationships with Words and Numbers
We often use sentences that include numbers to
describe relationships between objects. To the right
we have provided a diagram that is made up of
arrows, lightning bolts, and hearts. We can create
different sentences that describe the number of shapes
we have and the relationships between these shapes.
These are listed below.
• There are 3 arrows.
• There are 5 lightning bolts.
• There are 6 hearts.
• There are a total of 14 shapes.
• There is 1 more heart than lightning bolts.
• There is 1 less lightning bolt than hearts.
• There are 2 more lightning bolts than arrows.
• There are 2 less arrows than lightning bolts.
• There are 3 more hearts than arrows.
• There are 3 less arrows than hearts.
• There are twice as many hearts as arrows.
• There are half as many arrows as hearts.
Question:
Look at the diagram on the right. Use sentences with
numbers to describe what you see.
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More Describing Relationships with Words and Numbers
As previously mentioned, we often use sentences that include numbers to describe relationships
between objects.
Consider this situation. Jim has 4 keys on his key chain,
and Meera has 12 keys on her key chain. We could write
the following sentences using this information.
• Jim and Meera have a total of 16 keys.
• Meera has 8 more keys than Jim.
• Jim has 8 fewer keys than Meera.
• Meera has three times as many keys as Jim.
Questions
Read the sentence. Make new sentences using this information.
1. Lei has 6 grandchildren, and Nasrin has 12 grandchildren.
2. In the first week, Ryan saved $20 after his expenses, but in the second week he saved an
additional $100.
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Recognizing the Important Information
In this section, we will be given written statements and then asked a series of questions. None of
these questions will require any mathematical calculations; all of the answers can be pulled
directly from the statement. By doing this, we are learning how to recognize the important
information given in a written statement. It is all about understanding that statement.
Example 1
Carlos started a three month exercise program. He devoted two hours a
day to this program, seven days a week. He originally weighed 195
pounds but dropped 10 pounds over the three month period such that his
new weight was 185 pounds.
(a) How many hours per day did he spend on this exercise program?
(b) How much weight did Carlos lose during this exercise program?
(c) How much did Carlos weigh at the end of the exercise program?
(d) How long did this exercise program run?
(e) How much did Carlos weigh at the beginning of the exercise program?
Answers:
(a) 2 hours per day
(b) 10 pounds
(c) 185 pounds
(d) 3 months
(e) 195 pounds
Reason: "He devoted two hours a day to this program"
Reason: "dropped 10 pounds over the three month period"
Reason: "such that his new weight was 185 pounds."
Reason: "started a three month exercise program."
Reason: "He originally weighed 195 pounds"
Questions
Answer the series of questions that accompany the written statements. No calculations are
needed, and you are not required to explain how your arrived at the answer. Just write the
answer in the supplied box.
1. Janice has one child who will be returning to elementary school in September. She spent
$120 on new clothing and $80 on school supplies. She was pleased that she spent $200
because she had originally thought that everything would have cost $250.
(a)
How much did Janice spend in total?
(b)
How much did Janice spend on clothing for her child?
(c)
How much did Janice think she was going to have to spend?
(d)
How much did Janice spend on school supplies for her child?
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2. A particular cut of meat normally sells for $8 per kilogram. A grocery store has a seven day
sale and sells that same meat for $6 per kilogram. That means that on a three kilogram cut,
which normally sells for $24, would now sell for $18.
(a)
(b)
How much does three kilograms of meat sell for during the
sale?
How much per kilogram does the meat sell for during the sale?
(c)
How long is the sale?
(d)
How much does three kilograms of meat sell for during the
regular pricing period?
3. Rana was hired to do yard work on a two acre property that had been neglected for several
years. She charged the owner of the property $600 for this four day job. She based this price
on the fact that it took her 30 hours to complete the work. That means that she was making
$20 per hour.
(a)
What was Rana's hourly wage?
(b)
How much money did Rana make on this job?
(c)
How many hours did Rana work on this job?
(d)
How large was the property?
4. Jorell went on a 490 kilometer road trip. For the first two hours, his vehicle's speed was 110
kilometres per hour. For the last three hours, his vehicle was travelling at 90 kilometres per
hour. This trip took him a total of five hours.
(a)
At what speed was Jorell travelling on the last leg of the trip?
(b)
How long did the road trip take?
(c)
How many kilometres did Jorell travel in total?
(d)
How long was the vehicle travelling at 110 kilometres per
hour?
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Does It Make Sense?
A statement has been supplied. You decide whether the statement makes sense. If not, explain
why.
1. Jim purchased a song from iTunes for $9. Before the purchase he had $20 in his iTunes
account; after the purchase, he had $11 in the account.
2. Janice purchased 2 litres of homogenized milk, 4 litres of skim milk, and 1 litre of chocolate
milk. In total she purchased 8 litres of milk.
3. Tanya and her two sisters decided to equally share the $210 bill for their parents' anniversary
gift. That meant that each girl had to pay $70.
4. Kiana ran 6 kilometres per day over 7 days. In that period of time she ran 36 kilometres.
5. Yisha had 3 quarters, 1 dime, and 1 nickel. That means that he was 10 cents short of one
dollar.
6. The roommates in an apartment decided to purchase a $600 flat screen television and share in
the expense equally. If there were 100 roommates, then that meant that each had to pay $6.
7. Sapphire bought 6 sweaters, each costing $15 before taxes. Her total bill for the sweaters
before taxes would be $90.
8. There were 26 students in the class at the beginning of the year. Over the course of the year,
4 transferred out and 1 transferred in. That means that by the end of the year, there were 21
students in the class.
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9. Faris only works Monday through Wednesday. Each day he works 8 hours. Over 2 weeks,
he works a total of 48 hours.
10. There were initial 20 people signed up for the workshop. Over the next week that number
doubled. At the last minute, 6 people said that they would be unable to attend. In the end, 34
people attended the workshop.
11. Jacob bought 3 bags of potatoes and 2 bags of carrots. The each bag of potatoes had a mass
of 5 kg, and a bag of carrots had a mass of 1 kg. The total mass of carrots and potatoes that
were purchased by Jacob was 30 kg.
12. Each day, Anne drove at 100 km/h for 6 hours. If she maintains this, then she will travel
18 000 km over 3 days.
13. The brand new 50 inch flat screen television cost $100 but the taxes came to $15, meaning
that the total bill was $115.
14. The time it takes Arthur to get ready for work is 1 hour and 15 minutes. His shower takes 20
minutes. Breakfast and watching the morning news takes 25 minutes. The remaining 30
minutes is spent doing things like brushing his teeth, making his bed, getting dressed, and
using the washroom.
15. There were 12 apples and 18 oranges. If they were shared equally among 3 people, then each
person would get 4 oranges and 6 apples.
16. Helen was selling beverages at the fair for $2. Half of the money on each beverage was
profit. If she sold 230 beverages, then her profit was $460.
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Insert Your Own Numbers and Words
Below you have been given a written statement that is missing numbers (smaller blanks) and
words (larger blanks). Your job is to add your own numbers and words so that the written
statement makes sense. Reasonable numbers and words must be used. Naturally there are an
infinite numbers of acceptable answers.
e.g. Kimi purchased _______ _________________________, each costing _______. These
items came to a total of _______.
Learner #1's Acceptable Answer:
Kimi purchased 7 pairs of socks, each costing $3. These items came to a total of $21.
Learner #2's Acceptable Answer:
Kimi purchased 2 jars of peanut butter, each costing $4.50. These items came to a total of
$9.
1. Tammy and Peter have _______ _________________________ and _______
_________________________. That means that they have a total of _______
_________________________.
2. The bag of _________________________ had a mass of _______ kg. If _______ kg is
removed, then that means _______ kg remains.
3. Nita split the _______ _________________________ evenly amongst her _______ friends.
That meant that each friend got ______.
4. The _______ children each had _______ _________________________. That means that
altogether they had _______ _________________________.
5. The temperature of the _________________________ was initially _______ degrees
Celsius. Over time, the temperature dropped by _______ oC, so that it ended up being
_______ oC.
6. At the beginning of the day Jorell had _______ litres of _________________________. He
used _______ litres and a friend later returned _______ litres she had taken a few weeks ago.
Jorell now had _______ litres.
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Complete the Statement: Addition and Subtraction
With each question, you have been given a brief statement and you must fit the numbers
(supplied in the boxes to the right) into the appropriate blanks so that it all makes sense. A
number can only be used once and there is one extra number provided in the boxes that should
not be used.
1.
Alex and Tylena have _____ chocolates. Alex, who is not very
good at sharing, eats _____ chocolates, leaving only _____
2
13
5
8
30
20
60
50
18
2
16
4
50
30
90
60
2
9
4
7
14
9
3
5
2
15
11
13
chocolates for Tylena.
2.
Micheline ran _____ kilometres on the first day. The second day
she ran even further covering _____ kilometres. Over the two
days she ran a total of _____ kilometres.
3.
The tank of water was initially full. _____ litres of water was then
drained from the _____ litre tank. With such a small amount of
water being drained, it meant that _____ litres remained.
4.
Samir borrows _____ dollars from his two friends. Jun, who
supplies most of the money, lends Samir _____ dollars. Nita, his
other close friend, lends him _____ dollars.
5.
Jacob was hoping to get _____ hours of sleep. He ended up
getting _____ hours of sleep, just _____ hours short of his desired
number of hours of sleep.
6.
Nashi has only purchased or made _____ gifts for the holidays.
She still needs to purchase or made _____ gifts if she wishes to
give a gift to each of her _____ family members or friends.
7.
Asra received a bonus and had enough money to take ______ of
her ______ friends out to dinner. Unfortunately she did not have
enough to pay for a few friends, specifically ______ friends.
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Complete the Statement: Multiplication and Division
With each question, you have been given a brief statement and you must fit the numbers
(supplied in the boxes to the right) into the appropriate blanks so that it all makes sense. A
number can only be used once and there is one extra number provided in the boxes that should
not be used.
1.
A class of _____ students wishes to have a guest speaker deliver a
lecture. If the speaker charges _____ dollars for the lecture, then
75
3
20
25
20
5
60
3
3
90
30
20
6
20
80
4
27
9
3
11
5
90
75
15
3
24
12
2
each student will have to pay _____ dollars.
2.
The _____ candies are to be shared equally amongst the _____
roommates. That means that each roommate will receive _____
candies.
3.
If the cyclist is travelling at an average speed of _____ kilometres
per hour for _____ hours, then the she will cover a distance of
_____ kilometres in that time.
4.
The ______ siblings inherit a ______ acre plot of land. To be
fair, each sibling will receive a parcel of land measuring ______
acres.
5.
Kate, who overuses sick time, only has _____ sick days left in her
bank. Lei, who rarely uses sick time, has _____ days left. Lei has
_____ times the number of sick days as compared to Kate.
6.
The boxed set of all seasons of Seinfeld is on sale for _____
dollars. If _____ close friends wish to share the cost of the boxed
set, each will pay _____ dollars.
7.
Kamala wants to increase her hourly wage by a factor of _____.
If her present hourly wage is _____ dollars per hour, then her
desired hourly wage is _____ dollars per hour.
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Complete the Statement (A)
With each question, you have been given a brief statement and you must fit the numbers
(supplied in the boxes to the right) into the appropriate blanks so that it all makes sense. A
number can only be used once.
1.
Lei and Jun are siblings. Lei is _____ years younger than Jun. If Jun is
_____ years old, then Lei is _____ years old.
2.
24
15
12
3
35
5
40
12
36
3
60
180
3
15
25
40
120
40
3
800
4
200
The mechanic ordered _____ containers of engine oil at a cost of
______ dollars per container. The total cost was ______ dollars.
10.
3
Ryan, who prefers running, ran for _____ minutes and biked for _____
minutes. That means he trained for a total of _____ minutes.
9.
8
If there are _____ minutes in an hour, then we know that there are
_____ minutes in _____ hours.
8.
2
Bashir had _____ candies to split evenly between his _____ children.
Each child got _____ candies; enough to ruin their supper.
7.
7
Anne had _____ dollars but spent most of her money on a _____ dollar
top (after taxes). She now has _____ dollars left in her purse.
6.
55
Normally _____ people attend the neighborhood watch meeting. That
number increased slightly by _____ such that _____ people attended.
5.
31
Very few men attended the show. If there were _____ men and _____
women, then there ______ times as many women as men.
4.
4
Marcus has dime and nickels in his pocket. If he has _____ nickels and
_____ dimes, then he has a total of _____ cents.
3.
27
______ room mates got together to purchase an ______ dollar couch. If
they all paid the same amount, then each pays ______ dollars.
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Complete the Statement (B)
With each question, you have been given a brief statement and you must fit the numbers
(supplied in the boxes to the right) into the appropriate blanks so that it all makes sense. A
number can only be used once.
1.
Tanya has _____ dimes and _____ quarters in her purse. That means
she has _____ cents of change in her purse.
2.
3
7
2
21
11
19
30
48
24
2
3
18
21
250
50
300
4
24
32
50
2
25
There are 8 SUVs and _____ cars in the lot. Therefore there are _____
times as many cars as SUVs, or _____ more cars than SUVs.
10.
8
There were ______ millilitres of water in a container. If only ______
millilitres is poured out, then the container still has ______ millilitres.
9.
12
The room temperature was _____ degrees Celsius. If it is turned up
slightly by _____ degrees, then the new temperature is _____ degrees.
8.
47
If there are _____ hours in one day, then there are _____ hours in _____
days.
7.
3
Ryan had _____ dollars but spent most of his money on a _____ dollar
DVD (after taxes). He now has _____ dollars left.
6.
50
Three friends equally share the cost of a _____ dollar pizza that was
divided into 6 pieces. Each pays _____ dollars and gets _____ pieces.
5.
3
Kim bought _____ apples and 4 oranges. Therefore she bought _____
times as many apples as oranges, or _____ more apples than oranges.
4.
80
Bill and Ajay are friends. Bill is _____ years older than Ajay. If Bill is
_____ years old, then Ajay is _____ years old.
3.
2
The store owner ordered ______ packages of printer paper at a cost of
______ dollars per package. His bill (before taxes) was ______ dollars.
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Complete the Statement (C)
With each question, you have been given a brief statement and you must fit the numbers
(supplied in the boxes to the right) into the appropriate blanks so that it all makes sense. A
number can only be used once and there is one extra number provided in the boxes that should
not be used.
1.
The cereal, the more expensive item, cost _____ dollars, and the
dish soap cost _____ dollars. The total cost was _____ dollars.
2.
10
20
60
3
10
8
18
12
30
40
90
3
8
7
2
9
8
15
6
7
3
65
2
90
94
70
24
54
15
4
80
20
The friends on his Facebook account increased by ____, going
from _____ to _____
10.
10
Hinto had _____ nickels and _____ quarters. He had a total of
_____ cents in nickels and quarters.
9.
40
Kim worked _____ hours on Monday and less on Tuesday. If
she got _____ hours on Tuesday, then her total was _____ hours.
8.
60
Tom drove for _____ more hours than Ed. Tom drove _____
hours, and Ed drove for _____ hours.
7.
30
The corner store owner sold _____ bottles of pop. If each sold
for _____ dollars, then his total pop sales were _____ dollars.
6.
8
The DVD cost _____ dollars. The socks cost _____ dollars.
The DVD was _____ dollars more expensive than the socks.
5.
2
_____ kg of flour must be divided evenly amongst ______
families. Each family was pleased to get _____ kg.
4.
5
The jar contained _____ candies. If you ate _____ , which is
most of the candy, then that would leave _____ in the jar.
3.
3
There _____ times as many children at the movie compared to
adults. There were _____ adults and _____ children.
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Complete the Statement (D)
With each question, you have been given a brief story and you must fit the numbers (supplied in
the box below) into the appropriate blanks in the story so that it all makes sense. A number can
only be used once.
1. A cinema in a movie theatre can hold _______ people. Unfortunately
that day, only half of the cinema was full meaning only _______ people
are viewing the movie. The theatre charges ________ dollars for child
tickets and _________ dollars for adult tickets. The total earnings for
that showing in that cinema were _________ dollars.
6
120
456
10
60
2. Attendance for the annual blues concert is normally ________ people. This year, the number
attending grew by ________, meaning that a total of ________ people attended. If
individual tickets sold for ________, the promoters expected to bring in _________ dollars
more than last year just in ticket sales.
925
75
3000
850
40
3. Taylor works at a hardware store where he makes _________
dollars per hour. Typically he works ________ hours per week,
just shy of full time hours, and brings in ________ dollars (before
deductions). If he works an additional ________ hours a week, he
will make ________ dollars more (before deductions).
38
75
5
570
15
4. Tanya has _________ teenage children. Montez, the oldest, is _________ years old. Tylena,
the youngest, is _________ years old. Kiana, the middle child, is _________ years older
than Tylena, making her _________ years old. Tanya, the mother, is ________ years old.
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5. A group of seniors wants to charter buses to go on a trip. They
check with the local charter company and learn that each bus can
take _________ people and that the company charges _________
dollars a day for the bus and driver. Since _________ seniors wish
to take the trip, then that means that they will need to charter _________ buses.
Unfortunately that means that __________ seats on the buses will be unused. If the seniors
are planning on taking a two day trip, the total cost for chartering the buses is _________
dollars (before taxes).
10
50
4800
800
3
140
6. A tank initially held _________ litres of water. A pump that removes water from the tank at
a rate of _________ litres per minute is switched on ten minutes. That means that
_________ litres have been removed, leaving _________ litres in the tank. If someone
comes after the pump was switched off and pours _________ litres of water into the tank, the
tank will now hold _________ litres of water.
20
400
450
200
50
600
7. Two brothers, Brian and Dave, work for the same company. Brian makes _________ dollars
more per hour than Dave. Since Brian makes _________ dollars per hour, that means that
Dave makes _________ dollars per hour. That means that in a __________ hour work week,
Brian will make _________ dollars before deductions, and David will make ________
dollars before deductions.
640
16
3
40
19
760
8. There were __________ times as many people at the Rolling Bones concert than at the
Tragically Flipped concert. If __________ people were at the Flip concert, then that means
that __________ people were at the Bones concert. The Bones charged __________ dollars
per ticket, while the Flip only charged __________ dollars per ticket. That means that the
Bone brought in ___________ dollars more in ticket sales for their concert.
7000
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Not Enough Information is Provided
In each situation below, you are asked to solve a problem but not quite enough information is
provided. Explain what is needed to complete the question.
e.g. For a Christmas bonus, the owner of a company is going to rent a bus to take his 110
employees to a concert. How much is it going to cost to rent the buses?
Answer:
We need to know if the buses are the same size, how many passengers each bus holds, and
how much the bus company charges for each bus for this particular round trip.
1.
The jar of marbles has a mass of 40 grams. What is the mass of each marble?
2.
Manish bought six cans of apple juice. How much did he pay?
3.
Marcus has a $600 to pay his friends for helping him shingle his cottage roof. How much
should each get?
4.
If a large container of water has water being removed from it at a constant rate of 2 litres per
minute, then how much water will be left in the container?
5.
If at the fire hall fund raiser, volunteers are selling hotdogs for $1, chips for $1.25, and pop
for $0.75, then how much money would they make by the end of the day?
6.
Jun's children would like to spend a few days at an overnight outdoor adventure camp this
summer. The camp charges $40 per day per child. This includes meals. How much will
Jun have to pay?
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Word Problems with Too Much Information
With each word problem, identify the extra information (i.e. number) that is not needed to solve
the problem, and then identify the correct solution from the multiple choice selections.
e.g. There are 3 children in Tammy's family and she needs to purchase 2
litres of orange juice, 5 litres of milk, and 4 litres of apple juice each
week for them. How many litres in total of fluids does she purchase
each week for her children?
(b) (5 + 4 ) ÷ 3 = 3
(a) 3 + 2 + 5 = 10
(c) 5 + 4 − 2 = 7
(d) 2 + 5 + 4 = 11
Answer:
Extra Information: There are 3 children in Tammy's family
Correct Solution: (d) 2 + 5 + 4 = 11
e.g. There are 18 kg of potatoes and 12 kg of turnip. If these vegetables are to be shared
equally by 3 families, how many kilograms of turnip does each family get?
(b) 12 ÷ 3 = 4
(a) 18 + 12 = 30
(d) 3 × 12 = 36
(c) 18 ÷ 3 = 6
Answer:
Extra Information: 18 kg of potatoes
Correct Solution: (b) 12 ÷ 3 = 4
1. A stapler costs $5, a notebook costs $3, and a calendar costs $10. How much more is the
calendar compared to the stapler?
(b) 10 − 3 = 7
(a) 10 − 5 = 5
(d) 5 × 3 = 15
(c) 10 + 5 = 15
Extra Information: __________________________________________
2. A bottle of pop costs $2 and a bag of potato chips costs $3. How much does it cost to
purchase 8 bottles of pop?
(b) 2 + 3 = 5
(a) 3 + 8 = 11
(c) 8 × 2 = 16
(d) 8 ÷ 2 = 4
Extra Information: __________________________________________
3. Tyrus is taking a flight. His carry-on bag weighs 10 kg, and his two check-in bags weigh 16
kg and 24 kg. How much do his check-in bags weigh in total?
(a) 24 − 16 = 8
(b) 24 + 16 = 40
(c) 16 − 10 = 6
(d) 16 + 10 = 26
Extra Information: __________________________________________
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4. Kiana has 4 children, 6 nieces, and 8 nephews. What is the difference between the number
of nieces and the number of nephews Kiana has?
(a) 6 − 4 = 2
(b) 8 ÷ 4 = 2
(c) 6 × 4 = 24
(d) 8 − 6 = 2
Extra Information: __________________________________________
5. There are 20 pens and 10 pencils to be shared equally among 5 employees. How many pens
does each employee get?
(b) 10 ÷ 5 = 2
(a) 20 − 10 = 10
(c) 20 ÷ 5 = 4
(d) 20 × 10 = 200
Extra Information: __________________________________________
6. There are 7 men and 8 women in a running club. If each woman ran 6 kilometres that day,
how far did the women travel in total?
(b) 8 + 6 = 14
(a) 7 × 6 = 42
(c) 7 + 8 = 15
(d) 6 × 8 = 48
Extra Information: __________________________________________
7. After 3 hours, Ryan completed 2 pages of math homework and 6 pages of English
homework. How many pages of homework did he complete within that period of time?
(a) 6 ÷ 3 = 2
(b) 2 + 6 = 8
(c) 3 × 5 = 15
(d) 6 + 3 = 9
Extra Information: __________________________________________
8. Dave has $30 to spend at the flea market. No sales tax is charged at the flea market. He is
trying to decide if he should buy a $12 DVD or a $23 sweatshirt. If he purchases the DVD,
how much change will he have?
(b) 12 + 23 = 35
(a) 30 − 12 = 18
(d) 23 − 12 = 11
(c) 30 + 12 = 42
Extra Information: __________________________________________
9. In 3 hours, Anne could read 90 pages of a novel or run 15 km. How many pages of a novel
can Anne read in an hour?
(b) 90 ÷ 3 = 30
(a) 3 × 15 = 45
(c) 3 + 90 = 93
(d) 90 − 15 = 75
Extra Information: __________________________________________
10. When Lei jogs, she travels at 6 kilometres per hour. When she cycles, she travels at 20
kilometres per hour. How far can she jog in 4 hours?
(a) 4 + 6 = 10
(b) 20 − 6 = 14
(c) 20 ÷ 4 = 5
(d) 6 × 4 = 24
Extra Information: __________________________________________
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Create Your Own Math Statement
In each case you are provided with a mathematical operation and its corresponding solution.
Your mission is to create your own real world statement that corresponds to that operation. Use
complete sentences. Try to be creative. Do not use the same type of application more than once
(e.g. don't create two math statements involving how much money is left after making a
purchase). Naturally there will be a wide range of acceptable answers.
e.g. 16 - 5 = 11
Possible Answer:
There were 16 people attending Jeff's party. Five people had to leave at 10:00 p.m., leaving
11 people still at the party.
(a) 7 + 9 = 16
(b) 24 - 9 = 15
(c) 3
6 = 18
(d) 35
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(e) 30 - 12 = 18
(f) 6
7 = 42
6=3
(g) 18
(h) 4 + 5 = 9
(i) 7 + 5 - 2 = 10
(j) 6
4 + 3 = 27
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Word Problems (A)
Answer the following questions. Show your work (i.e. Show the operation you used).
e.g. Marcy went to the flea market and bought each of her 3 nephews the same
toy truck. Each toy cost $8. How much did she spend on toys?
Answer: 3 × 8 = 24
She spent $24 on toys for her nephews.
1. Last week Micheline trained 5 hours for the iron man competition, and this week she trained
for 6 hours. How many hours in total did she train over this two week period?
2. Nita purchased a small bottle of over-the-counter medication. The bottle
initially contained 20 tablets. A few weeks later, there were only 6 tablets left
in the bottle. How many tablets had she used?
3. Rhonda ran 5 kilometres each day for 7 days. How far did she run during that week?
4. Four friends have $28 to split evenly between them. How much does each person get?
5. Ajay had $35 cash but then spent $13 on a paperback novel and coffee. How
much cash did he have left?
6. A building is 27 metres high. If each story is 3 metres high, how many stories does the
building have?
7. Thomas grew tomato plants in his backyard. Two months later, he picked 8
tomatoes from one of the plants and 7 tomatoes from another. How many
tomatoes did he pick altogether on that day?
8. A vendor is charging $3 for a hot dog and pop. If 40 customers purchased this combination,
how much money did he bring in from the sales of the hot dog and pop combo.
9. Two departments in a company were combined to create one new department. If there were
10 people in the first department, and 9 people in the second department, how many are now
in the new department. Assume that no one was fired or laid off.
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10. Three room mates pooled their money to buy a $900 flat screen television.
How much did each pay, assuming they all paid the same amount?
11. Two neighbors live along the same lake. Kiana has 15 metres of beachfront, while Lei has
28 metres of beachfront. How much more beachfront does Lei have compared to Kiana?
12. Two people are cycling. One cyclist is travelling at a speed of 12 km/h, and
the other at 19 km/h. How much faster is the second cyclist?
13. There are 5 construction sites, and each site requires 9 workers. How many workers are
needed in total?
14. A century is 100 years. A decade is 10 years. How many decades are in a century?
15. Anne has $60. Dave has $39 more than Anne. How much money does Dave have?
16. The owner of a bookstore ordered 60 copies of a new hardcover book. If each
book costs her $9, how much will she have to pay for all the books?
17. On Thursday, the low temperature of the day was 4oC, while the high temperature was 17oC.
What is the difference between the high and low temperature for that day?
18. Manish went for a hike with his friend. Both carried a pack; Manish's pack
weighed 12 kg, and his friend's weighed 14 kg. Part way through the hike,
the friend injured his ankle, so Manish had to carry out both packs. How
much did Manish have to carry out on the return trip?
19. A local rock band was organizing its own concert. They were hoping to raise $480. If the
tickets to the show cost $6, how many tickets need to be sold?
20. The doctor instructed the patient to take 4 pills per day for 10 days. How many pills did the
doctor prescribe to this patient?
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Word Problems (B)
Answer the following questions. Show all your work in the space provided.
1. A skydiver jumps from the
aircraft at an altitude of 3640
metres. At an altitude of
1780 metres, she deploys the
parachute. How far did she
fall before deploying the
chute?
2. Sapphire had 237 books in
her collection. She went to a
used bookstore and bought
19 more. How many books
does she now have?
3. Six campaign workers have to
hand out information booklets
to the 498 households in a
neighborhood. If each worker
must distribute the same
number of booklets, how many booklets should
each worker distribute?
4. Each container of drywall
compound weights 27
kilograms. If a contractor
needs to purchase 15
containers, what will the
total weight be?
5. Jeff's band knows 24 rock
songs, 19 blues songs, and 5
country songs. How many
songs do they know in total?
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6. Ryan's take-home pay this
month was $1920. If his
rent was $650, how much
is left over for other
expenses?
7. A vehicle travelled 207
kilometres on 9 litres of
gasoline. How far could the
vehicle travel on one litre of
gasoline?
8. The local professional hockey
team drew 8454 people to their
first game, and 7461 people to
their second game. How many
people in total attended the
first two games?
9. Sixteen people decided
to go to an outdoor
concert. If each ticket
cost $85, how much was
spent in total by the
sixteen people on
tickets?
10. Three friends are driving
to a vacation spot in the
same car. If they have to
travel 1365 kilometres
and decide to share the
driving equally, how far
does each have to drive?
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11. Jacob has 89 DVD movies.
If Sasha has 47 more DVD
movies, how many does
she have?
12. There are 257
employees at the
company. If 118 of
them are women,
how many are men?
13. Originally there was 196 litres of
liquid in the barrel. If 68 litres is
removed, how much remains in
the barrel?
14. A local charity wants to
raise $1260 by selling $9
tickets to the dance at the
fire hall. How many
tickets need to be sold so
that the charity reaches its
desired goal?
15. A doggie daycare company
has 31 sites across the
country. If each site can
serve 25 dogs on any one day,
what is the maximum number
of dogs that can be served by
the company on one day?
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Same Numbers, Similar Context, Different Math (A)
Word problems that use similar numbers in similar contexts have been grouped together. The
mathematics to solve these grouped word problems, however, is quite different. Your mission is
to solve each of these problems, showing the number sentence (e.g. 5 + 16 = 21) that was used
solve the question. If there are any word problems involving the purchasing of product, do not
worry about the taxes associated with those purchases.
1. (a) Five women went out for lunch and each spent $20. How much was
spent in total?
(b) Five women wished to purchase a $20 cheese plate for their afternoon party. If they
shared in the cost of the cheese plate equally, how much did each woman have to pay?
(c) Twenty women were supposed to attend the lunch but five cancelled out at the last
moment. How many women attended the lunch?
(d) Restaurant reservations were made for twenty women but five women turned up
unexpectedly. How many women attended the lunch?
2. (a) Marcus has two dogs; one weighs 40 kg and the other weighs 8 kg. How many times
heavier is the large dog compared to the small dog?
(b) Jacob's overweight dog originally weighed 40 kg, but after being put on a new diet and
exercise program, the dog's weight dropped by 8 kg. How much does the dog weigh?
(c) Hanna runs a large kennel and presently owns 40 dogs. If each dog eats 8 kg of dry dog
food per month, how many kilograms of dog food does she need each month?
(d) At one year, Manish's dog weighed 40 kg. Over the next year the dog gained another 8
kg. How much does the dog weigh?
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3. (a) Before starting his exercise program, Jim estimated that he only did
physical activity for 5 hours per month. After starting his exercise
program, he estimates that he is doing 30 hours per month. How many
additional hours of exercise are being received under this new program
compared to when Jim had no program?
(b) Rana was exercising for approximately 30 hours per month, and then decided to increase
that by an additional 5 hours per month. How many hours per month of exercise is she
now receiving?
(c) Each month Grace exercises for 30 hours. Each month Tanya only exercises for 5 hours.
How many times larger is Grace's exercise program compared to Tanya's?
(d) Each month Bashir exercises for 30 hours. How many hours of exercise would he get
over 5 months?
4. (a) Caledonia Elementary School was taking 50 children on a field trip. School board policy
requires 10 chaperones. How many people in total should be attending the trip?
(b) A college was organizing a ski day for its learners. They decided that they would need
10 buses to transport learners. If each bus can take 50 learners, how many learners in
total can the college take to the ski hill?
(c) Parker Middle School was going to take 50 students on a field trip, but 10 were unable to
attend due to illness. How many students were able to make the trip?
(d) Sampson High School was initially planning on using a bus to transport 50 students to the
science fair. A bus was unavailable so they decided to rent vans. If each van can
transport 10 students, how many vans would be needed?
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Similar Numbers, Similar Context, Different Math (B)
Word problems that use similar numbers in similar contexts have been grouped together. The
mathematics to solve these grouped word problems, however, is quite different. Your mission is
to solve each of these problems, showing the number sentence (e.g. 5 + 16 = 21) that was used
solve the question. If there are any word problems involving the purchasing of product, do not
worry about the taxes associated with those purchases.
1. (a) Janice bought a $40 item at the store. If she received $10 change, how much money did
she initially pass to the cashier?
(b) Janice was hosting a celebration for her family and needed to purchase several $40 items.
If she purchased 10 of these items, how much did she pay?
(c) Janice wanted to purchase a $40 item but only had $10 on her. Her sister lent her the rest
of the money to buy the item. How much money did her sister lend her?
(d) Janice received a $40 gift from her 10 friends. If the friends equally shared in the cost of
the gift, how much did each person pay?
(e) If Janice bought two $40 items and three $10 items, what was the total bill?
(f) In January Janice still owed $40 on the purchase of an item. Since that time she has
made two monthly payments, each of $10. How much does she still owe?
(g) Janice wanted to buy three $40 items but only had nine $10 bills in her purse. How much
more money does she need to make the purchase?
2. (a) If Samir has to take 2 pills per day for 20 days, then how many pills did he ultimately
have to take?
(b) Samir's pill bottle contained 20 pills. If he was instructed to take 2 pills per day, how
long would his supply of pills last?
(c) Samir's pill bottle contained 20 pills. If he took 2 pills on the first day, how many pills
are left for the remaining days?
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(d) Samir's new pill bottle contains 20 pills and his old pill bottle only contains 2 pills. How
many pills does he have in total?
(e) Samir has initially had 20 pills. If he takes 2 pills per day for three days, how many pills
will be left?
(f) Samir had three bottles, each containing 20 pills. He also had four sample packages, each
containing 2 pills. How many pills does he have in total?
(g) Samir has 20 pills and must take 2 pills twice a day. How long will his supply of pills
last?
3. (a) If a business has 5 employees in one department and 30 employees in another
department, then how many people do they have in the two departments?
(b) If a business has 5 departments, each with 30 employees, then how many employees do
they have in total?
(c) A business had 30 employees but unfortunately had to lay off 5 people. How many
employees do they now have?
(d) A business has 30 employees that are shared equally amongst the 5 departments. How
many employees does each department have?
(e) A business runs three shifts each day. Each shift is made up of 5 managers and 30
assembly line workers. How many employees does this business need each day?
(f) The 5 managers each make twenty dollars per hour. The 30 assembly line workers each
make ten dollars per hour. Assuming that all employees are at work during the day, how
much does the business pay out per hour for employee wages?
(g) A business originally had 30 employees but proceeded to hire four teams, each made up
of 5 employees. How many employees does the company now have?
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More than One Question
Example
Frank collected famous autographs. He had 5 autographs from baseball
players, 8 from football players, and 10 from hockey players.
(a) How many more autographs does he have football players compared to
baseball players?
(b) How many autographs does he have in total?
(c) If Frank had not collected football player autographs, how many
autographs would he have in total?
(d) Suppose Frank also wanted to collect autographs of basketball players
and set a goal of having 3 times as many of autographs of basketball
players as compared to hockey players. If he reached this goal, how
many basketball player autographs would he have?
Answers:
(a) 8 − 5 =
3 autographs
(b) 5 + 8 + 10 =
23 autographs
(c) 5 + 10 =
15 autographs
(d) 3 ×10 =
30 autographs
Questions
1. The large container initial held 12 litres of water. Meera first removed 5 litres of water, then
removed 3 litres of water.
(a) How much water did Meera remove in total?
(b) How much more water did Meera first remove compared to the amount she removed the
second time?
(c) In the end, how much water remained in the large container?
(d) If Meera had not removed water for the second time, how much would have remained in
the large container.
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2. Harris retired and purchased a hobby farm. On this farm he several
types of animals which included 2 cows and 14 chickens. He also had
sheep and pigs. The number of sheep was 3 times the number of cows.
He had half as many pigs as chickens.
(a) How many sheep did Harris have?
(b) How many more chickens did he have compared to cows?
(c) How many pigs did Harris have?
(d) How many animals did he have in total?
(e) If he sold all his pigs, how many animals would he have?
3. Candice restores antique automobiles and motorcycles. She
presently has 3 automobiles, and 4 times as many motorcycles.
(a) How many motorcycles does she have?
(b) How many antique vehicles does she have in total?
(c) Assuming that each car has a spare tire, how many tires does she have for her antique
automobiles?
(d) How many more antique motorcycles does she have compared to antique automobiles?
(e) If she purchased another automobile, but the number of motorcycles remained the same,
how many times more motorcycles would she have compared to automobiles?
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Food Chart (A)
The following chart shows the amounts of protein, fat and carbohydrates in different servings of
foods.
Food and Serving
Protein
(grams)
3
3
2
2
5
24
16
6
1
2
Watermelon (1 slice)
Apple Pie (1 slice)
White Bread (1 slice)
Tomato Juice (1 cup)
Peanut Butter (1 tablespoon)
Canned Tuna in Oil (3 ounces)
Beef and Vegetable Stew (1 cup)
Poached Egg (1 egg)
Blueberries (1 cup)
Corn Chips (1 ounce)
Fat
(grams)
2
18
1
0
8
7
11
5
1
9
Carbohydrates
(grams)
35
60
12
10
3
0
15
1
20
16
Answer each of the following using the information supplied in the above chart. Include the
number sentence (e.g. 10 - 6 = 4) that you used to find your answer.
1. If you were to have one slice of watermelon and one slice of white bread, then how many
grams of protein would you have ingested (i.e. eaten)?
2. What's the difference in the number of grams of fat between one tablespoon of peanut butter
and one poached egg?
3. If you ate two cups of blueberries, then how many grams of carbohydrates would you have
ingested?
4. How many times larger is the number of grams of protein in one cup of beef and vegetable
stew compared to one cup of tomato juice?
5. If you ate a poached egg, a slice of white bread, and one cup of tomato juice, then how many
grams of carbohydrates would you have ingested?
6. How many times larger is the number of grams of fat in two slices of apple pie compared to
one ounce of corn chips?
7. What is the difference in the number of grams of protein between one 3 ounce can of tuna (in
oil) and five slices of watermelon?
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Food Chart (B)
The following chart shows the amounts of protein, fat and carbohydrates in different servings of
foods.
Food and Serving
Watermelon (1 slice)
Apple Pie (1 slice)
White Bread (1 slice)
Tomato Juice (1 cup)
Peanut Butter (1 tablespoon)
Canned Tuna in Oil (3 ounces)
Beef and Vegetable Stew (1 cup)
Poached Egg (1 egg)
Blueberries (1 cup)
Corn Chips (1 ounce)
Food Energy
(kilocalories)
155
405
65
40
95
165
220
75
80
155
Sodium
(milligrams)
10
476
129
881
75
303
292
140
9
233
Carbohydrates
(grams)
35
60
12
10
3
0
15
1
20
16
Answer each of the following using the information supplied in the above chart. Show your
work.
1. If you drank a cup of tomato juice and ate one poached egg, how many
kilocalories of food energy would you have ingested?
2. How many times larger is the number of grams of carbohydrates in two
cups of beef and vegetable stew compared to one cup of tomato juice?
3. What is the difference in the number of milligrams of sodium in one
slice of apple pie and one poached egg?
4. How many milligrams of sodium are there in four slices of white
bread?
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5. If you ate one slice of watermelon, one tablespoon of peanut butter, and
one ounce of corn chips, how many kilocalories would be ingested?
6. What is the difference in the number of milligrams of sodium in three
ounces of corn chips and two tablespoons of peanut butter?
7. How many grams of carbohydrates would be ingested if you ate two
slices of pie, three slices of watermelon, and one cup of blueberries?
8. How many times larger is the amount of sodium in two poached eggs
compared to four slices of watermelon?
9. What is the difference in the number of kilocalories of a meal
comprised of one cup of beef and vegetable stew and one cup of tomato
juice, and a meal comprised of a three ounce can of tuna and a slice of
white bread.
10. How many times larger is the amount of carbohydrates in a meal
comprised of one slice of apple pie and two cups of blueberries,
compared to a meal comprised of one cup of beef and vegetable stew
and one slice of watermelon?
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Keeping Track of New Stock (A)
The store just started selling five new products. A large order of each item was made at the
beginning of week one and all of these items were placed on their shelves. The spreadsheet
below shows how many units of each product are present on the shelves at the beginning of each
week. Assume that the shelves were not restocked with the new items.
Week 1
Week 2
Week 3
Delicate
Chocolates
30
24
10
Deluxe Dog
Food
40
34
22
Extra Soft
Toilet Tissue
20
13
4
Greek Olives
Ceramic Pots
20
18
15
10
4
0
Use the information in the spreadsheet to answer the following questions. Show the number
sentence (e.g. 19 - 8 = 11) that you used to solve the question.
1. How many times larger was the number of units of deluxe dog food compared to the number
of units of ceramic pots at the beginning of week one?
2. How many more units of delicate chocolates were there on the shelves at the beginning of
week one compared to units of extra soft toilet tissue?
3. In total, how many units of Greek olives and ceramic pots were on the shelves at the
beginning of week two?
4. If they had tripled their order of extra soft toilet paper at the beginning of week one, how
many units would they have ordered?
5. How many units of delicate chocolates were sold between the beginning of week one and the
beginning of week three?
6. How many times larger is the number of units of extra soft toilet paper on week one
compared to week three?
7. In total, how many units of delicate chocolates, extra soft toilet paper and Greek olives were
on the shelves at the beginning of week three?
8. If five customers wanted to buy all the units of Greek olives that were present at the
beginning of week three and each would purchase the same amount, how many units would
each customer get?
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Keeping Track of New Stock (B)
The store just started selling five new products. A large order of each item was made at the
beginning of week one and all of these items were placed on their shelves. The spreadsheet
below shows how many units of each product are present on the shelves at the beginning of each
week. Assume that the shelves were not restocked with the new items.
Week 1
Week 2
Week 3
Delicate
Chocolates
30
24
10
Deluxe Dog
Food
40
34
22
Extra Soft
Toilet Tissue
20
13
4
Greek Olives
Ceramic Pots
20
18
15
10
4
0
Use the information in the spreadsheet to answer the following questions. Show the number
sentence (e.g. (45 - 33) + (30 - 25) = 17) that you used to solve the question.
1. If the store doubled the number of units of deluxe dog food and ceramic pots they ordered on
week one, how many units of these two items would they have in total at that time?
2. How many units of deluxe dog food and Greek olives were sold in total between the
beginning of week two and the beginning of week three?
3. How many more units of deluxe dog food were sold between weeks one and two compared
to units of Greek olives sold in the same time period?
4. How many more units of delicate chocolates were sold between weeks two and three
compared to units of extra soft toilet tissue sold in the same time period?
5. How many more combined units of deluxe dog food and extra soft toilet tissues are on the
shelves at the beginning of week two compared to the number of units of delicate chocolates
at that same time?
6. If the units sold of delicate chocolates tripled between the beginning of week one and the
beginning of week two, then how many units would have been sold at that time?
7. How many times larger was the number of units sold of ceramic pots between the beginning
of week one and the beginning of week three compared to units sold of Greek olives over the
same period?
8. How many times larger was the number of units sold of deluxe dog food between the
beginning of week two and the beginning of week three compared to units sold of the same
product between the beginning of week one and the beginning of week two?
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C. D. Pilmer
Answers
Comparing Quantities (page 2)
1. Most families have:
• fewer cars as compared to bicycles
• fewer windows as compared to doors (remember to include closet doors)
• more pillows as compared to beds
• more toothbrushes as compared to hair dryers
• fewer forks as compared to spoons
• fewer running shoes as compared to socks
• more dogs as compared to hamsters
• the same number of fingers as compared to toes
2. Most cars have:
• more headlights as compared to bumper stickers
• the same number of headrests as compared to seatbelts
• fewer steering wheels as compared to windows
• more tires as compared to rear view mirrors
• more floor mats as compared to horns
• fewer gas caps as compared to tail lights
Expanded Form (A) (page 3)
1. (a) 40 + 2
(c) 3000 + 900 + 80 + 5
(e) 70 + 8
(g) 800 + 60 + 7
(i) 6000 + 400 + 90 + 7
(k) 800 + 20 + 6
(m) 50 + 9
(o) 4000 + 800 + 8
(q) 6000 + 400 + 20
(s) 800 + 10
(b)
(d)
(f)
(h)
(j)
(l)
(n)
(p)
(r)
(t)
600 + 90 + 4
500 + 60 + 9
4000 + 200 + 80 + 1
30 + 1
500 + 20 + 8
5000 + 900 + 20 + 3
3000 + 40 + 5
700 + 3
5000 + 90 + 9
9000 + 600 + 3
2. (a) 58
(c) 5874
(e) 8387
(g) 493
(i) 619
(k) 850
(m) 1708
(o) 5340
(q) 58
(b)
(d)
(f)
(h)
(j)
(l)
(n)
(p)
(r)
629
286
35
721
7082
503
6089
8657
352
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(s) 1259
(u) 8367
(w) 6084
(t) 649
(v) 7580
(x) 8035
Expanded Form (B) (page 4)
1. (a)
(c)
(e)
(g)
(i)
50 000 + 4000 + 900 + 80 + 2
700 000 + 40 000 + 6000 +100 + 70 +3
300 000 + 6000 + 700 + 80 + 1
300 000 + 70 000 + 2000 + 80
200 000 + 70 000 + 400 + 80
(b)
(d)
(f)
(h)
(j)
5000 + 600 + 80 + 5
20 000 + 7000 + 900 + 50 + 9
40 000 + 3000 + 900 + 8
50 000 + 700 + 30 + 6
900 000 + 8000 + 700 + 4
2. (a)
(c)
(e)
(g)
(i)
(k)
67 591
41 278
602 892
530 905
29 756
290 468
(b)
(d)
(f)
(h)
(j)
(l)
562 713
850 374
95 043
684 710
735 269
907 135
(b)
(d)
(f)
(h)
(j)
(l)
(n)
(p)
(r)
(t)
(v)
(x)
49
10
27
13
11
23
99
83
52
31
63
68
Write the Number (A) (page 5)
1. (a) 82
(c) 16
(e) 46
(g) 74
(i) 56
(k) 38
(m) 8
(o) 17
(q) 12
(s) 90
(u) 7
(w) 15
2. (a)
(b)
(c)
(d)
(e)
(f)
(g)
fifty-nine
forty-two
eighteen
thirty-seven
sixty-one
ninety-five
twenty-one
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Write the Number (B) (page 6)
1. (a) 932
(c) 712
(e) 79
(g) 107
(i) 490
(k) 811
(m) 702
(o) 12
(q) 670
(s) 186
(u) 216
(w) 720
2. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(b)
(d)
(f)
(h)
(j)
(l)
(n)
(p)
(r)
(t)
(v)
(x)
246
360
621
589
42
276
319
531
9
900
65
462
(b)
(d)
(f)
(h)
(j)
(l)
(n)
(p)
(r)
(t)
(v)
4 683
411
3 429
1 078
5 983
9 046
8 274
4 928
38
6 900
760
five hundred seventy-eight
three hundred fifty-two
seventy-nine
two hundred seventeen
nine hundred six
seven hundred forty
five hundred forty-one
Write the Number (C) (pages 7 and 8)
1. (a) 8 321
(c) 7 513
(e) 9 512
(g) 2 950
(i) 207
(k) 6 807
(m) 597
(o) 15
(q) 1 318
(s) 3 076
(u) 7 308
(w) 9570
2. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
four thousand, five hundred sixty-seven
two thousand, three hundred fifty-one
five hundred forty-seven
nine thousand, one hundred eighty-nine
six thousand, nine hundred eleven
sixty-three
eight thousand, sixty-three
one thousand, nine hundred four
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(i) seven hundred eight
(j) seven thousand, eight hundred fifty
Write the Number (D) (pages 9 and 10)
1. (a) 56 746
(c) 40 371
(e) 523 090
(g) 93
(i) 901
(k) 536 000
(m) 9 460
(o) 713 391
(q) 630
(s) 80 570
(u) 306 111
(w) 807 002
2. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(b)
(d)
(f)
(h)
(j)
(l)
(n)
(p)
(r)
(t)
(v)
239 115
3 605
60 208
305 068
13 715
407 052
50 609
12 096
200 516
10 400
915
thirty-four thousand, seven hundred eighty-one
two hundred forty-five thousand, three hundred fifty-nine
seven hundred eighty
twelve thousand, six hundred ninety-two
three hundred four thousand, five hundred sixty-two
seven thousand, twenty-three
seventy thousand, six hundred fifty
six hundred thirty-four thousand, nine hundred four
fifty-three thousand, eleven
nine hundred forty thousand, sixty
Write the Number (E) (pages 11 and 12)
1. (a) 10 906 802
(b) 1 205 016
(c) 734 000 000
(d) 80 529 070
(e) 412 670 000
(f) 85 015 900
(g) 97 812
(h) 627 000 750
(i) 40 065 090
(j) 506 070 900
(k) 302 028
(l) 11 003047
(m) 90 313 004
(n) 520 000 672
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2. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
seven million, three hundred five thousand, four hundred eleven.
twenty-three million, seventy-eight thousand, six hundred
three hundred twenty-eight million, one hundred nine thousand
thirteen million, four hundred thirty-six thousand, five hundred
six million, nine thousand, seven hundred forty
four hundred ninety-eight thousand, three hundred fifteen
five hundred forty million, six hundred seventy-nine thousand, twenty
ninety-five million, eight hundred eleven thousand, two
Place Value (page 13)
1. (a) 4
(b) 2
(c) 5
2. (a) 6
(b) 7
(c) 5
3. (a) 0
(b) 7
(c) 8
4. (a) 1
(b) 4
(c) 5
5. (a) 2
(b) 4
(c) 9
6. (a) 9
(b) 0
(c) 1
7. (a) 5
(b) 8
(c) 0
8. (a) 3
(b) 0
(c) 8
9. (a) 6
(b) 1
(c) 8
10. (a) 2
(b) 7
(c) 1
11. (a) 5
(b) 2
(c) 3
12. (a) 5
(b) 2
(c) 1
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C. D. Pilmer
Before, After, or Between (A) (page 14)
Number
Word Description
1.
What number is after 6?
7
seven
2.
What number is before 11?
10
ten
3.
What number is between 18 and 20?
19
nineteen
4.
What number is after 15?
16
sixteen
5.
What number is between 23 and 25?
24
twenty-four
6.
What number is before 27?
26
twenty-six
7.
What number is between 28 and 30?
29
twenty-nine
8.
What number is after 34?
35
thirty-five
9.
What number is before 37?
36
thirty-six
10.
What number is after 11?
12
twelve
11.
What number is between 46 and 48?
47
forty-seven
12.
What number is after 59?
60
sixty
13.
What number is before 70?
69
sixty-nine
14.
What number is between 81 and 83?
82
eighty-two
15.
What number is after 42?
43
forty-three
16.
What number is between 90 and 92?
91
ninety-one
17.
What number is before 77?
76
seventy-six
18.
What number is after 99?
100
one hundred
19.
What number is between 52 and 54?
53
fifty-three
20.
What number is before 80?
79
seventy-nine
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Before, After, or Between (B) (page 15)
Number
Word Description
1.
What number is after 325?
326
three hundred twenty-six
2.
What number is before 421?
420
four hundred twenty
3.
What number is between 188 and 190?
189
one hundred eighty-nine
4.
What number is after 239?
240
two hundred forty
5.
What number is between 356 and 358?
327
three hundred fifty-seven
6.
What number is before 650?
649
six hundred forty-nine
7.
What number is between 286 and 288?
287
two hundred eighty-seven
8.
What number is before 700?
699
six hundred ninety-nine
9.
What number is before 998?
997
nine hundred ninety-seven
10.
What number is after 437?
436
four hundred thirty-six
11.
What number is between 638 and 640?
639
six hundred thirty-nine
12.
What number is after 399?
400
four hundred
13.
What number is before 900?
899
eight hundred ninety-nine
14.
What number is between 513 and 515?
514
five hundred fourteen
15.
What number is after 661?
662
six hundred sixty-two
16.
What number is before 712?
711
seven hundred eleven
17.
What number is between 600 and 602?
601
six hundred one
18.
What number is after 807?
808
eight hundred eight
19.
What number is after 999?
1000
one thousand
20.
What number is between 499 and 501?
500
five hundred
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Closer to, and Odd or Even (A) (page 16)
(a)
Word Description
eighteen
(b)
seventy-six
76
100
even
(c)
eighty-three
83
100
odd
(d)
forty
40
0
even
(e)
twenty-four
24
0
even
(f)
thirty-nine
39
0
odd
(g)
ninety-three
93
100
odd
(h)
sixty-five
65
100
odd
(i)
forty-six
46
0
even
(j)
seventy-one
71
100
odd
(k)
eleven
11
0
odd
(l)
thirty-eight
38
0
even
(m)
forty-nine
49
0
odd
(n)
seventy-two
72
100
even
(o)
ninety-six
96
100
even
(p)
twenty-seven
27
0
odd
(q)
sixteen
16
0
even
(r)
twelve
12
0
even
(s)
forty-four
44
0
even
(t)
eighty-five
85
100
odd
(u)
fifty-eight
58
100
even
(v)
thirty-seven
37
0
odd
NSSAL
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Number
18
Closer to 0 or 100
0
Odd or Even
even
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Draft
C. D. Pilmer
Closer to, and Odd or Even (B) (page 17)
(a)
Word Description
two hundred forty-nine
Number
249
Closer to 0 or 1000
0
Odd or Even
odd
(b)
six hundred twenty-three
623
1000
odd
(c)
seven hundred ninety-four
794
1000
even
(d)
one hundred eighty-six
186
0
even
(e)
five hundred twelve
512
1000
even
(f)
three hundred seven
307
0
odd
(g)
four hundred sixty
460
0
even
(h)
ninety-nine
99
0
odd
(i)
two hundred seventy-six
276
0
even
(j)
five hundred forty-seven
547
1000
odd
(k)
one hundred eighty-four
184
0
even
(l)
nine hundred eight
908
1000
even
(m)
fifty-three
53
0
odd
(n)
three hundred fifty
350
0
even
(o)
nine hundred ninety-six
996
1000
even
(p)
six hundred forty-five
645
1000
odd
(q)
one thousand, thirty-one
1031
1000
odd
(r)
two hundred seventy-seven
277
0
odd
(s)
three thousand, ten
3010
1000
even
(t)
seven hundred sixty-nine
769
1000
odd
(u)
two thousand, three hundred
2300
1000
even
(v)
eight hundred fifteen
815
1000
odd
Find the Odd or Even Numbers (A) (page 18)
(a)
(b)
(c)
(d)
8, 10, 12
7, 9, 11, 13, 15
10, 12, 14, 16
11, 13, 15, 17
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(e) 20, 22, 24
(f) 15, 17, 19, 21
(g) 38, 40, 42, 44, 46
(h) 53, 55, 57, 59, 61
(i) 28, 30, 32, 34, 36, 38
(j) 89, 91, 93, 95
(k) 40, 42, 44, 46, 48, 50
(l) 47, 49, 51, 53, 55
(m) 74, 76, 78, 80
(n) 67, 69, 71, 73, 75, 77, 79
(o) 56, 58, 60, 62, 64, 66
(p) 87, 89, 91, 93, 95, 97, 99
(q) 94, 96, 98
Find the Odd or Even Numbers (B) (page 19)
(a) 130, 132, 134, 136
(b) 93, 95, 97, 99, 101, 103
(c) 216, 218, 220, 222, 224
(d) 69, 71, 73, 75, 77, 79, 81
(e) 460, 462, 464, 466, 468, 470
(f) 325, 327, 329, 331, 333
(g) 584, 586, 588, 590
(h) 797, 799, 801, 803
(i) 698, 700, 702, 704, 706
(j) 199, 201, 203, 205, 207, 209
(k) 998, 1000, 1002, 1004
(l) 2397, 2399, 2401
(m) 6546, 6548, 6550
(n) 4993, 4995, 4997, 4999, 5001
(o) 1684, 1686, 1688, 1690
(p) 3921, 3923, 3925, 3927, 3929
(q) 5990, 5992, 5994, 5996
(r) 7689, 7691, 7693, 7695
(s) 8030, 8032, 8034
Order the Numbers (page 22)
1. (a)
(b)
(c)
(d)
(e)
(f)
2, 9, 17, 32, 39
7, 29, 32, 73, 87
19, 49, 54, 56, 91
8, 28, 35, 37, 70, 74
4, 12, 39, 61, 65, 85
3, 25, 28, 43, 47, 96
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(g) 8, 15, 27, 37, 80, 86
(h) 9, 16, 19, 49, 54, 67
(i) 6, 12, 21, 26, 34, 41, 49
(j) 18, 23, 30, 52, 61, 73, 78
(k) 29, 33, 37, 46, 51, 72, 93
(l) 25, 27, 45, 62, 67, 86, 91
(m) 13, 32, 35, 58, 65, 78, 82
(n) 6, 12, 42, 46, 60, 95, 98
2. (a) 7, 12, 17, 22, 27, 32
(b) 14, 17, 20, 23, 26, 29
(c) 27, 31, 35, 39, 43, 47, 51
(d) 46, 48, 52, 54, 56, 58, 60
3. Answers will vary.
Give an Example (page 23)
Answers will vary. We have supplied possible answers.
1. 10 years in a decade, 10 millimetres in a centimetre, a dime is worth 10 cents
2. 12 inches in a foot, 12 eggs in a carton of a dozen eggs, 12 beers in a case
3. age legally recognized as an adult, 18 holes of golf
4. 24 hours in a day, 24 inches in 2 feet, 2-4 of beer
5. a quarter is worth 25 cents, quarter of a century, silver wedding anniversary
6. 30 days in September, April, June and November
7. 50 years in half a century, golden wedding anniversary
8. 60 seconds in a minute, 60 minutes in an hour
9. 100 years in a century, 100 centimetres in one metre
10. 1000 years in a millennium, 1000 metres in a kilometre, 1000 millilitres in a litre
NSSAL
©2012
253
Draft
C. D. Pilmer
Closer To (A) (page 24)
1.
Is 6 closer to 5 or 8?
2.
Is 3 closer to 1 or 7?
3.
Is 4 closer to 0 or 6?
4.
Is 7 closer to 4 or 9?
5.
Is 5 closer to 1 or 8?
6.
Is 8 closer to 4 or 10?
7.
Is 6 closer to 3 or 10?
8.
Is 2 closer to 0 or 3?
9.
Is 9 closer to 7 or 12?
10.
Is 8 closer to 3 or 12?
11.
Is 6 closer to 0 or 11?
12.
Is 7 closer to 5 or 11?
13.
Is 9 closer to 6 or 13?
14.
Is 10 closer to 7 or 15?
15.
Is 11 closer to 10 or 14?
16.
Is 10 closer to 6 or 12?
17.
Is 15 closer to 13 or 19?
18.
Is 13 closer to 10 or 15?
19.
Is 16 closer to 14 or 20?
20.
Is 12 closer to 8 or 15?
21.
Is 19 closer to 15 or 21?
22.
Is 18 closer to 16 or 22?
23.
Is 24 closer to 20 or 30?
24.
Is 27 closer to 20 or 30?
25.
Is 39 closer to 30 or 40?
26.
Is 46 closer to 40 or 50?
27.
Is 73 closer to 70 or 80?
28.
Is 94 closer to 90 or 100?
29.
Is 28 closer to 20 or 30?
30.
Is 60 closer to 0 or 100?
31.
Is 40 closer to 0 or 100?
32.
Is 70 closer to 0 or 100?
33.
Is 30 closer to 20 or 50?
34.
Is 70 closer to 50 or 100?
NSSAL
©2012
254
Draft
C. D. Pilmer
Closer To (B) (page 25)
1.
Is 36 closer to 30 or 40?
2.
Is 44 closer to 40 or 50?
3.
Is 67 closer to 60 or 70?
4.
Is 50 closer to 20 or 60?
5.
Is 60 closer to 10 or 90?
6.
Is 80 closer to 70 or 100?
7.
Is 100 closer to 80 or 150?
8.
Is 150 closer to 130 or 160?
9.
Is 270 closer to 250 or 300?
10.
Is 420 closer to 400 or 500?
11.
Is 200 closer to 0 or 300?
12.
Is 500 closer to 400 or 800?
13.
Is 700 closer to 600 or 750?
14.
Is 600 closer to 550 or 700?
15.
Is 640 closer to 600 or 700?
16.
Is 870 closer to 800 or 900?
17.
Is 81 closer to 20 or 100?
18.
Is 67 closer to 30 or 80?
19.
Is 58 closer to 0 or 90?
20.
Is 37 closer to 0 or 100?
21.
Is 99 closer to 0 or 200?
22.
Is 230 closer to 100 or 300?
23.
Is 341 closer to 300 or 350?
24.
Is 789 closer to 750 or 800?
25.
Is 699 closer to 600 or 750?
26.
Is 219 closer to 100 or 250?
27.
Is 224 closer to 200 or 300?
28.
Is 547 closer to 500 or 550?
29.
Is 839 closer to 800 or 900?
30.
Is 658 closer to 650 or 700?
31.
Is 2399 closer to 2000 or 3000?
32.
Is 1837 closer to 1000 or 2000?
33.
Is 5643 closer to 5000 or 6000?
34.
Is 2845 closer to 2000 or 4000?
What Number Does the Star Represent? (Addition) (page 30)
(a)
(c)
(e)
(g)
(i)
4
2
10
9
8
NSSAL
©2012
(b)
(d)
(f)
(h)
(j)
255
4
7
7
8
7
Draft
C. D. Pilmer
(k) 5
(m) 2
(o) 12
(q) 1
(s) 3
(u) 10
(w) 7
(y) 1
(l)
(n)
(p)
(r)
(t)
(v)
(x)
(z)
13
5
3
11
8
3
13
3
Adding Multi-Digit Numbers (pages 31 to 34)
1.
(a)
(b)
1
+
4
2
7
8
6
4
(c)
+
1
(d)
5
6
1
4
1
5
1
1
6
7
4
3
7
1
(e)
+
(g)
3
2
5
2
5
7
+
1
+
1
(h)
1
6
7
5
9
7
1
8
2
8
0
0
+
1
+
1
9
3
2
8
6
9
3
+
1
(b) 75
(e) 981
(h) 873
(c) 136
(f) 1390
(i) 1637
3. (a) 163
(d) 1172
(b) 128
(e) 655
(c) 1238
(f) 1403
(a)
(b)
+
NSSAL
©2012
4
5
9
0
6
6
(c)
1
+
1
8
7
6
256
9
9
8
7
9
6
1
+
1
2. (a) 97
(d) 669
(g) 811
4.
3
8
2
(f)
(i)
2
5
7
3
7
1
7
9
6
1
5
7
1
2
5
4
3
3
6
0
5
3
4
2
6
9
7
4
0
1
+
1
8
3
2
7
6
3
Draft
C. D. Pilmer
(d)
(e)
+
3
4
7
4
2
6
(g)
7
1
8
+
1
3
7
1
(h)
+
2
7
5
9
1
5
0
2
7
(f)
1
5
7
2
+
2
+
1
(i)
1
8
9
4
2
3
5
8
3
6
4
3
+
1
1
1
8
6
5
9
4
4
2
1
3
4
5
4
6
7
6
0
7
8
5
6
2
3
1
5. Answers will vary.
NSSAL
©2012
257
Draft
C. D. Pilmer
Subtraction Search (page 39)
14
20
6
6
9
11
8
8
12
9
4
12
15
4
11
2
9
3
3
7
6
1
9
17
10
6
8
5
1
10
9
20
20
8
8
12
9
9
Subtraction Facts:
1. 14 - 6 = 8
3. 6 - 4 = 2
5. 9 - 5 = 4
7. 12 - 4 = 8
9. 7 - 1 = 6
11. 6 - 1 = 5
13. 8 - 5 = 3
15. 15 - 9 = 6
17. 20 - 8 = 12
19. 10 - 8 = 2
6
16
10
15
0
4
7
13
17
5
7
5
8
18
7
2
0
2.
4.
6.
8.
10.
12.
14.
16.
18.
20.
20 - 9 = 11
12 - 5 = 7
18 - 10 = 18
9-3=6
15 - 4 = 11
16 - 9 = 7
13 - 6 = 7
20 - 10 = 10
17 - 0 = 17
9-9=0
Subtracting Multi-Digit Numbers (pages 40 to 45)
1.
(a)
(b)
-
NSSAL
©2012
4
1
3
8
2
6
-
5
14
6
3
2
4
9
5
258
(c)
-
8
11
9
2
6
1
5
6
Draft
C. D. Pilmer
(d)
(e)
-
3
2
1
(g)
9
5
4
7
1
6
-
8
2
5
5
7
8
(i)
14
8
5
2
2
4
8
9
7
4
12
8
10
11
6
4
1
4
7
6
0
8
2
8
3
4
5
8
6
2
5
7
9
7
1
0
5
4
1
6
5
6
11
7
2
4
1
4
7
-
(b) 47
(e) 156
(a)
34
66
320
605
148
7
3
4
9
2
7
4
12
5
1
3
2
9
3
8
12
9
2
6
2
7
5
(e)
7
1
6
(g)
9
8
1
5
3
2
-
(h)
16
58
38
351
756
359
(c) 386
(f) 168
-
(d)
-
(c)
(f)
(i)
(l)
(o)
(b)
-
-
9
5
4
12
10
3. (a) 26
(d) 124
-
5
3
2
7
3
(b)
(e)
(h)
(k)
(n)
-
(f)
5
2. (a) 35
(d) 15
(g) 365
(j) 163
(m) 358
4.
15
(h)
13
-
7
(c)
-
(f)
9
5
4
-
5
2
3
(i)
13
3
15
4
0
3
5
6
9
9
4
6
13
5
3
12
2
10
14
5
2
2
7
7
9
3
9
4
6
5
4
8
5
2
6
6
3
1
1
0
5
4
4
5
9
-
-
5. Answers will vary.
NSSAL
©2012
259
Draft
C. D. Pilmer
Multiples Puzzles (page 59)
(a)
(c)
(e)
4
70
49
56
21
14
50
24
36
22
10
6
63
5
12
20
16
18
5
14
35
14
9
30
32
30
2
25
8
28
8
20
27
15
28
21
15
35
7
42
12
18
3
24
18
8
3
9
40
10
45
35
21
6
22
14
2
48
40
6
36
24
54
14
7
10
8
9
12
16
24
64
81
63
30
63
60
12
16
27
6
3
33
48
99
18
12
56
28
30
3
18
15
20
18
8
15
54
27
18
24
54
10
35
21
12
6
9
36
4
45
6
45
15
25
5
24
44
8
28
16
40
9
63
99
14
9
33
21
4
55
88
121
77
44
81
77
42
24
55
3
16
33
3
18
27
66
45
63
18
27
30
40
36
11
20
32
12
7
18
49
28
7
11
25
12
28
8
21
33
49
72
9
22
55
44
40
48
24
77
63
21
56
77
110
88
5
15
45
32
16
72
8
35
14
27
72
45
15
10
35
9
56
6
2
25
14
63
16
8
42
35
7
49
21
4
NSSAL
©2012
(b)
(d)
(f)
260
Draft
C. D. Pilmer
Factors (page 60)
1. (a) 1, 2, 3, 6
(c) 1, 2, 4, 5, 10, 20
(e) 1, 2, 3, 4, 6, 8, 12, 24
(g) 1, 2, 3, 4, 6, 9, 12, 18, 36
(i) 1, 5, 7, 35
(b)
(d)
(f)
(h)
(j)
1, 2, 7, 14
1, 2, 3, 6, 9, 18
1, 2, 3, 5, 6, 10, 15, 30
1, 2, 4, 5, 8, 10, 20, 40
1, 3, 7, 9, 21, 63
2. (a) 1, 2, 4
(c) 1, 2, 4
(b) 1, 2, 5, 10
(d) 1, 2, 3, 6
3. Answers will vary. We have provided some acceptable answers.
(a) 12, 24, 36, 48, 60, 72, …
(b) 15, 30, 45, 60, 75, …
(c) 20, 40, 60, 80, 100, …
(d) 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …
Factor Flowers (page 61)
1
12
8
24
7
3
6
3
12
4
5
8
18
2
2
2
10
5
40
20
6
10
16
8
1
6
6
3
5
21
24
30
4
8
20
12
15
5
48
4
1
4
8
16
36
6
3
6
1
35
9
6
4
35
9
21
32
4
1
7
2
7
2
2
2
9
2
3
18
42
10
10
5
21
6
NSSAL
©2012
261
Draft
C. D. Pilmer
Random Multiplication Facts Quizzes (pages 62 and 63)
Quiz A1
Name: ________________
Quiz A2
Name: ________________
×
2
7
5
8
4
×
5
2
0
1
9
1
2
7
5
8
4
7
35
14
0
7
63
5
10
35
25
40
20
0
0
0
0
0
0
0
0
0
0
0
0
8
40
16
0
8
72
9
18
63
45
72
36
6
30
12
0
6
54
2
4
14
10
16
8
3
15
6
0
3
27
Quiz A3
Name: ________________
Quiz A4
Name: ________________
×
9
6
3
4
7
×
1
5
9
2
0
9
81
54
27
36
63
6
6
30
54
12
0
2
18
12
6
8
14
8
8
40
72
16
0
1
9
6
3
4
7
1
1
5
9
2
0
5
45
30
15
20
35
3
3
15
27
6
0
0
0
0
0
0
0
7
7
35
63
14
0
Quiz B1
Name: ________________
Quiz B2
Name: ________________
×
8
3
2
7
6
×
4
9
0
3
5
5
40
15
10
35
30
8
32
72
0
24
40
3
24
9
6
21
18
2
8
18
0
6
10
4
32
12
8
28
24
4
16
36
0
12
20
1
8
3
2
7
6
7
28
63
0
21
35
9
72
27
18
63
54
6
24
54
0
18
30
NSSAL
©2012
262
Draft
C. D. Pilmer
Quiz B3
Name: ________________
Quiz B4
Name: ________________
×
8
6
7
1
3
×
3
5
1
4
9
9
72
54
63
9
27
7
21
35
7
28
63
2
16
12
14
2
6
6
18
30
6
24
54
4
32
24
28
4
12
2
6
10
2
8
18
5
40
30
35
5
15
4
12
20
4
16
36
3
24
18
21
3
9
8
24
40
8
32
72
Quiz C1
Name: ________________
Quiz C2
Name: ________________
×
8
7
5
6
3
×
6
9
2
8
7
6
48
42
30
36
18
4
24
36
8
32
28
4
32
28
20
24
12
8
48
72
16
64
56
8
64
56
40
48
24
6
36
54
12
48
42
2
16
14
10
12
6
9
54
81
18
72
63
7
56
49
35
42
21
7
42
63
14
56
49
Quiz C3
Name: ________________
Quiz C4
Name: ________________
×
1
6
9
7
3
×
2
7
6
9
3
5
5
30
45
35
15
3
6
21
18
27
9
8
8
48
72
56
24
7
14
49
42
63
21
4
4
24
36
28
12
4
8
28
24
36
12
7
7
42
63
49
21
8
16
56
48
72
24
6
6
36
54
42
18
9
18
63
54
81
27
What Number Does the Star Represent? (Multiplication) (page 64)
(a) 4
(c) 40
NSSAL
©2012
(b) 3
(d) 7
263
Draft
C. D. Pilmer
(e) 2
(g) 3
(i) 5
(k) 6
(m) 1
(o) 5
(q) 8
(s) 3
(u) 2
(w) 8
(y) 2
(f)
(h)
(j)
(l)
(n)
(p)
(r)
(t)
(v)
(x)
(z)
1
4
9
32
3
56
5
6
7
3
6
Put the Number in the Right Box (page 69)
1.
Multiple of 2
Not a Multiple of 2
Multiple of 3
6, 18, 12
15, 9, 21
Not a Multiple of 3
4, 10, 14
11, 19, 7
2.
Multiple of 2
Not a Multiple of 2
Multiple of 5
10, 30, 40
15, 25, 45
Not a Multiple of 5
16, 22, 18
19, 9, 21
3.
Multiple of 3
Not a Multiple of 3
Multiple of 5
15, 30, 45
20, 25, 40
Not a Multiple of 5
24, 9, 27
8, 28, 26
4
Multiple of 4
Not a Multiple of 4
Multiple of 6
24, 12, 36
30, 18, 6
Not a Multiple of 6
16, 8, 28
21, 9, 15
5.
Multiple of 5
Not a Multiple of 5
Multiple of 4
20, 40, 80
12, 32, 28
Not a Multiple of 4
25, 10, 35
27, 12, 19
NSSAL
©2012
264
Draft
C. D. Pilmer
Multiplying by Multiples of 10, 100, and 1000 (pages 71 and 72)
1.
Answer in Written Form
(a)
2000 × 600 = 1 200 000
One million, two hundred thousand
(b)
9 × 4000 = 36 000
Thirty-six thousand
(c)
300 × 500 = 150 000
One hundred fifty thousand
(d)
80 × 400 = 32 000
Thirty-two thousand
(e)
60 × 60 = 3 600
Three thousand, six hundred
(f)
800 × 5 = 4 000
Four thousand
2. (a) 1 400
(c) 12 000
(e) 2 400
(g) 630 000
(i) 320
(k) 720
(m) 270 000
(o) 1 800 000
(q) 4 800
(s) 8 000
(u) 5 600 000
(w) 10 000 000
(y) 90 000
(b)
(d)
(f)
(h)
(j)
(l)
(n)
(p)
(r)
(t)
(v)
(x)
(z)
540
25 000
400 000
12 000 000
490 000
30 000
28 000
40 000
1 000 000
8 100
64 000
720
35 000
Multiplying Two Digit Numbers, Part 1 (Expanded Form) (pages 73 to 77)
60 + 3
× 90 + 5
1.
1
5
3
0
0
2
7
0
5
4
0
0
5
9
8
5
NSSAL
©2012
265
Draft
C. D. Pilmer
70 + 4
× 30 + 8
2.
3
2
5
6
0
1
2
0
2
1
0
0
2
8
1
2
3. (a)
(c)
(e)
(g)
4.
1118
2059
2844
2928
(b)
(d)
(f)
(h)
1995
4399
3403
5688
400 + 30 + 6
×
70 + 2
8
4
2 1
2 8 0
1
6
0
2
0
0
2
0
0
0
0
0
3 1 3 9 2
Multiplying Two Digit Numbers, Part 2 (Lattice Method) (pages 78 to 83)
1. (a) 2400
(c) 7452
(e) 1206
(b) 1476
(d) 1776
(f) 2088
2. 2166
3. (a) 1932
(c) 2808
(b) 3692
(d) 3108
Multiplying Multi-Digit Numbers (pages 84 to 87)
1 (a) 28 086
(c) 257 948
NSSAL
©2012
(b) 139 895
266
Draft
C. D. Pilmer
2. (a) 28 035
(b) 309 628
Division Search (page 89)
48
32
28
15
4
8
7
24
64
8
5
9
5
3
10
3
2
27
5
2
3
4
5
56
8
9
1
42
16
4
4
5
7
36
NSSAL
©2012
9
9
25
8
Division Facts:
1. 28 ÷ 4 = 7
3. 48 ÷ 6 = 8
5. 18 ÷ 9 = 2
7. 32 ÷ 4 = 8
9. 10 ÷ 5 = 2
11. 64 ÷ 8 = 8
13. 16 ÷ 4 = 4
15. 20 ÷ 4 = 5
17. 36 ÷ 6 = 6
19. 36 ÷ 9 = 4
18
45
8
20
6
6
7
5
4
1
9
8
8
18
36
6
6
2
2.
4.
6.
8.
10.
12.
14.
16.
18.
20.
267
9
27 ÷ 3 = 9
15 ÷ 5 = 3
9÷9 = 1
45 ÷ 5 = 9
24 ÷ 8 = 3
25 ÷ 5 = 5
42 ÷ 6 = 7
8÷8 = 1
56 ÷ 8 = 7
18 ÷ 2 = 9
Draft
C. D. Pilmer
More Divisibility (A) (page 91)
1.
(a)
64
(c)
81
(e)
40
(g)
105
(i)
208
(k)
915
(m)
720
(o)
1245
(q)
4109
(s)
9130
(u)
8374
2
3





5
2













(b)
35
(d)
90
(f)
42
(h)
307
(j)
635
(l)
410
(n)
816
(p)
2036
(r)
7281
(t)
3075
(v)
7320






3
5












2. Answers will vary.
NSSAL
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C. D. Pilmer
More Divisibility (B) (pages 92 and 93)
1.
(a)
12
(b)
45
(c)
30
(d)
14
(e)
36
(f)
19
(g)
430
(h)
114
(i)
207
(j)
96
(k)
225
(l)
600
(m)
704
(n)
425
(o)
408
(p)
570
(q)
615
(r)
1078
(s)
2310
(t)
7131
(u)
8706
(v)
5603
(w)
4700
(x)
6125
(y)
3210
2
3












5
6
10
15


























































2. Answers will vary.
NSSAL
©2012
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C. D. Pilmer
3. (a) 14
(b) 33
(c) 30
Division with Remainders (pages 95 to 98)
1. (a) 13 ÷ 5 =
2, R:3
(b) 7 ÷ 2 =
3, R:1
(c) 14 ÷ 3 =
4, R:2
(d) 18 ÷ 3 =
6
(e) 33 ÷ 7 =
4, R:5
(f) 43 ÷ 8 =
5, R:3
(g) 25 ÷ 6 =
4, R:1
(h) 19 ÷ 2 =
9, R:1
(i) 29 ÷ 3 =
9, R:2
(j) 40 ÷ 6 =
6, R:4
(k) 42 ÷ 7 =
6
(l) 22 ÷ 5 =
4, R:2
(m) 60 ÷ 9 =
6, R:6
(n) 31 ÷ 8 =
3, R:7
(o) 13 ÷ 4 =
3, R:1
(p) 38 ÷ 5 =
7, R:3
(q) 48 ÷ 9 =
5, R:3
(r) 32 ÷ 4 =
8
(s) 18 ÷ 4 =
4, R:2
(t) 19 ÷ 6 =
3, R:1
(u) 54 ÷ 5 =
10, R:4
(v) 35 ÷ 8 =
4, R:3
(w) 50 ÷ 7 =
7, R:1
(x) 69 ÷ 9 =
7, R:6
2. 20 ÷ 6 =
3, R:2
Each person gets 3 apples, and 2 apples are left over.
Long Division (Partial Quotient Method) (pages 99 to 106)
1.
637
2.
516
3.
752
4.
254
5.
372 R: 1
6.
928 R: 2
7.
386 R: 5
8.
658 R: 3
9.
583
10.
872 R: 4
11.
958 R: 1
12.
967 R: 2
13.
2741
14.
4217
15.
5736 R: 1
16.
4763 R: 3
17.
3592
18.
6523 R: 5
19.
5217 R: 2
20.
7305
NSSAL
©2012
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C. D. Pilmer
Prime Factorization (pages 107 and 108)
1. (a) 6= 2 × 3
(b) 21= 3 × 7
(c) 10= 2 × 5
(d) 35= 5 × 7
(e) 49= 7 × 7
(f) 26= 2 ×13
(g) 33= 3 ×11
(h) 20 = 2 × 2 × 5
(i) 44 = 2 × 2 ×11
(j) 42 = 2 × 3 × 7
(k) 45 = 3 × 3 × 5
(l) 66 = 2 × 3 ×11
(m) 30 = 2 × 3 × 5
(n) 70 = 2 × 5 × 7
(o) 27 = 3 × 3 × 3
(p) 63 = 3 × 3 × 7
(q) 110 = 2 × 5 ×11
(r) 18 = 2 × 3 × 3
(s) 16 = 2 × 2 × 2 × 2
(t) 100 = 2 × 2 × 5 × 5
(u) 36 = 2 × 2 × 3 × 3
(v) 250 = 2 × 5 × 5 × 5
(w) 81 = 3 × 3 × 3 × 3
(x) 24 = 2 × 2 × 2 × 3
Find the Center (page 111)
(a) 6
(b) 8
(c) 5
(d) 9
(e) 12
(f) 10
(g) 7
(h) 14
(i) 20
Name the Preceding or Next (page 112)
1.
(a) 49
(b) 65
(c) 31
2.
(a) 34
(b) 56
(c) 48
3.
(a) 30
(b) 15
(c) 50
4.
(a) 70
(b) 40
(c) 100
5.
(a) 24
(b) 18
(c) 30
6.
(a) 32
(b) 24
(c) 12
7.
(a) 12
(b) 6
(c) 24
8.
(a) 32
(b) 24
(c) 8
NSSAL
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C. D. Pilmer
9.
(a) 30
(b) 48
(c) 24
One of these Things is Not Like the Others (pages 113 and 114)
Note: There is often more than one acceptable answer for these questions.
1. The 7 does not belong because it is an odd number and the remaining numbers, 10, 12, and 4,
are all even numbers.
2. The 5 does not belong because unlike the remaining numbers, 13, 15, and 9, it is not divisible
by 3.
3. The 11 does not belong because it is a prime number and the remaining numbers, 15, 6, and
8, are all composite numbers.
4. The sequence 1, 2, 4, 8, 16 does not belong because you are multiplying a number to one
term to get the next term. With the remaining sequences you are adding a number to one
term to get the next term.
5. The 6 does not belong because it is not a perfect square while the remaining numbers, 25
(52), 9 (32), and 16 (42), are.
6. The 22 does not belong because unlike the remaining numbers, 15, 40, and 55, it is not
divisible by 5.
7. The 9 does not belong because it is a composite number and the remaining numbers, 17, 7,
and 19, are all prime numbers.
8. The sequence 66, 63, 60, 57 does not belong because each term is decreasing by 3, while the
remaining sequences have terms that are increasing by 3.
9. The 14 does not belong because it is a number other than 12. The words or mathematical
expressions are equal to 12.
10. The one hundred five does not belong because it is a three digit number. The remaining
numbers are all two digit numbers.
11. The 10 - 5 does not belong because it is equal to 5, while the rest are equal to 15.
12. The 9 × 9 does not belong because it is equal to 81, while the rest equal 99.
NSSAL
©2012
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C. D. Pilmer
Provide the Other Members of the Fact Family (page 117)
1. (a) 35 ÷ 5 =
7
5× 7 =
35
7×5 =
35
(b) 7 × 6 =
42
42 ÷ 6 =
7
42 ÷ 7 =
6
(c) 5 + 8 =
13
13 − 8 =
5
13 − 5 =
8
(d) 16 − 7 =
9
7+9 =
16
9+7 =
16
(e) 50 + 60 =
110
110 − 60 =
50
110 − 50 =
60
(f) 240 ÷ 80 =
3
3 × 80 =
240
80 × 3 =
240
(g) 800 − 300 =
500
300 + 500 =
800
500 + 300 =
800
(h) 40 × 7 =
280
280 ÷ 40 =
7
280 ÷ 7 =
40
(i) 1500 ÷ 30 =
50
50 × 30 =
1500
30 × 50 =
1500
2. (a)
(b)
(c)
(d)
(e)
9× 4 =
36 , 4 × 9 =
36 , 36 ÷ 9 =
4 , 36 ÷ 4 =
9
140 − 60 =
80 , 140 − 80 =
60 , 60 + 80 =
140 , 80 + 60 =
140
2100 ÷ 700 =
3 , 2100 ÷ 3 =
700 , 3 × 700 =
2100 , 700 × 3 =
2100
1300 , 1300 − 900 =
900
400 + 900 =
1300 , 900 + 400 =
400 , 1300 − 400 =
5400 , 5400 ÷ 90 =
60 , 5400 ÷ 60 =
90
90 × 60 =
5400 , 60 × 90 =
What Number Does the Star Represent? (page 118)
(a) 8
(c) 3
(e) 5
(g) 5
(i) 6
(k) 45
(m) 5
(o) 8
(q) 9
(s) 3
(u) 4
(w) 72
(y) 1
NSSAL
©2012
(b)
(d)
(f)
(h)
(j)
(l)
(n)
(p)
(r)
(t)
(v)
(x)
(z)
273
9
4
15
3
3
4
11
8
7
9
10
6
9
Draft
C. D. Pilmer
Pathways (page 119)
Everything along this pathway equals 8.
Everything along this pathway equals 6.
2× 4
16 ÷ 2
12 ÷ 2
6+2
4× 2
3× 2
10 − 4
6 ×1
12 − 4
3+5
32 ÷ 4
18 ÷ 3
5 +1
3+3
30 ÷ 5
0+6
8 ×1
9 −1
15 − 9
Everything along this pathway equals 12.
9× 2
6+6
21 − 3
8+4
3× 4
12 ÷ 1
20 − 8
4+2
18 × 1
9+9
12 × 1
8−2
Everything along this pathway equals 18.
15 − 3
18 − 6
9−3
24 − 6
2×6
13 + 5
0 + 12
6×3
9+3
7 + 11
18 + 0
Two of These Boxes Just Don't Belong (A) (page 120)
1.
2.
3.
4.
5.
6.
1st, 2nd, and 5th box equal 5
1st, 3rd, and 5th boxes equal 7
2nd, 3rd, and 4th boxes equal 6
2nd, 4th, and 5th boxes equal 10
2nd, 3rd, and 5th boxes equal 8
1st, 3rd, and 4th boxes equal 11
NSSAL
©2012
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Draft
C. D. Pilmer
7.
8.
9.
10.
2nd, 4th, and 5th boxes equal 15
1st, 3rd, and 5th boxes equal 12
2nd, 3rd, and 4th boxes equal 20
1st, 4th, and 5th boxes equal 18
Two of These Boxes Just Don't Belong (B) (page 121)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
1st, 3rd, and 5th boxes equal 9
2nd, 4th, and 5th boxes equal 40
1st, 2nd, and 4th boxes equal 16
1st, 3rd, and 4th boxes equal 14
2nd, 3rd, and 5th boxes equal 30
1st, 4th, and 5th boxes equal 50
2nd, 3rd, and 4th boxes equal 60
3rd, 4th, and 5th boxes equal 80
1st, 2nd, and 5th boxes equal 0
3rd, 4th, and 5th boxes equal 90
Equivalent (page 122)
Part 1
(a) 8
(d) 6
(b) 5
(e) 7
(c) 4
(f) 2
Part 2
(a) 4
(d) 8
(b) 4
(e) 5
(c) 7
(f) 15
Part 3
(a) 4
(d) 10
(b) 10
(e) 3
(c) 4
(f) 6
Part 4
(a) 2
(d) 30
(b) 12
(e) 5
(c) 4
(f) 40
Part 5
(a) 8
(d) 3
(g) 6
(j) 2
(m) 5
(p) 9
(s) 8
(b)
(e)
(h)
(k)
(n)
(q)
(t)
(c)
(f)
(i)
(l)
(o)
(r)
(u)
NSSAL
©2012
5
4
1
7
4
31
45
275
4
6
18
4
7
6
5
Draft
C. D. Pilmer
(v) 2
(w) 1
(x) 17
Greater Than, Less Than or Equal To; Whole Number Operations (page 123)
1. (a) 3+7= 5 + 5
(c) 12 ÷ 4 < 10 ÷ 2
(e) 3 × 4 = 2 × 6
(g) 24 ÷ 3 > 12 − 7
(i) 5 + 2 < 63 ÷ 7
(k) 50 + 40 > 8 ×10
(m) 30 + 90 = 30 × 40
(o) 160 ÷ 8 > 13 + 6
(q) 3 × 70 = 70 + 70 + 70
(s) 60 − 20 > 2 × 4 × 3
(u) 130 − 40 < 20 + 50 + 30
(b)
(d)
(f)
(h)
(j)
(l)
(n)
(p)
(r)
(t)
(v)
4 × 5 > 3× 6
12 − 7 > 9 − 5
6 + 9 < 11 + 5
11 + 11 > 3 × 7
4 × 7 = 30 − 2
80 − 50 < 120 ÷ 3
120 − 50 > 9 × 6
180 ÷ 90 < 17 − 12
20 + 30 = 5 × 2 × 5
280 ÷ 7 < 3 × 3 × 5
6 × 2 ×1 > 240 ÷ 80
2. (a)
(c)
(e)
(g)
(b)
(d)
(f)
(h)
2 or 3
4
2, 3 or 4
6
3
5 or 6
5 or 6
5
3. Answers will vary.
Find the Digit Based on the Reasonable Estimate (page 124)
1. 1
2. 4
3. 9
4. 2
5. 3
6. 4
7. 8
8. 7
9. 4
10. 3
11. 9
12. 3
13. 7
14. 8
NSSAL
©2012
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C. D. Pilmer
Venn Diagrams and Whole Numbers (pages 125 and 126)
1.
9
4
14
15
Divisible by 2
8
10
20
30
Divisible by 5
5
25
22
21
2.
11
27
6
10
25
15
Divisible by 3
9
12
18
Divisible by 5
30
20
16
3.
4
10
15
25
Odd
9
21
3
7
13
2
Prime
23
20
18
NSSAL
©2012
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C. D. Pilmer
4.
3
25
4
Perfect
Square
16
30
Even
10
36
7
15
18
28
9
27
5.
48
7
18
15
45
55
Multiple of 9
Multiple of 5
90
9
10
30
36
13
Whole Number Crossword Puzzle (A) (page 127)
3
8
6
0
7
4
2
6
3
5
5
1
NSSAL
©2012
2
8
6
2
1
3
0
2
4
3
4
3
4
3
2
7
9
8
2
7
7
5
9
0
6
6
4
6
1
278
5
Draft
C. D. Pilmer
Whole Number Crossword Puzzle (B) (page 128)
4
5
3
4
4
1
7
2
3
3
4
6
6
4
8
4
8
3
6
8
0
2
8
3
1
0
9
4
2
5
4
1
6
1
5
9
8
4
7
3
8
6
Whole Number Crossword Puzzle (C) (page 129)
5
6
0
8
3
1
4
3
5
9
3
3
NSSAL
©2012
7
4
8
9
7
0
5
4
8
8
1
9
6
1
0
6
2
3
6
1
4
9
2
0
2
3
6
2
4
279
8
Draft
C. D. Pilmer
Whole Number Crossword Puzzle (D) (page 130)
7
7
1
1
8
7
1
6
3
5
7
7
6
4
3
6
4
8
3
2
0
8
1
2
9
8
5
4
8
7
6
9
2
1
8
8
2
0
5
8
7
2
KenKen Puzzles (A) (pages 131 and 132)
(a)
(b)
2
3
1
2
3
1
3
1
2
1
2
3
1
2
3
3
1
2
(c)
NSSAL
©2012
(d)
3
2
1
1
3
2
1
3
2
3
2
1
2
1
3
2
1
3
280
Draft
C. D. Pilmer
(e)
(f)
3
2
1
1
2
3
1
3
2
3
1
2
2
1
3
2
3
1
(g)
(h)
1
3
2
2
1
3
3
2
1
3
2
1
2
1
3
1
3
2
(i)
NSSAL
©2012
(j)
3
2
1
2
3
1
1
3
2
3
1
2
2
1
3
1
2
3
281
Draft
C. D. Pilmer
KenKen Puzzles (B) (page 133)
(a)
(b)
2
1
3
3
1
2
1
3
2
2
3
1
3
2
1
1
2
3
(c)
(d)
1
2
3
2
3
1
3
1
2
1
2
3
2
3
1
3
1
2
(e)
NSSAL
©2012
(f)
3
1
2
3
2
1
1
2
3
2
1
3
2
3
1
1
3
2
282
Draft
C. D. Pilmer
KenKen (C) (page 134)
(a)
(b)
4
5
3
3
5
4
3
4
5
4
3
5
5
3
4
5
4
3
(c)
(d)
5
3
4
3
4
5
4
5
3
5
3
4
3
4
5
4
5
3
(e)
NSSAL
©2012
(f)
5
4
3
4
3
5
3
5
4
3
5
4
4
3
5
5
4
3
283
Draft
C. D. Pilmer
KenKen Puzzle (D) (page 135)
(a)
(b)
5
6
7
7
5
6
7
5
6
6
7
5
6
7
5
5
6
7
(c)
(d)
6
5
7
6
7
5
5
7
6
5
6
7
7
6
5
7
5
6
(e)
NSSAL
©2012
(f)
7
6
5
5
7
6
5
7
6
6
5
7
6
5
7
7
6
5
284
Draft
C. D. Pilmer
KenKen Puzzles (E) (page 136)
(a)
(b)
2
3
1
4
3
1
2
4
4
1
2
3
2
4
3
1
3
2
4
1
4
2
1
3
1
4
3
2
1
3
4
2
(c)
NSSAL
©2012
(d)
1
2
4
3
2
1
3
4
3
4
2
1
4
3
1
2
4
1
3
2
3
4
2
1
2
3
1
4
1
2
4
3
285
Draft
C. D. Pilmer
KenKen Puzzles (F) (pages 137 and 138)
(a) 1, 2, 3, 4 Puzzle
(b) 1, 2, 3, 4 Puzzle
1
3
4
2
2
3
1
4
4
1
2
3
4
1
2
3
2
4
3
1
3
2
4
1
3
2
1
4
1
4
3
2
(c) 2, 3, 4, 5 Puzzle
NSSAL
©2012
(d) 2, 3, 4, 5 Puzzle
4
5
2
3
5
3
4
2
2
3
5
4
3
2
5
4
5
4
3
2
2
4
3
5
3
2
4
5
4
5
2
3
286
Draft
C. D. Pilmer
(e) 3, 4, 5, 6 Puzzle
(f) 3, 4, 5, 6 Puzzle
6
4
3
5
4
3
6
5
5
6
4
3
6
4
5
3
3
5
6
4
5
6
3
4
4
3
5
6
3
5
4
6
(g) 4, 5, 6, 7 Puzzle
NSSAL
©2012
(h) 4, 5, 6, 7 Puzzle
4
6
5
7
7
4
6
5
7
5
4
6
5
6
7
4
5
7
6
4
4
7
5
6
6
4
7
5
6
5
4
7
287
Draft
C. D. Pilmer
(i) 5, 6, 7, 8 Puzzle
(j) 6, 7, 8, 9 Puzzle
8
6
7
5
7
9
6
8
5
7
8
6
6
8
7
9
6
8
5
7
9
6
8
7
7
5
6
8
8
7
9
6
Find the Two Numbers (page 139)
(a)
(d)
(g)
(j)
(m)
(p)
(s)
(v)
3, 5
12, 1
4, 4
6, 2
2, 11
4, 10
7, 5
1, 8
(b)
(e)
(h)
(k)
(n)
(q)
(t)
(w)
10, 2
3, 6
2, 15
1, 10
6, 7
7, 9
2, 25
20, 3
(c)
(f)
(i)
(l)
(o)
(r)
(u)
(x)
4, 7
8, 5
4, 9
5, 5
2, 12
8, 4
4, 6
10, 10
Which Combination of Numbers Works? (page 140)
(a)
4
×
5
-
1
=
19
The first two numbers can be interchanged.
(b)
2
×
5
+
7
=
17
The first two numbers can be interchanged.
(c)
3
+
8
-
6
=
5
The first two numbers can be interchanged.
(d)
5
×
6
+
4
=
34
The first two numbers can be interchanged.
(e)
3
×
7
-
6
=
15
The first two numbers can be interchanged.
(f)
20
÷
4
+
5
=
10
NSSAL
©2012
288
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C. D. Pilmer
(g)
7
+
9
-
3
=
13
The first two numbers can be interchanged.
(h)
5
×
7
-
8
=
27
The first two numbers can be interchanged.
(i)
8
÷
4
+
12
=
14
(j)
20
÷
5
-
2
=
2
(k)
6
+
10
-
7
=
9
The first two numbers can be interchanged.
(l)
6
×
7
+
3
=
45
The first two numbers can be interchanged.
(m)
15
÷
5
+
3
=
6
(n)
16
÷
8
+
24
=
26
(o)
6
×
9
-
5
=
49
The first two numbers can be interchanged.
Magic Squares (page 141)
(a)
(b)
1
6
3
8
1
7
2
9
3
5
7
2
4
6
8
6
4
4
9
2
7
0
5
3
10
5
2
9
4
9
4
11
10
3
8
7
5
3
10
8
6
5
7
9
6
1
8
5
12
7
6
11
4
(d)
NSSAL
©2012
(c)
8
(e)
(f)
289
Draft
C. D. Pilmer
(g)
(h)
(i)
5
10
3
6
11
4
12
5
10
4
6
8
5
7
9
7
9
11
9
2
7
10
3
8
8
13
6
Addition Pyramids (pages 142 and 143)
1.
2.
10
4
3.
6
4.
8
8.
7
2
10.
10
18
11.
3
4
13.
7
4
8
2
2
1
7
290
6
6
15.
0
24
14
10
5
4
5
13
12
7
9
19
11
9
9
16
12.
21
12
3
6
10
1
7
29
14.
19
8
9
18
8
9
9.
3
6
11
7
6.
9
9
10
3
18
12
13
4
5
21
3.
14
2
9
NSSAL
©2012
5
10
7.
9
4
6
8
Draft
C. D. Pilmer
16.
17.
29
11
18
6
6
5
0
13
19.
8
3
12
5
17
3
4
11
3
13
5
3
24.
2
6
10
6
22
18
2
8
40
12
8
6
5
7
20
19
6
10
2
8
28
8
6
13
5
2
15
39
9
1
23
9
3
19
10
7
3
20
7
3
21.
15
23.
36
9
6
27
2
10
50
7
5
22.
1
12
8
5
2
20.
25
13
7
3
30
16
10
5
18.
26
5
1
10
12
7
3
Factor Rows and Factor Columns (page 144)
(a)
2
5
10
4
3
12
8
15
3
9
27
7
1
7
21
9
9
2
18
4
5
20
36
10
(d)
(g)
NSSAL
©2012
(b)
(e)
(h)
6
3
18
1
2
2
6
6
6
10
60
4
2
8
24
20
3
6
18
5
9
45
15
54
291
(c)
(f)
(i)
4
5
20
2
6
12
8
30
4
7
28
1
3
3
4
21
8
5
40
1
7
7
8
35
Draft
C. D. Pilmer
(j)
3
8
24
9
2
18
27
16
9
6
54
8
5
40
72
30
(m)
(k)
(n)
5
4
20
6
7
42
30
28
7
8
56
3
2
6
21
16
(l)
(o)
7
2
14
4
8
32
28
16
5
7
35
3
9
27
15
63
Letter and Number Sentences (page 145)
1.
2.
3.
4.
5.
6.
7.
8.
(a)
(a)
(a)
(a)
(a)
(a)
(a)
(a)
4
7
4
5
9
3
9
4
(b)
(b)
(b)
(b)
(b)
(b)
(b)
(b)
2
9
8
18
12
6
12
11
(c)
(c)
(c)
(c)
(c)
(c)
(c)
(c)
6
15
2
7
4
15
7
7
(d)
(d)
(d)
(d)
(d)
(d)
(d)
(d)
8
13
24
60
17
1
20
5
Number Sentences (A) (page 147)
1. 4 + 8 = 12
8 + 4 = 12
12 − 8 = 4
12 − 4 = 8
2. 5 + 10 = 15
10 + 5 = 15
15 − 5 = 10
15 − 10 = 5
3. 2 × 8 = 16
8 × 2 = 16
16 ÷ 2 = 8
16 ÷ 8 = 2
4. 4 + 5 = 9
5+4 =9
9−5 = 4
9−4=5
5. 4 × 7 = 28
7 × 4 = 28
28 ÷ 7 = 4
28 ÷ 4 = 7
6. 5 × 9 = 45
9 × 5 = 45
45 ÷ 9 = 5
45 ÷ 5 = 9
7. 9 + 3 = 12
3 + 9 = 12
12 − 3 = 9
12 − 9 = 3
8. 3 × 5 = 15
5 × 3 = 15
15 ÷ 3 = 5
15 ÷ 5 = 3
9. 20 + 10 = 30
10 + 20 = 30
30 − 10 = 20
30 − 20 = 10
10. 30 × 6 = 180
6 × 30 = 180
180 ÷ 6 = 30
180 ÷ 30 = 6
NSSAL
©2012
292
Draft
C. D. Pilmer
Number Sentences (B) (page 148)
1. 4 × 5 = 20
5 × 4 = 20
20 ÷ 4 = 5
20 ÷ 5 = 4
3 × 9 = 27
9 × 3 = 27
27 ÷ 3 = 9
27 ÷ 9 = 3
4+5=9
5+4 =9
9−4=5
9−5 = 4
2. 2 × 4 = 8
4× 2 = 8
8÷2 = 4
8÷4 = 2
3 × 8 = 24
8 × 3 = 24
24 ÷ 3 = 8
24 ÷ 8 = 3
3 × 4 = 12
4 × 3 = 12
12 ÷ 4 = 3
12 ÷ 3 = 4
4 + 8 = 12
8 + 4 = 12
12 − 4 = 8
12 − 8 = 4
2 × 12 = 24
12 × 2 = 24
24 ÷ 2 = 12
24 ÷ 12 = 2
3. 2 × 9 = 18
9 × 2 = 18
18 ÷ 2 = 9
18 ÷ 9 = 2
3 × 6 = 18
6 × 3 = 18
18 ÷ 6 = 3
18 ÷ 3 = 6
2 × 6 = 12
6 × 2 = 12
12 ÷ 2 = 6
12 ÷ 6 = 2
3 + 9 = 12
9 + 3 = 12
12 − 3 = 9
12 − 9 = 3
3+6 = 9
6+3=9
9−3= 6
9−6=3
2×3 = 6
3× 2 = 6
6÷3 = 2
6÷2=3
Order of Operations (A) (pages 149 to 151)
(a) 11
(d) 23
(g) 11
(j) 26
(m) 8
(p) 30
(s) 16
(v) 7
(b) 4
(e) 42
(h) 27
(k) 14
(n) 23
(q) 1
(t) 10
(w) 45
(c)
(f)
(i)
(l)
(o)
(r)
(u)
(x)
23
15
13
2
8
19
13
34
(c)
(f)
(i)
(l)
(o)
(r)
6
10
16
7
1
59
Order of Operations (B) (pages 152 to 154)
(a) 26
(d) 37
(g) 43
(j) 28
(m) 44
(p) 34
NSSAL
©2012
(b)
(e)
(h)
(k)
(n)
(q)
39
28
12
35
2
4
293
Draft
C. D. Pilmer
Order of Operations (C) (pages 155 to 157)
(a) 26
(d) 39
(g) 40
(j) 81
(m) 6
(p) 12
(b)
(e)
(h)
(k)
(n)
(q)
32
72
7
3
23
29
(c)
(f)
(i)
(l)
(o)
(r)
4
50
24
4
94
7
What's the Pattern (A) (page 159)
(a)
, , 
(b)
, , 
(c)
, , 
(d)
, , 
(e)
, , 
(f)
G, , H
(g)
, , 
(h)
, , 
(i)
, , 
(j)
P, , R
(k)
, , 
(l)
, , 
, 
(n)
T, t, R
, , 
(p)
, , 
(m) ,
(o)
What's the Pattern? (B) (page 160)
Very Challenging (Only give to your strongest learners)
(a)
, , 
(b)
, , 
(c)
, , 
(d)
, , 
(e)
, , 
(f)
, , 
(g)
, , 
(h)
, , 
(i)
, , 
(j)
, , 
(k)
, , 
(l)
, , 
(n)
, , 
(m) ,
NSSAL
©2012
E, 
294
Draft
C. D. Pilmer
(o)
, , 
Toothpick Patterns (pages 161 and 162)
Note: The next figures have not been provided in this answer key.
1. (b) 3, 5, 7, 9, 11, 13
(c) Start at 3 and keep adding 2.
2. (b) 8, 12, 16, 20, 24
(c) Start at 8 and keep adding 4.
3. (b) 19, 15, 11, 7, 3
(c) Start at 19 and keep subtracting 4.
4. (b) 8, 15, 22, 29, 36
(c) Start at 8 and keep adding 7.
5. (b) 31, 25, 19, 13, 7
(c) Start at 31 and keep subtracting 6.
6. (b) 10, 16, 22, 28, 34
(c) Start at 10 and keep adding 6.
7. (b) 9, 13, 17, 21, 25
(c) Start at 9 and keep adding 4
8. (b) 4, 14, 24, 34, 44
(c) Start at 4 and keep adding 10.
Create the Pattern (A) (page 163)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
Start at 9 and go up by 2 each time.
Start at 24 and go down by 1 each time.
Start at 8 and go up by 3 each time.
Start at 4 and go up by 5 each time.
Start at 33 and go down by 3 each time.
Start at 29 and go down by 2 each time.
Start at 11 and go up by 4 each time.
Start at 30 and go down by 2 each time.
Start at 3 and go up by 6 each time.
Start at 2 and go up by 4 each time.
Start at 11 and go up by 5 each time.
Start at 38 and go down by 3 each time.
NSSAL
©2012
9, 11, 13, 15, 17
24, 23, 22, 21, 20
8, 11, 14, 17, 20
4, 9, 14, 19, 24
33, 30, 27, 24, 21
29, 27, 25, 23, 21
11, 15, 19, 23, 27
30, 28, 26, 24, 22
3, 9, 15, 21, 27
2, 6, 10, 14, 18
11, 16, 21, 26, 31
38, 35, 32, 29, 26
295
Draft
C. D. Pilmer
(m) Start at 36 and go down by 4 each time.
(n) Start at 23 and go up by 6 each time.
(o) Start at 40 and go down by 5 each time.
(p) Start at 13 and go up by 10 each time.
(q) Start at 72 and go down by 2 each time.
(r) Start at 99 and go down by 10 each time.
(s) Start at 41 and go up by 4 each time.
36, 32, 28, 24, 20
23, 29, 35, 41, 47
40, 35, 30, 25, 20
13, 23, 33, 43, 53
72, 70, 68, 66, 64
99, 89, 79, 69, 59
41, 45, 49, 53, 57
Create the Pattern (B) (page 164)
(a) Start at 63 and go up by 7 each time.
(b) Start at 234 and go down by 2 each time.
(c) Start at 81 and go up by 3 each time.
(d) Start at 79 and go down by 3 each time.
(e) Start at 126 and go up by 5 each time.
(f) Start at 540 and go down by 10 each time.
(g) Start at 352 and go up by 20 each time.
(h) Start at 47 and go up by 4 each time.
(i) Start at 68 and go down by 4 each time.
(j) Start at 275 and go up by 25 each time.
(k) Start at 134 and go up by 6 each time.
(l) Start at 456 and go down by 100 each time.
(m) Start at 99 and go down by 11 each time.
(n) Start at 347 and go up by 3 each time.
(o) Start at 605 and go down by 5 each time.
(p) Start at 710 and go up by 30 each time.
(q) Start at 670 and go down by 20 each time.
(r) Start at 412 and go up by 6 each time.
(s) Start at 364 and go down by 3 each time.
63, 70, 77, 84
234, 232, 230, 228
81, 84, 87, 90
79, 76, 73, 70
126, 131, 136, 141
540, 530, 520, 510
352, 372, 392, 412
47, 51, 55, 59
68, 64, 60, 56
275, 300, 325, 350
134, 140, 146, 152
456, 356, 256, 156
99, 88, 77, 66
347, 350, 353, 356
605, 600, 595, 590
710, 740, 770, 800
670, 650, 630, 610
412, 418, 424, 430
364, 361, 358, 355
Number Patterns (A) (page 165)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
6, 8, 10, 12, 14, 16, 18, 20, 22
23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12
5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27
30, 28, 26, 24, 22, 20, 18, 16, 14, 12
0, 3, 6, 9, 12, 15, 18, 21, 24
10, 15, 20, 25, 30, 35, 40, 45, 50
40, 36, 32, 28, 24, 20, 16, 12, 8
31, 28, 25, 22, 19, 16, 13, 10, 7, 4
7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57
45, 47, 49, 51, 53, 55, 57, 59, 61, 63
19, 22, 25, 28, 31, 34, 37, 40, 43, 46
51, 47, 43, 39, 35, 31, 27, 23, 19, 15, 11
NSSAL
©2012
296
Draft
C. D. Pilmer
(m) 67, 62, 57, 52, 47, 42, 37, 32, 27, 22
(n) 44, 43, 42, 41, 40, 39, 38, 37, 36
(o) 23, 25, 27, 29, 31, 33, 35, 37, 39
(p) 30, 27, 24, 21, 18, 15, 12, 9, 6
(q) 17, 21, 25, 29, 33, 37, 41, 45, 49, 53
(r) 9, 14, 19, 24, 29, 34, 39, 44, 49
(s) 30, 33, 36, 39, 42, 45, 48, 51, 54, 57
(t) 51, 46, 41, 36, 31, 26, 21, 16, 11, 6
(u) 91, 92, 93, 94, 95, 96, 97
Number Patterns (B) (page 166)
(a) 2, 4, 6, 8, 10, 12, 14, 16, 18
(b) 4, 7, 10, 13, 16, 19, 22, 25
(c) 29, 27, 25, 23, 21, 19, 17, 15
(d) 55, 50, 45, 40, 35, 30, 25, 20
(e) 3, 10, 17, 24, 31, 38, 45, 52, 59
(f) 64, 56, 48, 40, 32, 24, 16, 8, 0
(g) 0, 6, 12, 18, 24, 30, 36, 42, 48, 54
(h) 1, 12, 23, 34, 45, 56, 67, 78, 89
(i) 44, 40, 36, 32, 28, 24, 20, 16, 12, 8
(j) 100, 104, 108, 112, 116, 120, 124, 128, 132
(k) 675, 680, 685, 690, 695, 700, 705, 710, 715, 720, 725
(l) 190, 210, 230, 250, 270, 290, 310, 330, 350, 370, 390
(m) 326, 324, 322, 320, 318, 316, 314, 312, 310
(n) 6, 10, 14, 18, 22, 26, 30, 34, 38
(o) 40, 37, 34, 31, 28, 25, 22, 19, 16
(p) 56, 61, 66, 71, 76, 81, 86, 91, 96, 101, 106
(q) 21, 25, 29, 33, 37, 41, 45, 49, 53, 57
(r) 251, 245, 239, 233, 227, 221, 215, 209, 203
(s) 16, 24, 32, 40, 48, 54, 64, 72, 80
(t) 300, 296, 292, 288, 284, 280, 276, 272, 268
(u) 420, 440, 460, 480, 500, 520, 540
Row, Column, and Diagonal Pattern (page 167)
1.
2.
3.
4.
5.
6.
Row Pattern
Add 3
Subtract 2
Add 10
Subtract 4
Add 2
Subtract 5
NSSAL
©2012
Column Pattern
Add 9
Subtract 6
Add 30
Subtract 8
Subtract 6
Add 15
297
Diagonal Pattern
Add 12
Subtract 8
Add 40
Subtract 16
Subtract 4
Add 10
Draft
C. D. Pilmer
What's the Relationship? (pages 168 and 169)
(a) Even Numbers
Divisible by 2
Multiples of 2
Start at 2 and keep adding 2
Start at 50 and keep subtracting 2
(b) Divisible by 3
Multiples of 3
Start at 3 and keep adding 3
Start at 48 and keep subtracting 3
(c) Divisible by 5
Multiples of 5
Start at 5 and keep adding 5
Start at 50 and keep subtracting 5
(d) Start at 1 and keep adding 4
Start at 49 and keep subtracting 1
(e) Start at 4 and keep adding 9
Start at 49 and keep subtracting 9
(f) Perfect Squares
(g) Start at 1 and keep adding 11
Start at 45 and keep subtracting 11
(h) Odd Numbers
Start at 1 and keep adding 2
Start at 49 and keep subtracting 2
(i) Divisible by 4
Multiples of 4
Start at 4 and keep adding 4
Start at 48 and keep subtracting 4
(j) Composite Numbers
(k) Start at 2 and keep adding 12
Start at 50 and keep subtracting 12
(l) Divisible by 7
Multiples of 7
Start at 7 and keep adding 7
Start at 49 and keep subtracting 7
Input Output (A) (page 170)
(a) add 4
(d) subtract 5
(g) divide by 4
(b) multiply by 2
(e) multiply by 5
(h) add 6
(c) divide by 3
(f) subtract 3
(i) multiply by 4
(b) subtract 7
(e) multiply by 6
(h) subtract 10
(c) add 9
(f) divide by 8
(i) multiply by 8
Input Output (B) (page 171)
(a) multiply by 9
(d) divide by 7
(g) add 7
NSSAL
©2012
298
Draft
C. D. Pilmer
Input Output (C) (page 172)
(a) add 30
(d) divide by 4
(g) subtract 70
(b) multiply by 50
(e) subtract 200
(h) add 300
(c) subtract 60
(f) multiply by 400
(i) divide by 2
Input Output (D) (page 173)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
multiply by 2, then add 5
multiply by 3, then add 1
multiply by 2, then subtract 3
multiply by 4, then subtract 2
multiply by 3, then add 5
multiply by 5, then add 1
multiply by 4, then add 1
multiply by 5, then subtract 2
multiply by 3, then subtract 2
Filling or Draining (pages 174 to 178)
1. Written Description:
Table of Values:
Time
Litres
0
18
1
15
2
12
3
9
2. Written Description:
Table of Values:
Time
Litres
0
6
1
8
2
10
3
12
NSSAL
©2012
The container initially had 18 L of water in it and it is being drained
at a rate of 3 litres per minute.
Equation: L = 18 -2T
Empty: 6 minutes
The container initially had 6 L of water in it and it is being filled at a
rate of 2 litres per minute.
Equation: L = 6 + 2T
Full: 7 minutes
299
Draft
C. D. Pilmer
3. Written Description:
Table of Values:
Time
Litres
0
5
1
8
2
11
3
14
4. Written Description:
Table of Values:
Time
Litres
0
20
1
16
2
12
3
8
5. Written Description:
Table of Values:
Time
Litres
0
13
1
12
2
11
3
10
6. Written Description:
Table of Values:
Time
Litres
0
0
1
5
2
10
3
15
NSSAL
©2012
The container initially had 5 L of water in it and it is being filled at a
rate of 3 litres per minute.
Equation: L = 5 + 3T
Full: 5 minutes
The container initially had 20 L of water in it and it is being drained
at a rate of 4 litres per minute.
Equation: L = 20 - 4T
Empty: 5 minutes
The container initially had 13 L of water in it and it is being drained
at a rate of 1 litre per minute.
Equation: L = 13 - 1T
Empty: 13 minutes
The container was initially empty and it is being filled at a rate of 5
litres per minute.
Equation: L = 5T
Full: 4 minutes
300
Draft
C. D. Pilmer
7. Written Description:
Table of Values:
Time
Litres
0
10
1
8
2
6
3
4
8. Written Description:
Table of Values:
Time
Litres
0
14
1
15
2
16
3
17
The container initially had 10 L of water in it and it is being drained
at a rate of 2 litres per minute.
Equation: L = 10 - 2T
Empty: 5 minutes
The container initially had 14 L of water in it and it is being filled at a
rate of 1 litre per minute.
Equation: L = 14 + 1T
Full: 6 minutes
Travelling Towards or Away From Home (pages 179 to 185)
1. Written Description:
Table of Values:
Time
Distance
from Home
0
9
1
11
2
13
3
15
4
17
NSSAL
©2012
Montez is initially 9 metres from home and runs away from home at a
rate of 2 metres per second.
Equation: d = 9 + 2t
At t = 7 seconds, d = 23 m
301
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C. D. Pilmer
2. Written Description:
Table of Values:
Time
Distance
from Home
0
4
1
7
2
10
3
13
4
16
3. Written Description:
Table of Values:
Time
Distance
from Home
0
30
1
26
2
22
3
18
4
14
4. Written Description:
Table of Values:
Time
Distance
from Home
0
24
1
21
2
18
3
15
4
12
NSSAL
©2012
Montez is initially 4 metres from home and runs away from home at a
rate of 3 metres per second.
Equation: d = 4 + 3t
At t = 7 seconds, d = 25 m
Montez is initially 30 metres from home and runs towards home at a
rate of 4 metres per second.
Equation: d = 30 - 4t
At t = 7 seconds, d = 2 m
Montez is initially 24 metres from home and runs towards home at a
rate of 3 metres per second.
Equation: d = 24 - 3t
At t = 7 seconds, d = 3 m
302
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C. D. Pilmer
5. Written Description:
Table of Values:
Time
Distance
from Home
0
14
1
12
2
10
3
8
4
6
6. Written Description:
Table of Values:
Time
Distance
from Home
0
20
1
21
2
22
3
23
4
24
7. Written Description:
Table of Values:
Time
Distance
from Home
0
35
1
30
2
25
3
20
4
15
NSSAL
©2012
Montez is initially 14 metres from home and runs towards home at a
rate of 2 metres per second.
Equation: d = 14 - 2t
At t = 7 seconds, d = 0 m
Montez is initially 20 metres from home and runs away from home at
a rate of 1 metre per second.
Equation: d = 20 + t
At t = 7 seconds, d = 27 m
Montez is initially 35 metres from home and runs towards home at a
rate of 5 metres per second.
Equation: d = 35 - 5t
At t = 7 seconds, d = 0 m
303
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C. D. Pilmer
8. Written Description:
Table of Values:
Time
Distance
from Home
0
0
1
6
2
12
3
18
4
24
9. Written Description:
Table of Values:
Time
Distance
from Home
0
32
1
26
2
20
3
14
4
8
Montez is initially 0 metres from home and runs away from home at a
rate of 6 metres per second.
Equation: d = 6t
At t = 7 seconds, d = 42 m
Montez is initially 32 metres from home and runs towards home at a
rate of 6 metres per second.
Equation: d = 32 - 6t
At t = 7 seconds, d = 0 m (actually arrived at home
between t = 5 and t = 6 seconds)
Weight of the Water (pages 186 to 190)
1. Written Description:
Table of Values:
Time
Weight
0
2
1
5
2
8
3
11
4
14
NSSAL
©2012
The container initially contained water weighting 2 newtons and then
water was added such that the weight increased by 3 newtons per
second.
Equation: w = 2 + 3t
At t = 6 seconds, the weight is 20 N.
304
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C. D. Pilmer
2. Written Description:
Table of Values:
Time
Weight
0
1
1
5
2
9
3
13
4
17
3. Written Description:
Table of Values:
Time
Weight
0
29
1
26
2
23
3
20
4
17
4. Written Description:
Table of Values:
Time
Weight
0
30
1
25
2
20
3
15
4
10
NSSAL
©2012
The container initially contained water weighting 1 newton and then
water was added such that the weight increased by 4 newtons per
second.
Equation: w = 1 + 4t
At t = 6 seconds, the weight is 25 N.
The container initially contained water weighting 29 newtons and
then water was removed such that the weight decreased by 3 newtons
per second.
Equation: w = 29 - 3t
At t = 6 seconds, the weight is 11 N.
The container initially contained water weighting 30 newtons and
then water was removed such that the weight decreased by 5 newtons
per second.
Equation: w = 30 - 5t
At t = 6 seconds, the weight is 0 N.
305
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C. D. Pilmer
5. Written Description:
Table of Values:
Time
Weight
0
0
1
2
2
4
3
6
4
8
6. Written Description:
Table of Values:
Time
Weight
0
23
1
20
2
17
3
14
4
11
The container initially contained water weighting 0 newtons (i.e. the
container was initially empty) and then water was added such that the
weight increased by 4 newtons every 2 seconds, or 2 newtons every
second.
Equation: w = 2t
At t = 6 seconds, the weight is 12 N.
The container initially contained water weighting 23 newtons and
then water was removed such that the weight decreased by 6 newtons
every 2 seconds, of 3 newtons every second.
Equation: w = 23 - 3t
At t = 6 seconds, the weight is 5 N.
Describing the Relationships with Words (page 192)
There are 4 moons.
There are 5 hearts.
There are 8 suns.
There is a total of 17 shapes.
There is 1 more heart than moons.
There is 1 less moon than hearts.
There are 4 more suns than moons.
There are 4 less moons than suns.
There are 3 more suns than hearts.
There are 3 less hearts than suns.
There are twice as many suns as moons.
There are half as many moons as suns.
NSSAL
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List the Numbers Based on the Written Description (page 193)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
9, 10, 11, 12, 13, 14
10, 12, 14, 16, 18, 20, 22
10, 20, 30, 40, 50, 60, 70, 80, 90, 100
15, 20, 25, 30, 35, 40
11, 22, 33, 44, 55, 66, 77, 88, 99
15, 18, 21, 24, 27, 30
5, 7, 11, 13, 17
3, 9, 15, 21
10, 20, 30, 40
16, 25, 34, 43, 52, 61, 70
36, 45, 54, 63, 72, 81, 99
8, 16, 24, 32
9, 10, 12, 14, 15, 16, 18, 20, 21
6, 12, 18, 24, 30
15, 25, 35, 45, 55
NSSAL
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C. D. Pilmer
Addition and Subtraction Crossword (pages 194 and 195)
t
w
o
f
e
t
n
h
t
h
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r
w
e
l
i
x
t
e
e
t
e
e
e
n
o
u
v
e
n
t
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v
l
h
e
n
o
n
t
i
g
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t
t
e
t
s
o
i
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t
w
e
t
w
e
x
f
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308
i
t
y
l
t
e
h
i
r
n
e
e
v
e
e
y
NSSAL
©2012
n
e
l
h
r
f
e
i
v
e
v
n
g
e
e
z
o
r
e
y
r
h
v
f
e
t
e
i
t
f
s
l
g
e
r
l
e
e
w
t
x
y
y
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n
i
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f
o
f
e
s
v
i
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e
n
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n
e
Draft
C. D. Pilmer
Multiplication and Division Crossword (page 196 to 197)
t
e
i
g
t
t
t
y
s
i
i
t
x
w
t
t
w e
y
e
l
e
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v
t
i
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-
w
g
e
o
h
w o
t
h
i
e
n
t
s
h
y
f
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i
i
r
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y
x
f
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r
-
n
t
f
f
y
i
o
-
v
r
g
h
s
t
w e
l
v
e
v
s
e
n
i
n
e
v
-
e
s
n
t
w o
l
i
h
u
v
x
r
r
e
e
e
NSSAL
©2012
n
f
i
t
e
o
i
u
r
r
v
n
e
h
i
i
x
t
o
u
t
y
h
t
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v
e
e
n
f
n
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f
e
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r
i
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309
f
n
i
f
-
f
t
-
f
r
x
e
y
n
f
e
y
f
s
s
v
y
i
y
f
f
t
n
e
e
t
y
o
n
n
w o
l
e
t
t
n
w
i
y
e
t
o
t
e
v
r
t
e
e
i
f
e
t
f
t
i
y
v
e
s
h
e
e
i
g
h
t
y
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f
s
-
o
e
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f
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v
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o
u
r
t
u
w
r
o
e
e
i
n
x
-
x
n
Draft
C. D. Pilmer
Operations Crossword (pages 198 to 199)
t
t
w
h
e
f
n
n
i
t
y
-
o
n
w
i
o
n
r
f
i
v
g
h
t
y
-
o
n
e
n
i
i
n
e
t
e
e
n
h
z
e
r
o
e
s
e
v
e
n
t
y
n
t
o
u
t
y
t
e
n
e
y
h
x
n
-
i
t
f
y
o
t
-
u
e
r
e
s
o
t
n
r
t
h
r
r
t
n
e
v
e
y
-
t
w
o
z
l
v
e
i
r
g
o
n
i
f
i
g
i
t
f
h
v
w
t
t
e
e
-
n
h
e
y
s
n
o
i
t
i
t
n
n
e
n
z
w
e
NSSAL
©2012
n
e
v
r
o
i
l
n
e
s
i
x
310
t
s
e
r
f
y
i
x
w
o
n
u
e
h
e
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e
-
i
o
u
y
t
w
o
r
t
n
n
f
v
x
e
e
f
n
t
v
-
e
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t
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-
s
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g
l
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v
e
n
Draft
C. D. Pilmer
Word Sentence to Number Sentence to Answer (A) (page 200)
Word Sentence
Number Sentence
Answer
(a)
What is the sum of three and eight?
3+8
11
(b)
What is six multiplied by five?
6×5
30
(c)
What do you get when you double three?
2×3
6
(d)
What is nine decreased by four?
9−4
5
(e)
What is half of ten?
10 ÷ 2
5
(f)
Given seven and five, what is their total?
7+5
12
(g)
What is eight times three?
8×3
24
(h)
What is sixteen divided by two?
16 ÷ 2
8
(i)
What is seven increased by six?
7+6
13
(j)
What do you get when you triple six?
3× 6
18
(k)
What do you get when eight is taken away from ten?
10 − 8
2
(l)
What is seven less four?
7−4
3
(m) What is three plus eleven?
3 + 11
14
(n)
What do we get when ten is broken into five equal parts?
10 ÷ 5
2
(o)
How much more is nine compared to two?
9−2
7
(p)
What is a quarter of eight?
8÷4
2
(q)
What is the product of nine and two?
9× 2
18
(r)
What is six combined with eight?
6+8
14
(s)
What is six taken from thirteen?
13 − 6
7
(t)
What is ten increased by six?
10 + 6
16
(u)
How many threes fit into fifteen?
15 ÷ 3
5
(v)
What do you get when nine is removed from ten?
10 − 9
1
NSSAL
©2012
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Draft
C. D. Pilmer
Word Sentence to Number Sentence to Answer (B) (page 201)
Word Sentence
Number Sentence
Answer
(a)
What do you get when you double eleven?
2 × 11 2
22
(b)
What is thirty-five divided by seven?
35 ÷ 7
5
(c)
What is seven times six?
7×6
42
(d)
What is seventeen increased by eight?
17 + 8
25
(e)
What do you get when nine is taken away from fifteen?
15 − 9
6
(f)
What is the sum of twelve and seven?
12 + 7
19
(g)
What do we get when ten is broken into two equal parts?
10 ÷ 2
5
(h)
What is sixteen plus ten?
16 + 10
26
(i)
What is the product of nine and seven?
9×7
63
(j)
What is half of twenty-four?
24 ÷ 2
12
(k)
What is thirty-seven less five?
37 − 5
32
(l)
What is six multiplied by eight?
6×8
48
(m) What is three taken from forty-nine?
49 − 3
46
(n)
What is twenty-six increased by eleven?
26 + 11
37
(o)
What do you get when six is removed from twenty-three?
23 − 6
17
(p)
How many nines fit into eighty-one?
81 ÷ 9
9
(q)
What is forty-five decreased by six?
45 − 6
39
(r)
What is eighteen combined with nine?
18 + 9
27
(s)
What do you get when you triple twelve?
3× 12
36
(t)
Given seven and seventeen, what is their total?
7 + 17
24
(u)
How much more is thirty-six compared to three?
36 − 3
33
(v)
What is a third of twenty-seven?
27 ÷ 3
9
NSSAL
©2012
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Draft
C. D. Pilmer
Word Sentence to Number Sentence to Answer (C) (page 202)
Word Sentence
Number Sentence
Answer
500 + 200
700
50 × 3
150
(a)
Given five hundred and two hundred, what is their total?
(b)
What is fifty multiplied by three?
(c)
What is three hundred increased by four hundred twenty?
300 + 420
720
(d)
What is forty removed from one hundred?
100 − 40
60
(e)
What is a third of nine thousand?
900 ÷ 3
300
(f)
How many sevens fit into three hundred fifty?
350 ÷ 7
50
(g)
What is seventy times eighty?
70 × 80
5600
(h)
What is the sum of eleven and eighty?
11 + 80
91
(i)
How much more is ninety compared to thirty?
90 − 30
60
(j)
What is five thousand plus eight thousand?
5000 + 8000
13000
(k)
What is two hundred sixty decreased by twenty?
260 − 20
240
(l)
What do you get when you double four thousand?
2 × 4000
8000
600 + 230
830
3× 40
120
(m) What is six hundred increased by two hundred thirty?
(n)
What do you get when you triple forty?
(o)
What is one thousand nine hundred less eight hundred?
1900 − 800
1100
(p)
What is half of sixteen thousand?
16000 ÷ 2
8000
(q)
What is two hundred eighty divided by seven?
280 ÷ 7
40
(r)
What do you get when ten is taken away from ninety?
90 − 10
80
(s)
What is sixty combined with eighty?
60 + 80
140
(t)
What do we get when six is broken into six equal parts?
6÷6
1
(u)
What is seventy taken from ninety-six?
96 − 70
26
(v)
What is the product of six and seven thousand?
6 × 7000
42000
What are the Possibilities? (A) (pages 203 to 205)
1. 0 and 5, 1 and 4, 2 and 3
2. 1 and 18, 2 and 9, 3 and 6, 1 and 20, 2 and 10, 4 and 5
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©2012
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C. D. Pilmer
3. 4 and 2, 6 and 3, 8 and 4
4. 9 and 4, 8 and 3, 7 and 2, 6 and 1, 5 and 0
5. 1 and 4, 2 and 2, 1 and 5, 1 and 6, 2 and 3
6. 6 and 9, 7 and 8, 7 and 9, 8 and 8, 8 and 9, 9 and 9
7. 12 and 3, 16 and 4, 20 and 5, 24 and 6
8. 6 and 4, 6 and 3, 5 and 3, 5 and 2, 4 and 2, 4 and 1, 3 and 1, 3 and 0, 2 and 0
9. 2 and 5, 1 and 8, 1 and 6, 2 and 3, 1 and 4, 1 and 2
10. 1 and 7, 3 and 5, 1 and 9, 3 and 7, 5 and 5
11. 18 and 9, 14 and 7, 10 and 5
12. 10 and 6, 8 and 4, 6 and 2, 4 and 0
What are the Possibilities? (B) (pages 206 and 207)
1. 2 and 8, 4 and 4, 4 and 6, 4 and 8, 6 and 6, 6 and 8, 8 and 8
2. 1 and 3, 1 and 5, 1 and 7, 1 and 9, 3 and 5, 3 and 7, 3 and 9
3. 3 and 11, 5 and 9, 7 and 7
4. 8 and 5, 7 and 4, 6 and 3, 5 and 2
5. 2 and 12, 3 and 8, 4 and 6
6. 2 and 6, 2 and 8, 2 and 10, 2 and 12, 4 and 4, 4 and 6
Describing the Relationships with Words (page 208)
There are 5 moons.
There are 4 hearts.
There are 8 suns.
There is a total of 17 shapes.
There is 1 more moon than hearts.
There is 1 less heart than moons.
There are 4 more suns than hearts.
There are 4 less hearts than suns.
NSSAL
©2012
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C. D. Pilmer
There are 3 more suns than moons.
There are 3 less moons than suns.
There are twice as many suns as hearts.
More Describing Relationships with Words and Numbers (page 209)
1. Lei has 6 grandchildren, and Nasrin has 12 grandchildren.
• Lei and Nasrin have a total of 18 children.
• Nasrin has 6 more grandchildren than Lei.
• Lei has 6 fewer grandchildren than Nasrin.
• Nasrin has twice as many grandchildren as Lei.
2. In the first week, Ryan has only able to save $20 after all his expenses, however, in the
second week he able to save an additional $100.
• Over the two weeks Ryan saved a total of $120.
• Ryan saved $80 more in the second week as compared to the first week.
• Ryan saved $80 less in the first week as compared to the second week.
• Ryan saved five times more money in the second week as compared to the first week.
Recognizing the Important Information (pages 210 and 211)
1. (a) $200
(c) $250
(b) $120
(d) $80
2. (a) $18
(c) 7 days
(b) $6
(d) $24
3. (a) $20 per hour
(c) 30 hours
(b) $600
(d) 2 acres
4. (a) 90 kilometres per hour
(c) 490 kilometres
(b) 5 hours
(d) 2 hours
Does It Make Sense? (pages 212 and 213)
1. Does not make sense; one song from iTunes costs around $1, not $9
2. Does not make sense; she purchased 7 litres of milk, not 8 litres.
3. Makes sense
4. Does not make sense; she ran 42 kilometres, not 36 kilometres
NSSAL
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C. D. Pilmer
5. Makes sense.
6. Does not make sense; can't have 100 roommates in an apartment.
7. Make sense.
8. Does not make sense; by the end of the year there were 23 students, rather than 21
9. Makes sense.
10. Makes sense.
11. Does not make sense; the total mass was 17 kg, not 30 kg.
12. Does not make sense; the total distance is 1800 km, not 18 000 km.
13. Does not make sense; a brand new 50 inch flat screen television would be anywhere from
$700 to $1200, not $100.
14. Makes sense.
15. Does not make sense; it should be 4 apples and 6 oranges, rather than 4 oranges and 6 apples.
16. Does not make sense; she may have brought in $460, but her profit was $230.
Complete the Statement: Addition and Subtraction (page 215)
1. Alex and Tylena have 13 chocolates. Alex, who is not very good at sharing, eats 8
chocolates, leaving only 5 chocolates for Tylena.
2. Micheline ran 20 kilometres on the first day. The second day she ran even further covering
30 kilometres. Over the two days she ran a total of 50 kilometres.
3. The tank of water was initially full. 2 litres of water was then drained from the 18 litre tank.
With such a small amount of water being drained, it meant that 16 litres remained.
4. Samir borrows 90 dollars from his two friends. Jun, who supplies most of the money, lends
Samir 60 dollars. Nita, his other close friend, lends him 30 dollars.
5. Jacob was hoping to get 9 hours of sleep. He ended up getting 7 hours of sleep, just 2 hours
short of his desired number of hours of sleep.
6. Nashi has only purchased or made 5 gifts for the holidays. She still needs to purchase or
made 9 gifts if she wishes to give a gift to each of her 14 family members or friends.
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7. Asra received a bonus and had enough money to take 11 of her 13 friends out to dinner.
Unfortunately she did not have enough to pay for a few friends, specifically 2 friends.
Complete the Statement: Multiplication and Division (page 216)
1. A class of 25 students wishes to have a guest speaker deliver a lecture. If the speaker charges
75 dollars for the lecture, then each student will have to pay 3 dollars.
2. The 60 candies are to be shared equally amongst the 3 roommates. That means that each
roommate will receive 20 candies.
3. If the cyclist is travelling at an average speed of 30 kilometres per hour for 3 hours, then the
she will cover a distance of 90 kilometres in that time.
4. The 4 siblings inherit a 80 acre plot of land. To be fair, each sibling will receive a parcel of
land measuring 20 acres.
5. Kate, who overuses sick time, only has 3 sick days left in her bank. Lei, who rarely uses sick
time, has 27 days left. Lei has 9 times the number of sick days as compared to Kate.
(Note: One could argue that the 3 and 9 be interchanged in this answer.)
6. The boxed set of all seasons of Seinfeld is on sale for 75 dollars. If 5 close friends wish to
share the cost of the boxed set, each will pay 15 dollars.
7. Kamala wants to increase her hourly wage by a factor of 2. If her present hourly wage is 12
dollars per hour, then her desired hourly wage is 24 dollars per hour.
Complete the Statement (A) (page 217)
1. Lei and Jun are siblings. Lei is 4 years younger than Jun. If Jun is 31 years old, then Lei is
27 years old.
2. Marcus has dime and nickels in his pocket. If he has 7 nickels and 2 dimes, then he has a
total of 55 cents.
3. Very few men attended the show. If there were 3 men and 24 women, then there 8 times as
many women as men.
4. Normally 12 people attend the neighborhood watch meeting. That number increased slightly
by 3 such that 15 people attended.
5. Anne had 40 dollars but spent most of her money on a 35 dollar top (after taxes). She now
has 5 dollars left in her purse.
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6. Bashir had 36 candies to split evenly between his 3 children. Each child got 12 candies;
enough to ruin their supper.
7. If there are 60 minutes in an hour, then we know that there are 180 minutes in 3 hours.
8. Ryan, who prefers running, ran for 25 minutes and biked for 15 minutes. That means he
trained for a total of 40 minutes.
9. The mechanic ordered 40 containers of engine oil at a cost of 3 dollars per container. The
total cost was 120 dollars.
10. 4 room mates got together to purchase an 800 dollar couch. If they all paid the same amount,
then each pays 200 dollars.
Complete the Statement (B) (page 218)
1. Tanya has 3 dimes and 2 quarters in her purse. That means she has 80 cents of change in her
purse.
2. Bill and Ajay are friends. Bill is 3 years older than Ajay. If Bill is 50 years old, then Ajay is
47 years old.
3. Kim bought 12 apples and 4 oranges. Therefore she bought 3 times as many apples as
oranges, or 8 more apples than oranges.
4. Three friends equally share the cost of a 21 dollar pizza that was divided into 6 pieces. Each
pays 7 dollars and gets 2 pieces.
5. Ryan had 30 dollars but spent most of his money on a 19 dollar DVD (after taxes). He now
has 11 dollars left.
6. If there are 24 hours in one day, then there are 48 hours in 2 days.
7. The room temperature was 18 degrees Celsius. If it is turned up slightly by 3 degrees, then
the new temperature is 21 degrees.
8. There were 300 millilitres of water in a container. If only 50 millilitres is poured out, then
the container still has 250 millilitres.
9. There are 8 SUVs and 32 cars in the lot. Therefore there are 4 times as many cars as SUVs,
or 24 more cars than SUVs.
10. The store owner ordered 25 packages of printer paper at a cost of 2 dollars per package. His
bill (before taxes) was 50 dollars.
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Complete the Statement (C) (page 219)
1. The cereal, the more expensive item, cost 5 dollars, and the dish soap cost 3 dollars. The
total cost was 8 dollars.
2. The jar contained 40 candies. If you ate 30, which is most of the candy, then that would
leave 10 in the jar.
3. The flour, which weighs 60 kg, must be divided evenly amongst 3 families. Each family was
pleased to get 20 kg.
4. The DVD cost 18 dollars. The socks cost 10 dollars. The DVD was 8 dollars more
expensive than the socks.
5. The corner store owner sold 30 bottles of pop. If each sold for 3 dollars, then his total pop
sales were 90 dollars.
6. Tom drove for 2 more hours than Ed. Tom drove 9 hours, and Ed drove for 7 hours.
7. Kim worked 8 hours on Monday and less on Tuesday. If she got 7 hours on Tuesday, then
her total was 15 hours.
8. Hinto had 3 nickels and 2 quarters. He had a total of 65 cents in nickels and quarters.
9. The friends on his Facebook account increased by 24, going from 70 to 94.
10. There 4 times as many children at the movie compared to adults. There were 20 adults and
80 children.
Complete the Statement (D) (pages 220 and 221)
1. A cinema in a movie theatre can hold 120 people. Unfortunately that day, only half of the
cinema was full meaning only 60 people are viewing the movie. The theatre charges 6
dollars for child tickets and 10 dollars for adult tickets. The total earnings for that showing
in that cinema were 456 dollars.
2. Attendance for the annual blues concert is normally 850 people. This year, the number
attending grew by 75, meaning that a total of 925 people attended. If individual tickets sold
for 40, the promoters expected to bring in 3000 dollars more than last year just in ticket sales.
3. Taylor works at a hardware store where he makes 15 dollars per hour. Typically he works 38
hours per week, just shy of full time hours, and brings in 570 dollars (before deductions). If
he works an additional 5 hours a week, he will make 75 dollars more (before deductions).
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4. Tanya has 3 teenage children. Montez, the oldest, is 18 years old. Tylena, the youngest, is
13 years old. Kiana, the middle child, is 2 years older than Tylena, making her 15 years old.
Tanya, the mother, is 43 years old.
5. A group of seniors wants to charter buses to go on a trip. They check with the local charter
company and learn that each bus can take 50 people and that the company charges 800
dollars a day for the bus and driver. Since 140 seniors wish to take the trip, then that means
that they will need to charter 3 buses. Unfortunately that means that 10 seats on the buses
will be unused. If the seniors are planning on taking a two day trip, the total cost for
chartering the buses is 4800 dollars (before taxes).
6. A tank initially held 600 litres of water. A pump that removes water from the tank at a rate
of 20 litres per minute is switched on ten minutes. That means that 200 litres have been
removed, leaving 400 litres in the tank. If someone comes after the pump was switched off
and pours 50 litres of water into the tank, the tank will now hold 450 litres of water.
7. Two brothers, Brian and Dave, work for the same company. Brian makes 3 dollars more per
hour than Dave. Since Brian makes 16 dollars per hour, that means that Dave makes 19
dollars per hour. That means that in a 40 hour work week, Brian will make 760 dollars
before deductions, and David will make 640 dollars before deductions.
8. There were 9 times as many people at the Rolling Bones concert than at the Tragically
Flipped concert. If 7000 people were at the Flip concert, then that means that 63 000 people
were at the Bones concert. The Bones charged 100 dollars per ticket, while the Flip only
charged 40 dollars per ticket. That means that the Bone brought in 6 020 000 dollars more in
ticket sales for their concert.
Not Enough Information is Provided (page 222)
1.
We need to know if all of the marbles are the same size and made of the same material (i.e.
they all have the same mass). We also need to know how many marbles are in the jar.
2.
We need to know the cans are sold individually or in a six-pack. If they are individual cans,
we need to know the cost of each can (They would all be of the same cost if they are the
same brand and same size). If they are in a six-pack, then we just look at the price on the
six-pack.
3.
We need to know how many friends helped out. We also need to know if they all worked
the same number of hours and did similar tasks. If so then each would be entitled to an
equal share of the $600.
4.
We need to know two things. How much water did the container initially hold? For what
length of time was the water being removed?
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5.
We need to know how many hotdogs, pops, and chips are sold. We also need to know the
fire fall's cost for those items; we can't figure out how much is made (i.e. profits) without
considering the initial costs.
6.
We need to know how many children Jun has and how many days each is planning on
attending the camp.
Word Problems with Too Much Information (pages 223 and 224)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Correct Solution
(a) 10 − 5 = 5
(c) 8 × 2 = 16
(b) 24 + 16 = 40
(d) 8 − 6 = 2
(c) 20 ÷ 5 = 4
(d) 6 × 8 = 48
(b) 2 + 6 = 8
(a) 30 − 12 = 18
(b) 90 ÷ 3 = 30
(d) 6 × 4 = 24
Extra Information
notebook costs $3
bag of potato chips costs $3
carry-on bag weighs 10 kg
4 children
10 pencils
7 men
3 hours
$23 sweatshirt
run 15 km
cycles at 20 kilometres per hour
World Problems (A) (pages 227 and 228)
1.
3.
5.
7.
9.
11.
13.
15.
17.
19.
5 + 6 = 11
5 × 7 = 35
35 - 13 = 12
8 + 7 = 15
10 + 9 = 19
28 - 15 = 13
5 × 9 = 45
60 + 39 = 99
17 - 4 = 13
480 ÷ 6 = 80
2.
4.
6.
8.
10.
12.
14.
16.
18.
20.
20 - 6 = 14
28 ÷ 4 = 7
27 ÷ 3 = 9
3 × 40 = 120
900 ÷ 3 = 300
19 - 12 = 7
100 ÷ 10 = 10
60 × 9 = 540
12 + 14 = 26
4 × 10 = 40
2.
4.
6.
8.
10.
12.
14.
237 + 19 = 256
27 × 15 = 405
1920 - 650 = 1270
8454 + 7461 = 15 915
1365 ÷ 3 = 455
257 - 118 = 139
1260 ÷ 9 = 140
World Problems (B) (pages 229 to 231)
1.
3.
5.
7.
9.
11.
13.
15.
3640 - 1780 = 1860
498 ÷ 6 = 83
24 + 19 + 5 = 48
207 ÷ 9 = 23
85 × 16 = 1360
89 + 47 = 136
196 - 68 = 128
31 × 25 = 775
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Same Numbers, Similar Context, Different Math (A) (pages 232 and 233)
1. (a)
(b)
(c)
(d)
5 × 20 =
100 dollars
20 ÷ 5 =
4 dollars
20 − 5 =
15 women
20 + 5 =
25 women
2. (a)
(b)
(c)
(d)
40 ÷ 8 =
5 times as heavy
40 − 8 =
32 kg
40 × 8 =
320 kg
40 + 8 =
48 kg
3. (a)
(b)
(c)
(d)
30 − 5 =
25 hours
30 + 5 =
35 hours
30 ÷ 5 =
6 times
30 × 5 =
150 hours
4. (a)
(b)
(c)
(d)
50 + 10 =
60 people
10 × 50 =
500 learners
50 − 10 =
40 students
50 ÷ 10 =
5 vans
Same Numbers, Similar Context, Different Math (B) (pages 234 and 235)
1. (a)
(c)
(e)
(g)
40 + 10 = $50
40 - 10 = $30
(2 × 40) + (3 × 10) = $110
(3 × 40) - (9 × 10) = $30
(b) 10 × 40 = $400
(d) 40 ÷ 10 = $4
(f) 40 - (2 × 10) = $20
2. (a)
(c)
(e)
(g)
20 × 2 = 40 pills
20 - 2 = 18 pills
20 - (3 × 2) = 14 pills
20 ÷ (2 × 2) = 5 days
(b) 20 ÷ 2 = 10 days
(d) 20 + 2 = 22 pills
(f) (3 × 20) + (4 × 2) = 68 pills
3. (a)
(c)
(e)
(g)
5 + 30 = 35 employees
30 - 5 = 25 employees
3 × (5 + 30) = 105 employees
30 + (4 × 5) = 50 employees
(b) 5 × 30 = 150 employees
(d) 30 ÷ 5 = 6 employees
(f) (5 × 20) + (30 × 10) = $400
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More than One Question (pages 236 and 237)
1. (a)
(b)
(c)
(d)
5+3 =
8 litres
5−3 =
2 litres
12 − 8 =
4 litres
12 − 5 =
7 litres
2. (a)
(b)
(c)
(d)
(e)
3× 2 =
6 sheep
14 − 2 =
12
14 ÷ 2 =
7 pigs
2 + 14 + 6 + 7 =
29 animals
2 + 14 + 6 =
22 animals or 29 − 7 =
22 animals
3. (a)
(b)
(c)
(d)
(e)
4×3 =
12 motorcycles
3 + 12 =
15 vehicles
5× 3 =
15 tires
12 − 3 =
9
12 ÷ 4 =
3 times
Food Chart A (page 238)
1.
3.
5.
7.
3 + 2 = 5 grams
2 × 20 = 40 grams
1 + 12 + 10 = 23 grams
24 - (5 × 3) = 9 grams
2. 8 - 5 = 3 grams
4. 16 ÷ 2 = 8 times larger
6. (2 × 18) ÷ 9 = 4 times larger
Food Chart B (pages 239 and 240)
1.
3.
5.
7.
9.
2. (2 × 15) ÷ 10 = 3 times larger
4. 4 × 129 = 516 milligrams
6. (3 × 233) - (2 × 75) = 549 milligrams
8. (2 × 140) ÷ (4 × 10) = 7 times larger
10. (60 + (2 × 20)) ÷ (15 + 35) = 2 times larger
40 + 75 = 115 kilocalories
476 - 140 = 336 milligrams
155 + 95 + 155 = 405 kilocalories
(2 × 60) + (3 × 35) + 20 = 245 grams
(220 + 40) - (165 + 65) = 30 grams
Keeping Track of New Stock (A) (page 241)
1.
3.
5.
7.
40 ÷ 10 = 4 times larger
18 + 4 = 22 units
30 - 10 = 20 units
10 + 4 + 15 = 29 units
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4.
6.
8.
323
30 - 20 = 10 units
3 × 20 = 60 units
20 ÷ 4 = 5 times larger
15 ÷ 5 = 3 units
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C. D. Pilmer
Keeping Track of the New Stock (B) (page 242)
1.
3.
5.
7.
(2 × 40) + (2 × 10) = 100 units
(40 - 34) - (20 - 18) = 4 units
(34 + 13) - 24 = 23 units
(10 - 0) ÷ (20 - 15) = 2 times larger
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4.
6.
8.
324
(34 - 22) + (18 - 15) = 9 units
(24 - 10) - (13 - 4) = 5 units
3 × (30 - 24) = 18 units
(34 - 22) ÷ (40 - 34) = 2 times larger
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C. D. Pilmer