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TABLE OF CONTENTS
Letter to the Student
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
New Jersey
Core Curriculum
Content Standards
Standard 4.1: Number and Numerical Operations
Lesson
Lesson
Lesson
Lesson
Lesson
Factors and Multiples . . . . . . . . . . . . . . . . . . . . . . . . . .8
Prime and Composite Numbers . . . . . . . . . . . . . . . . .12
Interpreting Fractions . . . . . . . . . . . . . . . . . . . . . . . . . .15
Fraction and Decimal Equivalences . . . . . . . . . . . . . .18
Comparing and Ordering Fractions and
Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
Lesson 6: Adding and Subtracting Decimals . . . . . . . . . . . . . . .28
Lesson 7: Adding and Subtracting Fractions with Unlike
Denominators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32
Lesson 8: Using Mental Arithmetic to Add or Subtract
Fractions and Decimals . . . . . . . . . . . . . . . . . . . . . . . .37
Progress Check for Lessons 1–8 . . . . . . . . . . . . . . . . . . . . . . . . . . .39
Lesson
Lesson
Lesson
Lesson
1:
2:
3:
4:
5:
9:
10:
11:
12:
Properties of Numbers . . . . . . . . . . . . . . . . . . . . . . . .45
Dividing by Two-Digit Divisors . . . . . . . . . . . . . . . . . .48
Computing with Money . . . . . . . . . . . . . . . . . . . . . . . .52
Solving Problems with Whole Number
Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58
Lesson 13: Recognizing Reasonable Answers . . . . . . . . . . . . . . .62
Lesson 14: Using Estimation to Solve Problems . . . . . . . . . . . . .65
4.1.5.A.5
4.1.5.A.5
4.1.5.A.1
4.1.5.A.4
4.1.5.A.3, 4.1.5.A.6
4.1.5.B.2
4.1.5.B.2
4.1.5.B.2, 4.1.5.B.4
4.1.5.B.6
4.1.5.B.3
4.1.5.A.2
4.1.5.B.1
4.1.5.B.5
4.1.5.C.1, 4.1.5.C.2,
4.1.5.C.3, 4.1.5.C.4
Progress Check for Lessons 9–14 . . . . . . . . . . . . . . . . . . . . . . . . . .70
Standard 4.2: Geometry and Measurement
Lesson 15: Properties of Lines, Segments, Rays, and Angles . . .78
Lesson
Lesson
Lesson
Lesson
Lesson
Lesson
Lesson
Lesson
Lesson
16:
17:
18:
19:
20:
21:
22:
23:
24:
Two-Dimensional Figures . . . . . . . . . . . . . . . . . . . . . .85
Special Triangles and Quadrilaterals . . . . . . . . . . . . .91
Similar Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96
Congruent Triangles . . . . . . . . . . . . . . . . . . . . . . . . . .100
Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104
Translations, Rotations, and Reflections . . . . . . . . .109
Plotting Ordered Pairs . . . . . . . . . . . . . . . . . . . . . . . .116
Measurement in the Metric System . . . . . . . . . . . . .121
Measurement in the Customary System . . . . . . . . .124
NOTICE: Duplicating any part of this book is forbidden by law .
4.2.5.A.1, 4.2.5.D.1,
4.2.5.E.1
4.2.5.A.2
4.2.5.A.2
4.2.5.A.3
4.2.5.A.4
4.2.5.A.4
4.2.5.B.1, 4.2.5.B.2
4.2.5.C.1
4.2.5.D.2, 4.2.5.D.4
4.2.5.D.2, 4.2.5.D.4
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New Jersey ASK Coach, Gold Edition, Mathematics, Grade 5
Lesson 25: Equivalent Measures in the Metric and
Customary Systems . . . . . . . . . . . . . . . . . . . . . . . . . .126
Lesson 26: Perimeter and Area . . . . . . . . . . . . . . . . . . . . . . . . . .129
4.2.5.D.3
4.2.5.D.1, 4.2.5.E.2
4.2.5.E.3, 4.2.5.E.4
Progress Check for Lessons 15–26 . . . . . . . . . . . . . . . . . . . . . . . .133
Standard 4.3: Patterns and Algebra
Lesson
Lesson
Lesson
Lesson
27:
28:
29:
30:
Extending Number Patterns . . . . . . . . . . . . . . . . . . .144
One- and Two-Operation Function Tables . . . . . . . . .149
Graphing Points That Satisfy Functions . . . . . . . . . .154
Translating Word Problems into Expressions
and Sentences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .160
Lesson 31: Using Graphs to Make Predictions . . . . . . . . . . . . . .167
Lesson 32: Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .171
Progress Check for Lessons 27–32 . . . . . . . . . . . . . . . . . . . . . . . .175
4.3.5.A.1
4.3.5.B.1
4.3.5.B.2
4.3.5.C.1
4.3.5.C.2
4.3.5.D.1
Standard 4.4: Data Analysis, Probability, and Discrete
Mathematics
Lesson 33: Interpreting Information from Data Displays . . . . .184
Lesson 34: Range, Median, and Mean . . . . . . . . . . . . . . . . . . . .192
Lesson 35: Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .198
Lesson 36: Outcomes of Several Independent Events . . . . . . .203
Lesson 37: Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .208
Progress Check for Lessons 33–37 . . . . . . . . . . . . . . . . . . . . . . . .212
4.4.5.A.1, 4.4.5.A.2
4.4.5.A.3
4.4.5.A.2, 4.4.5.A.3
4.4.5.B.1, 4.4.5.B.2,
4.4.5.B.3
4.4.5.B.3, 4.4.5.C.1
4.4.5.C.2
Standard 4.5: Mathematical Processes
Lesson 38: Understanding the Language of Problems . . . . . . .220
Lesson 39: Strategies for Problem Solving . . . . . . . . . . . . . . . . .224
Lesson 40: Estimating to Solve Problems . . . . . . . . . . . . . . . . . .228
Lesson 41: Using Different Ways to Solve Problems . . . . . . . . .231
Lesson 42: Missing Information and Too Much Information . . .234
Extended Constructed Responses . . . . . . . . . . . . . . . . . . . . . . . . .238
4.3.5.C.1, 4.5
4.3.5.C.1, 4.5
4.1.5.C.1, 4.5
4.1.5.B.1, 4.5
4.5
NJ ASK Grade 5 Practice Test 1 . . . . . . . . . . . . . . . . . . . . . . . . . . .241
NJ ASK Grade 5 Practice Test 2 . . . . . . . . . . . . . . . . . . . . . . . . . . .257
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .273
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LETTER TO THE STUDENT
This book will help you prepare for the New Jersey Grade 5 ASK (Assessment of
Skills and Knowledge) Test.
The Coach will:
•
Show you what math questions on the ASK Test are like
•
Tell you what you need to know to do well on the test
•
Give you practice on the kind of math that will be on the test
The ASK Test in Math has many multiple-choice questions. They are like the ones you
will work with in this book. After each question there are four possible answers. Only
one is correct. The others are wrong. You must mark the one correct answer after
each question.
The ASK Test also has short constructed and extended constructed responses.
On these questions, you will have to write a short sentence, or draw a diagram,
or give a full explanation of why something works.
Here are some tips that will help when you work in this book and take the test:
•
Read each question carefully.
•
Work as carefully as you can.
•
Make sure you answer the question that is asked.
•
Ask yourself if the answer makes sense.
•
On multiple-choice questions, if you cannot decide on the answer, make the
best guess you can. There is no penalty for guessing.
•
Answer as many questions as you can.
•
On short constructed and extended constructed responses, answer the
questions as completely as you can.
NOTICE: Duplicating any part of this book is forbidden by law .
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Standard 4.1
4.1.5.A.5:
1
Number and Numerical Operations
Develop and apply number theory concepts in problem solving situations.
• Primes, factors, multiples
FACTORS AND MULTIPLES
Factors
The factors of 6 are the numbers you multiply to get 6.
For example, 2 ⫻ 3 ⫽ 6, so 2 and 3 are factors of 6.
The complete set of factors of 6 is: 1, 2, 3, and 6.
The factors of a number are those numbers that divide the number evenly.
EXAMPLE 1
What are the factors of 15?
STRATEGY:
Find the pairs of numbers that you multiply to get 15.
STEP 1:
Always start with 1 and the number itself.
Since 1 ⫻ 15 ⫽ 15, 1 and 15 are factors of 15.
STEP 2:
What other pairs of numbers do you multiply to get 15?
The other two numbers are 3 and 5, since 3 ⫻ 5 ⫽ 15.
SOLUTION:
8
The factors of 15 are 1, 3, 5, and 15.
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Lesson 1: Factors and Multiples
Multiples
The multiples of 7 are the same as the numbers you get by counting by 7s:
7, 14, 21, 28, 35, and so forth.
These are the same as 1 ⫻ 7, 2 ⫻ 7, 3 ⫻ 7, 4 ⫻ 7, 5 ⫻ 7, and so forth.
The multiples of a number are the numbers you get when you multiply that number
by the counting numbers 1, 2, 3, 4, and so on.
EXAMPLE 2
What are the first five multiples of 8?
STRATEGY:
Multiply 8 by the first five counting numbers.
Multiply 8 by 1, then 2, 3, 4, and 5.
1 ⫻ 8 ⫽ 8; 2 ⫻ 8 ⫽ 16; 3 ⫻ 8 ⫽ 24; 4 ⫻ 8 ⫽ 32; 5 ⫻ 8 ⫽ 40
SOLUTION:
The first five multiples of 8 are: 8, 16, 24, 32, and 40.
NOTICE: Duplicating any part of this book is forbidden by law .
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SAMPLE TEST QUESTIONS
1.
What ar e all the factors of 24?
5.
1, 2, 6, 12, 24
8
1, 2, 4, 6, 12, 24
16
1, 2, 3, 4, 6, 8, 12, 24
24
1, 2, 3, 4, 6, 8, 12
2.
What ar e the first thr ee multiples
of 9?
28
6.
________________
3.
Which number is a multiple of
4 and 7?
Which of the following is a factor
of 21?
Mrs. Stark can evenly divide her
class into gr oups of 3 students or
gr oups of 4 students. Which of the
following is a possible size of her
class?
22
2
23
3
24
5
25
6
7.
4.
Which of the following is NOT a
factor of 6?
What is the smallest number that is a
multiple of both 3 and 5?
________________
2
3
4
6
10
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Lesson 1: Factors and Multiples
8.
How many factors does 16 have?
________________
9.
Noel has five favorite numbers. His
favorite numbers ar e the first five
multiples of the number 6. What ar
Noel’s favorite numbers?
e
6, 7, 8, 9, 10
6, 8, 10, 12, 14
6, 12, 18, 24, 30
12, 18, 24, 30, 36
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Standard 4.1
4.1.5.A.5:
2
Number and Numerical Operations
Develop and apply number theory concepts in problem solving situations.
• Primes, factors, multiples
PRIME AND COMPOSITE
NUMBERS
Prime Numbers
A number is a prime number if its only factors are 1 and itself.
For example, 7 is a prime number because its only factors are 1 and 7.
1⫻7⫽7
Composite Numbers
A number that is not a prime number is called a composite number.
8 is a composite number because it has more than two factors.
The factors of 8 are 1, 2, 4, and 8.
1⫻8⫽8
2⫻4⫽8
Note: 2 is the only even prime number.
1 is special. It is neither a prime number nor a composite number.
12
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Lesson 2: Prime and Composite Numbers
EXAMPLE 1
Is 17 a prime number?
STRATEGY:
Use the definition of a prime number.
Find the factors of 17.
1 ⫻ 17 ⫽ 17
Check other numbers between 1 and 17.
Can any of them be multiplied to get 17? No.
The only factors of 17 are 1 and 17.
SOLUTION:
Yes. 17 is a prime number.
EXAMPLE 2
Which of the following is a composite number?
7
11
27
13
STRATEGY:
Test each number.
What are the factors of each number?
7: The only factors of 7 are 1 and 7. It is a prime number.
11: The only factors of 11 are 1 and 11. It is a prime number.
27: The factors of 27 are 1, 3, 9, and 27. It has more than two factors.
It is a composite number.
13: The only factors of 13 are 1 and 13. It is a prime number.
SOLUTION:
27 is a composite number, so C is the correct answer.
NOTICE: Duplicating any part of this book is forbidden by law .
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SAMPLE TEST QUESTIONS
1.
Which of the following is a prime
number?
5.
22
What ar e all the prime numbers
between 30 and 40?
________________
21
20
6.
19
How many prime numbers ar
between 60 and 70?
e ther e
0
2.
Which number is prime?
1
72
2
3
71
70
69
3.
Which of the following is a
composite number?
7.
What ar e all the prime numbers
between 70 and 80?
________________
17
19
23
33
4.
Look at the pairs of numbers. Which
pair includes a prime number and a
composite number?
21, 27
41, 43
5, 46
23, 29
14
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Standard 4.1
4.1.5.A.1:
3
Number and Numerical Operations
Use real-life experiences, physical materials, and technology to construct
meanings for numbers (unless otherwise noted, all indicators for grade 5 pertain
to these sets of numbers as well).
• All fractions as part of a whole, as subset of a set, as a location on a number
line, and as divisions of whole numbers
INTERPRETING FRACTIONS
One way to interpret a fraction is to think of the fraction as part of a whole.
EXAMPLE 1
What fraction of this circle is shaded?
STRATEGY:
Compare the part that is shaded to the whole figure.
STEP 1:
Identify the whole.
The whole is the circle (8 equal regions).
STEP 2:
Identify the shaded part.
The shaded part is made up of 3 regions.
STEP 3:
Write a fraction comparing the part to the whole.
3
8
SOLUTION:
ᎏ3ᎏ
8
➝ shaded part (3 regions)
➝ whole circle (8 regions)
of the circle is shaded.
NOTICE: Duplicating any part of this book is forbidden by law .
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