Exploring
Arrays and Facts
Explorations
Objectives To provide opportunities for children to estimate
the
t number of dots in a large array, solve a problem involving
factors of whole numbers, and practice multiplication facts.
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Assessment
Management
Common
Core State
Standards
Teaching the Lesson
Key Concepts and Skills
• Generate multiplication fact families. [Operations and Computation Goal 3]
• Use multiplication facts to estimate the number of dots in a large array. [Operations and Computation Goal 3]
• Use arrays to solve a multidigit multiplication problem. [Operations and Computation Goal 4]
• Explore the inverse relationship between multiplication and division. [Patterns, Functions, and Algebra Goal 2]
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Ongoing Learning & Practice
1 2
4 3
Playing Beat the Calculator (Multiplication)
Math Journal 1, p. 156
Student Reference Book, p. 279
per group: calculator
Children practice multiplication skills.
Ongoing Assessment:
Recognizing Student Achievement
Use an Exit Slip (Math Masters, page 398). [Operations and Computation Goal 3]
Key Activities
Math Boxes 4 8
Exploration A: Children estimate the number of dots in an array. Then they
find the exact number using calculators.
Math Journal 1, p. 94
Exploration B: Children find the smallest number of chairs that can be
arranged in equal rows to satisfy certain conditions.
Math Masters, p. 108
Exploration C: Children use a fact platter to generate sets of multiplication
and division facts within fact families.
Ongoing Assessment: Informing Instruction See page 285.
Home Link 4 8
Differentiation Options
ENRICHMENT
Materials
Exploring with Arrays
Home Link 47
per partnership: counters, paper
Exploration A: Per group:
Math Journal 1, p. 92
22 square pattern blocks calculator
Exploration B: Per group:
Math Journal 1, p. 93
Math Masters, p. 106
at least 35 counters
EXTRA PRACTICE
Minute Math +
Minute Math +®, p. 89
Exploration C: Per partnership:
Math Masters, p. 107
fact platters
Advance Preparation
Tape fact platters (see Planning Ahead, page 282) to the board. You may also draw fact platters on the board.
Teacher’s Reference Manual, Grades 1–3 pp. 92, 186–189, 196–199
Lesson 4 8
283
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP7
Content Standards
Getting Started
3.OA.1, 3.OA.7, 3.OA.9, 3.MD.6, 3.MD.7a
Mental Math and Reflexes
Math Message
Pose multiplication facts. Suggestions:
How many dots are in this array? Be prepared to
explain how you got your answer.
1 × 5 5 2 × 5 10 10 × 5 50
2 × 4 8 3 × 4 12 5 × 4 20
3 × 6 18 4 × 6 24 6 × 6 36
PROBLEM
P
PR
PRO
RO
ROBL
BLEM
BLE
B
BL
LEM
LE
L
EM
SO
SOL
S
SOLVING
OL
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LV
VING
VI
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IN
NG
G
Home Link 4 7 Follow-Up
Before turning in their work, have children compare
their answers with a partner.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
There are 18 dots in the array. Expect that few children will count
the dots. Most children will probably use the strategy from Lesson
4-2, finding the number of rows 3 rows and the number of dots per
row. 6 dots per row Multiply to find the total number of dots.
3 × 6 = 18
The explorations in this lesson use arrays and multiplication facts.
Exploration A: Estimating
a Number of Dots
Student Page
Date
Exploration A: How Many Dots?
48
First children use any method to estimate the number of dots
in the array on journal page 92.
□ square pattern blocks
Materials
□ calculator
1. Estimate how many dots are
Then they make a second estimate with the help of square
pattern blocks: They count the number of dots that are covered
by one square block.
in the array at the right.
Answers vary.
About
dots
Make another estimate.
Follow these steps.
2. Cover part of the array with a
square pattern block. About how
many dots does one block cover?
Next they tile the array with square blocks and use the total
number of blocks to estimate the number of dots in the whole
array.
25–30 dots
3. Cover the array. Use as many
square pattern blocks as you
can. Do not go over the borders
of the array. How many blocks
did you use?
20
Possible solution strategy: Multiply the number of blocks covering
the array by the number of dots covered by one block. Then, add
the approximate number of dots left uncovered.
blocks
4. Use the information in Steps 2
and 3 to estimate the total
number of dots in the array. About
500–600
dots
The Try This problem asks children to use a calculator to find the
exact number of dots in the array and to explain their strategy.
Possible solution strategy: Multiply the number of rows by the
number of dots per row.
Try This
5. Find the exact number of dots in the array.
Use a calculator to help you. Total number of dots =
504
Follow-Up
Describe how you found the exact number of dots.
Sample answer:
I counted the number of dots in one column and one row. Then
I used my calculator to multiply.
Math Journal 1, p. 92
79-101_EMCS_S_SMJ_G3_U04_576353.indd 92
284
SOLVING
(Math Journal 1, p. 92)
Time
LESSON
SMALL-GROUP
ACTIVITY
Unit 4 Multiplication and Division
1/26/11 9:11 AM
Teaching Master
Name
Date
Ongoing Assessment: Informing Instruction
Watch for children who do not have a strategy for solving the Try This problem.
Remind them of the strategy they used to find the total number of dots in the
Math Message.
Time
Exploration B: Setting Up Chairs
LESSON
48
䉬
Materials
䊐 Math Journal 1, page 93
䊐 35 pennies or other counters
Problem
A teacher was setting up the chairs in his classroom for
parent night. He wanted them to be in rows with the
same number of chairs in each row. Use the clues to find
out how many chairs were in the room.
Clues
1. When the teacher tried to set up the chairs in rows of 2, there was
1 leftover chair.
Exploration B: Setting Up Chairs
(Math Journal 1, p. 93; Math Masters, p. 106)
SMALL-GROUP
ACTIVITY
PROBLEM
PRO
PR
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
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LV
VIN
ING
2. He also had 1 leftover chair when he tried to set them up in rows of 3.
3. One of his students suggested that he try rows of 4. There was still
1 leftover chair.
4. Finally, he tried rows of 5. This worked fine. There were no leftover
chairs.
5. The number of chairs in the room was the smallest possible number
that fits all of the clues.
Children solve the following problem using counters.
What to Do
1. Experiment with counters to build rows. Hint: Could the answer be
A teacher is setting up chairs in equal rows for parent night. If he
tries to arrange them in equal rows of 2, 3, or 4, there is always
1 leftover chair. If he tries to arrange them in equal rows of 5, he
can make equal rows. What is the smallest number of chairs that
can be arranged in equal rows of 5 but not in equal rows of 2, 3, or
4? 25 chairs After children have solved the problem, they record
their work on journal page 93.
Exploration C: Practicing
5 chairs? 10 chairs (2 rows of 5 chairs each)? 15 chairs (3 rows of
5 chairs each)?
2. When you have found the answer, record it on journal page 93.
Next, circle dots to show the chairs in rows of 2, 3, and 4, each with
1 chair left over. Then show the chairs in rows of 5.
Math Masters, p. 106
PARTNER
ACTIVITY
Multiplication and Division
Facts with a Fact Platter
Adjusting the Activity
Have children find another number
of chairs that fits all of the clues, except for
being the smallest number. 85 chairs
(Math Masters, p. 107)
AUDITORY
As partners work together at the board,
KINESTHETIC
TACTILE
VISUAL
One partner writes any factor from 1 through 10 in the center
of the fact platter.
The other partner multiplies the factor in the center by each
factor on the platter and writes the products on the board
around the rim of the platter.
Student Page
Date
Time
LESSON
Partners help and check each other.
4 8
䉬
Exploration B: Setting Up Chairs
1. Record the answer to the problem about setting up chairs from Math Masters,
They also write at least one division fact for each multiplication
fact on the board.
Then they erase the board, trade jobs, and keep going until
each person has had several turns picking a factor for the
center of the platter.
page 106.
There were
25
chairs in the room.
2. Circle dots below to show how you set up the chairs for each of the clues.
Rows of 2
Rows of 3
Rows of 4
1 left over
1 left over
1 left over
Rows of 5
Adjusting the Activity
Have children use numbers 1–5 as the center factor. Have children use
factors larger than 5.
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
V I S U A L
0 left over
Math Journal 1, p. 93
Lesson 4 8
285
Teaching Master
Name
Date
Time
Exploration C: Fact Platters
LESSON
48
䉬
䊐 fact platter
Materials
䊐 chalk
䊐 eraser
42
Work with a partner at the board.
10
1. Take turns. On your turn, pick a number from
1 to 10. Write it in the middle with a
multiplication sign.
6
it by each number on the circle, writing the
products along the outside of the platter.
7
3
6⫻
2
2. Your partner takes that number and multiplies
4
8
3. Check the products together. Make corrections.
5
9
1
30
4. Then, each of you writes one division fact for
every multiplication fact around the circle.
42 ÷ 6 = 7
A fact platter
5. Check each other’s work.
6. Erase the board. Trade roles. Start again.
7. Keep going until each of you has had several turns picking a
number for the center.
2 Ongoing Learning & Practice
Playing Beat the Calculator
Math Masters, p. 107
SMALL-GROUP
ACTIVITY
(Student Reference Book, p. 279;
Math Journal 1, p. 156)
Children develop automaticity with multiplication facts by playing
Beat the Calculator. Have children record a check mark on the
Fact Power Table (journal page 156) for each fact that they solve
correctly when playing the role of the Brain. For Fact Power Table
directions, see Lesson 4-5. For game directions, see page 279 in the
Student Reference Book.
Ongoing Assessment:
Recognizing Student Achievement
Student Page
Date
Math Boxes
4 8
䉬
1.
Make equal groups.
30 days make
2
with
4
6
2
Write three things that you think
are very likely to happen.
Sample answer: I will
days left over.
56 pennies make
with
2.
weeks
home after school. School
quarters
will start on time tomorrow.
pennies left over.
Fill in the circle for the best answer.
The perimeter of the square is
B. 16 cm
4 cm
C. 8 cm
4.
Complete the Fact Triangle. Write
the fact family.
3 ⫻
7 ⫻
21 ⫼
21 ⫼
A. 12 cm
4 cm
D. 20 cm
7
3
3
7
21
⫽ 21
⫽ 7
⫽ 3
•
21
⫽
5. 56,937
Which digit is in the tens place?
3
6
3,065
23,605
Which digit is in the hundreds
32,605
9
place?
50,007
Which digit is in the ones
place?
⬍
⬎
⬎
⬍
3,605
20,365
23,605
50,700
7
18 19
94
Math Journal 1, p. 94
286
7
Use ⬎, ⬍, or ⫽.
Which digit is in the thousands
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 94)
55
6.
Math Boxes 4 8
⫻, ⫼
3
150 151
place?
[Operations and Computation Goal 3]
have lunch today. I will go
73
3.
Use an Exit Slip (Math Masters, page 398) to assess children’s progress toward
demonstrating automaticity with multiplication facts through 10 × 10. Children
record the facts from the Fact Power Table for which they have earned at least
one check mark. Children are making adequate progress if they record at least
5 facts from the unshaded portion of the Fact Power Table. Some children may
record more than 5 facts.
Time
LESSON
Exit Slip
Unit 4 Multiplication and Division
13 20
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 4-10. The skills in Problems 5 and
6 preview Unit 5 content.
Writing/Reasoning Have children write an answer to the
following: For Problem 1, show how you found one of your
answers. Explain your thinking. Sample answer: I know
there are 7 days in 1 week. I made an array with 7 dots in each
row. I made 4 rows. That was 28 days. 2 days were left over.
Home Link 4 8
Home Link Master
INDEPENDENT
ACTIVITY
(Math Masters, p. 108)
Name
HOME LINK
40 08
䉬
Family
Note
Home Connection Children write number models and
find the areas of rectangles, using grids and unit squares.
Date
Time
Arrays and Areas
Your child uses the same procedure for finding the area of a rectangle that is used for finding
the number of dots in an array. For Problem 3 it does not matter whether your child draws
an array with 4 rows of 8 dots or 8 rows of 4 dots. What is important is that the array
has two sides that have 4 dots and two sides that have 8 dots. The same concept is true
for Problem 4.
64 65
Please return this Home Link to school tomorrow.
Make a dot inside each small square in one row. Then fill in the blanks.
1. Number of rows:
7
Number of squares in a row:
⫽
Number of squares in a row:
7
Area:
3 Differentiation Options
Number model:
Exploring with Arrays
PARTNER
ACTIVITY
42
⫻
35
square units
2. Number of rows:
Area:
ENRICHMENT
35
7
5
5
Number model:
6
6
⫻
7
⫽
42
square units
Mark the dots to show each array. Then fill in the blanks.
3. Make a 4-by-8 array.
Number model:
4
⫻
8
⫽
32
⫻
5
⫽
45
15–30 Min
To further explore arrays, have children use arrays to prove that
the sum of two odd numbers is always an even number. Pose the
following problem to the children:
Use arrays to show that the sum of two odd numbers is always an
even number. Explain. Show your work on paper. Use counters
to help.
4. Make a 9-by-5 array.
Number model:
9
Math Masters, p. 108
If children have difficulty getting started, you might suggest that
they make arrays with 2 rows.
EXTRA PRACTICE
Minute Math+
SMALL-GROUP
ACTIVITY
5–15 Min
To offer children more experience with multiplication and division,
see the following page in Minute Math+:
Number Stories: page 89.
Planning Ahead
For Part 1 in Lesson 4-9, collect a variety of maps.
For the Readiness activity in Lesson 4-9, have children bring in
toy animals.
Lesson 4 8
287