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576 Unit 7 Multiplication and Division
Advance Preparation
Teacher’s Reference Manual, Grades 1–3 pp. 204, 205
Patterns in ProductsObjective To review square-number facts, multiplication, and
division patterns.d
��������
Key Concepts and Skills• Identify factors, products, square numbers,
and patterns in the Multiplication/Division
Facts Table.
[Operations and Computation Goal 3]
• Use the Multiplication/Division Facts
Table to generate fact families.
[Operations and Computation Goal 3]
• Use arrays to find square products.
[Operations and Computation Goal 6]
• Use the turn-around rule (Commutative
Property of Multiplication) to generate
multiplication facts.
[Patterns, Functions, and Algebra Goal 4]
Key ActivitiesChildren identify patterns in a sequence of
square numbers and in the Multiplication/
Division Facts Table.
Ongoing Assessment: Recognizing Student Achievement Use journal page 157. [Operations and Computation Goal 3]
Key Vocabularyproduct � square product � square number �
factor
MaterialsMath Journal 2, p. 157
Student Reference Book, p. 52
slate
Playing Name That Number Student Reference Book, pp. 299
and 300
per partnership: 4 each of number
cards 0–10 and 1 each of number
cards 11–20 (from the Everything Math
Deck, if available).
Children practice finding equivalent
names for a number.
Math Boxes 7�1Math Journal 2, p. 158
Children practice and maintain skills
through Math Box problems.
Home Link 7�1Math Masters, p. 206
Children practice and maintain skills
through Home Link activities.
READINESS
Building Square and Rectangular ArraysMath Masters, p. 207
cm cubes
Children build square and rectangular arrays
and look for patterns.
ENRICHMENTExploring a Pattern in a Sequence of ProductsMath Masters, p. 208
Student Reference Book, pp. 198 and 199
Children look for patterns in a sequence
of rectangular arrays and in the products
they represent.
ELL SUPPORT
Building a Math Word BankDifferentiation Handbook, p. 132
Children add the terms product and factor
to their Math Word Banks.
Teaching the Lesson Ongoing Learning & Practice
132
4
Differentiation Options
eToolkitePresentations Interactive Teacher’s
Lesson Guide
Algorithms Practice
EM FactsWorkshop Game™
AssessmentManagement
Family Letters
CurriculumFocal Points
Common Core State Standards
576_EMCS_T_TLG_G3_U07_L01_576892.indd 576576_EMCS_T_TLG_G3_U07_L01_576892.indd 576 2/23/11 10:53 AM2/23/11 10:53 AM
Adjusting the Activity
Lesson 7�1 577
Array showing square products
2 × 2 = 4
3 × 3 = 9 (add 5 dots)
Part A
Product PatternsLESSON
7�1
Date Time
Math Message
Complete the facts.
1. 1 × 1 = 1
2. 2 × 2 = 4
3. 3 × 3 = 9
4. 4 × 4 = 16
5. 5 × 5 = 25
6. 6 × 6 = 36
7. 7 × 7 = 49
8. 8 × 8 = 64
9. 9 × 9 = 81
10. 10 × 10 = 100
A Two’s Product Pattern
Multiply. Look for patterns.
11. 2 × 2 = 4 12. 2 × 2 × 2 = 8
13. 2 × 2 × 2 × 2 = 16 14. 2 × 2 × 2 × 2 × 2 = 32
15. 2 × 2 × 2 × 2 × 2 × 2 = 64
Use the Two’s Product Pattern for Problems 11 through 15. Multiply.
16. 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128
Part B
�
Try This
EM3MJ2_G3_U07_157-179.indd 157 1/18/11 3:34 PM
Math Journal 2, p. 157
Student Page
Interactive whiteboard-ready
ePresentations are available at
www.everydaymathonline.com to
help you teach the lesson.
1 Teaching the Lesson
� Math Message Follow-Up WHOLE-CLASSDISCUSSION
(Math Journal 2, p. 157)
Review answers to the Math Message problems. Ask children to share how they found answers to facts they have not yet memorized. Some children may suggest a count-by strategy: for example, for 5 × 5, count by 5s five times. 5, 10, 15, 20, 25
Ongoing Assessment: Journal
Page 157 �Recognizing Student Achievement
Use journal page 157, Part A to assess children’s progress toward learning
the multiplication facts. Children are making adequate progress if they use
strategies to correctly complete the facts in Problems 1 through 10. Some
children will demonstrate automaticity with the facts.
[Operations and Computation Goal 3]
Point out how the array diagram highlights the number of dots that are added to an array to obtain the next array. For example, 5 dots are added to the 2-by-2 array to obtain the 3-by-3 array. This shows that 3 × 3 = 9 is 5 more than 2 × 2 = 4.
Draw square arrays for 4, 9, and 16 on the board without lines. Ask
children to imagine what a 5-by-5 array looks like. Have a volunteer draw the
5-by-5 array on the board.
4, 9, and 16 are square products
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
ELL
Getting Started
Math MessageTurn to page 157 in your new journal. Find the products in Problems 1 through 10.
Mental Math and ReflexesHave children practice quick recall of basic multiplication facts. Suggestions:
2 × 4 8 3 × 4 12 3 × 5 15 4 × 5 20
3 × 6 18 4 × 6 24 4 × 7 28 5 × 7 35
3 × 8 24 3 × 9 27 4 × 8 32 6 × 6 36
EM3cuG3TLG2_577-581_U07L01.indd 577EM3cuG3TLG2_577-581_U07L01.indd 577 1/23/11 12:25 PM1/23/11 12:25 PM
578 Unit 7 Multiplication and Division
Adjusting the Activity
NOTE In Lesson 4-6, children used the
Multiplication/Division Facts Table to generate
fact families. They looked for patterns in the
table with teacher guidance. In this activity,
expect that children will be able to describe
many of the patterns on their own.
52 fifty-two
Basic Facts for Multiplication and DivisionSolving problems is easier when you know the basic number facts. Here are some examples of basic multiplication and division facts:
Basic multiplication facts:6 � 4 � 24, 10 � 7 � 70, 1 � 8 � 8, 3 � 9 � 27
Basic division facts:24 � 6 � 4, 70 � 10 � 7, 8 � 1 � 8, 27 � 3 � 9
The facts table shown below is a chart with rows and columns. It can be used to find all of the basic multiplication and division facts.
Operations and Computation
1 2 3
1 2 3 4 5 6 7 8 9 10
1 4 5 6 7 8 9 10
2 2 4 6 8
7 7
8 8
9 9
10 10
32 40 48 56
36 45 54 63 72
40 50 60 70 80 90
3 3 6 9
4 4 8 12 16
5 5 10 15 20 25
6 6 12 18 24 30 36
14 21 28 35 42 49
16 24
18 27
20 30
64
81
100
10 12 14 16 18 20
12 15 18 21 24 27 30
20 24 28 32 36 40
30 35 40 45 50
42 48 54 60
56 63 70
72 80
90
Multiplication/Division Facts Table
Student Reference Book, p. 52
Student Page
With the help of the children, list the number that is added to each square product to obtain the next square product. The numbers below the arrows name the number of dots that are added to each succeeding array. Point out this pattern on the dot array on journal page 157.
1 4 9 16 25 36 49 64 81 100
+3 +5 +7 +9 +11 +13 +15 +17 +19
Write a list of square products on the board. Remind children that these are called square products or square numbers. Draw arrays to illustrate the square numbers. Then ask why these numbers might be called square numbers. Starting with 2 × 2, each product can be represented by a square array.
Find 11 × 11 without using a calculator. Add 21 to continue the pattern
shown above. 11 × 11 = 10 × 10 + 21 = 100 + 21 = 121
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
� Finding Patterns in the SMALL-GROUP ACTIVITY
Multiplication/DivisionFacts Table(Student Reference Book, p. 52)
Algebraic Thinking Divide the class into small groups and ask children to turn to the Multiplication/Division Facts Table on page 52 in their Student Reference Book. Remind them that the shaded numbers across the top and down the left side of the table are called factors and that the rest of the numbers are the products of the factors. Ask each group to look for patterns in the table and record them on a sheet of paper. After a few minutes, bring the class together to share the patterns they found. To support English language learners, write the patterns on the board as the children describe them.
Examples:
● The products in the row for a factor are counts by that factor. For example, the products in the 3s row are counts by 3: 3, 6, 9, 12. The same is true of the products in the column for a factor.
● The numbers on the diagonal (from the upper-left to the lower-right corner) are square numbers.
● The square numbers divide the table into two parts that are mirror images of each other.
● All products in even-factor rows and columns are even numbers.
● Products in odd-factor rows and columns alternate between even and odd numbers.
PROBLEMBBBBBBBBBBOOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEEEEEMMMBLEBLBLELBLEBLELLLLBLEBLEBLEBLEBLEBLEEEMMMMMMMMMMMMOOOOOOOOOOBBBBBBLBLBLBLBBLBLLLLLPROPROPROPROPROPROPROPROPROPROPRPPRPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROROOROROROROOPPPPPP MMMMMMMMMMMMMMMMMMMMMMMEEEEEEEEEEEEEEEEELELEEEEEEEEEEELLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRPROBLEMSOLVING
BBBBBBBBBBBBBBBBBBBB EEELEMMMMMMMMOOOOOOOOOBBBLBLBLBLBBLBROOOOROROROROROROROROROO LELELELEEEEEELEEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGLLLLLLLLLLLLLVINVINVINVINVINVINNNVINVINVINNVINVINVINVINV GGGGGGGGGGGOLOOOLOOOLOLOO VINVINVINVLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINVINVINNGGGGGGGGGGGOOOLOLOLOLOLOLOO VVVVLLLLLLLLLLVVVVVVVVVOSOSOOSOSOSOSOSOSOSOOSOSOSOSOOOOSOOSOSOSOSOSSOOSOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVVVLLLLLVVVVVVVVLLVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSSOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIIISOLVING
ELL
EM3cuG3TLG2_577-581_U07L01.indd 578EM3cuG3TLG2_577-581_U07L01.indd 578 1/19/11 6:28 PM1/19/11 6:28 PM
5. Complete the Fact Triangle.
Write the fact family.
6 × 6 = 36
36 ÷ 6 = 6
3. Draw and label three parallel line
segments. Draw and label a line that
intersects all three line segments.
Date Time
2. Draw an array with 25 Xs arranged
in 5 rows.
How many Xs in each row? 5 Write a number model for the array.
5 × 5 = 25
6. Divide the rectangle into
4 equal parts.
1. This is a picture of a triangular
pyramid. This shape has
4 faces
6 edges
4 vertices
Math BoxesLESSON
7�1
116 64 65
55
N O
H
IK
J L
M
Sample answer:
99 100
4. Fill in the circle next to the correct
answer.
A 251 B 1,751
C 1,795 D 1,805
777
+ 1,028
57–59
6 6
36
×,÷
Sample answer:
EM3MJ2_G3_U07_157-179.indd 158 1/18/11 3:34 PM
Math Journal 2, p. 158
Student Page
Name Date Time
Which Way Out?HOME LINK
7�1
Today your child explored patterns in square products, such as 3 × 3 and 4 × 4. The activity below provides practice in identifying square products. Have your child start at the picture of the Minotaur and use a pencil so he or she can erase wrong turns. If it would be helpful, suggest that your child mark each square product before attempting to find a path.
Please return this Home Link to school tomorrow.
Family Note
199
According to Greek mythology, there was a monster called the Minotaur
that was half bull and half human. The king had a special mazelike
dwelling built, from which the Minotaur could not escape. The dwelling,
called a labyrinth (la buh rinth), had many rooms and passageways
that formed a puzzle. Whoever went in could not find their way out
without help. One day, a Greek hero, Theseus, decided to slay the
monster. To find his way out of the labyrinth, his friend Ariadne gave
him a very, very long ball of string to unwind as he walked through the
passageways. After Theseus slew the Minotaur, he followed the string
to escape.
Pretend you are
Theseus. To find your
way out, you may
go through only
those rooms
numbered with
square products.
Start at the Minotaur’s
chambers and draw
a path to the exit.
Possible paths:
EM3MM_G3_U07_206-236.indd 206 1/18/11 1:03 PM
Math Masters, p. 206
Home Link Master
Lesson 7�1 579
● The 1s products are consecutive counting numbers.
● The 2s products end in 2, 4, 6, 8, or 0.
● The 5s products end in 0 or 5.
● The 10s products end in 0.
● The sum of the digits in each of the 9s products is 9. For example, 4 × 9 = 36, and 3 + 6 = 9.
NOTE Zero can be divided by any nonzero number, but no number can be
divided by zero. Because this table is also used for division, the zero facts are
omitted from it.
� Exploring Multiplication Patterns INDEPENDENTACTIVITY
(Math Journal 2, p. 157)
Algebraic Thinking Have children work for about 5 minutes to solve the problems in Part B on journal page 157. Bring the class together to share solution strategies. Mention that each product is twice as much as the product before it. To find the product in the Try This problem, children can double 64 (or add 64 + 64).
2 Ongoing Learning & Practice
� Playing Name That Number PARTNER ACTIVITY
(Student Reference Book, pp. 299 and 300)
Children practice finding equivalent names for a number as they play Name That Number. Encourage them to use as many operations as they can to name numbers. See Lesson 1-6 or pages 299 and 300 in the Student Reference Book for detailed instructions.
� Math Boxes 7�1 INDEPENDENTACTIVITY
(Math Journal 2, p. 158)
Mixed Practice The Math Boxes in this lesson are paired with the Math Boxes in Lesson 7-3. The skill in Problem 6 previews Unit 8 content.
� Home Link 7�1 INDEPENDENTACTIVITY
(Math Masters, p. 206)
Home Connection Children read about the Greek myth of the Minotaur. They will then trace a path through a labyrinth, or maze, moving from one square product to another.
EM3cuG3TLG2_577-581_U07L01.indd 579EM3cuG3TLG2_577-581_U07L01.indd 579 1/20/11 12:38 PM1/20/11 12:38 PM
580 Unit 7 Multiplication and Division
LESSON
7�1
Name Date Time
Product Patterns
Find each product. Then look for patterns.
1. 1 × 2 = 2
2.
2 × 3 = 6
3.
3 × 4 = 12
4.
4 × 5 = 20
5.
5 × 6 = 30
6.
6 × 7 = 42
7. What happens when you subtract each product
from the next larger product?
Each difference is two more than the
previous difference.
Read pages 198 and 199 in the Student Reference Book to learn more
about patterns in multiplication.
206-236_EMCS_B_MM_G3_U07_576957.indd 208 3/29/11 2:42 PM
Math Masters, p. 208
Teaching Master
LESSON
7�1
Name Date Time
Square and Rectangular Arrays
Follow these steps:
1. Use centimeter cubes to build arrays for each fact.
2. Record the arrays on the grids.
3. Name the shapes of the arrays.
4. Write the number models that match the arrays.
1. 3 × 3 2. 3 × 4
3. 4 × 5 4. 4 × 4
The shape is a square .
Number model: 3 × 3 = 9
The shape is a rectangle .
Number model: 3 × 4 = 12
The shape is a rectangle .
Number model: 4 × 5 = 20
The shape is a square .
Number model: 4 × 4 = 16
Compare the shapes and the number models for different arrays. What
patterns do you see?
The number models with two factors that are the same make
square arrays. When factors are different, rectangular arrays
are made.
Sample answers:
EM3MM_G3_U07_206-236.indd 207 1/18/11 1:03 PM
Math Masters, p. 207
Teaching Master
3 Differentiation Options
READINESS
INDEPENDENTACTIVITY
� Building Square and 5–15 Min
Rectangular Arrays(Math Masters, p. 207)
To provide experience with square and rectangular arrays, have children use centimeter cubes to build arrays for given factors. They record their work on Math Masters, page 207. When the children have completed the page, have them share the patterns they found.
ENRICHMENT INDEPENDENTACTIVITY
� Exploring a Pattern in a 5–15 Min
Sequence of Products(Math Masters, p. 208; Student Reference Book, pp. 198 and 199)
To further explore multiplication patterns, have children look for patterns in a sequence of multiplication problems in which one factor is 1 more than the other factor (1 × 2, 2 × 3, 3 × 4, and so on).
Possible patterns:
● Each array has one more row and one more column than the preceding array.
● Each array has one more column than a square array. Therefore, each product can be expressed in the form n × n + n:
1 × 2 = 1 × 1 + 1 = 2
2 × 3 = 2 × 2 + 2 = 6
3 × 4 = 3 × 3 + 3 = 12
4 × 5 = 4 × 4 + 4 = 20
5 × 6 = 5 × 5 + 5 = 30
6 × 7 = 6 × 6 + 6 = 42
577-581_EMCS_T_TLG_G3_U07_L01_576892.indd 580577-581_EMCS_T_TLG_G3_U07_L01_576892.indd 580 3/30/11 11:26 AM3/30/11 11:26 AM
Lesson 7�1 581
● If you subtract each product from the next larger product, each difference is 2 more than the preceding difference.
1 × 2 = 2
6 - 2 = 4
2 × 3 = 6
12 - 6 = 6
3 × 4 = 12
20 - 12 = 8
4 × 5 = 20
30 - 20 = 10
5 × 6 = 30
ELL SUPPORT
SMALL-GROUP ACTIVITY
� Building a Math Word Bank 5–15 Min
(Differentiation Handbook, p. 132)
To provide language support for multiplication, have children use the Word Bank template found on Differentiation Handbook, page 132. Ask the children to write the terms product and factor, draw a picture representing each term, and write other related words. See the Differentiation Handbook for more information.
REMINDER Have children copy the sunrise/sunset data on journal page 125
to the graph on journal page 279.
Have children continue recording the sunrise, sunset, and length of day for your
location in their new journals on pages 279–281.
Have children copy their body measures from journal page 64
onto journal page 251. They will revisit their body measures in Lesson 10-7.
NOTE The data on journal page 43 will be graphed in Lesson 7-8. You may
want children to keep Journal 1 accessible until the data are used in Lesson 7-8,
or you might choose to make the data available by making copies of the class
record of temperature differences kept on Math Masters, page 48. Children will
continue to record the national high and low temperatures for the rest of the year
on journal page 175 as they did on journal page 43.
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