Multiplication Arrays - Everyday Math - Login 248 Unit 4 Multiplication and Division Advance...
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248 Unit 4 Multiplication and Division
Advance PreparationGather a large supply of pennies or other counters. Choose a place for an Arrays Exhibit. See Home
Link 4�2 for details. Display the Guide to Solving Number Stories on Math Masters, page 406 and a
multiplication/division diagram.
Key Concepts and Skills• Use basic facts to solve multiplication
number stories.
[Operations and Computation Goal 3]
• Use arrays to model multiplication.
[Operations and Computation Goal 6]
• Write number sentences to model
number stories.
[Patterns, Functions, and Algebra Goal 2]
• Model the turn-around rule for multiplication
(Commutative Property of Multiplication)
using an array model.
[Patterns, Functions, and Algebra Goal 4]
Key ActivitiesChildren use arrays to represent multiples of
equal groups. Children solve multiplication
number stories using arrays, multiplication/
division diagrams, and number models.
Ongoing Assessment: Recognizing Student Achievement Use journal page 81. [Operations and Computation Goal 6]
Key Vocabularyarray � factor � product
MaterialsMath Journal 1, p. 81
Home Link 4�1
transparency of Math Masters, pp. 406 and
419 (optional) � per person: 40 counters �
slate � number grid � calendar � number line
(optional) � calculator (optional)
Measuring Perimeters of PolygonsMath Journal 1, p. 82
ruler
Children measure perimeters
of polygons and draw figures
with given perimeters.
Math Boxes 4�2Math Journal 1, p. 83
Children practice and maintain skills
through Math Box problems.
Home Link 4�2Math Masters, p. 87
Children practice and maintain skills
through Home Link activities.
READINESS
Building ArraysMath Masters, pp. 88 and 416
centimeter cubes or other counters �
1 six-sided die
Children build arrays.
ENRICHMENTExploring Square NumbersMath Masters, pp. 89 and 416
per group: centimeter cubes or pennies
(or both), tape
Children explore square arrays.
ELL SUPPORT
Building a Math Word Bank Differentiation Handbook, p. 132
Children add the term array to their Math
Word Banks.
Teaching the Lesson Ongoing Learning & Practice Differentiation Options
������� Multiplication ArraysObjective To provide opportunities to use arrays, multiplication/
division diagrams, and number models to represent multiplication
number stories.
d
eToolkitePresentations Interactive Teacher’s
Lesson Guide
Algorithms Practice
EM FactsWorkshop Game™
AssessmentManagement
Family Letters
CurriculumFocal Points
Common Core State Standards
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Lesson 4�2 249
rows
6
trombones per row
4
trombonesin all
24
24 = 6 × 4
6 × 4 = 24 6× 4
244
6
rows
4
trombones per row
6
trombones in all
24
4
6
24 = 4 × 64 × 6 = 24
4× 6
24
1 Teaching the Lesson
� Math Message Follow-Up WHOLE-CLASS ACTIVITY
Have a volunteer draw an array on the board showing how he or she arranged the counters. Use circles or Xs to represent trombone players.
Ask others to draw different arrangements on the board. Explain that the pictures show the number of trombone players in arrays. An array is a group of objects arranged in rows and columns. Each row has the same number of objects, and each column has the same number of objects. Arrays provide a way to model multiplication. Possible arrays: 4-by-6, 6-by-4, 3-by-8, 8-by-3, 2-by-12, 12-by-2, 1-by-24, and 24-by-1
1. Draw an array on the board representing 4 rows of trombone players with 6 players in each row.
2. Display a multiplication/division diagram, and have a volunteer fill in the known quantities.
3. Write 24 in the bottom right box.
4. Call attention to the fact that the numbers do not have the same units or labels, as they do in addition and subtraction stories.
5. Ask a volunteer to write a number model for this array: 4 × 6 = 24.
6. Next, draw an array to represent 6 rows of trombone players with 4 players in each row.
Getting Started
Home Link 4�1 Follow-Up Ask several children to share the multiples of equal groups they found. They will share some of their stories later in this lesson.
Math Message There are 24 trombone players in a big parade.
Use counters to represent trombone players. Arrange the counters to show them in equal rows.
Mental Math and ReflexesHave children count chorally. They may use number grids, number lines, calculators, or calendars as needed.
Skip count forward and backward by 2s, 5s, and 10s.
Skip count forward and backward by 6s.
Skip count forward and backward by 8s.
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250 Unit 4 Multiplication and Division
NOTE A factor is a number that is multiplied
by another number. A product is the result of
multiplying factors. It is beneficial to use these
words when discussing multiplication number
stories, but do not insist that children use them.
Repeat Steps 1 through 5: Fill in a multiplication/division diagram and write a number model. Compare the two arrays, emphasizing that although the total number of players is the same in each array, the number of rows and the number of players per row are reversed. Remind children of the turn-around rule for multiplication (Commutative Property of Multiplication), which states that two numbers can be multiplied in either order without changing the product: 4 × 6 = 6 × 4
� Solving Multiplication WHOLE-CLASS ACTIVITY
Number Stories Using Arrays(Math Masters, p. 419)
Children draw arrays on their slates to solve simple multiplication number stories. Pose multiplication number stories in which arrays are obvious, such as the following.
� 4 boxes of crayons, 8 crayons per box
� 6 shoe boxes, 1 pair of shoes per box
� 5 rows of chairs, 3 chairs per row
� 5 shelves of books, 8 books per shelf
� 2 cartons of eggs, 12 eggs per carton
For each story:
1. Write a brief summary of the story on the board.
2. Fill in a multiplication/division diagram, going over the number of rows and the number of items per row (the factors) and the total number of things (the product) as children tell you what information to enter into the diagram. To support English language learners, distinguish between the common meaning and mathematical meaning of product.
3. Ask a volunteer to write a number model for the story on the board while the other children write number models on their slates. Children may use either a ? or a letter variable to represent the unknown quantity.
4. Have children draw arrays on their slates or arrange counters on their desks to model the story. As they do this, have a volunteer draw an array on the board or arrange counters on the overhead projector.
5. Have children solve the problem and share their solution strategies. Remind them to include the unit in the answer.
6. Ask children to discuss the reasonableness of their answers and whether their answers make the number model true. Finally, have children write summary number models on their slates for the story.
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BBBBBBBBBBBBBBBBBBBB EEELEMMMMMMMOOOOOOOOOBBBLBLBLBLBBLBOOROROROOROROROROROROROO LELELELEEEEEELEEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGLLLLLLLLLLLLLVVINVINVINVINNNNVINVINVINVINVINVINVINGGGGGGGGGGOLOOOLOLOLOOLOO VINVINVLLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINVINVINNGGGGGGGGGGGOOOLOLOLOLOLLOOO VVVLLLLLLLLLLVVVVVVVVVVOSOSOOSOSOSOSOSOSOSOOSOSOSOOSOOOOOSOSOSOSOSOSOSOOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVVLLLLLLVVVVVVVVVLLVVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSSOOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIISOLVING
Adjusting the Activity
Hold up two egg cartons to provide a visual
for 24 eggs in 2 cartons or 2 dozen eggs.
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
Name Date Time
Multiplication/Division Diagram
per in all
per in all
per in all
Math Masters, p. 419
Teaching Aid Master
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Lesson 4�2 251
More Multiplication Number StoriesLESSON
4�2
Date Time
� Fill in the multiplication/division diagram. Write a number model.
� Make an array with counters. Mark the dots to show the array.
� Find the answer. Write the unit with your answer.
1. Mrs. Kwan has 3 boxes of scented markers. Each box has 8 markers.
How many markers does she have?
markers markers boxes
per box in all
3 8 ?
Number model: 3 × 8 = ? Answer: 24 markers
(unit)
2. Monica keeps her doll collection in a case with 5 shelves. On each
shelf there are 6 dolls. How many dolls are in Monica’s collection?
dolls dolls
shelves per shelf in all
5 6 ?
Number model: 5 × 6 = ? Answer:
(unit)
3. During the summer Jack mows lawns. He can mow 4 lawns per day.
How many lawns can he mow in 7 days?
lawns lawns
days per day in all
7 4 ?
Number model: 7 × 4 = ? Answer: 28 lawns
(unit)
30 dolls
�
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Math Journal 1, p. 81
Student Page
Measuring PerimeterLESSON
4�2
Date Time
82 i ht t
Measure the perimeter of each shape in inches.
Perimeter: 6 inches
Perimeter: 3 3
_ 4 inches
Perimeter: 7 inches
5. Draw any shape with
a perimeter of
20 units.
What is the area of
your shape?
square units
in.
2 in.
2
in.1in.1
in.2
in.1in.1
in.1in.
12
in.11 2
in.1
in.1in.1
in.1
in.3
in.1
in.1in.1
Sample answer:
in.11 4
in.114 in.11
4
Perimeter: 10 inches
4.
1. 2.
3.
150 151
Try This
Sample answer: 16
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Math Journal 1, p. 82
Student Page
� Representing Multiplication WHOLE-CLASS ACTIVITY
Situations with Arrays(Math Masters, p. 419)
Have children share stories from Home Link 4-1 about multiples of equal groups. Follow steps 1–5 from the previous activity. Make the point that arrays can be used to represent multiplication situations, even when the real-life groups do not look like arrays. Arrays are easy to draw, and they simplify problem solving.
� Solving More PARTNER ACTIVITY
MultiplicationNumber Stories(Math Journal 1, p. 81)
Partners or individuals make arrays and use diagrams to solve multiplication number stories. Children may use counters if they wish. Circulate and ask guiding questions such as those on the Guide to Solving Number Stories.
When reviewing the answers to the journal page, ask follow-up questions so children can practice more difficult multiplication facts. For example:
● Suppose Mrs. Kwan has 7 boxes of markers. How many markers would she have now? 56 markers
● If Monica had 60 dolls, how many shelves would she need? 10 shelves
Ongoing Assessment: Journal
Page 81 �Recognizing Student Achievement
Use journal page 81 to assess whether children are making progress toward
using arrays and multiples of equal groups to demonstrate the meaning of
multiplication. Children are making adequate progress if they are able to draw
arrays and find the answer for Problems 1, 2, and 3. Some children may be able
to fill in the diagrams and write the number models.
[Operations and Computation Goal 6]
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BBBBBBBBBBBBBBBBBB ELELEELEMMMMMMMMOOOOOOOOOBLBLBLBLBLBLBLBLBLROOOROROROROROROROROROROO LELELELEEEEEELEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGGLLLLLLLLLLLLLVINVINVINVINNNVINVINVINVINNVINVINVINVINVV GGGGGGGGGGGOLOOOOLOLOOLOO VINVINVVLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINNGGGGGGGGGGGOLOLOLOLOLOLOLOOOLO VVVVVLLLLLLLLLLVVVVVVVVOOSOSOSOSOSOSOSOSOSOOSOSOSOOOOOOSOSOSOSOSOSOSOSOOOSOOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVVLLLLLVVVVVVVVVLLLLVVVVVVVVLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIIISOLVING
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252 Unit 4 Multiplication and Division
Unit3. Solve. Make a ballpark
estimate to checkthat the answermakes sense.
� 648 � 209
estimate: Sample answer:650 � 200 � 850
857
Date Time
2. 10 packs of gum on the shelf in thecandy store. 8 sticks of gum perpack. How many sticks of gum inall?
Answer: 80 sticks
4. Solve.
3 � 5 �
3 nickels � ¢
� 4 � 5
¢ � 4 nickels2020
1515
1. Complete the bar graph.
86 87 259
10
8
6
4
2
0A B C D
Players
Poin
ts
Point Totals Player A scores4 points.Player B scores8 points.Player C scores3 points.Player D scores9 points.
192
sticks of stickspacks gum per of gum
pack in all
10 8 ?
6. 1,798
Which digit is in the tens place? Which digit is in the hundreds place?
Which digit is in the ones place?
Which digit is in the thousands place? 1
8
79
200 201 18 19
5. Fill in the empty frames.
�1
�31 23
6515
52 5356
Math BoxesLESSON
4�2
Math Journal 1, p. 83
Student Page
NOTE Some teachers include a small
section in their display titled Almost an Array.
This section may include children’s
contributions that are close but not quite
correct. It is important to discuss why such
contributions are considered almost an array
but not a true array.
Write these problems on the back of this page. Solve. Show your work.
4. 331 � 27 � 5. � 187 � 17
6. 907 � 479 � 428
204358
Name Date Time
ArraysHOME LINK
4�2
Your child is learning how to represent multiplication problems using pictures called arrays.An array is a group of items arranged in equal rows and equal columns. Help your child usecounters, such as pennies or macaroni, to build the array in each problem. Yourchild should record each solution on the dots next to the problem.
Please return this Home Link to school tomorrow.
FamilyNote
64 65
For the next few weeks, look for pictures
of items arranged in equal rows and
columns, or arrays. Look in newspapers or
magazines. Have people in your family help
you. Explain that your class is making an
Arrays Exhibit.
Make an array with counters. Mark the dots to show the array.
1. 4 rows with 6 counters per row
a 4-by-6 array
counters
2. 3 rows with 8 counters per row
a 3 � 8 array
counters
3. 2 rows with 12 counters per row
a 2 � 12 array
counters24
24
24
This is a 5-by-6 array.
There are 5 rows. There
are 6 dots in each row.
There are 30 dots in all,
since 5 � 6 � 30.
Unit
Practice
Math Masters, p. 87
Home Link Master
2 Ongoing Learning & Practice
� Measuring Perimeters INDEPENDENTACTIVITY
of Polygons(Math Journal 1, p. 82)
This journal page reviews content covered in Unit 3. Problems 1–4 provide children with practice measuring line segments. Children measure the perimeters of polygons to the nearest 1 _ 4 inch. Problem 5 involves children drawing a figure with a given perimeter and then counting unit squares to find the area.
� Math Boxes 4�2 INDEPENDENTACTIVITY
(Math Journal 1, p. 83)
Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 4-4. The skill in Problem 6 previews Unit 5 content.
Writing/Reasoning Have children write an answer to the following: Why are estimates helpful in Problem 3? Sample answer: Estimates help you know whether your answer makes sense.
� Home Link 4�2 INDEPENDENTACTIVITY
(Math Masters, p. 87)
Home Connection Children use arrays to solve multiplication stories. They find pictures from newspapers, magazines, and other sources that show
arrays of items. You may want to use these pictures to build an Arrays Exhibit.
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LESSON
4�2
Name Date Time
Building Arrays
Materials � cm cubes or other counters
� 1 six-sided die
� centimeter grid paper (Math Masters, p. 416)
1. Roll the die 2 times. The first number tells how many rows to put in
the array. The second number tells how many counters to put in
each row.
Example
You roll a 1 first and then a 5. You make this figure.
2. Draw the arrays you make on centimeter grid paper.
3. Then, fill in the Array Table below.
How manyHow many
How manyrows?
cubes in eachcubes in all?row?
1
2
3
4
5
Math Masters, p. 88
Teaching Master
LESSON
4�2
Name Date Time
Square Products
Work in a small group.
Materials � centimeter grid paper (Math Masters, p. 416)
� centimeter cubes or pennies (or both)
� tape
Directions
1. Each person chooses a different number from 2 to 10.
2. Build an array that shows your number multiplied by itself. Usecentimeter cubes or pennies.
3. Draw each array on centimeter grid paper. Write each numbermodel under each array.
4. Continue to build arrays with other numbers. Draw the arrays ongrid paper. You may need to tape pieces of grid paper together forthe larger arrays.
5. Look at the arrays you made. Why do you think the products of thenumber models you wrote are called square products?
Sample answer: All of the arrays are squares.
2 � 2 � 4
4 � 4 � 16
3 � 3 � 9
Math Masters, p. 89
Teaching Master
Lesson 4�2 253
3 Differentiation Options
READINESS INDEPENDENTACTIVITY
� Building Arrays 5–15 Min
(Math Masters, pp. 88 and 416)
To provide children with experience forming arrays, children build arrays with counters, following directions on Math Masters, page 88. They record the arrays on centimeter grid paper, Math Masters, page 416, and fill in the Array Table on page 88.
ENRICHMENT SMALL-GROUP ACTIVITY
� Exploring Square Numbers 15–30 Min
(Math Masters, pp. 89 and 416)
Algebraic Thinking To further explore arrays, have children build square arrays with counters for as many numbers as possible, beginning with 2. They record their arrays on centimeter grid paper (Math Masters, page 416) and write a number model under each array. As children build arrays for larger products, they might need to tape sheets of grid paper together. Have children arrange the number models and products for each array in order from smallest to largest. Ask: What patterns do you see? Sample answer: The differences between consecutive products are odd numbers, and they increase by 2.
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 children to write the word array, draw a picture representing the word, and write other related words. See the Differentiation Handbook for more information.
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