Exploring Arrays, Equilateral
Triangles, and Strength of Paper
Explorations
Objective To provide opportunities for children to explore
2-digit multiplication, number patterns, and the rigidity
2
of triangles.
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Focal Points
Interactive
Teacher’s
Lesson Guide
Ongoing Learning & Practice
Key Concepts and Skills
Sharing Money
• Use arrays to model multiplication. [Number and Numeration Goal 6]
• Describe number patterns. [Patterns, Functions, and Algebra Goal 1]
Math Journal 2, p. 231
per partnership: at least 2 each of number
cards 0–9 (from the Everything Math Deck, if
available), six-sided die, play money and tool-kit
coins (optional)
Key Activities
Math Boxes 9 10
Exploration D: Children make arrays with base-10 blocks to model partial
products in problems involving 2-digit multiples of 10 and other 2-digit numbers.
Math Journal 2, p. 232
• Collect and organize data in a table. [Data and Chance Goal 1]
• Model and compare polygons. [Geometry Goal 2]
Exploration E: Children figure out and record the number of equilateral
triangles inside larger equilateral triangles and look for patterns.
Ongoing Assessment:
Recognizing Student Achievement
Use Math Boxes, Problem 5. Exploration F: Children test the effect of folding on the strength of paper.
[Measurement and Reference Frames Goal 1]
Key Vocabulary
Home Link 9 10
equilateral triangle
Math Masters, p. 301
Materials
Home Link 9 9
slate base-10 blocks half-sheet of paper
Exploration D: Per group:
Math Journal 2, pp. 229 and 230
Math Masters, pp. 273, 274, 295, and 296
green, red, and blue crayons 4 flats, 25 longs, 28 cubes
Exploration E: Per partnership:
Math Masters, pp. 297–299
Pattern-Block Template triangle pattern blocks paper
Exploration F: Per group:
Math Masters, p. 300
8_12 " by 11" sheet of paper scissors centimeter ruler 2 books of equal
thickness paper small objects varying in weight spring or kitchen scale
766
Unit 9
Multiplication and Division
Differentiation Options
ENRICHMENT
Finding Patterns with Triangular Numbers
Math Masters, p. 302
EXTRA PRACTICE
Frames-and-Arrows Problems
Math Masters, p. 27 or 80
Mathematical Practices
SMP1, SMP2, SMP4, SMP5, SMP6, SMP7
Getting Started
Content Standards
3.MD.2, 3.MD.5b, 3.MD.7a, 3.MD.7b, 3.MD.7c, 3.MD.7d
Mental Math and Reflexes
Math Message
Pose problems like the following. Children
answer on slates.
Use the fewest possible number of
base-10 blocks to model the numbers 150 and 237.
Use the base-10 shorthand
and record your
answers on half-sheets of paper.
4 [8s] 32
40 [8s] 320
400 [8s] 3,200
Home Link 9 9 Follow-Up
9 [70s] 630
90 [70s] 6,300
900 [70s] 63,000
Briefly review answers. Have partners discuss
which multiplication algorithm they prefer.
80 [50s] 4,000
800 [50s] 40,000
8,000 [50s] 400,000
Continue as time allows.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
With the fewest blocks possible, 150 is displayed with 1 flat and
5 longs. 237 is displayed with 2 flats, 3 longs, and 7 cubes.
Pose follow-up questions as needed. For example:
Ask: How could I show 270 with the fewest number of blocks
possible? 2 flats, 7 longs
389? 3 flats, 8 longs, 9 cubes
306? 3 flats, 6 cubes
Lesson 9 10
767
Teaching Master
Name
Date
Exploration D: Modeling
Time
LESSON
9 10 Array Multiplication 2
Multiplication with Arrays
and Base-10 Blocks
Exploration D:
Work in a group of 2 to 4.
Materials
SMALL-GROUP
ACTIVITY
□ array grid
□ base-10 blocks (at least 3 flats and 24 longs)
(Math Journal 2, pp. 229 and 230; Math Masters,
pp. 273, 274, 295 and 296)
□ Math Journal 2, p. 229
□ green and red crayons or coloring pencils
1. Cover a 20-by-13 array of squares on the array grid
using as few base-10 blocks as possible.
This Exploration is an extension of Exploration A in Lesson 9-3.
Instructions are given on Math Masters, pages 295 and 296; work
is recorded on journal pages 229 and 230. All children should do
this Exploration, as these activities are followed up in Lessons
9-11 and 9-12. Plan to work with the children as they rotate
through this Exploration.
Start at the lower-left corner.
Use flats first and then longs.
2. Make a picture of this array in Problem 1 on journal page 229.
Color the squares covered by flats green.
Color the squares covered by longs red.
3. Record the result next to the picture.
4. Cover an 18-by-30 array of squares on the array grid using as few
base-10 blocks as possible.
Start at the lower-left corner.
p
Use flats first and then longs.
g
5. Make a picture of this array in Problem 2 on journal page 229.
Color the squares covered by flats green.
py g
Color the squares covered by longs red.
6. Record the result next to the picture.
Math Masters, p. 295
267-318_EMCS_B_MM_G3_U09_576957.indd 295
2/18/11 7:37 PM
In Array Multiplication 2, children use flats and longs to model
partial products in multiplication of 2-digit numbers by 2-digit
multiples of 10. In Array Multiplication 3, children use flats,
longs, and cubes to model partial products in multiplication of
2-digit numbers. They cover squares on the array grid (Math
Masters, pages 273 and 274) with as few blocks as possible.
On journal pages 229 and 230, children use a green crayon to
represent the squares covered by flats, red to represent the
squares covered by longs, and blue to represent the squares
covered by cubes.
Adjusting the Activity
Have children record their results by first outlining areas with the correct
colors and then filling them in.
A U D I T O R Y
K I N E S T H E T I C
Student Page
Date
9 10
1.
Time
LESSON
Array Multiplication 2
How many squares are in a 20-by-13 array?
Student Page
Date
Time
LESSON
T A C T I L E
9 10
Total squares =
20 × 13 =
260
260
Array Multiplication 3
How many squares are in a 17-by-34 array?
1.
Total squares =
17 × 34 =
578
578
20
20
10
10
0
0
2.
10
How many squares are in an 18-by-30 array?
Total squares =
18 × 30 =
20
540
540
20
How many squares are in a 22-by-28 array?
2.
30
Total squares =
22 × 28 =
616
616
20
10
10
0
10
20
0
30
Math Journal 2, p. 229
204-239_EMCS_S_MJ2_G3_U09_576418.indd 229
768
10
20
Unit 9 Multiplication and Division
10
20
30
Math Journal 2, p. 230
2/18/11 1:54 PM
204-239_EMCS_S_MJ2_G3_U09_576418.indd 230
4/11/11 3:26 PM
V I S U A L
Exploration E:
Teaching Master
PARTNER
ACTIVITY
Finding Number Patterns
by Building Equilateral Triangles
Name
Date
LESSON
Time
Equilateral Triangles
9 10
continued
Exploration E: continued
Fill in the table.
(Math Masters, pp. 297–299)
Length of each
side in inches
1
2
3
Number of 1-in.
triangles inside
1
4
9
4
5
6
7
8
9
10
16 25 36 49 64 81 100
Describe number patterns that you see in the table.
Children work with a partner or in a small group to follow
the steps on Math Masters, page 297. They figure out how
many equilateral triangles, each 1 inch on a side, fit
into each triangle on Math Masters, page 299. They build five
more successively larger equilateral triangles with pattern blocks
or by tracing triangle shapes from their templates on blank paper.
They record results and descriptions of the patterns they find on
Math Masters, page 298.
Exploration F: Building Bridges
Sample answers: The numbers in the second
row are squares of the numbers in the first
row; the differences between consecutive
numbers in the second row increase by 2 as
the numbers become larger.
SMALL-GROUP
ACTIVITY
and Testing Their Strength
(Math Masters, p. 300)
Math Masters, p. 298
267-318_EMCS_B_MM_G3_U09_576957.indd 298
2/18/11 7:37 PM
Science Link Children work in small groups to build
bridges out of equal-size rectangles. They place small
objects on the bridges to test how much weight they can hold
before collapsing. Then they use a spring or kitchen scale to weigh
in grams the objects that each bridge holds and record their
findings. (See a sample record sheet on Math Masters, page 300.)
Children do not fold the first rectangle before testing its strength.
They test a second rectangle for strength after folding it fanlike
8 or 9 times. They test a third rectangle’s strength after folding it
fanlike 12 to 14 times.
When most children are finished, bring the class together to
compare results. Ask: Do the sizes of the folds affect how much
the bridge holds? Sample answer: Yes. The smaller the folds, the
stronger the bridge. What shapes do you see in the fan-folded
rectangles? Sample answer: Triangle shapes Discuss that the
strength of the folded bridges has to do with the rigidity of
triangles.
Ask children to recall their experiments with straw polygons. They
were able to change the shape of other polygons by pushing or
pulling at one of the corners, but they could not change the shapes
of triangles. This rigidity gives the fan-folded bridges their
strength.
Teaching Master
Name
LESSON
9 10
Date
Time
Building Bridges
Exploration F:
Work alone or with a partner.
Materials
1
□ 8_
-by-11 sheet of paper
2
□ centimeter ruler; scissors
□ 2 equal-size books
□ paper for recording
□ light objects—paper clips, rubber bands, straws, pencils,
crayons, erasers, calculators, and so on
□ spring or kitchen scale for weighing objects
1. Fold the paper into fourths as shown. Cut along
the folds.
2. Make a bridge between two books with one of
the rectangles as shown. This is Bridge #1.
Check to see if this bridge will hold any of the
light objects. Record the objects the bridge holds
and their total weight in grams.
Bridge #1
3. Fold another rectangle into 8 or 9 equal-size
fan-folds as shown. This will make the folds a
little more than 1 cm apart. This is Bridge #2.
Place it between the two books. Test the bridge
by placing light objects, and then heavier objects,
on it. Record the objects this bridge can hold and
their total weight.
Fan-folded Rectangle
Bridge #2
Bridge Objects
4. Fold another rectangle into 12 to 14 equal-size
1
folds (each a little more than _
centimeter apart).
2
This is Bridge #3. Test what this bridge can hold.
Record the objects and their total weight.
Weight
(in grams)
1
2
3
Sample record sheet
Math Masters, p. 300
267-318_EMCS_B_MM_G3_U09_576957.indd 300
3/10/11 2:43 PM
Lesson 9 10
769
Student Page
Date
Time
LESSON
2 Ongoing Learning & Practice
Sharing Money
9 10
Work with a partner.
□ number cards 0–9 (at least 2 of each)
Materials
Sharing Money
□ 1 die
□ $10 bills, $1 bills, and tool-kit coins (optional)
Draw 2 number cards. Form a 2-digit number to show how much
money will be shared. Roll 1 die to show how many friends will share
the money. Fill in the blanks below.
(Math Journal 2, p. 231)
Answers vary.
1. $
is shared equally by
friends.
a.
Number model:
b.
How many $10 bills does each friend get?
c.
How many $1 bills does each friend get?
d.
How many $1 bills are left over?
e.
If the leftover money is shared equally,
how many cents does each friend get?
f.
Each friend gets a total of $
Children shuffle the 0–9 number cards, take the two top
cards, and form a 2-digit number. This number represents
the amount of money to be shared. Next, children roll a
die. This determines the number of people who will share the
money. Children record the problems, number models, and their
answers on journal page 231.
.
Repeat. Draw the next 2 cards. Roll the die. Fill in the blanks below.
2. $
PARTNER
ACTIVITY
is shared equally by
friends.
a.
Number model:
b.
How many $10 bills does each friend get?
c.
How many $1 bills does each friend get?
d.
How many $1 bills are left over?
e.
If the leftover money is shared equally,
how many cents does each friend get?
f.
Each friend gets a total of $
Math Boxes 9 10
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 232)
.
Math Journal 2, p. 231
204-239_EMCS_S_MJ2_G3_U09_576418.indd 231
3/11/11 1:45 PM
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 9-12. The skill in Problem 6
previews Unit 10 content.
Ongoing Assessment:
Recognizing Student Achievement
Math Boxes
Problem 5
Use Math Boxes, Problem 5 to assess children’s ability to describe angle
rotations. Children are making adequate progress if they are able to complete
Problem 5 successfully. Some children may be able to draw the other three
angles as described.
[Measurement and Reference Frames Goal 1]
Home Link 9 10
Student Page
Date
Math Boxes
9 10
1.
Draw a shape with an area of
10 square centimeters.
2.
1,872
3 9
1 31 4
1 2 6
2 7 8
8 4 2
7 2
What is the perimeter of your
shape?
14 centimeters
150 151
154 155
$15.00
about
$4.00
191
191
5.
70–72
4. Estimate. Reuben buys 3 bags of
Make an estimate. About how
baby carrots at $2.19 per bag. He
much money, without tax, will you
gives the cashier a $10 bill. About
need for 5 gallons of milk that cost
how much change should he get?
$3.09 each? Sample answer:
about
Home Connection Children show someone at home how
to find products using the lattice method and the
partial-products algorithm.
Practice lattice multiplication.
39 × 48 =
Sample answer:
3.
(Math Masters, p. 301)
Time
LESSON
Fill in the circle for
the best answer.
The turn of
the angle is
6.
Circle the tool you would use to find
the length of a pen:
ruler
compass
scale
the weight of a dime:
1
A. greater than a _
4 turn.
ruler
1
B. less than a _
4 turn.
D. a full turn.
compass
scale
the way to get home:
ruler
1
C. greater than a _
2 turn.
compass
scale
167 168
Math Journal 2, p. 232
204-239_EMCS_S_MJ2_G3_U09_576418.indd 232
770
INDEPENDENT
ACTIVITY
Unit 9 Multiplication and Division
2/18/11 2:06 PM
Home Link Master
Name
3 Differentiation Options
Date
9 10
䉬
Family
Note
PARTNER
ACTIVITY
ENRICHMENT
Finding Patterns with
The class continues to practice the partial-products algorithm and the lattice method.
Encourage your child to try these problems both ways and to compare the answers to be
sure they are correct.
Show someone at home how to use both the lattice method and the
partial-products algorithm.
1. 3 57 5
(Math Masters, p. 302)
3. 8 204 2
INDEPENDENT
ACTIVITY
EXTRA PRACTICE
Frames-and-Arrows Problems
5–15 Min
171
2. 4 91 7
1 2
1 5 1
7 1
To further explore number patterns, have children identify
patterns in triangular numbers. They record their work on
Math Masters, page 302.
68 – 72
Please return this Home Link to school tomorrow.
15–30 Min
Triangular Numbers
Time
Multiplication Two Ways, Part 2
HOME LINK
9
57
3
3
1,632
0
1
3 0
3 6 4
6 4
150
21
171
4. 9 480 4
364
91
4
4
360
4
364
4,320
480
4 8 0
9
31 7 0 9 3,600
4 6 2 0
720
3 2 0
4,320
204
8
1 0 3 8
1 6 0 2 1,600
32
6 3 2
1,632
(Math Masters, p. 27 or 80)
Algebraic Thinking To offer children more practice with number
patterns, have them make up and solve one- or two-rule Framesand-Arrows problems. They record their work on Math Masters,
page 27 or 80. Encourage children to trade problems with a
partner and then solve.
Math Masters, p. 301
Teaching Master
Name
LESSON
1 12
1.
Date
Teaching Master
Name
Time
LESSON
Frames and Arrows
9 10
䉬
Date
Time
Triangular Numbers
Rule
1
3
6
10
15
1. How many dots are in each of the triangular arrays? List the number
2.
3.
of dots on the line below each triangular array. These are triangular
numbers. What patterns do you see?
Rule
Sample answer: To make the next larger triangle, the same
triangle was drawn and a bottom row of dots with 1 more
dot was added. Each triangle starts with 1 dot and increases
by 1 dot in each row. The second triangle has 2 more dots than
the first; the third triangle has 3 more dots than the second;
the fourth triangle has 4 more dots than the third; and so on.
Rule
2. Show the tenth triangular number. How many dots are in the
triangular array? Explain how you figured it out.
4.
Rule
55
Sample answer: I followed the pattern and kept adding
rows of dots until I got to the tenth triangle. I knew the
number pattern for the first 5 triangles was 1 2 3;
3 3 6; 6 4 10; 10 5 15, so I continued it until
I got to the 10th triangle: 15 6 21; 21 7 28;
28 8 36; 36 9 45; 45 10 55. The tenth
triangle number is 55.
Math Masters, p. 27
EM3cuG3MM_U01_001-035.indd 27
Math Masters, p. 302
1/5/11 3:38 PM
Lesson 9 10
771
Name
LESSON
9 10
Date
Time
Building Bridges
Exploration F:
Work alone or with a partner.
Materials
1
□ 8_
-by-11 sheet of paper
2
□ centimeter ruler; scissors
□ 2 equal-size books
□ paper for recording
□ light objects—paper clips, rubber bands, straws, pencils,
crayons, erasers, calculators, and so on
□ spring or kitchen scale for weighing objects
1. Fold the paper into fourths as shown. Cut along
the folds.
2. Make a bridge between two books with one of
the rectangles as shown. This is Bridge #1.
Check to see if this bridge will hold any of the
light objects. Record the objects the bridge holds
and their total weight in grams.
Bridge #1
3. Fold another rectangle into 8 or 9 equal-size
fan-folds as shown. This will make the folds a
little more than 1 cm apart. This is Bridge #2.
Place it between the two books. Test the bridge
by placing light objects, and then heavier objects,
on it. Record the objects this bridge can hold and
their total weight.
Fan-folded Rectangle
Bridge Objects
4. Fold another rectangle into 12 to 14 equal-size
1
folds (each a little more than _
centimeter apart).
2
This is Bridge #3. Test what this bridge can hold.
Record the objects and their total weight.
Weight
(in grams)
1
2
3
Sample record sheet
300
Copyright © Wright Group/McGraw-Hill
Bridge #2