Exploring Arrays,
Coins, and Division
Explorations
Objectives To develop readiness for multiplication; to guide
children
in finding coin combinations equivalent to $1.00; and to
c
explore one meaning of division.
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Teaching the Lesson
Common
Core State
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Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Ongoing Learning & Practice
Key Concepts and Skills
Making a Picture Graph
• Count dots in an array. [Number and Numeration Goal 1]
Math Journal 1, pp. 146A and 146B
• Create equal-size groupings. [Operations and Computation Goal 4]
Math Boxes 6 6
• Use geoboards to create arrays. [Operations and Computation Goal 4]
Math Journal 1, p. 147
• Create complements of $1.00 using nickels, dimes, and quarters. [Measurement and Reference Frames Goal 4]
Home Link 6 6
Math Masters, p. 177
Key Activities
Exploration A: Children make arrays on geoboards, record these arrays on
dot paper, and sort arrays having the same number of dots into groups.
Exploration B: Children make and record different combinations of nickels,
dimes, and quarters that are equivalent to $1.00.
Exploration C: Children determine how many groups of n objects can be made
from a specified number of objects and how many objects are left over.
Ongoing Assessment: Recognizing Student Achievement
Use an Exit Slip (Math Masters, page 415). [Measurement and Reference Frames Goal 4]
Materials
Home Link 65
slate
Differentiation Options
ENRICHMENT
Solving Dollar Riddles
Math Masters, p. 178
tool-kit coins
EXTRA PRACTICE
Minute Math +
Minute Math ®+, pp. 64–67
Exploration A: Per child (except where noted):
Math Masters, p. 172; p. 173 or 174
geoboard rubber band overhead geoboard (optional) per group: scissors,
large sheet of paper (optional), glue or paste (optional)
Exploration B: Per child:
Math Masters, pp. 175 and 415
20 nickels, 10 dimes, 4 quarters 2 half-dollars (optional) paper
Exploration C: Per child (except where noted):
Math Journal 1, p. 146
Math Masters, p. 176
per group: container with about 50 pennies or other counters, 1 six-sided die
Advance Preparation
Plan to spend most of your time working on Exploration A with children. Math Masters, pages 172,
175, and 176 give directions for the Explorations.
Teacher’s Reference Manual, Grades 1–3 pp. 90–92
Lesson 6 6
407
Mathematical Practices
SMP1, SMP2, SMP5, SMP6, SMP7
Content Standards
Getting Started
2.OA.4, 2.MD.8, 2.MD.10
Mental Math and Reflexes
Home Link 6 5 Follow-Up
Write subtraction problems like the following on the
board. Have children write ballpark estimates and the
number models they used to make them on their slates.
As children share their answers, circulate and note
how they crossed out (subtracted) cubes.
Some children will not cross out the cubes they are subtracting in
the simplest way. For example, in Problem 3, a child may cross
out 25 cubes by crossing out 5 from each of the first 3 longs and
10 from the final long–leaving 5, 5, 5, 0, and 3 cubes.
98 - 42 100 - 40 = 60
45 - 22 45 - 20 = 25; or 40 - 20 = 20
173 - 39 170 - 40 = 130
Because 25 is 2 tens and 5 ones, it can be represented by 2
longs and 5 cubes. The simplest approach is to cross out 2 longs,
then the 3 cubes, and finally 2 more cubes on a remaining long.
Math Message
How many dots?
1 Teaching the Lesson
How many rows?
How many dots
in each row?
How many dots in all?
Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
Ask:
1.
2.
3.
4.
In Exploration A, children use either a 5 × 5
geoboard (Math Masters, page 173), or a
7 × 7 geoboard (Math Masters, page 174.)
LESSON
66
Date
Time
Geoboard Arrays
Materials □ geoboard dot paper for each person
□ geoboard for each person
□ rubber band for each person
□ scissors for the group
□ glue or paste for the group (optional)
□ large sheet of paper for the group (optional)
How many dots are in each row? 5
●
How many dots are there in all? 10
(Math Masters, p. 172; and p. 173 or 174)
geoboard. The pegs inside and the pegs that touch the
rubber band make an array.
2. Draw your array on the geoboard
Directions for making rectangles on a geoboard are found on Math
Masters, page 172. Before proceeding, check the size of your
class’s geoboards. If the geoboards in your classroom are 5 × 5,
use Math Masters, page 173. If the geoboards are 7 × 7, use Math
Masters, page 174.
dot paper.
3. Write about your array at the bottom
of the geoboard dot paper. Tell how
many rows are in your rectangle, how
many dots are in each row, and how
many dots in all are in your rectangle.
There are 2 rows of 5 pegs.
10 pegs are in the array.
5. Cut apart the dot-paper records of your 4 arrays.
Work with your group to complete Step 6.
6. Sort your group’s arrays into piles that have the same
number of dots. You might want to use the arrays in each
pile to make a display about that number.
Math Masters, p. 172
156-194_EMCS_B_G2_MM_U06_576949.indd 172
408
PARTNER
ACTIVITY
Geoboard Arrays
1. Use one rubber band to make a rectangle on your
Follow steps two and three.
●
Exploration A: Making
Work by yourself to complete Steps 1–5.
4. Make 3 more arrays—all different.
How many rows of dots are there? 2
Use an overhead geoboard to show children how to use a rubber
band to enclose 10 pegs in a 2 × 5 rectangle or demonstrate this
on a regular geoboard and pass it around the classroom. Explain
that in one of the Explorations in this lesson, children will use
rubber bands to form rectangles and then count the number of
pegs in the enclosed array including the ones touching the
rubber bands.
Teaching Master
Name
●
1/26/11 3:45 PM
Unit 6 Whole-Number Operations and Number Stories
Teaching Master
Exploration B: Making
Name
SMALL-GROUP
ACTIVITY
a Dollar
Date
LESSON
Time
Making a Dollar
66
䉬
Work together in a small group.
Materials 䊐 20 nickels
(Math Masters, p. 175)
䊐 10 dimes
䊐 4 quarters
Before children begin, ask them to think about an
organized way to determine the different groups of coins
equivalent to $1.00. Children are given a hint: Find ways
of using 3 quarters and other coins.
Children plan how to find the coin combinations and then record
the groups of coins using Â, Í, and ‰.
䊐 paper and pencil
Directions
1. Use the coins to find as many different ways as you can to
make $1.00.
2. Before you begin, THINK about how to do this. Hint: First,
make a dollar using 3 quarters and some other coins.
3. Plan how you will record the different ways to make $1.00.
4. On a sheet of paper, record the different ways you find to
make $1.00. Use Â, Í, and ‰ to show the coins.
Follow-Up
䉬
How many ways did you find to make $1.00? Check with
other groups to see if they thought of any ways that your
group didn’t find.
䉬
Did you have a plan to find all the combinations? Compare
your plan with the plan used by another group.
Math Masters, p. 175
Coin combinations
Date
Time
LESSON
How Many Children Get n Things?
6 6
䉬
Adjusting the Activity
Follow the directions on Math Masters, page 176 to fill in the table.
What is the
total number
of counters?
Add two half-dollars to the set of coins.
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
Ongoing Assessment:
Recognizing Student Achievement
How many counters
are in each group?
(Roll a die to find out.)
How many
groups are
there?
How many
counters are
left over?
V I S U A L
Exit Slip
Children record totals in Exploration C on Math
Journal, page 146.
Use an Exit Slip (Math Masters, page 415) to assess children’s ability to make
bill and coin exchanges. Have children record as many combinations for $1.00
as they can. Children are making adequate progress if they can draw four or
more combinations. Some children may be able to find all possible combinations.
[Measurement and Reference Frames Goal 4]
Teaching Master
Name
LESSON
66
䉬
Date
Time
How Many Children Get n Things?
Materials 䊐 Math Journal 1, p. 146 (per person)
䊐 one container with about 50 pennies or other
counters (per group)
Exploration C: Finding How
Many Children Get n Things
䊐 1 six-sided die (per group)
SMALL-GROUP
ACTIVITY
(Math Journal 1, p. 146; Math Masters, p. 176)
Use counters to make up and solve problems like this one:
Your group has been given 32 crayons.
Each person is to get 8 crayons.
How many of you will get 8 crayons?
Are there any crayons left over?
Now make up your own problems. Follow these steps:
1. Each person takes a handful of counters. Put all the
counters together in a pile.
Algebraic Thinking Using instructions from Math Masters,
page 176, children make as many equal piles as possible. Then,
on journal page 146, they record the total number of counters,
the number of counters in each group, and the total number of
groups. They also record how many counters are left over; that is,
how many are not in a full group.
How many counters are in the pile? Count them and
record the number on the journal page.
2. Make equal-size groups of counters. One person rolls
the die. The number that lands faceup tells how many
counters to put in each group.
Record this number on the journal page.
3. Make as many groups as you can with the counters in
the pile.
4. Record on the journal page how many groups you made.
If any counters are left over, record that number, too.
NOTE You may want to vary Step 1 of this activity by having children estimate
the number of counters that are in the pile before they begin counting. Have them
compare their estimates to the actual number of counters.
5. Put the counters back in the container. Repeat Steps 1–4.
Math Masters, p. 176
Lesson 6 6
409
Student Page
Date
Time
LESSON
6 6
Links to the Future
Making a Picture Graph
A zoo has a collection of birds. One month, they recorded the number of
eggs the birds laid. The number of eggs a bird lays at one time is called a
clutch. Use the information in the table below to create a picture graph.
Type of
Bird
American
Robin
Canada
Goose
Flamingo
Mallard
Duck
Toucan
Number
of Eggs in
a Clutch
6
10
1
12
4
In Lesson 5-1, children had experience with equal-sharing division situations. In
this activity, children work with equal-grouping division situations. It isn’t
important that children be able to distinguish between the two meanings of
division at this point. However, it is important that they have had informal
experience with both meanings. In Lesson 6-10, children will begin to formalize
their understanding of equal-sharing and equal-grouping. This is an early
exposure to the concept of division. Demonstrating automaticity with all
multiplication facts and proficiency with related division facts are Grade 4 Goals.
Bird Eggs Laid in 1 Month
12
11
10
9
8
Number of
7
Eggs in
a Clutch 6
5
2 Ongoing Learning & Practice
4
3
Making a Picture Graph
2
1
American
Robin
KEY:
Canada
Goose
Flamingo
Mallard
Duck
Toucan
INDEPENDENT
ACTIVITY
ELL
(Math Journal 1, pp. 146A and 146B)
= 1 egg
Math Journal 1, p. 146A
Children use data to make a picture graph and answer questions.
EM3MJ1_G2_U06_131_158.indd 146A
1/29/11 10:57 AM
To support English language learners, write the word clutch on the
board and discuss the various meanings it has. Sometimes a person
will clutch something to hold onto it. A bird clutch is the collection of
eggs that a bird lays at one time.
Math Boxes 6 6
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 147)
Mixed Practice Math Boxes in this lesson are linked with
Math Boxes in Lessons 6-8 and 6-10. The skill in Problem
6 previews Unit 7 content.
Student Page
Date
Student Page
Time
Date
LESSON
6 6
Making a Picture Graph
continued
Time
LESSON
Math Boxes
66
Use the information from the picture graph on journal page 146A to
solve the number stories.
1. 639 has
1. The American Robin and Toucan are tree birds. What is the total
number of eggs laid by the tree birds?
Answer:
10
eggs
Number model:
Sample answer:
6 + 4 = 10
2. Draw the line of symmetry.
6
hundreds
3
tens
9
ones
2. The Canada Goose, Flamingo, and Mallard Duck are water birds.
How many eggs did the water birds lay in total?
Answer:
23
Number model:
10 11
Sample answer:
10 + 1 + 12 = 23
60
eggs
3. Share 1 dozen cookies equally
4. Use counters to make a 5-by-2
among 5 children. Draw a
picture.
3. How many more eggs did the Mallard Duck lay than the American
array. Draw the array.
Robin?
Answer:
6
Number model:
eggs
Sample answer:
12 - 6 = 6
Each child gets
4. Use the information from the graph to write your own number story.
There are
over.
Have a partner solve the problem.
Answer:
Answers vary.
2
2
5. The temperature is
Number model:
cookies.
How many counters in all?
cookies left
52
°F.
10
counters
6. Fill in the pattern.
°F
60
50
Math Journal 1, p. 146B
131_158_EMCS_S_MJ1_G2_UO6_576345.indd 146B
Math Journal 1, p. 147
3/9/11 12:39 PM
EM3MJ1_G2_U06_131_158.indd 147
410
Unit 6 Whole-Number Operations and Number Stories
1/29/11 9:43 AM
Home Link Master
Writing/Reasoning Have children draw their answers to
the following to extend Problem 2: Draw a shape that has
one line of symmetry. Next, draw another shape that has
more than one line of symmetry. Sample answers: An isosceles
trapezoid; a circle or square
Name
Date
HOME LINK
66
Family
Note
How Many?
Your child has been working with arrays—rectangular arrangements of objects having the
same number of objects in each row—to develop readiness for multiplication. Because this
is a readiness activity, children have not yet written number models for multiplication, such
as 4 × 5 = 20. Your child will do this in later lessons in this unit.
Please return this Home Link to school tomorrow.
1. Show someone
Home Link 6 6
X
X
X
X
at home
this array.
INDEPENDENT
ACTIVITY
(Math Masters, p. 177)
How many rows?
How many Xs
in each row?
Home Connection Children draw arrays with specified
numbers of Xs. For each array, children give the number
of rows and the number of Xs in each row.
Time
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
4
Sample answers:
2. Draw an array of 16 Xs.
XXXXXXXX
XXXXXXXX
How many Xs
in each row?
5
How many Xs in all?
4. Draw a different array
of 24 Xs.
XXXXXXXX
XXXXXX
XXXXXX
XXXXXX
XXXXXX
XXXXXXXX
XXXXXXXX
How many Xs
in each row?
3 Differentiation Options
ENRICHMENT
Solving Dollar Riddles
3
8
4
How many rows?
How many Xs
in each row?
PARTNER
ACTIVITY
8
20
3. Draw an array of 24 Xs.
How many rows?
2
How many rows?
6
Math Masters, p. 177
156-194_EMCS_B_G2_MM_U06_576949.indd 177
1/26/11 3:45 PM
15–30 Min
(Math Masters, p. 178)
To apply children’s understanding of coins and coin combinations,
have them solve dollar riddles. Have children share strategies for
solving some of the more difficult problems.
EXTRA PRACTICE
Minute Math +
SMALL-GROUP
ACTIVITY
5–15 Min
To offer children more experience with problems including money,
see the following pages in Minute Math+: pages 64–67.
Teaching Master
Name
LESSON
66
Date
Time
Solving Dollar Riddles
1. To make a dollar, use all four types of coins and create a
coin combination where there are two times as many of one
type of coin as another.
Sample answers: 10 pennies, 5 dimes,1 quarter, 3 nickels;
10 nickels, 5 pennies, 1 quarter, 2 dimes
2. To make a dollar, use all four types of coins. Use half as
many of one type of coin.
Sample answers: 4 nickels, 2 quarters, 2 dimes, 10 pennies;
2 quarters, 1 dime, 7 nickels, 5 pennies
3. To make a dollar, use only nickels and dimes and create a
coin combination where one type of coin is used twice as
much as the other type. Ten nickels and five dimes
4. Using only three types of coins, make a dollar with the
least number of coins you could use. 3 quarters, 2 dimes,
1 nickel
Using only three types of coins, make a dollar with the
greatest number of coins you could use. 85 pennies, 1 dime,
1 nickel
Try This
Use pennies, nickels, dimes, and quarters. Make a combination
that is worth one dollar where you have one of some kind of
coin, double of another, double that of another, and some
number of the last coin.
Sample answers: 4 dimes, 2 quarters, 1 nickel, 5 pennies;
4 nickels, 2 quarters,1 dime, 20 pennies
Math Masters, p. 178
156-194_EMCS_B_G2_MM_U06_576949.indd 178
1/26/11 3:45 PM
Lesson 6 6
411