frames and arrows having two rules - everyday math · solving frames-and-arrows puzzles math...

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Lesson 3�6 213

Advance PreparationFor the Math Message, make one copy of Math Masters, page 75 for every two children. Cut the copies apart

and place them near the Math Message.

For Part 1, make overhead transparencies of Math Masters, pages 75–77 and 431, or draw a two-rule

Frames-and-Arrows diagram on the board.

Teacher’s Reference Manual, Grades 1–3 pp. 205, 206

Key Concepts and Skills• Use dollars-and-cents notation.

[Number and Numeration Goal 2]

• Calculate values of coin and bill

combinations.

[Operations and Computation Goal 2]

• Identify and use rules for a function

involving addition and subtraction of coins.

[Patterns, Functions, and Algebra Goal 1]

Key ActivitiesChildren make up and solve Frames-and-

Arrows problems about coins and use coins

to solve Frames-and-Arrows problems

having two rules.

Ongoing Assessment: Informing Instruction See page 216.

Ongoing Assessment: Recognizing Student Achievement Use Math Masters, page 431. [Patterns, Functions, and Algebra Goal 1]

MaterialsMath Journal 1, pp. 69 and 70

Home Link 3�5

Math Masters, pp. 75–77 and 431

transparencies of Math Masters, pp. 75–77

and 431 (optional) � per child: 5 quarters,

5 dimes, 5 nickels, and 5 pennies

Practicing Complements of 100 by Playing Dollar RummyMath Journal 1, p. 71

Math Masters, pp. 454 and 455

scissors

Children find as many different

combinations of $1.00 as possible.

Math Boxes 3�6Math Journal 1, p. 72

Children practice and maintain skills

through Math Box problems.

Home Link 3�6Math Masters, p. 78

Children practice and maintain skills

through Home Link activities.

READINESS

Counting Up and Back on a Number LineMath Masters, p. 79

Children find the rules for skip counts on a

number line.

ENRICHMENTSolving Frames-and-Arrows PuzzlesMath Masters, p. 80

red and blue crayons

Children explore two-rule Frames-and-Arrows

diagrams.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

Frames and ArrowsHaving Two Rules

Objective To guide children as they solve Frames-and-Arrows

problems having two rules.

��

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

213_EMCS_T_TLG1_G2_U03_L06_576949.indd 213213_EMCS_T_TLG1_G2_U03_L06_576949.indd 213 2/11/11 9:20 AM2/11/11 9:20 AM

214 Unit 3 Place Value, Money, and Time

Name Date Time

LESSON

3�6 Frames-and-Arrows Diagrams

Name Date Time

LESSON

3�6 Frames-and-Arrows Diagrams

Rule

Rule

Math Masters, p. 75

Teaching Master

1 Teaching the Lesson

� Math Message Follow-Up WHOLE-CLASS ACTIVITY

(Math Masters, p. 75)

Algebraic Thinking Choose two of the children’s Frames-and-Arrows problems. Review filling in frames and finding a missing rule. Share solution strategies with the class on an overhead transparency of Math Masters, page 75 or on the board. Make sure everyone understands these one-rule Frames-and-Arrows diagrams.

� Solving Frames-and-Arrows WHOLE-CLASS ACTIVITY

Diagrams Having Two Rules(Math Masters, pp. 76, 77, and 431)

Algebraic Thinking Display the example on Math Masters, page 76. Ask what is different about this Frames-and-Arrows diagram from Frames-and-Arrows diagrams that children have seen before. Sample answer: There are two kinds of arrows, one dashed and one solid. Explain that the two kinds of arrows stand for two different rules.

Have children act out the example with you. Start with a dime as indicated in the first frame, add a nickel (solid-arrow rule) to get the result shown in the second frame, add a dime (dashed-arrow rule) to get the result shown in the third frame, and so on.

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BBBBBBBBBBBBBBBBBBBB EELEMMMMMMMOOOOOOOOOBBBLBLBLBLBLBROOROROOROROROROROROROROO LELELELEEEEEELEEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGLLLLLLLLLLLLLVINVINVINVINNNVINVINVINNVINVINVINVINVV GGGGGGGGGGOLOLOOLOLOOLOOO VINVINVLLLLLLLLLLVINVINVINVINNVINVINVINVINVINVINVINNGGGGGGGGGGGOOLOLOLOLOLOLOOOO VVLLLLLLLLLLVVVVVVVVSOSOSOOSOSOSOSOSOSOOSOSOSOOOOOSOOSOSOSOSOSOSOOSOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVVLLLLLVVVVVVVVVLLVVVVVVVVLLLLLLLVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIISOLVING

Getting Started

Math MessageHere is a Frames-and-Arrows problem that uses nickels:

Rule

Add 5¢ 10¢ 15¢ 20¢ 35¢

Take a blank Frames-and-Arrows diagram from Math Masters, page 75. Make up a

problem that uses nickels, dimes, or quarters. Hand in your problem.

Home Link 3�5 Follow-Up Have small groups of children compare their bar graphs by asking questions like the following:

• What is the greatest number of pockets shown by any graph? The least number?

• Use all of your group’s graphs. What is the total number of people who had zero pockets? 1 pocket? (Continue with other numbers.)

Mental Math and Reflexes Pose comparison number stories, which practice -9 and -8 subtraction facts or +9 and +8 addition facts, depending on which number model a child uses; for example, 4 + 8 = 12 or 12 - 4 = 8. Have children share strategies. Suggestions:

Damian is 6 years old. Maya is 9 years older. How old is Maya? 15 years old

Jenna has $4. Martina has $12. How many more dollars does Jenna need in order to have the same amount of money as Martina? $8

Jackson has 14 sports magazines in his collection. Luis has 9. How many more magazines does Luis need in order to have the same number of magazines as Jackson? 5 magazines

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Adjusting the Activity

Adjusting the Activity

Have children write the appropriate rule above each arrow to help them

navigate filling in the frames. For example, write +10¢ and -5¢ above the

arrows in Problem 1 on Math Masters, page 76.

10¢ 20¢ 15¢ 25¢

+10› –5› +10›

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

Repeat the procedure with the other problems on Math Masters, pages 76 and 77. Use the Frames-and-Arrows blank diagrams on Math Masters, page 431 to make up more problems as necessary.

In Problems 4 and 5, children find one of the rules and complete the

frames. To find the rule, children should use coins to act out the pattern. Children

can then fill in the empty frames.

Focus on Problems 1–3. With additional practice on journal pages and in Math

Boxes throughout the year, children will be better prepared for Problems like

4 and 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

Name Date Time

76

Example:

1.

2.

LESSON

3�6 Two-Rule Frames and Arrows

Rule

Rule

10¢ 15¢ 25¢ 30¢

55¢ 45¢ 40¢

Add 5¢

Add 10¢

Rule

Rule

10¢ 20¢ 15¢ 25¢Add 10¢

Subtract 5¢ 25¢ 30¢ 20¢

Rule

Rule

90¢ 80¢ 85¢ 75¢Subtract 10¢

Add 5¢ 75¢ 70¢ 80¢

Math Masters, p. 76

Teaching Master

Name Date Time

Two-Rule Frames and Arrows

Rule

Rule

Rule

Rule

Rule

Rule

Math Masters, p. 431

Teaching Aid Master

Lesson 3�6 215

77

Name Date Time

3.

4.

5.

LESSON

3�6 Two-Rule Frames and Arrows cont.

Rule

Rule

15¢ 20¢ 30¢ 35¢Add 5¢

Add 10¢ 60¢ 50¢ 45¢

Rule

Rule

10¢ 20¢ 25¢ 35¢Add 10¢

Add 5¢ 55¢ 50¢ 40¢

Rule

Rule

50¢ 55¢ 45¢ 50¢Add 5¢

Subtract 10¢ 35¢ 45¢ 40¢

Math Masters, p. 77

Teaching Master



Try This

Two-Rule Frames and Arrows continuedLESSON

3 � 6

Date Time

4. Fill in the frames. Use coins to help you.

Fill in the frames and find the missing rules. Use coins to help you.5.

6.

Rule

Rule

Add 10¢

Subtract1¢

3¢ 13¢

30¢

12¢

21¢31¢

22¢

Rule

Rule

Add 2¢

Add 1¢

1¢ 3¢ 4¢ 6¢

7¢9¢10¢

Rule

Rule

Add 5¢

Add 10¢

$1.00 $1.05

$1.45

$1.20$1.15

$1.35 $1.30

Math Journal 1, p. 70

Student Page

Fill in the frames. Use coins to help you.

1.

2.

3.

Two-Rule Frames and Arrows LESSON

3� 6

Date Time

Rule

Rule

Add 5¢

Add 10¢

5¢ 10¢ 20¢ 25¢

50¢ 40¢ 35¢

Rule

RuleSubtract

5¢

Add 10¢ 20¢ 15¢

25¢ 30¢ 20¢

25¢10¢

Rule

Rule

Add 25¢

Add 5¢

50¢25¢ 80¢

$1.15 $1.10 85¢

55¢

Try This


Student Page

� Solving Frames-and- INDEPENDENTACTIVITY

Arrows Problems(Math Journal 1, pp. 69 and 70)

Algebraic Thinking Children solve Frames-and-Arrows problems and use coins to help them. Encourage children to check each problem after they have completed it by applying the rules to the completed frames. For example, to check Problem 2, children begin with their answer of 10¢. The solid-arrow rule is Add 10¢. The second frame is 20¢. Does 10¢ + 10¢ = 20¢? Yes, so the answer is correct.

Ongoing Assessment: Informing Instruction

Watch for children who do not have a strategy for completing Frames-and-

Arrows problems where the first frame is blank. Suggest that the children write

the rule above the arrows. Use either a number grid or a number line to

demonstrate how to count up and back.

� Creating Frames-and- INDEPENDENTACTIVITY

Arrows Problems(Math Masters, p. 431)

Algebraic Thinking Children create and solve their own two-rule Frames-and-Arrows problems. Encourage children to vary the operations and numbers they use for the rules.

You may want to have children trade papers with a partner to solve each other’s problems. Children should check each other’s work.

Ongoing Assessment: Math Masters

31 �Recognizing Student Achievement

Use Math Masters, page 431 to assess children’s ability to create

number patterns and rules in Frames-and-Arrows problems. Children

are making adequate progress if they can make up a problem and complete it

accurately. Some children may be able to include more difficult numbers for

the rules.

[Patterns, Functions, and Algebra Goal 1]

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2 Ongoing Learning & Practice

� Practicing Complements of 100 PARTNER ACTIVITY

by Playing Dollar Rummy(Math Journal 1, p. 65; Math Masters, pp. 454 and 455)

Explain the rules of Dollar Rummy on journal page 65. Using game cards cut from Math Masters, page 454, have children find as many different combinations of $1.00 as they can. For another version, use cards cut from Math Masters, page 455.

� Math Boxes 3�6 INDEPENDENTACTIVITY

(Math Journal 1, p. 72)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 3-8. The skills in Problems 5 and 6 preview Unit 4 content.

� Home Link 3�6 INDEPENDENTACTIVITY

(Math Masters, p. 78)

Home Connection Children use coins to solve Frames-and-Arrows problems that have two rules.

Dollar RummyLESSON

3 � 6

Date Time

Materials □ Dollar Rummy cards (Math Masters, p. 454)

□ scissors to cut out cards

□ cards from Math Masters, p. 455 (optional)

Players 2

Skill Find complements of 100

Object of the Game To have more $1.00 pairs

Directions

1. Deal 2 Dollar Rummy cards to each player.

2. Put the rest of the deck facedown between the players.

3. Take turns. When it’syour turn, take the topcard from the deck. Layit faceup on the table.

4. Look for two cards that add up to $1.00. Use any cards that are in your hand or faceup on the table.

5. If you find two cards that add up to $1.00, lay these two cards facedown in front of you.

6. When you can’t find any more cards that add up to $1.00, it is the other player’s turn.

7. The game ends when all of the cards have been used or when neither player can make a $1.00 pair.

8. The winner is the player with more $1.00 pairs.

EM3cuG2MJ1_U03_53-79.indd 71 12/18/10 11:46 AM


Student Page

Name Date Time

Frames-and-Arrows diagrams show sequences of numbers—numbers that follow one afterthe other according to a rule.

The problems on this Home Link are a variation of the Frames-and-Arrows problems yourchild brought home in the last unit. In each of the problems below, two different rules arerepresented by two different arrows.

Please return this Home Link to school tomorrow.

FamilyNote

HOME LINK

3�6 Frames-and-Arrows Problems

Show someone at home how to solve these Frames-and-Arrows problems. Use coins to help you.

1.

2.

Rule

Rule

10¢ 20¢ 25¢ 35¢Add 10¢

Add 5¢ 55¢ 50¢ 40¢

Rule

Rule

40¢ 35¢ 45¢ 40¢Subtract 5¢

Add 10¢ 55¢ 45¢ 50¢

98 99

Math Masters, p. 78

Home Link Master

Lesson 3�6 217

Math BoxesLESSON

3 � 6

Date Time

Unit

¢

1. Write the amount. Fill in the circle next to the best answer.

Â Â ‰ Î Í Í A 56¢

B 55¢

C 30¢

D 46¢

2. Solve.

70 + = 100

100 = 20 + 30 + = 50 + 40 + 10

100 = + 10 + 60 88 89

5. I had 17 tulips. I planted 20 more. How many do I

have now?

Start EndChange

50 10 15Children

20

DogsCatsBirdsFish

Children’s Pets 4. Tomorrow I will get dressed for the day. (Circle one.)

Likely

Unlikely

Impossible

Unit 6. Solve.

= 70 - 60

= 88 - 80

63 - 20 = = 92 - 30

3. How many children have cats?

How many

children have fish?

56 3050

10030

108

6243

17 +20 ?

15 children

5 children

Answer: tulips37

EM3cuG2MJ1_U03_53-79.indd 72 12/18/10 11:46 AM


Student Page



Name Date Time

Each puzzle has 2 rules. Figure out where to place the rules tosolve the problems. Sample answers:1.

2. 3.

4.

LESSON

3�6 Frames-and-Arrows Puzzles

Try This

�4 �4 �4�1 �1 �1

3 187 8 12 13 17

Rule�4

Rule�1

�3�10 �10

27 17 14 4

Rule�10

Rule�3

�8 �8�5 �8

10 18 13 21 29

Rule�8

Rule�5

�5 �3 �5�5 �5 �3

7 2112 17 14 19 24

Rule�5

Rule�3

Math Masters, p. 80

Teaching Master

Name Date Time

1.

2.

Star

t at 3

. Red

rule

is A

dd 2

. Blu

e ru

le is

Add

4.

3.

Star

t at 1

. Red

rule

is A

dd 4

. Blu

e ru

le is

Add

2.

4.

Star

t at 1

9. R

ed ru

le is

Sub

tract

4. B

lue

rule

is S

ubtra

ct 2

.

LESSON

3�6 Counting on the Number Line

12

34

56

78

910

1112

1314

1516

1718

1920

red

�2

�4

�2

�4

�2

red

red

blue

blue

12

34

56

78

910

1112

1314

1516

1718

1920

blue

�2

�4

�2

blue

red

�2

blue

�4 red

�4

red

12

34

56

78

910

1112

1314

1516

1718

1920

�4

�2

blue

red

�2

blue

�2

blue

�4

red

�4

red

12

34

56

78

910

1112

1314

1516

1718

1920

Try

This

Math Masters, p. 79

Teaching Master

3 Differentiation Options

READINESS SMALL-GROUP ACTIVITY

� Counting Up and Back 5–15 Min

on a Number Line(Math Masters, p. 79)

To explore navigating on a number line, have children find the rules for skip counts on the number line. Introduce the activity by doing several counts with children on the first number line. They should follow along with their fingers and say each stopping point together. Suggestions:

� Start at 5 and count up by 2s on the number line. Say the stops together. 7, 9, 11, 13, 15, 17, 19 Ask them if they can think of a rule that would mean the same thing as “Count up by 2s.” Add 2 Write the rule on the board.

� Start at 20 and count back by 3s on the number line. Say the stops together. 17, 14, 11, 8, 5, 2 Ask them if they can think of a rule that would mean the same thing as “Count up by 3s.” Subtract 3 Write the rule on the board.

Now work on a couple of two-rule problems together, having children use their fingers to follow along on the number line. Tell children that they have been using only one rule, but that now they are going to do counts with two rules. Suggestions:

� Start at 1. Combine the “Add 3” rule and the “Add 2” rule. Say the stops together. 4, 6, 9, 11, 14, 16, 19

� Have children repeat this two-rule count. This time, have them draw the hops on the number line. Use the red crayon when they add 3 (3 hops) and use the blue crayon when they add 2 (2-hops).

Children work with a partner to complete the page.

ENRICHMENT SMALL-GROUP ACTIVITY

� Solving Frames-and- 5–15 Min

Arrows Puzzles(Math Masters, p. 80)

Algebraic Thinking To further explore two-rule Frames-and-Arrows diagrams, have children solve frames-and-arrows puzzles on Math Masters, page 80. The arrows will not always alternate ABAB. Note that the order in which children place the rules will not matter as long as each rule is used the correct number of times. You may want to have children color-code their rules with crayons—underline the first rule in red and the second rule in blue for each problem; then draw the appropriate color arrow between the frames. Have children explain their solution strategies. Sample answer: I always take the biggest rule first until I am close to my number.


frames and arrows having two rules - everyday math · solving frames-and-arrows puzzles math...

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