Foundations of Algebraic Thinking - ode.state.or.us · Foundations of Algebraic Thinking: ... Number & Operations & Algebra Standards Continuum/Trajectory ... Each table group will

Foundations of Algebraic Thinking:

Addition, Subtraction, Patterning

K-2

Building Instructional Leadership Across Oregon

“Developing Algebraic Thinking”

Session 2

Winter 2010

K-2 Goal:

I can describe early numeracy skills that build foundations for algebraic thinking (patterning, number properties and operations)

and how these skills continue to develop with and through state standards through grade 8.

Foundations of Algebraic Thinking:

Addition, Subtraction, Patterning K-2

February Follow Up K‐8 Math Institute

Foundations for Algebraic Thinking Vocabulary 3

Activities Activity Descriptions, Note Pages 5

Participant Activity Planner 9

Number Talk: What Do You See? 15

A. Number & Operations & Algebra Standards Activity 17

B. Calendar Pattern (Bridges, Grade 2) 19

C. Make a Pattern/Non‐pattern 24

D. What Comes Before/After? 25

E. Sixes & Sevens: Unifix Cube Equations 28

2

Addition, Subtraction, Patterning K-2 Vocabulary

Properties

Associative Property (Addition, Multiplication) For a given number sentence that combines three quantities (2 at a

time); the initial pairing of the quantities is arbitrary. The way quantities associate will not change the result of the operation. (a + b) + c = a + (b + c) and (a x b) x c = a x (b x c).

Commutative Property (Addition, multiplication) The order of factors or multiples doesn’t change the sum or product.

Identity Property (Addition: 0, Multiplication: 1) Adding 0 to another number (or multiplying by 1) cannot change the original quantity.

Models: Addition & Subtraction

Discrete objects (counters) blocks, bears, frogs, buttons, etc. Generally used when focusing on counting and counting strategies.

Length‐based cube trains, (Cuisinaire) rods, etc Hundreds charts

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Number Lines Strategies: Addition & Subtraction

Compensation Turning a number into a ‘friendly’ number to perform an operation, then adjusting for this change.

Students often use compensation intuitively once they have a good understanding of quantity and number order. Example:

Compose/decompose Numbers can be broken down into their smaller parts to allow children to have greater facility

with numbers and operations. Students may use place value or landmark (friendly, benchmark) numbers to break numbers apart, depending on the numbers and context. This will transfer to understandings with larger numbers in subsequent grades.

3

Difference (comparison) Some contexts do not lend themselves to modeling with a “take‐away” model. How many

more? How many less? Both of these call for comparing to arrive at a difference. The following comparison reflects the original question as no one is taking away or removing cars. Begin with young students using concrete models. In this example, young students would be using small toy cars, line them up, compare and count how many to see the difference. As children develop mathematically, we move them to more abstract models for understanding operations (example below with number lines). The ultimate goal is to move toward fluency with abstract numbers.

Sam has 12 cars. Josie has 16 cars. How many more does Josie Have?

Take‐Away Some contexts will involve ‘taking away’ a specified amount and ask about the remaining amount.

4

Number Talk: What Do You See? A dot pattern is shown to the group for a few seconds. Private think time: What did you see? How did you see it? Ideas are shared and recorded until all new thinking has been brought to light.

Activity A: Number & Operations & Algebra Standards Continuum/Trajectory Activity Materials

• Various 2007 standards K‐8 (Text only, not identified by grade level) One or two per table group • Chart paper with K‐8 Continuum

This activity will be completed by table groups prior to working other activities. Each table group will have 1‐2 algebra standards. Read each standard and (before looking at standards) form a conjecture about what grade level that standard would be found at. Then, locate the specific grade level that standard is found at and place the standard on the grade level continuum on the wall. Quickly review whole group, then move on to activities.

Activity B: Bridges Grade 2 January Number Corner The first week of the calendar pattern is shown. Work with a partner(s), using the posted questioning strategies, to discuss and conjecture about the pattern(s) you see and what might be coming up. Materials

• January Calendar Cards • Pocket Chart

• Questioning Strategies • Calendar Grid Recording Sheet

5

Activity C: Make a Pattern/Make a Non‐pattern Good Questions Great Ways to Differentiate Mathematics Instruction (2009) Small, M, p.123. Participants work in pairs, using 24 counters (in small, resealable plastic bags) to create a pattern with 12 and a non‐pattern with the remaining 12 counters. They will record their findings and must be able to justify and explain why each is or is not a pattern. Materials

• 24 pattern blocks (4 shapes) • 24 2‐color counters • 30 Unifix cubes (3 colors) • 24 numeral cards (students may select 12 from deck to make a pattern, 12 others to make a non‐pattern – or teacher can determine a selection of 24 cards)

• Blank paper to record pattern/non pattern The number of objects can be varied to differentiate for students.

Activity D: What Comes Before? After? Good Questions Great Ways to Differentiate Mathematics Instruction (2009) Small, M, p.123, 124 Participants select a pattern card and discuss with a partner what might complete the patterns on the cards by identifying what could have come before and after the identified element in the series. Each participant will record the completed patterns on paper. Materials

• Several different pattern cards showing a series of 5‐6 elements in a pattern (pattern blocks, shapes, numbers, colors, cubes, etc.) Only one element in the pattern is revealed – the others elements are replaced with placeholders)

• Blank paper strips for recording patterns • Manipulatives to help visualize patterns

6

Activity E: Unifix Cube Equations Developed by the Math Learning Center as a supplemental activity for grade 1. It is called A4 Number and Operations: Equivalent Names and a complete copy of the two‐day lesson is included in your electronic copies. (Additional activities are available free, online at http://www.mathlearningcenter.org/resources/materials/or.asp ) Participants select a number to work with and find the corresponding record sheet. (The full activity incorporates 5‐10, but we will focus on six and seven today). Using cubes of 2‐3 colors, participants will find four different ways to make that target number, recording their work on paper. Participants directly model with cubes, represent their model on paper with color crayons and numbers, then practice adding and subtracting and are given a number line to help participants visualize and solve the number problems. Materials

• Unifix Cube Equation record sheets (6, 7) • Unifix cubes • Crayons (colors to match cubes)

7

http://www.mathlearningcenter.org/resources/materials/or.asp

8

NAME DATE

K-2 Addition, Subtraction, Patterning (Follow Up: Algebraic Thinking February 2010)

A. Standards Activity

B. Calendar Pattern

C. Make a Pattern & Non Pattern

D. What Comes Before? After?

E. Unifix Cube Equations: 6’s & 7’s

9

Calendar Pattern Record Sheet

10

Bridges in Mathematics Grade 1 Supplement • a4.11© The Math Learning Center

name date

set a4 number & Operations: equivalent names Blackline Run 10–15 copies back-to-back with page A4.12.

unifix Cube equations, 6’s page 1 of 2

1 Color in the unifix cubes and write an equation to match each train.

a

__________________________ = 6

b

6 = __________________________

c

__________________________ = 6

d

6 =__________________________

2 Circle T or F.

a 1 + 4 = 6 T or F b 6 = 2 + 2 + 3 T or F

c 6 = 3 + 3 T or F d 4 + 2 = 6 T or F

(continued on back)11

© The Math Learning Centera4.12 • Bridges in Mathematics Grade 1 Supplement

name date



3 Add 3 0 3 4 0 2 + 3 + 6 + 1 + 0 + 5 + 3 ____ ____ ____ ____ ____ ____

3 2 4 4 2 5 + 2 + 4 + 2 + 1 + 2 + 1 ____ ____ ____ ____ ____ ____

3 + 3 =________ 2 + 1 + 2 = ________ 0 + 6 = ________

4 Subtract 5 6 5 6 5 6 – 2 – 0 – 4 – 1 – 3 – 5 ____ ____ ____ ____ ____ ____

6 4 6 4 6 6 – 2 – 2 – 4 – 3 – 3 – 6 ____ ____ ____ ____ ____ ____

6 – 3 =_________ 5 – 2 = _________ 4 – 3 = _________

Can I help you?

0 1 2 3 4 5 6 7 8 9 10

12


name date




a

__________________________ = 7

b

7 = __________________________

c

__________________________ = 7

d

7 =__________________________

2 Circle T or F.

a 3 + 4 = 7 T or F b 7 = 2 + 3 + 1 T or F

c 7 = 3 + 4 T or F d 7 + 0 = 7 T or F



name date



3 Add 3 3 2 4 7 2 + 4 + 3 + 5 + 2 + 0 + 3 ____ ____ ____ ____ ____ ____

2 4 4 6 6 5 + 2 + 3 + 1 + 1 + 0 + 2 ____ ____ ____ ____ ____ ____

3 + 4 =________ 2 + 2 + 2 = ________ 5 + 2 = ________

4 Subtract 7 7 6 7 6 7 – 7 – 0 – 4 – 1 – 3 – 5 ____ ____ ____ ____ ____ ____

7 5 7 5 7 7 – 2 – 2 – 4 – 3 – 3 – 6 ____ ____ ____ ____ ____ ____

7 – 2 =_________ 6 – 4 = _________ 7 – 4 = _________

Can I help you?

0 1 2 3 4 5 6 7 8 9 10

14

15

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A. Standards Activity Copy and cut the cards apart (without identifying grade levels). Give each table group a card or two. Without looking at copies of standards documents, give groups a couple of minutes to determine at which grade level the standard belongs. Once they have reached consensus with their group they may look up the actual standard to see if they identified the correct grade level. Facilitator: Did we have a Kindergarten standard? … go up through grade levels. Have a representative from each group read aloud the standard as each grade level is called. Once they read it, post it on the grade level continuum. Options: If you have more time and want to focus on standards, each group could be given a set of cards and place them in order. Discuss as a group to identify developmental stages and key developmental milestones in standards. Whole group discussion following this would bring out and summarize these points.

Identify, duplicate, and extend simple number patterns and

sequential and growing patterns (e.g., patterns made with simple

shapes.)

Identify, create, extend, and supply a missing element in

number patterns involving addition or subtraction by a single-digit number

Kindergarten Grade 1

Develop fluency with efficient procedures for adding and

subtracting multi-digit whole numbers and understand why they work on the basis of place value

and number properties.

Develop fluency with efficient procedures for multiplying multi-digit whole numbers and justify why the procedures work on the basis of place value and number

properties.

Grade 2 Grade 4

Develop fluency with efficient procedures for dividing multi-digit

whole numbers and justify why the procedures work on the basis

of place value and number properties.

Solve one-step equations by using number sense, properties of operations, and the idea of

maintaining equality on both sides of an equation.

Grade 5 Grade 6

17

Apply properties of rational numbers and algebra to write and

solve linear equations in one variable.

Use linear functions and equations

to represent, analyze and solve problems, and to make predictions

and inferences.

Grade 7 Grade 8

18

B. Calendar Patterning Activity Grade 2 Calendar pieces from Bridges in Mathematics (The Math Learning Center) are used here. Other teacher or class created calendar patterns could easily be substituted here. Post the pieces in the calendar grid. Show markers 1‐12. Place 13‐31 backwards in the pocket chart so the pictures are not readily visible. Materials

• January Calendar Cards • Pocket Chart • Questioning Strategies • Calendar Grid Recording Sheet

Write the month, year and number your calendar recording sheet. Work with a partner using the questions that are posted to help examine the pieces for any patterns. Record what you notice on your record sheet. Predict the next 5‐7 pieces and write it on your sheet. What do you predict for the week after that? For any specific date? After you have formed a conjecture and have discussed it with your partner you may turn over one calendar marker at a time to check your prediction. When you have looked at a piece to check your prediction make sure to replace it in the chart the way you found it!

19

20

21

22

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C. Make a Pattern/Non Pattern This activity inspired by one found in Good Questions Great Ways to Differentiate Mathematics Instruction (2009) Small, M., p. 123.

Students select a collection of objects in a resealable plastic bag. They are to use half the objects to create a pattern, the other half of the objects to create a non pattern. Once they have created their patterns they should represent their thinking by recording patterns and non patterns on the pattern strip paper provided (blank 3‐4” x 14” paper strips). Materials

• 24 pattern blocks (4 shapes) • 24‐30 2‐color counters (could be collections of two types of counters ‐ frog & insects, buttons & cubes) • 30 Unifix cubes (3 colors) • 24 numeral cards (students may select 12 from deck to make a pattern, 12 others to make a non‐pattern – or teacher can determine a selection of 24 cards)

• Blank paper to record pattern/non pattern The number of objects can be varied to differentiate for students. A tub or container of additional manipulatives could be available if students ‘get stuck’ and need additional pieces. With the red/yellow dots, however challenge students to stick with the 24!

Make a Pattern…Make a Non Pattern!

1. Select a collection in a bag. Spread the objects out. 2. Split the collection in half. 3. Make a pattern with one half of the objects. Convince a partner

it’s a pattern. Draw a picture of your pattern on paper. 4. Make a non pattern with the other half.

24

D. What Comes Before? After? This activity inspired by one found in Good Questions Great Ways to Differentiate Mathematics Instruction (2009) Small, M., p. 123 ‐ 124. Materials

• Several different pattern cards showing a series of 5‐6 elements in a pattern (pattern blocks, shapes, numbers, colors, cubes, etc.) Only one element in the pattern is revealed – the other elements are replaced with placeholders.

• Blank paper strips for recording patterns • Manipulatives to help visualize patterns

Pairs of students select a pattern card and discuss what they think could have come before and after in a series containing the given element. There will be more than one possibility. If students quickly identify a possibility, challenge them to see if they can identify another possible series! Have students represent their thinking on blank paper strips. Students could also be challenged to create their own mystery series and create a card showing just one of the elements. Make manipulatives available for students who would like concrete support.

? ? ? ?

?

? ? ?

? ? ? ?

25

?

? ? ?

? ? ? ?

?

? ? ?

? ?

? ?

26

Blank strips for students to create their own “What Comes Before? After?” strip

27

E. Sixes and Sevens Developed by the Math Learning Center as a supplemental activity for grade 1. It is called A4 Number and Operations: Equivalent Names and a complete copy of the two‐day lesson is included in your electronic copies. (Additional activities are available free, online at http://www.mathlearningcenter.org/resources/materials/or.asp ) Participants select a number to work with and find the corresponding record sheet. (The full activity incorporates 5‐10, but we will focus on six and seven today). Using cubes of 2‐3 colors, participants will find four different ways to make that target number, recording their work on paper. Participants directly model with cubes, represent their model on paper with color crayons and numbers, then practice adding and subtracting and are given a number line to help participants visualize and solve the number problems. Materials

• Unifix Cube Equation record sheets (6, 7) • Unifix cubes • Crayons (colors to match cubes)

1. Choose a Unifix Cube Equation sheet. 2. Build 4 trains to match the number. Use 2-3 colors of cubes only! 3. When you build trains make sure all the like colors are together. 4. Color the trains on your sheet to match your Unifix

cube trains. Write an equation for each train. 5. Complete the problem at the bottom of the page. 6. Complete the problems on the back of the page. Be sure to use the number line to help!

28

http://www.mathlearningcenter.org/resources/materials/or.asp

Grade 1 supplementset a4 Number & Operations: Equivalent Names

IncludesActivity 1: Sixes & Sevens, Day 1 A4.1

Activity 2: Sixes & Sevens, Day 2 A4.5

skills & ConceptsH fluently compose and decompose numbers to at least 10

H connect physical and pictorial representations to addition and subtraction equations

H use the equal sign and the word equals to indicate that two expressions are equivalent

H add three or more one-digit numbers using the commutative and associative properties

of addition

P080829

Bridges in mathematics Grade 1 supplement

set a4 Numbers & Operations: Equivalent Names

The Math Learning Center, PO Box 12929, Salem, Oregon 97309. Tel. 1 800 575–8130.

© 2008 by The Math Learning Center

All rights reserved.

Prepared for publication on Macintosh Desktop Publishing system.

Printed in the United States of America.

P0808

The Math Learning Center grants permission to classroom teachers to reproduce blackline

masters in appropriate quantities for their classroom use.

Bridges in Mathematics is a standards-based K–5 curriculum that provides a unique blend

of concept development and skills practice in the context of problem solving. It incorpo-

rates the Number Corner, a collection of daily skill-building activities for students.

The Math Learning Center is a nonprofit organization serving the education community.

Our mission is to inspire and enable individuals to discover and develop their mathematical

confidence and ability. We offer innovative and standards-based professional development,

curriculum, materials, and resources to support learning and teaching. To find out more,

visit us at www.mathlearningcenter.org.30

set a4 H Activity 1

Activity

sixes & sevens, day 1

OverviewEach student builds a train of 5 or 6 Unifix cubes in 2 or

3 different colors, and writes an addition expression to

match. The class examines the trains and expressions to

find equivalent equations.

skills & ConceptsH fluently compose and decompose numbers to at

least 10

H connect physical and pictorial representations to

addition and subtraction equations

H use the equal sign and the word equals to indicate

that two expressions are equivalent

H add three or more one-digit numbers using the com-

mutative and associative properties of addition

You’ll needH Numerals & Symbols cards (page A4.4, see Advance

Preparation)

H Unifix cubes (see Advance Preparation)

H 3" × 5" index cards, class set plus a few extra

H pocket chart

H Work Places currently in use

advance preparation Run 4 copies of the Numerals &

Symbols cards on cardstock and cut the cards apart. Have

students help you set up a container of cubes for each

table or group of 4 students. Each container should have

about 100 Unifix cubes in 4–5 different colors.

Instructions for sixes & sevens, day 11. Gather students to your discussion circle. Explain that they are going to use Unifix cubes today to learn some more about adding numbers. Tell them that in a minute, each of them is going to make a train of 6 or 7 Unifix cubes using 2 or 3 different colors. Demonstrate by making a train of 4 red and 2 red cubes. Note with students that the colors are grouped—all the reds are together and all the yellows are together.

2. Next, make a train of 7 cubes using 3 different colors, but don’t tell students what your total is before-hand. When you’re finished, give them a moment to examine your train carefully and share with the person next to them what they believe the total is. Then ask several volunteers to share their answer and their reasoning with the class.

Students I think it’s 7 because I counted them when Mr. S. was putting them together. It’s 7 because 2 white and 2 brown makes 4. There’s 3 in the middle, so that’s 5, 6, 7. Two and 3 is 5, and then 2 more at the end makes 7.

3. Send students back to their tables. Assign the students seated at half the tables to each make a train of 6 cubes. Have the students at the rest of the tables each make a train of 7 cubes. Encourage them to

set a4 number & Operations: equivalent names


31

make their trains different than yours and different from anyone sitting near them. Remind them that they can only use 2 or 3 colors, and ask them to keep the colors grouped together. That is, if they use 3 browns and 3 yellows, put all the browns together and all the yellows together.

4. As students finish, have them return to the discussion circle with their trains. Call them a few at a time to set their trains in the middle of the circle. Have them group the trains of six in one area, and the trains of seven in another.

5. Give students a minute or two to pair-share their observations, and then invite a few of them to share their ideas with the class. What do they notice about the trains?

Students They’re all 6 over here, and 7 over there. They’re all the same long in each pile, but they’re different colors. Mine’s on top. It’s 4 greens and 2 blues, see? Mine is the one in the middle of the 7’s. It has 4 greens and 3 blues.

6. Choose 5 trains from each set and put the rest aside for now. (Explain that you’ll come back to them tomorrow.) Then work with input from the class to write a matching expression on an index card for the each of the 10 trains you selected.

4 + 2

2 + 4

3 + 3

5 + 1

2 + 2 + 2

2 + 5

4 + 1 + 3

5 + 2

3 + 4

4 + 3



activity 1 Sixes & Sevens, Day 1 (cont.)

32

7. Choose one of the trains from the collection. Set it on the chalk ledge or on a small table next to your pocket chart. Use the matching expression card, along with the other cards you prepared for this ac-tivity to create an equation in the pocket chart. Start with the total, however. Ask students to read the equation with you. Invite their comments and observations. Some may feel that you’ve inserted the cards backwards, and that the equation should end with the total, rather than starting with it. Explain that the equals sign means “the same as”, and read the sentence that way with the class (i.e., 7 is the same as 3 + 4).

7 = 3 + 4

8. Repeat Step 7 several times, but change the order in which you arrange the cards, starting with the total sometimes and the expression others. Read each new equation with the class. Continue to use the phrase “is the same as” in place of equals.

9. Next, choose two of the 6 trains or two of the 7 trains. Set them on the chalk ledge or table, and solicit students’ agreement that they both have the same number of cubes. Then use your cards to create an equation that matches the trains. Read the equation with your students and ask volunteers to explain it to the class.

3 + 35 + 1 =

Teacher Is this true? Is 5 + 1 really the same as 3 + 3? Talk with the person next to you for a mo-ment, and then let’s have some volunteers share their thinking with the class.

Students They’re both 6, so they’re kind of the same. The numbers look different, but both trains have 6 in them. Five and 1 is 6, right? Then 3 and 3 is 6. So they’re the same. I don’t get it!

10. Repeat Step 9 until you’ve used all the trains and matching expression cards. Tell students you’ll re-turn to the activity the following day, and send them out to do Work Places.

Note Return the cubes from the 10 trains you used today to your tub of cubes. Save the other trains for use in the next activity.




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numerals & symbols cards


set a4 number & Operations: equivalent names Blackline Run 4 copies on cardstock. Cut the cards apart.

6Numerals & Symbols card




=Numerals & Symbols card

=Numerals & Symbols card

= Numerals & Symbols card

= Numerals & Symbols card

34



set a4 H Activity 2

Activity

sixes & sevens, day 2

OverviewStudents continue to explore equivalent equations during

this activity, and a new Work Place is introduced.

skills & ConceptsH fluently compose and decompose numbers to at

least 10

H connect physical and pictorial representations to

addition and subtraction equations

H use the equal sign and the word equals to indicate

that two expressions are equivalent

H add three or more one-digit numbers using the com-

mutative and associative properties of addition

You’ll needH 3" × 5" index cards (see Advance Preparation)

H Unifix Cube Equations, 5’s, 6’s, 7’s, 8’s, 9’s, and 10’s

(pages A4.9–A4.20, see Advance Preparation)

H resealable bag of crayons in colors to match the Unifix

cubes (see Advance Preparation)

H Unifix cube trains from Set A4, Activity 1

H Numerals & Symbols cards from Set A4, Activity 1

H pocket chart

H individual chalkboards/whiteboards, chalk/pens, and

erasers for each student

H Work Places currently in use

advance preparation Write an expression on an index

card to match each of the Unifix cube trains you saved

from the previous activity. Also, run 10–15 copies of each

pair of Unifix Cube Equations worksheets. Place these in

pocket folders. Put the pocket folders, along with several

hundred loose Unifix cubes, and the bag of crayons into a

tub to create a new Work Place. (This new Work Place can

be used in place of Work Place 2J, 50 or Bust!)

Instructions for sixes & sevens, day 21. Gather students to your discussion circle. Set the rest of the Unifix trains from the previous activity in the center of the circle, 6’s in one area and 7’s in another. Hold up one by one the expression cards you’ve prepared. Read each card with the students, and have a volunteer lay it beside the matching train on the rug.

3 + 1 + 2

1 + 5

2 + 2 + 2

5 + 1

1 + 4 + 1

6 + 1

3 + 2 + 2

4 + 1 + 2

5 + 2

2 + 5

35



2. When all the trains have been labeled, have a student or two help you gather up all the cards. (Leave the trains where they are in the middle of the circle.) As the cards are being gathered, ask helpers to hand out individual chalkboards, chalk, and erasers to everyone sitting in the circle.

3. Mix the expression cards thoroughly, and place them in a stack face-down on a small table near your pocket chart. Use your Numerals & Symbols cards to place a 7 and an equals sign in the pocket chart. Ask a student to come up and draw an expression from the top of the stack and place it to the right of the equals sign in the pocket chart.

4. Read the resulting equation with your students, using the term “is the same as” for the equals sign. Is it true? If so, ask students to write a “T” on their chalkboards. If it’s not true, ask students to write an “F” for false on their chalkboards. Have them hold up their boards when they’re finished, and then ask two or three students to explain their answers.

7 = 5 + 1

Students I put an F because 1 + 5 is 6. It’s not 7. I did too. They’re not the same! They could be the same if that card said 2 + 5.

5. Ask one of the students to find the train that matches the expression just posted, and hold it up or set it near the pocket chart so children can use it to confirm their responses. Then ask the students to erase their boards. Place an equals sign and a 6 in the next row on the pocket chart. Ask a volunteer to draw another expression card from the top of the stack, and place it in the pocket chart to the left of the equals sign. Read the resulting equation with your class and have students write a “T” or an “F” on their boards. Ask them to pair-share their answers, and then invite two or three of them to explain their thinking to the class. Again, have a student find the matching train and hold it up or set it near the pocket chart so children can confirm their responses.

2 + 2 + 2 = 6

Students Yep, that one’s true. I put a T for true. Two plus 2 is 4, and then 2 more is 6. You can see it’s right because there are 6 cubes on that train.

6. Repeat Step 5 several times. Use both 6’s and 7’s cards, and switch the positions they occupy in the equation, sometimes to the right of the equals sign, and sometimes to the left.

7. When you’re down to the last 4–6 expression cards, have helpers draw two cards from the stack and place them in the pocket chart on either side of the equals sign. Ask different helpers to find the match-


36




ing trains and set them near the pocket chart. Have students examine the equation and write a “T” or an “F” on their boards to indicate whether they think it’s true or false. Then call on volunteers to explain their reasoning.

= 6 + 15 + 2

8. When all the expression cards have been used, ask students to work together to correct the false equa-tions by switching some of the cards or using some of your extra 6’s and 7’s cards.

3 + 1 + 2

1 + 5

2 + 2 + 2

6 + 1

3 + 2 + 2 4 + 1 + 2

5 + 2

1 + 5

=7

6=

6 =

= 7

=

=

Teacher Do we have to fix all of these equations to make them true?

Students No! Just the wrong ones! Can I put a 6 card in for that one on the top? Then it would be right.

Teacher Sure! Are there any others that need to be fixed?

Students Yeah! 3 + 1 + 2 is 6 not 7! Can I fix it? Six is not the same as 5 + 2. Put in a 7 card for that one!

37



9. Introduce the new Unifix Cube Equations Work Place. Show students copies of the worksheets, and model the activities as needed. Demonstrate that they’ll need to choose a sheet and build 4 trains to match the number they selected. Remind them to use only 2 or 3 colors, with like colors grouped to make each train. Then they’ll need to color in the trains on the sheet to match, write an equation for each, and complete the problem at the bottom of the sheet. After that, they’ll turn the sheet over, and complete the problems on the back with the help of the number line. Note with them that there are worksheets for all the numbers, 5–10, so they can choose their own challenge level.

10. If time allows, send students out to do Work Places.

NAME DATE

Set A4 Number & Operations: Equivalent Names Blackline Run 10–15 copies back-to-back with page A4.14.

U Equations, 6’s page 1 of 2


a

__________________________ = 6

b

6 = __________________________

c

__________________________ = 6

d

6 =__________________________

2 Circle T or F.

a 1 + 4 = 6 T or F b 6 = 2 + 2 + 3 T or F

c 6 = 3 + 3 T or F d 4 + 2 = 6 T or F

NAME DATE

Set A4 Number & Operations: Equivalent Names Blackline Run 10–15 copies back-to-back with page A4.11.

Unifi x Cube Equations, 6’s page 2 of 2

3 Add 3 0 3 4 0 2 + 3 + 6 + 1 + 0 + 5 + 3 ____ ____ ____ ____ ____ ____

3 2 4 4 2 5 + 2 + 4 + 2 + 1 + 2 + 1 ____ ____ ____ ____ ____ ____

3 + 3 =________ 2 + 1 + 2 = ________ 0 + 6 = ________

4 Subtract 5 6 5 6 5 6 – 2 – 0 – 4 – 1 – 3 – 5 ____ ____ ____ ____ ____ ____

6 4 6 4 6 6 – 2 – 2 – 4 – 3 – 3 – 6 ____ ____ ____ ____ ____ ____

6 – 3 =_________ 5 – 2 = _________ 4 – 3 = _________

Can I help you?

0 1 2 3 4 5 6 7 8 9 10

Note When you do Friday’s Figuring in the Number Corner over the coming months, take the opportunity to reinforce the idea that equals means “the same as”. You can do this by placing the day’s date at the beginning of some of the equations you record on the charts, rather than always at the end. If you also read the equals sign as “equals” sometimes, and “is the same as” sometimes, students will make a strong connection between the two by the end of the year.


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name date




a

__________________________ = 5

b

5 = __________________________

c

__________________________ = 5

d

5 =__________________________

2 Circle T or F.

a 1 + 4 = 5 T or F b 5 = 1 + 2 + 2 T or F

c 5 = 2 + 2 T or F d 2 + 3 = 5 T or F



name date



3 Add 2 5 3 4 0 1 + 3 + 0 + 1 + 0 + 3 + 2 ____ ____ ____ ____ ____ ____

2 3 3 1 2 2 + 2 + 0 + 2 + 1 + 2 + 1 ____ ____ ____ ____ ____ ____

2 + 3 =_________ 2 + 2 = _________ 3 + 1 + 1 = _________

4 Subtract 4 5 4 5 3 5 – 2 – 0 – 4 – 1 – 3 – 5 ____ ____ ____ ____ ____ ____

5 3 5 4 5 3 – 2 – 0 – 4 – 1 – 3 – 2 ____ ____ ____ ____ ____ ____

5 – 3 =_________ 4 – 1 = _________ 5 – 4 = _________

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40


name date




a

__________________________ = 6

b

6 = __________________________

c

__________________________ = 6

d

6 =__________________________

2 Circle T or F.

a 1 + 4 = 6 T or F b 6 = 2 + 2 + 3 T or F

c 6 = 3 + 3 T or F d 4 + 2 = 6 T or F



name date



3 Add 3 0 3 4 0 2 + 3 + 6 + 1 + 0 + 5 + 3 ____ ____ ____ ____ ____ ____

3 2 4 4 2 5 + 2 + 4 + 2 + 1 + 2 + 1 ____ ____ ____ ____ ____ ____

3 + 3 =________ 2 + 1 + 2 = ________ 0 + 6 = ________

4 Subtract 5 6 5 6 5 6 – 2 – 0 – 4 – 1 – 3 – 5 ____ ____ ____ ____ ____ ____

6 4 6 4 6 6 – 2 – 2 – 4 – 3 – 3 – 6 ____ ____ ____ ____ ____ ____

6 – 3 =_________ 5 – 2 = _________ 4 – 3 = _________

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42


name date




a

__________________________ = 7

b

7 = __________________________

c

__________________________ = 7

d

7 =__________________________

2 Circle T or F.

a 3 + 4 = 7 T or F b 7 = 2 + 3 + 1 T or F

c 7 = 3 + 4 T or F d 7 + 0 = 7 T or F



name date



3 Add 3 3 2 4 7 2 + 4 + 3 + 5 + 2 + 0 + 3 ____ ____ ____ ____ ____ ____

2 4 4 6 6 5 + 2 + 3 + 1 + 1 + 0 + 2 ____ ____ ____ ____ ____ ____

3 + 4 =________ 2 + 2 + 2 = ________ 5 + 2 = ________

4 Subtract 7 7 6 7 6 7 – 7 – 0 – 4 – 1 – 3 – 5 ____ ____ ____ ____ ____ ____

7 5 7 5 7 7 – 2 – 2 – 4 – 3 – 3 – 6 ____ ____ ____ ____ ____ ____

7 – 2 =_________ 6 – 4 = _________ 7 – 4 = _________

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44


name date




a

__________________________ = 8

b

8 = __________________________

c

__________________________ = 8

d

8 =__________________________

2 Circle T or F.

a 5 + 1 = 8 T or F b 8 = 2 + 3 + 1 T or F

c 8 = 4 + 4 T or F d 3 + 5 = 8 T or F



name date



3 Add 4 4 3 1 8 3 + 4 + 3 + 5 + 2 + 0 + 3 ____ ____ ____ ____ ____ ____

3 5 7 6 2 2 + 2 + 3 + 1 + 2 + 5 + 6 ____ ____ ____ ____ ____ ____

4 + 3 =________ 5 + 3 = ________ 4 + 2 + 2 = ________

4 Subtract 7 8 8 8 7 8 – 5 – 0 – 4 – 1 – 3 – 5 ____ ____ ____ ____ ____ ____

8 7 8 8 8 8 – 2 – 2 – 8 – 7 – 3 – 6 ____ ____ ____ ____ ____ ____

8 – 5 =_________ 7 – 5 = _________ 8 – 4 = _________

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46


name date




a

__________________________ = 9

b

9 = __________________________

c

__________________________ = 9

d

9 =__________________________

2 Circle T or F.

a 5 + 4 = 9 T or F b 9 = 3 + 3 + 3 T or F

c 9 = 3 + 6 T or F d 2 + 7 = 9 T or F



name date



3 Add 5 4 3 2 9 4 + 4 + 4 + 6 + 2 + 0 + 3 ____ ____ ____ ____ ____ ____

7 5 8 6 4 2 + 2 + 2 + 1 + 2 + 5 + 6 ____ ____ ____ ____ ____ ____

4 + 3 =________ 5 + 2 + 2 = ________ 6 + 2 = ________

4 Subtract 8 9 8 9 7 9 – 5 – 0 – 4 – 1 – 3 – 5 ____ ____ ____ ____ ____ ____

9 7 9 9 9 8 – 2 – 2 – 8 – 9 – 3 – 6 ____ ____ ____ ____ ____ ____

9 – 4 =_________ 9 – 6 = _________ 9 – 7 = _________

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48


name date




a

__________________________ = 10

b

10 = __________________________

c

__________________________ = 10

d

10 =__________________________

2 Circle T or F.

a 3 + 5 = 10 T or F b 10 = 2 + 4 + 4 T or F

c 10 = 5 + 5 T or F d 3 + 6 = 10 T or F



name date



3 Add 5 4 3 2 10 5 + 5 + 5 + 7 + 3 + 0 + 3 ____ ____ ____ ____ ____ ____

8 5 9 6 4 1 + 2 + 2 + 1 + 3 + 6 + 6 ____ ____ ____ ____ ____ ____

3 + 4 + 2 =________ 2 + 8 = ________ 2 + 3 + 5 = ________

4 Subtract 9 10 8 10 9 10 – 5 – 0 – 4 – 1 – 3 – 5 ____ ____ ____ ____ ____ ____

10 7 10 10 10 10 – 2 – 3 – 8 – 7 – 3 – 10 ____ ____ ____ ____ ____ ____

10 – 4 =_________ 10 – 6 = _________ 10 – 9 = _________

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50

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Foundations of Algebraic Thinking - ode.state.or.us · Foundations of Algebraic Thinking: ... Number & Operations & Algebra Standards Continuum/Trajectory ... Each table group will