Foundations of Algebraic Thinking - ode.state.or.us · • Recording sheet ... Close to 100, Close to 1000 Groups of Ten Solve Two Ways! print . 10 What number properties and strategies

Foundations of Algebraic Thinking: Place Value K-2

Building Instructional Leadership Across Oregon

“Developing Algebraic Thinking”

Session 3 March 2010

K-2 Goal:

Students use number sense, place value, and patterning to construct concrete models for thinking algebraically. I can describe how students begin to represent their thinking K-2 and how this builds more abstract thinking, models, and

representations through grade 8.

Name Date

Counters Number Word

glass blobs

Tens

Singles buttons

Tens

Singles square tile

Tens

Singles toothpicks

Tens

Singles lima beans

Tens

Singles

2

Foundations of Algebraic Thinking: Place Value K-2

March Follow Up K-8 Math Institute

Foundations for Algebraic Thinking

Vocabulary 3 Activities

Activity Descriptions, Note Pages 5

Participant Activity Planner 9 Number Talk: Ten Frame Flash 10 A. Compatible Pairs 11 B. Groups of Ten 20 C. Three Other Ways 24 D. The Other Part of 100 29 E. Close to 100, Close to 1000 31 F. Solve Two Ways 34

3

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 final quantity. 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.

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Comparison (subtraction): 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 (subtraction): Some contexts will involve ‘taking away’ a specified amount and ask about the remaining amount.

A Note on Fluency: Fluency does not relate to a specific number of math facts correctly performed in a specific amount of time. (In fact, research continues to show the practice of ‘timed tests’ most often detrimental to student learning and understanding!) Students need to have conceptual understanding of operations. Once they have this understanding teachers can help them learn various ways of remembering these basic facts. Why does it work that way? What does the student already know that will help them remember new facts? Building a connection between concrete models and representations to more abstract numbers provides the foundation for conceptual understanding and computational fluency. Look for strategies embedded in teaching materials to help identify how students work toward fluency.

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Number Talk: Ten Frame Flash: Teaching Student‐Centered Mathematics; Grades K‐3 (2006) Van de Walle & Lovi, p.47 Materials

• Mini Ten Frames • Chart Paper

• Download blacklines at www.ablongman.com/vandewalleseries

Activity This activity helps children see groups of ten. This Math Talk provides a wonderful setting for checking for understanding, sharing ideas, and multiple strategies. Recording different ways of “seeing” and thinking (as we do in class) facilitates students’ understanding of a wider range of ideas and makes individual thinking available to others. When we don’t take the time to record the ideas, we lose out on these wonderful opportunities. Keeping the Math Talk posters will also give teachers ways to connect new knowledge and concepts to prior learning. Pull out a previous poster, “Remember when we…?” __________________________________________________________________________________________

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Activity A: Compatible Pairs Teaching Student‐Centered Mathematics; Grades K‐3 (2006) Van de Walle & Lovi, p.148 Materials • Direction Cards • Compatible Pairs Record Sheet

• Compatible Pairs Number Sheet

Activity Students work with a partner, take turns, and find a combination of numbers that equals 50, 100, 1000, or some other choice. Students justify their choice to their partner, and then cover the two numbers with a cube if they’re correct. Players continue until they’ve matched all the pairs. __________________________________________________________________________________________

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Activity B: Groups of Ten Teaching Student‐Centered Mathematics; Grades K‐3 (2006) Van de Walle & Lovi, p.130 Materials • Containers of Counters • Number word sheet

• Recording sheet • Counting Mat

Activity Students get a recording sheet and choose a container of counters. The student dumps out the contents, records the amount as a number word, and groups the objects in as many tens as possible. The final count is then recorded on the recording sheet. Children collect their objects and choose another container. __________________________________________________________________________________________

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Activity C: Three Other Ways Teaching Student‐Centered Mathematics; Grades K‐3 (2006) Van de Walle & Lovi, p.133 Materials • Number strips with written numbers • glue

• Recording sheets • Items to represent hundreds, tens & ones (toothpicks, dots, color tiles, etc.)

Activity Students, working in groups or pairs, get a recording sheet and choose a number strip. They glue the number strip on their recording sheet. Using standard representations (dots for ones, toothpicks for tens, & squares for hundreds), students record their number one way. Then they find three other ways to record them. __________________________________________________________________________________________

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Activity D: The Other Part of 100 Teaching Student‐Centered Mathematics; Grades K‐3 (2006) Van de Walle & Lovi, p.147 Materials • Mini ten‐frame cards Activity Two students work together with min ten‐frame cards. One student makes a two‐digit number. Then both students work mentally to determine what would go with the first number to make 100. Students justify their answer. Each person takes turns choosing the first number. __________________________________________________________________________________________

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Activity E: Close to 100, Close to 1000 Grade 3, Grade 4 Investigations, 1st Edition (Pearson) Materials • Game sheet • Numeral Cards 0‐9 Activity Six cards are dealt to each player. Each player uses any four of these cards to make two numbers, that when added, come as close as possible to a total of 100. __________________________________________________________________________________________

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Activity F: Solve Two Ways Materials • Sample problems • glue

• Blank paper

Activity Choose a problem. Glue it down (top) of a blank page. (Add name/date!) Estimate an answer. Jot this down – students should have an idea of a reasonable answer. Solve ‐ Is this a reasonable answer? Solve another way – reinforces multiple strategies and gives students another way to solve to verify. Write a story (context) to match the original

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Name Date

Counters Number Word glass blobs

Tens Singles

buttons

Tens Singles

square tile

Tens Singles

toothpicks

Tens Singles

lima beans

Tens Singles

NAME DATE Work with some of the activities. Try to get to at least 3. Identify the skill targeted (can you match this to a grade level or standard(s)?) Identify ways the activity can be modified (for both accelerated learning and depth).

Three Other Ways The Other Part of 100

Compatible Pairs

Close to 100, Close to 1000

Groups of Ten

Solve Two Ways!

print

10

What number properties and strategies are evident in this activity?

Number Talk: Ten Frame Flash

A. Compatible Pairs This activity can be done as a worksheet activity or with the full class using the overhead projector. Prepare a transparency or duplicate a page with a search task. For today you will select a mat to work with and follow the rules below:

Markers of any type can be used to cover number pairs…

11

Name Date Make __________

Compatible Pairs Record Sheet

=

Represent your thinking

=


=


=


12

Compatible Pairs Record Sheet Name Date Make __________

+

+

+

=Represent your thinking

=


+ =

=



13

Compatible Pairs! Make 50

14

Compatible Pairs! Make 1000

15

-

Compatible Pairs! Make ?

16

Compatible Pairs! Make ?

17

18

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B. Groups of Ten Choose a container of counters. Get a record sheet and write your name on it Dump out and count the contents

Number Words

eleven ten one

twelve twenty two

thirteen thirty three

fourteen forth four

fifteen fifty five

sixteen sixty six

seventeen seventy seven

eighteen eighty eight

nineteen ninety nine

Record the amount as a number word Group the counters in as many tens as possible (use a counting mat to help you!) Name Date

Counters Number Word

glass blobs

Tens

Singles buttons

Tens

Singles square tile

Tens

Singles toothpicks

Tens

Singles lima beans

Tens

Singles

Record this on your paper.

Return the counters to the container.

20

Number Words

eleven ten one

twelve twenty two

thirteen thirty three

fourteen forth four

fifteen fifty five

sixteen sixty six

seventeen seventy seven

eighteen eighty eight

nineteen ninety nine

Groups of Ten Reference Page 21

Name Date

Counters Number Word glass blobs

Tens Singles

buttons

Tens Singles

square tile

Tens Singles

toothpicks

Tens Singles

lima beans

Tens Singles

Groups of Ten Record Sheet 22

10-frame check mats 23

C. Three Other Ways Students work in groups or pairs.

or

1. Draw out a strip with a written number. Read it with your partner(s).

Five hundred seventy-five Five hundred seventy-five

2. Glue the strip of paper down on your record sheet. Five hundred seventy-five 3. Build the number using standard representation (with your model) 4. Find and record three more ways to show this number.

24

http://www.thehaydenhouse.com/Envelope 4.gif

Name Date

Three Other Ways Record Sheet (Tens, Units)

glue written number here

Tens ______ Tens ______ Tens ______ Tens ______ Ones ______ Ones ______ Ones ______ Ones ______

Build the number with standard representation. Find three more ways!





25

Name Date

Three Other Ways Record Sheet (Hundreds, Tens, Units)

Write Numeral ____________ + _____________ + ______________ = _______________ Use standard Notation and build to match. Next, Find at least three other ways to build the same number.

Hundreds _______ Hundreds _______ Hundreds _______ Hundreds _______Tens ______ Tens ______ Tens ______ Tens ______Ones ______ Ones ______ Ones ______ Ones ______





26

one hundred forty-four eight hundred twenty-two

five hundred seven three hundred eleven

two hundred eighty-nine six hundred nineteen

nine hundred twenty-one five hundred forty-seven

seven hundred fifty-one six hundred thirty-four

two hundred sixty-three three hundred twenty-eight

two hundred seventy-six six hundred thirty-four

one hundred eighty-nine one hundred forty-six

two hundred sixty-two four hundred thirty-five

one hundred six three hundred sixteen

one hundred ten five hundred sixty

eight hundred three hundred eighty-two

three hundred fifteen two hundred

three hundred forty six hundred fifty-two

six hundred ninety-six four hundred seventy

seven hundred fourteen seven hundred seventy-eight

five hundred thirty-nine five hundred twenty-four

five hundred seventy-seven seven hundred twenty-nine

four hundred ninety-four one hundred eighty-six

seven hundred fifty-five three hundred fifty-eight

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forty-four twenty-two

seven eleven

eighty-nine nineteen

twenty-one forty-seven

fifty-one thirty-four

sixty-three twenty-eight

seventy-six thirty-four

eighty-nine forty-six

sixty-two thirty-five

six sixteen

one hundred ten sixty

eight hundred eighty-two

fifteen forty-eight

forty fifty-two

ninety-six seventy

fourteen seventy-eight

thirty-nine twenty-four

d seventy-seven twenty-nine

ninety-four eighty-six

fifty-five fifty-eight

28

D. The Other Part of 100 Two students work together with a set of little ten-frame cards.

1. One student makes a two-digit number.

2. Then both students work mentally to determine what

goes with the ten-frame to make 100.

3. They write their solutions on paper and then check by making the

other part with the cards to see if the total is 100.

4. Students take turns making the starting number.

29

Name Date

The Other Part of 100 Record Sheet

1

Represent first number What’s the other part of 100?

Justify Write the numeral Write the numeral

2



3



4



30

Inve

stig

atio

ns, 1

st E

ditio

n, G

rade

3 (P

ears

on)

+ = ______ + = ______

32

+ = ______ + = ______

33

F. Solve Two Ways! Choose a problem and stick it to the top of your work paper. Estimate an answer… write it down Solve 2 different ways Hmmm…is my answer “reasonable?” Write a story (give the problem a context) to match the original problem. Note: Print number sentences on return address labels or cut apart (to be glued on to blank paper)

34

354 + 537 = 307 - 289 = 1025 - ? = 375

789 + 1038 = 6521 - 387 = 73 + ? = 1000

1578 + 199 = 1000 – 271 = 100 – ? = 33.75

325+ 327+ 324= 1001 – 552 = 6.75 – ? =3.21

2957 + 123 = 352 - 175 10.25 – ? = 6.82

55 + 175 + 350= 210 – 35 = 9974 – ? = 55

724 + 1268 = 149 – 94 = ?- 1,581 = 16,785

562+625+256= 333 – 39 = ? + 526 = 2583

253+ 257+ 255= 17,458 – 873= ? – 73.2 = 2052

642 + 658 = 2952- 766 = ? – 0.671 = 372.6 35

Documents

Foundations of Algebraic Thinking - ode.state.or.us · • Recording sheet ... Close to 100, Close to 1000 Groups of Ten Solve Two Ways! print . 10 What number properties and strategies