Patterns and Pre-Algebra

Kindergarten–Grade 3

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Why Teach Patterns and Pre-Algebra?

Simple Patterns Are Everywhere

There Are Different Types of Patterns—Numerical/Non-numerical

The Same Pattern Can Be Expressed in Different Ways

Your Questions

Break

There Are Different Types of Patterns—Repeating and Increasing/Decreasing

Your Questions

Lunch

Pattern Rules Generalize Relationships

Your Questions

Break

Equations Express Relationships Between Numbers

Your Questions and Feedback

Agenda

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Why Teach Patterns and Pre-Algebra?

Working with patterns enables students to make connections both within and

beyond mathematics.

Through the study of patterns, students come to interpret their world mathematically and value

mathematics as a useful tool.

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Why Teach Patterns and Pre-Algebra?

By generalizing patterns, students develop strategies that can be used to solve a wide range of problems.

Mathematics is seen as reasoning rather than solving one unrelated problem after another.

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Exploring patterns and pre-algebra in elementary school lays the foundation for the study of formal algebra. Rather than a

new topic, algebra becomes a natural extension of the elementary curriculum

and is often defined as generalized arithmetic and geometry.3 + = 5

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• Think for 20 seconds

• Write and draw silently for 60 seconds

• Switch papers with another table

• Start again

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Learning Task - Predictable Stories

Note: From Crystal Cochrane, St. Leo Catholic Elementary School, Grade 1, Edmonton, AB, 2006. 6

Kindergarten – actions, sound, colour, size, shape, orientation

Grade 1 – add diagrams and events

Grade 2 – focus on attributes and numbers

Grade 3 – expressed as concrete, pictorial, symbolic

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Learning Tasks

Chairs

½When Will We

Reach ½ ?

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Pattern Puzzles

•Select an attribute card

•Make a core unit with 3–5 elements, using this attribute

(big, big, small)

(square, triangle, triangle)

(yellow, blue, red)

•Repeat the pattern 2 more times

•Ask your partner to describe your pattern

Learning Tasks

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Non-numerical patterns can be translated into a letter code (ABBA)

and then extended to make predictions and solve problems.

A AB B

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Learning Tasks – Translating Patterns

Mix and Match

•Create a 2- to 4-element core, using your choice of materials; e.g., colour, orientation, size.

•Extend your pattern 2 more times.

•Find someone else in the room with the same pattern code.

These are both AABB patterns.

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Patterns can be repeating and made up of a core set of elements—a core unit that is iterated.

Patterns can be increasing or decreasing and created by orderly change.

9 7 5 3

32 16 8 4 212

Learning Tasks – Repeating Patterns

The Stamping Machine

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Learning Tasks – Repeating Patterns

Rows and ColumnsCyclical Patterns

http://standards.nctm.org/document/eexamples/chap4/4.1/index.htm

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Learning Tasks – Repeating Patterns

Cups

15Note: From Listening Kit Level 4 (pp. 22–24), by D. Gagné, 2001, Red Deer, AB: Themes & Variations. Copyright 2001 by the author. Adapted with permission.

Learning Tasks

Predicting Patterns

Making the link between repeating and increasing patterns

2 31

5 10 15

a) What would the 20th shape be?b) What would the 30th shape be?c) What would the 32nd shape be?

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Learning Tasks

5

2 31

10 15

30 31 322515105 20

2 322717127 22

30 32 33 34 352515105 20

What would the 32nd shape be?

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Learning Tasks

5

2 31

10 15

a) Create a pattern in which the 20th shape is a .b) Create a pattern in which the 12th shape is a .c) Create a pattern in which the 6th and 9th shapes are both .

Your Turn

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Learning Tasks – Increasing/Decreasing Patterns

Critters That Grow

Frame 1 Frame 2 Frame 3 Frame 4

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Learning Tasks – Increasing/Decreasing Patterns

Frame 1 Frame 2 Frame 3 Frame 4

legs 2 4 6 8 ?

body parts 1 2 3 4 5

Add 2 legs each time, skip count by 2 (recursive), legs go up by

twos, bodies go up by ones.

Look at relationships across categories (function), double

the body parts.

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Developing relational thinking

1. Build and extend to the next frame:

How many legs would the critter have the next year?

2. Keep extending the pattern, frame by frame:

The next year after that?

3. Identify a pattern (often a skip-counting pattern):

What do you notice?

4. Push students to generalize:

How many legs will the critter have in 10 years? In 100 years?

How can you find the number of legs at any age?21

Learning Tasks – Increasing/Decreasing Patterns

Legs and heads 3 5 ? ? ? ? ?

Body parts 1 2 3 4 5 10 100

What’s the relation?

Doubles plus 1

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Learning Tasks – Increasing/Decreasing Patterns

Note: Caterpillars, Worms and Pattern Block Trees are adapted from Lessons for Algebraic Thinking: Grades K–2, pp. 2–11, 89–98, 157–170, by Leyani von Rotz and Marilyn Burns. Copyright © 2002 by Math Solutions Publications.

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Pattern rules reveal mathematical relationships.

Pattern rules describe how a pattern grows and can be used to make logical predictions.

What changes?

What stays the same?

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A pattern rule must account for all elements of a pattern, including the first one.

Body Parts 4 7 10 13 ? ? ?

Age 1 2 3 4 5 10 100

Body parts: Start at 4 and add 3 each time

Age: Start at 1 and add 1 each time

Relationship: Body parts—3 times the age plus 125

Learning Tasks – Pattern Rules

Addition Charts, Hundred Charts, What Are the Clues?

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Two of Everything by Lily Toy HongIllustrations on slides 27 to 36 and text on slides 28 to 34 are reproduced from Two of Everything by Lily Toy Hong. Copyright ©1993 by Lily Toy Hong. Excerpts reprinted by permission of Albert Whitman & Company. All rights reserved. 27

28

29

30

31

32

33

34

35

Would you rather have a doubling pot and a loonie, if you could only use the pot ten times, or…$1 000?

Note: Excerpted and reprinted with permission from National Council of Teachers of Mathematics. (2003). Reflections. Retrieved November 20, 2006, from http://my.nctm.org/eresources/reflections, copyright 2003 by the National Council of Teachers of Mathematics. All rights reserved.

Create your own magic pot. Make up a pattern rule for your pot. Show

what happens on an in-out chart.Note: Adapted from Lessons for Algebraic Thinking: Grades K–2, by Leyani von Rotz and Marilyn Burns. Copyright © 2002 by Math Solutions Publications.

36

3 + 2 = 5

Equality (=) expresses a relationship of balance between numbers.

Inequality () expresses a relationship of imbalance.

3 + 1 ≠ 5

37

What do elementary

students think the equal sign

means?

38

Note: Excerpted and reprinted with permission from Fennel, F., & Rowan, T. (January 2001). Representation: An Important Process for Teaching and Learning Mathematics. Teaching Children Mathematics, 7(5), 288–292, copyright 2001 by the National Council of Teachers of Mathematics. All rights reserved.

39

Equality and inequality between quantities can be considered as:

• whole to whole relationships (5 = 5)• part–part to whole relationships (3 + 5 = 8)• whole to part–part relationships (8 = 5 + 3)• part–part to part–part relationships (4 + 4 = 3 + 5).

40

7 12 1712 and

17Other

Grades 1 and 2 5% 58% 13% 8% 16%

Grades 3 and 4 9% 49% 25% 10% 7%

Grades 5 and 6 2% 76% 21% 1% 0%

8 + 4 = + 5

Note: From Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School (p. 4), by T. P. Carpenter, M. L. Franke and L. Levi, 2003, Portsmouth, NH: Heinemann. Copyright 2003 by the authors. Reprinted with permission.

• The answer comes next: 8 + 4 = 12 + 5• Use all the numbers (overgeneralizing associative property): 8 + 4 = 17 + 5• Extending the problem: 8 + 4 = 12 + 5 = 17

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Robin: Second-grade student

18 + 27 = + 29

“Twenty-nine is 2 more than 27, so the number in the box has to be 2 less than 18 to make the 2 sides equal. So it’s 16.”

Note: From Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School (p. 4), by T. P. Carpenter, M. L. Franke and L. Levi, 2003, Portsmouth, NH: Heinemann. Copyright 2003 by the authors. Reprinted with permission.

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4 + 5 = + 3

Kevin

Note: From Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School, by T. P. Carpenter, M. L. Franke and L. Levi, 2003, Portsmouth, NH: Heinemann. Copyright 2003 by the authors. Reprinted with permission. 43

Learning Task – Double Dominos

44

Mini Lessons – True/False

3 + 5 = 88 = 3 + 58 = 83 + 5 = 5 + 33 + 5 = 4 + 4

Developing an understanding of

the equal sign

Note: From Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School (p. 4), by T. P. Carpenter, M. L. Franke and L. Levi, 2003, Portsmouth, NH: Heinemann. Copyright 2003 by the authors. Reprinted with permission.

45

Other True/False Contexts

9 + 5 = 149 + 5 = 14 + 09 + 5 = 0 + 149 + 5 = 14 + 19 + 5 = 13 + 1

Using zero to introduce part-part = part-part

equations

How could you change the false statements so that they are true?

Place Value

56 = 50 + 687 = 7 + 8093 = 9 + 3094 = 80 + 1494 = 70 + 24

46

ChallengeDetermine if these equations are

true or false without calculating the actual sum or difference. Use

relational thinking!

37 + 56 = 39 + 5433 – 27 = 34 – 26471 – 382 = 474 – 385674 – 389 = 664 – 379583 – 529 = 83 – 29

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Very Able To Do Variables

+ = 12

+ = 12

+ = 129

+ = 12948

Join

Result Unknown

Connie had 15 marbles. Juan gave her 28 more marbles. How many marbles does Connie have altogether?

Change Unknown

Connie has 15 marbles. How many more marbles does she need to have 43 marbles altogether?

Start Unknown

Connie had some marbles. Juan gave her 15 more marbles. Now she has 43 marbles. How many marbles did Connie have to start with?

Separate

Connie had 43 marbles. She gave 15 to Juan. How many marbles does Connie have left?

Connie had 43 marbles. She gave some to Juan. Now she has 15 marbles left. How many marbles did Connie give to Juan?

Connie had some marbles. She gave 15 to Juan. Now she has 28 marbles left. How many marbles did Connie have to start with?

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Learning Tasks – What’s In the Bag?

50

Learning Tasks – What’s In the Bag?

51

Equalization and

Compare

Difference Unknown

Connie has 43 marbles. Juan has 15 marbles. How many more marbles does Connie have than Juan? (Compare)How many more marbles does Juan need to have as many as Connie? (Equalize)

Quantity Unknown

Juan has 15 marbles. Connie has 28 more than Juan. How many marbles does Connie have?

Referent Unknown

Connie has 43 marbles. She has 15 more marbles than Juan. How many marbles does Juan have?

Part-Part-Whole

Quantity Unknown

Connie has 15 red marbles and 28 blue marbles. How many marbles does she have?

Part Unknown

Connie has 43 marbles. 15 are red and the rest are blue. How many blue marbles does Connie have?

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Mini Lessons – Open Number Sentences

The teacher writes an open-number sentence on the board and asks the students how to make the statement true. Students can justify

their responses; e.g., using balance models, comparing distances on a number line.

3 + 5 = 8 = 3 + 8 = 3 + 5 = + 33 + 5 = + 4

53

Emma

Note: From Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School, by T. P. Carpenter, M. L. Franke and L. Levi, 2003, Portsmouth, NH: Heinemann. Copyright 2003 by the authors. Reprinted with permission.

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Each problem that I solved became a rule which served afterwards

to solve other problems.

René Descartes

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