Chapter 14 Algebraic Thinking To accompany Helping Children Learn Math Cdn Ed, Reys et al. ©2010 John Wiley & Sons Canada Ltd.  

Guiding Questions What does algebraic thinking mean for elementary students? How do patterns help children develop algebraic thinking and ideas? How can you encourage algebraic thinking as children solve problems? How can you use children’s understanding of relations to help develop the concept of variables? How do modelling, generalizing, and justifying intertwine?

Algebra is… a study of patterns and relationships. a way of thinking. an art, characterized by order and internal consistency. a language that uses carefully defined terms and symbols. a tool.

Problems, Patterns, and Relations The teaching of algebra in elementary school should build on ideas that are an essential part of the curriculum. Problems, patterns, and relations are each an essential part of elementary school mathematics and are components of algebra.

Problems (Routine) Routine problems are often considered exercises for practicing computation, but they can also be used to build algebraic understanding. You will find routine problems like these in every textbook: Before her birthday, Jane had 8 toy trucks. At her birthday party, she was given some more trucks. That night she counted all her trucks and found she had 15. How many trucks was she given at her party? Pat took 4 pails to the beach. She had 3 shovels and 2 rakes in each pail. How many tools did she take to the beach?

Problems: Nonroutine Patterns Many nonroutine problems, especially look-for-a-pattern problems and number puzzles, lend themselves well to an algebraic approach. What would the tenth pattern in this series be?

Problems: Nonroutine Number Puzzles Calendar. Choose any 3 consecutive days (e.g., Tuesday, Wednesday, and Thursday) for any 3 consecutive weeks in a month (e.g., July 8–10, 15–17, and 22–24). How can you easily find the sum of all 9 dates? S M T W F 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

Patterns: Repeating Research has shown that children should begin recognizing and thinking about patterns by describing and extending patterns with pictorial models.

Patterns: Growing Sidewalk-garden problem Master 14-3: Using Patterns Draw pictures to illustrate Term 4 and Term 5.

Growing Patterns (cont.) Growing patterns, like repeating patterns can be thought of in different ways, and we need to help children look at different ways in which patterns grow. Consider the following numeric growing patterns. What do you think the next term will be and why? Share your thinking with others. Pattern A: 1, 2, 4,… Pattern B: 1, 4, 9, 16, 25, … Pattern C: 1, 1, 2, 3, 5 Master 14-3: Using Patterns

Relations: Properties of Number Which statement is not true? Why? If you add a number to a given number and then subtract that same number from the sum, then you get the given number. If you subtract 0 from a given number, add the given number to the difference, then the sum is twice the given number. If you multiply any two numbers, the product is larger than each of the two numbers. If you divide a positive whole number by a proper fraction, the quotient is larger than the positive whole number.

Relations: Functions A function is a way of expressing a relation when two sets of numbers are related in such a way that each number in the first set is related to one and only one number in the second set.

Language and Symbols of Algebra A child may describe the rule in the function table as “Double the input + 1.” Eventually, if the input was labeled x and the output was labeled y as shown in the table, you could write that the change was 2x + 1 or that y = 2x +1. x y 3 7 5 11 6 13 29 53

Equality and Inequality Balance scales are an effective way to teach equality and inequality as the scale will show when an equation is out of balance or not equal.

Variables Use of Variable Representation Characteristics Placeholder Specific value for a Generalization a – a = 0 All values of a make the sentence true Function H = 2 × B Each value of B produces one and only one value of H 4/20/2017

Expressions and Equations Think of an expression as representing a phrase and an equation as representing a complete sentence. X + 3 is an expression A + 7 =10 is an equation

Modelling Routine Problems It is important to have young children modelling through concrete objects or semi-concrete representations what is going on in simple word problems. Master 14-5: Palindromes

Modelling Routine Problems Consider the problem we looked at earlier: Before her birthday, Jane had 8 toy trucks. At her birthday party, she was given some more trucks. That night she counted all her trucks and found she had 15. How many trucks was she given at her party? The child drew 8 red trucks and then draws green trucks, one at a time, to determine that 7 more are needed to make 15. Master 14-5: Palindromes 18

Justifying Answers Having children justify their answers on their approaches to problems can help them understand the mathematics and gain confidence in their knowledge and skills. Master 14-5: Palindromes 19

Generalizing Answers Finding a rule that generalizes a pattern is a type of generalization. One of the most powerful ways to tie repeating patterns to algebra is to number the terms as you would for growing patterns. Then ask children questions to guide their thinking about patterns in a more general way. For example: What shape is the ninth term? The twelfth term? Master 14-5: Palindromes 20

Relations: Functions Functions may be treated as patterns. By making tables and discussing the patterns that students see, you can help students find the general rule (the function) that relates to the two quantities. Functions are also a good way to study change in subject areas other than mathematics, such as plant growth in science. This change can then be modelled through a graph to help children see the relationships. Master 14-5: Palindromes 21

Relations: Properties of Numbers Investigating properties of numbers helps students to develop their ability to make and justify generalizations and to test conjectures. Often older children can be helped to explain their generalizations using algebra. Master 14-5: Palindromes 22

An Algebraic Activity The following slides show an activity which links well to the study of algebra for upper intermediate students. Master 14-5: Palindromes 23

Palindromes These are word palindromes: WOW TOOT NOON Which of the following numbers are palindromes? 6556 1231 44 232 9887 How would you define a palindrome? Circle all the palindromes on a hundred chart. Master 14-5: Palindromes 24

Palindromes (cont.) Consider 57. Can we transform 57 to form a palindrome? Start with 57 + 75 = 132 (not a palindrome) Try 132 + 231 = 363 (Palindrome!) We call 57 a two-step palindrome. Master 14-5: Palindromes

Palindromes (cont.) Choose another number and see how many steps it takes to form a palindrome. Investigate other numbers on the hundred board. Mark one-step, two-step, three-step palindromes using different colours for each category. Is there a pattern? Master 14-5: Palindromes

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