Level 6: Generalising
Materials
Instructions
Place True and False Cards in front of the student and say, “Some of these statements are true and some are false. Without calculating, can you tell me which ones are true and which are false and why?” … Note choices and whether or not student appears to be calculating. Explore reasoning where not obvious.
Place the cards in front of the student and say, “Two T-shirts and two drinks costs $44. One T-shirt and 3 drinks cost $30. … How much does each item cost? …
If little or no response, and/or the student appears uncertain about how to proceed, ask, “What is seems to be the problem here? … What would make this easier?” Note response.
If student produces a solution but his/her reasoning/working is unclear, ask, “Can you explain how you worked that out please?” Note student’s response.
6.1 Advice Rudric
A major source of student difficulties in algebra is the failure to recognise equivalence between different forms of the same relationship, or between the left and right sides of an equation. Early experiences in arithmetic, such as the representation of basic facts as equations (eg, 6 + 8 = 14, 7 × 8 = 56, 24 ¸ 6 = 4 etc) and ‘find the missing number’ tasks such as 3 + ? = 11, undoubtedly contribute to the view that the equal sign means ‘makes’ or ‘gives’ and that what follows is ‘the answer’. That is, that symbolic expressions are seen as instructions to operate rather than as objects that can be operated on in their own right and expressed in a number of different ways. This view is widely regarded as one the major reasons why students experience difficulty in formalising and representing numerical or algebraic relationships in later years.
Students’ responses to this task indicate the extent to which they can work with the notion of equivalence and understand some of the properties and conventions that underpin arithmetic expressions (eg, recognising and using the commutative property and understanding why it does not apply to subtraction or division).
| Observed response |
Interpretation/Suggested teaching response |
|---|
| Not all True/False Cards identified correctly, may need to calculate. Little/no response to the T-shirt problem, may try to represent problem in another form but unable to solve |
May not appreciate the significance of order (commutativity) or how multiplication can be distributed over addition, and/or be able to recognise these when applied to fractions - Invite students to create equivalent expressions for well known facts, eg, 3x4 = 2×6 = 24 ÷ 2 = 8+4 = 14-2 = 2+3+2+3+2 = and so on. Use Fraction Walls, Number Line Diagrams (see partitioning strategies in Level 4 ) and/or Cuisenaire Rods (where smallest rod is given a fractional name) to explore equivalent expressions involving fractions and decimals
- Explore consequences of reversing numbers, review order of operations
- Encourage students to contrast and compare alternative models, eg, 18×27 could be represented/thought of as 18 rows of 27 or 27 rows of 18, which is clearly larger than 19 rows of 18
- Explore simple word problems that lend themselves to algebraic reasoning, eg, Sally bought 2 snack bars and a drink for $6.15. If the drink cost $1.85, how much did she pay for one snack bar? Encourage students to describe what they need to do in words and explore possible representations
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| Identifies most of the True/False Cards correctly, may calculate some. Identifies the cost of a T-shirt and drink using a trial and error (guess and check) strategy |
Suggests some understanding of underlying properties and conventions and an arithmetic approach to finding unknowns - Consider renaming problems such as 37 + 45 in an equivalent form to demonstrate how equivalent statements can be used to solve problems, eg, 37 + 45 = 40 + 42 = 82 apply to problems like 4582 + 287 and generalise (eg, p + q = (p + m) + (q – m). Explore other generalisations like this
- Explore properties such as commutativity, distributivity, and associativity explicitly and examine other relationships such as whether or not (a + b) - c = (a - c) + b or (a × b) ÷ c = (a ÷ c) × b and why
- Use calculators to consolidate order of operations and the use of brackets to avoid ambiguities
- Discuss solution strategies for ‘missing number’ problems that do not lend themselves to guess and check, eg, 11 + - 2 = 4 ×
- Explore the use of ‘if ... then’ reasoning in problems like Spiders and Beetles (pdf - 196.34kb)
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| Correctly classifies all True/False Cards (A, B and E true). Identifies the cost of each item relatively quickly and efficiently using some form of algebraic reasoning (eg, recognises 1 T-shirt and 1 drink is half of $44 and uses this information to deduce that 2 drinks must be $8, so 1 drink is $4 etc, or solves using simultaneous equations methods) |
Suggests an understanding of equivalence as it is applied in non-conventional arithmetic settings and access to informal or formal strategies more closely related to algebraic thinking - Explore and justify concatenation, that is, the convention that the multiplication sign is omitted in situations like 4 × a to 4a
- Consider introducing/reviewing formal algebraic solutions to problems like the one above using linking words to show the reasoning involved, eg:
If 11 + a – 2 = 4a then
9 + a = 4a so
9 = 3a (a subtracted from both sides) and a = 3 - Discuss the value and power of using a logical sequence of equivalent statements as a solution strategy
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