Level 6: Generalising
Materials
Instructions
Place the cards in front of the student and say, “Can you sort all of these cards into two groups please? … Please tell me about each group?” Note (or photograph) student’s response, then ask, “Is there another way you could sort the cards into two groups?” Note (or photograph) student’s response and his or her reasons for the categorisation.
If student does not classify on the basis of primes/non-primes or factors/multiples, replace cards in front of the student and say, “Can you find 4 or more cards that might be connected in some way?” … Note student’s response and explore his or her reasoning.
Say, “I’m thinking of a number … If I add 57 to it I will get 0. What number am I thinking about?” Note student’s response and then say, "I’m thinking of a number … If I multiply it by 8 I will get 3. What number am I thinking about?” Note student’s response and explore thinking as appropriate.
6.2 Advice Rubric
An inadequate understanding of arithmetic, both in terms of the concepts that underpin the various representations of the four operations, and the properties that govern how we work with these more formally, is a related source of student difficulty in relation to algebra.
Student responses to this task indicate the extent to which they are able to recognise key differences between numbers (eg, primes and non-primes) and how these might be described more formally (ie, in terms of factors, multiples, and general expressions). They also reveal the extent to which students are inclined to think algebraically, that is, they are aware of and prepared to work with inverses and identities, as opposed to relying on arithmetic strategies such as ‘guess and check’ to solve equations.
| Observed response |
Interpretation/Suggested teaching response |
|---|
| Predominantly sorts cards on the basis of fairly superficial features such as the number of digits, or magnitude (eg, >100), may sort into odds and evens or identify factors/multiples as a common feature for 4 cards selected. Hesitant response to ‘think of a number’ questions, m |
Suggests that multiplication (and division) not thought about in terms of factors, multiples, and divisors, relationship between fractions and division may not be fully understood - Explore problems involving different concepts for multiplication and division (see Level 3, Level 4 and Level 5)
- Review the link between fractions and division (ie, that a / b means a ÷ b) using activities like Fraction Sequences (Stacey & MacGregor, 1997)
- Review the area and ‘for each’ (or Cartesian Product) ideas for multiplication that underpin the notions of factors, multiples, divisors
- Engage in games and activities that focus on factors such as Multo (Maths 300, Curriculum Corporation, 2003) and Multiples (eg, Stacey & MacGregor, 1997)
- Review the definition of prime numbers and how ‘primeness’ might be tested
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| Sorts cards on a more substantive basis (eg, odds/evens or multiples of a particular number), provides at least three different sorts of this type, may sort on prime/composite basis. Correct response to first ‘Think of a number’ question (-57), may not be able to correctly respond to second question |
Suggests a reasonable understanding of multiplication in terms of factors for whole numbers, may not appreciate role of identities and inverses in arithmetic - Use calculators to explore the impact of negative and fractional factors on quantities and expressions, apply in missing value problems, eg, Concentrates is mixed with water in the ratio 2 to 15 to make cordial. How much cordial could be made from 3 litres of concentrate?
- Explore the notion of inverses and identities for addition and subtraction using a wide range of numerical examples of the form: x ± ? = x
–x + ? = 0,
? – x = 0,
x + ? = 0, and
? + x = 0,
for whole numbers, integers, fractions and decimals
- Use a similar range of numerical examples to explore the notion of inverses and identities for multiplication and division
- Explore to role of factors in cancellation techniques, justify in terms of inverses and identities
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| Provides more than 3 sorts based on multiples or factors, including prime/composite sort, able to describe bases for sorts using appropriate language. Answers last two questions correctly (ie, 3 eighths or ‘3 divided by 8’ or 0.375) |
Suggests generalised understanding of multiplication and division in terms of ‘factor.factor.product’ and an understanding of inverses, may not be able to apply inverse operations to solve equations more formally - Relate the use of inverse operations to solving problems of the type, ax ± b = c ± dx where a, b, c, d progressively move from small whole numbers to larger whole numbers, integers, fractions and decimals
- Emphasise the value and power of recording which uses a logical sequence of equivalent statements to arrive at a solution (see 6.2 Advice above)
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