Level 6: Generalising
Materials
Instructions
Place Card 1 in front of the student and ask, “Can you tell me what you think this means please? … Could you think of a situation that this might describe? … Can you write it another way?” Note student’s responses.
Place Card 2 in front of the student, point to 17 – 8 and say, “This is an expression for subtract 8 from 17. How would you express add 4 to 7n?”… Note student’s response, then say, “and what does the n mean here? Does it mean anything, does it stand for anything, is it just a letter, or what?” … Note student’s response and explore thinking as necessary.
Repeat with Card 3.
If answered confidently, place either Card 4 or Card 5 in front of the student, read the question slowly … Note student’s response and explore as appropriate.
Place Card 6 in front of the student, read the question slowly, and indicate the three possible answer forms. …Note student’s response and explore reasoning as necessary.
Repeat with Card 7, stop if the student appears unwilling or unable to proceed.
6.4 Advice Rubric
Student responses to these tasks indicate the extent to which they are able to recognise and work with the conventions of algebraic text. In particular, the tasks explore how letters and departures from arithmetic forms of expression, such as the omission of the multiplication sign (concatenation or co-joining), are understood and used. For instance, many students believe that a + b is a more acceptable interpretation of ab than a × b (possibly on the basis of place-value). While some of the tasks appear ambiguous (eg, perimeter of incomplete polygon and spaceship journey), they were used with a large sample of Year 8 to 10 students (13-16 year-olds) in the UK (eg, see Booth,1988) to identify the extent to which students could derive and accept general expressions as ‘answers’ to problems. This is an important, often under-recognised difficulty that significantly impacts students’ capacity to work with general expressions. Where students view operation signs as instructions to ‘do something’ and expect, as in arithmetic, to be able to perform those operation and arrive at a numerical answer, they will either make assumptions about what the numbers might be or just give up as they really do not see what the purpose or value is in using algebraic text. This does not mean to say that students can’t ‘play the game’ of algebra, that is, look like they understand when they do not. For instance, many will correctly identify that 5a stands for ‘5 times a number’ (Card 1), that 4 + 7n (or 7n + 4) represents “add 4 to 7n” (Card 2), and that 4m represents the sum of 4 m’s (Card 3). However, some students will want to give numerical answers or approximations to the problems presented in Cards 4 and 5, and/or will not be able to formulate a correct expression for the Card 6 task (some version of p(a+ m), and/or identify the conditions under which the statements in Cards 7 and 8 are true (that is, p = y and n ≤ 2 respectively).
| Observed response |
Interpretation/Suggested teaching response |
|---|
| May correctly respond to Cards 1, 2 and 3, but experiences difficulty with remaining Cards (eg, may count sides of polygon shown and multiply by 2 to get numerical answer) |
Some understanding of the use of letters in simple additive contexts, may not understand equivalence, little/no understanding of the notion of variable - Check and consolidate as needed the convention of writing a × b as ab, write such expressions in words, and practice translating expressions written in words into algebraic expressions
- Use tasks like the ones in Cards 4 and 5 to develop the notion of variable.
- Create and use variables in ‘real-world’ contexts, eg, investigate differences between different newspapers and magazines by identifying all the things that might be measured (eg, column widths/lengths, words/page, area of different text forms, pictures, illustrations or headings /page, etc), assigning a letter to each variable agreed upon, and then taking sample measures to compare and discuss
- Use activities like the ones described in 6.1 above to develop a deeper understanding of equivalence
|
| Correctly responds to Cards 1, 2, and 3, may respond appropriately to Cards 4 or 5 and 6, and/or partially recognise the conditions required for Card 7 (eg, add p to left hand side and y to right hand side) |
Suggests an emerging understanding of how algebraic text is used, may be relying on arithmetic solution strategies rather than algebraic reasoning to solve problems - Review the use of renaming strategies to solve arithmetic problems and express as general relationships (see Irwin and Britt, 2005 and 6.1 above), eg, xy = x(y + a) – xa from examples like 6 × 99, 8 × 789, or 25 × 9996
- Use ‘consecutive number’ number problems (eg, 9 consecutive numbers sum to 936, what are the numbers?), ‘think of my rule’ activities, and problems like the ones suggested in 6.3 above to consolidate the derivation of algebraic relationships
|
| Correctly responds to Cards 1, 2, 3, 4 or 5, and 6, may not be able to explain reasoning, recognises the conditions required for Card 7, but unable to do so for Card 8 |
Suggests reasonable understanding of algebraic conventions, may not understand inequalities and/or be able to manipulate/transform equations or inequalities - Check and consolidate as needed, students’ understanding of inverse relationships (see Level 6.2) and how these might be applied to solve simple linear equations
- Relate the use of inverse operations to solving problems of the form ax ± b = c ± d where a, b, c, and d progressively move from small whole numbers to larger whole numbers, integers, fractions and decimals
- Introduce/consolidate inequalities working from words to symbols using examples like the one shown in Card 8
|
| Correctly responds to all Cards |
Suggests sound understanding of algebraic notation and conventions - Extend pattern recognition and identification to non-linear relationships and graphical representations
- Use problems like the following to consolidate the understanding that ‘answers’ can be expressed in general terms: A bacterium can reproduce itself every 20 minutes. If a colony of bacteria was put into a breeding dish at 10:05 am, how many bacteria would there be by 3:45 pm?
- Explore a wider range of situations and problems involving inequalities
- Explore the range of strategies that might be used to solve problems such as Spiders and Beetles (pdf - 196.34kb) eg, guess and check, draw a diagram, make a table, or use ‘if … then’ reasoning. Use this and similar problems, to demonstrate the value of algebraic representation and reasoning (eg, can solve simultaneously and/or graphically)
|