Knowing the Unknown

How would you describe this brick design to a friend on the phone?

Knowing the Unknown (A1) begins by studying the brick patterns on garden walls. In reasoning about how to build particular wall patterns, students generate a range of algebraic sentences. In the context of a Fish and Chips shop (“I’d like four fish and chips, please”), the natural pairing of this classic takeout treat helps students to develop a rationale for bracket notation, equivalence and simplifying expressions.

The second half of this module approaches equations through the classic model of balance metaphors. Rather than acting as a quick means of accessing the language of equations, the contexts of a see-saw and a “human-balance” allow students to gradually develop their understanding of what it means when scales are balanced and how to level them out when they are not. Inside the context of measuring weights on a grocer’s scale, students compare strategies that go beyond the classic algebraic idiom of “do the same on both sides”. Building progressively to formal algebraic notation, these contexts are kept alive to think through equations with unknowns on both sides, equations with brackets and equations with negative solutions.

(See the weighing scales example on What makes our material special? for more information.)

Overview of Knowing The Unknown

Section A: Brick Patterns

Lessons 1: Garden Walls
Students work within the context of bricks and brick walls and begin to informally symbolise patterns (and vice versa, moving from shorthand notation back to a familiar context). Symbolisation is seen as a natural shorthand way of writing.

Lesson 2: Brick Patterns
Students explore a variety of ways of symbolising particularly in relation to working out lengths. They engage with basic algebraic challenges and look for the “sameness” of these challenges. Students substitute length measurements for symbols.

Lesson 3 & 4: Fish and Chips
Symbolising in contexts of working in a restaurant and accepting orders over the phone. Recognising that to work out the price the order can be written in different ways (noting aspects of equivalence, collecting like terms and appropriate recording). Looking at a variety of ways of seeing an order including orders using multiplicative formats. Starting to recognise that the order of operations is important for effective communication. Students work informally on problems involving simplifying expressions, substitution, expanding brackets and factorising expressions.

Section B: Making it Balance

Lesson 5: Human Balance
Students develop ideas about how balances work through the contexts of a seesaw, a human balance and a grocer’s weighing scales. Using a human balance, they look at the effect of removing items from one side of balanced scales and consider ways to make the scales balanced again. Initially students are comparing the relative weight of objects (heavier, lighter or the same) on either side of the scales, as denoted by the position of the scales (level or tilted). Later they use the fact that the grocer’s sales are balanced to work out the weight of various bags of flour as an amount of ounces.

Lesson 6: Weighing It Up
Students are asked to ‘say what you see’ in weighing scale pictures which have food items and ounces on both sides of the weighing scales. This leads to subtly different ways of describing the situation, using a variety of language and interpretations. Students see at least 3 distinct strategies for figuring out the weight of one item and practice their preferred method on a number of different weighing problems. They are also expected to try out the alternative strategies. By comparing sets of weighing scales to see which hold the most and the least weight (on Activity Sheet 7 – Weighing Food and Mobiles), students indirectly check that balanced scales actually do balance. The mobiles question provides another context where students can employ similar strategies.

Lesson 7 & 8: Shorthand for Balancing
Students draw and compare their own pictures to represent a weighing scales problem in terms of how much ‘shorthand’ is used. A formal algebraic equation is introduced as a shorthand representation of a weighing scales problem and students work across these two representations: they practice solving equations with the unknown on both sides by drawing the scales and applying their solving strategies from Lesson 6. By drawing a scales picture, students make meaning of equations containing brackets. They develop and gradually refine their strategies for solving a variety of equations and inequalities with the unknown on both sides. They start to consider the possibility of negative solutions.

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