Key Ideas - Chapter 2: Quantities and algebraic expressions
A list of resources to support the book and website "Key Ideas in the Teaching of Mathematics"
Perimeter Expressions
This NRICH activity 'Perimeter Expressions' provides a rich context for exploring the value of algebra as a reasoning tool. Students use perimeter as a context to express sums of known and unknown quantities. To access this activity students are given a piece of A4 paper to split up and are encouraged to explore and discover as much as they can about various lengths before any of the questions are posed.
The use of the familiar situation of length to provide an image of the meaning of the letters provides a bridge between the missing number problems used in primary mathematics and solving equations algebraically.
Performing Number Magic A9
These five tasks steer students through several algebraic manipulations of expressions in order to expose how the 'tricks' work.
To understand how each trick works, students have to use the distributive law and are required to to prove equivalence by transforming expressions.
The teacher notes start with a nice simple trick to be used to set up the first activity.
Consecutive Sum adds five consectutive numbers with students showing that the total is five times the middle number.
Pyramid requires students to express consecutive numbers in algebraic fiorm and manipulate expressions to explian how the trick works.
Routes is a trick which can be manipulated by the students who can use their algebra skills to make up their own trick. Students need to be able to multiply a constant oiver a bracket e.g. 3(x+5)
Adding pairs an extension to the consective sum trick by keeping the second number fixed or introducing a second variable.
Calendar activity enables students to explore how certain algebraic representations, such as letting the middle number be x, can be more efficient that others to explain how the trick works.
Evaluating Algebraic Expressions A4
This sequence of tasks enables students to compare equations, inequalities, and identities by various means. Substitution is used to find out when expressions are equal in value. It is more common in textbooks to use substitution merely as a means to practice using symbolic conventions, but here it has a role to play in understanding the underlying relations.
The tasks are designed to challenge common misconceptions about the relations between numbers and variables and unknowns. Students are required to decide whether statements are always true, sometimes true or never true. This task generates useful debate and highlights misconceptions.
Number and Algebra 3
Number Spirals is on pages 37-38 of the text (page 43 of the pdf) and is about the intriguing and unexpected patterns that arise from spirals on number grids. These are structural, not sequential patterns, and need careful explanation.
The patterns are not predictable from number patterns alone, so avoid a purely inductive approach to generalisation. Often these are solved by students going to and fro between number, algebra, spatial features, specific cases and generalised relations.
Using a formula: Exercises often ask students to reason inductively a formula for a given sequence, but the tasks on pages 42-43 of the text (page 48/49 of the pdf) turn this process on its head and offer the formula to generate the sequence. This way round, students stand a good chance of spotting the effects of different parameters in the formula. If not, they can change parameters for themselves and see how the generated sequence changes. This is an old resource which mentions programming in BASIC in some of the tasks, but they can be adapted for use with spreadsheets.
Algebra Makes Sense
Matching Mappings is on pages 16-17 of the text ( pages 17-18 of the pdf) and involves preparing cards for a matching exercise between words and symbols.
Expressing situations in words is a pre-cursor to using symbols, and students can then see that algebra is expressing succinctly what is sometimes long-winded and complicated in words.
By starting with words, the order of operations is easier to understand as a form of notation, rather than the notation coming first, and word problems become more accessible because students get used to mathematising them.
Equivalent Expressions is on pages 36-37 of the text (pages 37-38 of the pdf). Traditionally, most of the algebra curriculum used to be about learning to manipulate expressions by collecting like terms, factorising, multiplying brackets and so on. This activity achieves alternative forms of expressing the same relations between unknowns and variables. For algebra to be meaningful, students can see that these manipulations produce equivalent expressions to describe situations that the students will understand. Embedded in this kind of work, but usually left implicit, is the use of = to mean 'is equivalent to', i.e. whatever the value of the variables. This is in contrast to equations where non-equivalent expressions are temporarily equal for particular values of the variables.
Keeping the Pizza Hot
Keeping the Pizza Hot can be found on the Bowland Maths website by clicking on The Case Studies the scrolling down the list using the arrow at the bottom until you see a case study called Keeping the pizza hot.
The task requires students to solve a practical problem, probably over several lessons, using a purposeful modelling cycle. Students set up relations between variables and to use them to find out the effects of different design decisions. Students can experience the value of mathematical approaches in the realistic context of a cooling curve.