Lessons 13 Chocolate box

# Chocolate box

Overview Resources
Pupils explore linear and quadratic functions in a visual concrete context and then compare the two types of function using both tables and graphs. Each of the functions can be represented in various forms, which allows pupils to make their own constructions of the same functional relationship.

For advanced pupils who can handle algebraic forms, this also gives an opportunity for algebraic manipulation. The lesson follows on from TM8: Ladders and slides and TM12: Functions.
Square dotty paper
Worksheets 1 and 2 - these may be copied side by side on A3. Worksheet 3
Square dotty grid to be shown on board.
Pegboards (optional). Small pieces of paper each with 4 non-consecutive numbers (not including 5) on- to give pairs or groups of pupils as the square arrays which they will investigate.
• Explore multiplicative relations and functions,
• Relate word descriptions, tables, algebra and graphs representing the same patterns to each other, to highlight the power of mathematics.
• Generate sequences from practical contexts and describe the general
term in simple cases,
• Express simple functions in words and algebra,
• Begin to plot and interpret the graphs of simple functions arising from
real-life situations.
• EPISODE 1
Exploring square arrays of dots
Using square arrays of dots, pupils compare the number of dots on one side, on the outside (border) and the inside (square). The use of a context is helpful to provide a reason for looking at this problem; a tried and tested context is that of a square box of chocolates, where those on the outside are plain and those on the inside milk chocolate.

Depending on the way they see the problem, different expressions will be found for generalising the number of dots on the border and the inside. These ways of seeing and expressions are shared with the whole class. The algebraic expressions are compared and found to be equivalent. Fully understanding the equivalence of expressions requires prior understanding of multiplying algebraic expressions with brackets. But a demonstration and a light explanation can still be meaningful to some pupils.
EPISODE 2
Graphing the functions
Using the data generated by members of the class during Episode 1, pupils look at patterns in the table, and plot the graphs of the functions.

These graphs are then described and discussed, with the features referred to the algebra. The point at which the graphs cross is a focus in this episode, including what it means in the chocolate box context.