Lessons 13 Chocolate box

# Episode 2

Reasoning Resources: Worksheets 2 and 3
Whole class preparation
Table of values helps some pupils see patterns, but not others. Understanding the presentation of the graph and how they are to proceed.

Using all the class data, with cross checking by different pairs, the table on Worksheet 2 should, by now, be complete. Pupils should be able to describe some patterns in the table, for example ‘border dots increase by 4 each time;’square numbers; etc. The questions on Worksheet 2 also prompt them to link the side and border and side and inside dots.

Explain to pupils that they are going to use the data to draw two graphs on the same grid - one for the number of dots ‘in the border’ and one for the number of dots ‘on the inside’ If necessary remind them of how to do this — they are using the number of dots on one side as the x-coordinate and for the y-coordinate either the number of border dots or the number of dots on the inside.

Discuss how using different colours for the two graphs may be helpful.

Key question: How will you describe the graphs and the point where they cross?
Pair and group work
Pupils may be surprised that one of the graphs is not a straight line. They may be unsure as to whether they can join the points, having met the problem of continuity in other situations.

Relating the graph back to the original problem helps them to understand what they have drawn and its meaning, particularly the crossing point. Seeing that different types of expressions (linear and quadratic) lead to different types of graph. Relating the ‘squaredness’ of the quadratic to the rapid growth.
Pupils plot the graphs of the functions on Worksheet 3, using the class results. They should then write a sentence to answer each of the questions on Worksheet 3.

Ask pairs to find the side length of a box where the number of milk chocolates is twice that of the plain chocolates, and to explain how they worked it out. What other kinds of boxes are there? For example, more plain than milk and vice versa.
Whole class sharing and discussion
Relating the graphs to the algebra, and to real life.

Linear vs non-linear graphs.
Draw out, in pupils’ words, the different growth patterns. For example, the border number starts higher and grows steadily, the number on the inside starts smaller but grows more rapidly. Discuss reasons for the difference.

Relate each graph to its formula, so that the class appreciates that for the border dots the gradient 4 is visible on the graph through a rise of 4 for each increase of 1 in the side. And compare that with the increasing increase for the inside dots.
End of Lesson Reflection
Relate the difference in the graphs to the expressions. Draw out that the expression for the inside has a square term in it. Introduce the term ‘quadratic; explaining that they will meet more equations like this (with squared terms) as they progress into years 8 and 9.

More general issues like usefulness of graphs to see relationships, equivalent algebraic expressions, and connections to other lessons are other possible discussion topics.