Lessons 13 Chocolate box

Episode 1

Reasoning Resources: Worksheet 1 on OHT and copies for pupils. Worksheet 2 - table on board or OHT, square dotty paper in OHT (optional), pegboards (optional)
Whole class preparation
Understanding the context and using the squares arrays of dots to represent the problem.

The number of dots on the border is related to the number of dots on one side of the square. Deceptively this looks to be four times the number of dots on the side. A simple cognitive conflict early in this lesson.

The table for all values is filled by different pairs

Show the OHT of square dotty paper on Worksheet 1. Explain that a chocolate manufacturer wants to make square boxes of chocolates in ‘nests; with plain chocolate in the border nests and milk chocolate in the inside nests. To add interest, ask for a show of hands to see if there is equal preferences for these types.

A couple want to know which size box will give them equal numbers of plain and milk as one only likes plain and the other milk.

Draw round a 5 X 5 array of dots, to represent a square box of chocolates. Show it briefly, then cover it and ask Are there more chocolates around the border or on the inside? Uncover, and label dots on the border P for plain and cots on the inside M for milk.

Count the dots on the border (P), inside (M) and total. Begin a table on the board/OHP like that on Worksheet 2.

Ask pupils to investigate how many P and M dots are produced by squares of sides from 3 to 12 dots. You could ask pairs of pupils to investigate two or three different squares each. Give each pair a copy of Worksheet 1 for their working and to record their results.

Key question: Are there ways, other than counting dots one by one, to find how many dots there will be on the inside and the border of any size box?
Pair and small group work
Pupils investigate and record their results and ideas on the board and Worksheet 1. Some pupils may benefit from previously drawn squares around the square array they are working on. Look for different ideas and ask pupils to show on a diagram how they see the problem.

Identify which groups/individuals to ask to share their ideas. (Giving a group an OHT of dotty paper to record their work can facilitate the sharing of ideas.) Encourage pupils to show how the side of the square as a whole or in parts can help in finding the total for the border, or outside dots.
Whole class sharing and discussion
The idea that different ways of seeing the problem lead to different expressions, which turn out to be equivalent, is an important idea for pupils to take away from this part of the lesson.

Each of the different ways are presented verbally, visually then algebraically.

Pupils intuitively accept that different expressions are equivalent if they know how fo manipulate algebra.
Collect in answers for squares of different side length. Write the results in the table on Worksheet 2. Next concentrate on how to find the number of border dots without counting.

Some of the following different ways of looking at the outside can be shown on a pegboard or with coloured pens. (You may not need all of these, and you could leave the algebraic expressions after listing all the methods in words and pictures.)

  • (4 times the number along the side) - 4 because the corners have been counted twice
  • ([4n - 4] where 7 is the number of dots along one side.)
  • 4times (the number along the side ~ 1) because where the corner ones belong is sorted out before counting up [4(i — 1)]
  • Twice the sides, then twice the top and bottom that are left (2H + 21 - 2))
  • One side, two sides less 1, then one side less 2 n+ (1-1) 4 (n- 1) + (n-2)
  • Four lots of side less 2 then add the four corners [4(n ~ 2) + 4]
  • Take the inner square from the full square.

    Ask: How come there are different expressions that give the same answer for any side length? Pupils should handle the idea that they are equivalent, and should be able to see that in a couple of examples. It may be enough to say: After we've done some more algebra we'll be able to show that these expressions are equivalent as well.
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