Lesson 26 Chunking in algebra

# Episode 3

Reasoning Resources: Worksheet 2
Whole class preparation
This part of the lesson deals with chunking quadratic expressions, and with the possibility of having many variables in an expression. This is a different use of two pairs of brackets. They are not nested, but multiplied by each other.

In Worksheet 2 the challenge is to put it all together, but as in all other TM lessons it is pupils’ sharing of their attempts to do this that is important, rather than the expectation that they will succeed with all aspects of the activity.
As an introduction you could give an example of area as a combined variable, For example, some artists use heavy expensive canvas for their paintings, they must order sheets of canvas longer and wider than the rectangles they need for the paintings so that the edges can be folded into the picture frames. The folded strip is always the same width, regardless of the lengths and widths of the pictures painted. The weight of the sheets depends on the area, which in turn depends on the width and length. The cost depends on the weight.

Effectively this example uses four variables: length, width, weight per unit area, and price per unit weight. The length is made into a compound variable through being made in two parts, a fixed amount for the holding strip plus a variable amount for the end product. The same is true for the width.
The context of the scarf in Worksheet 2 is similar, although some pupils may have confusion about the starting and final sizes of the scarf. You may need to clarify that the tassels are not added after the scarf has been made, but are made from the ends of the starting rectangle and similarly that the side edges are hemmed with the extra width of the starting rectangle.
Short periods of paired work interspersed with whole class discussion
A philosophical requirement is that a person demonstrates not merely the ability to give the definition of a concept, but also that he/she can show the use of the concept in all relevant contexts and show how that use is related to the definition. The strategy behind TM26 is to give full weight to the amount of construction the pupils have to do before bracketing becomes part of their everyday use of algebraic language, so that they can manipulate operations themselves within the system, rather than just particular numbers or values. After question 3, discuss the different possible methods for chunking. You may want to work through the extra questions below either in pairs, as extension work, or as a whole class exercise.

4. The price of silk itself ranges between 25p and 90p for each 100 square cm depending on
its quality. Estimate a price for silk of an average quality. Now choose a final width and length and calculate the cost of the starting size of the rectangle of silk.

5. The workshop charges £8 per scarf for working, and adds this to the price of the silk. Write the expression for the price of 12 of your chosen scarves.

6. Which of the numbers in your expression can be made into variables?

After question 6 discuss the issue of seeing many quantities as variables (that is, amounts that vary or change or can have different values); the pupils can end up with an expression where all the terms are letters.
End of Lesson Reflection
Regardless of how far the pupils have progressed in the agenda of the lesson, allow for a few minutes at the end for a relaxed review of the flow of the work. Encourage pupils to talk about any ‘novel for them’ ideas and what felt important for them personally.