Lesson 1 Algebra: Roofs – isometric trapezia

Episode 2 How the sides of a trapezium relate to each other (10-20 mins)

Reasoning Resources
Isometric dotty grid on board or projector,Worksheet 1
Whole class preparation
Following rules. using measures of lengths.

Coordinating set of numbers with corresponding intervals on the grid.

Demonstrate the drawing of a trapezium (roof, skirt, upturned boat) from code 2, 1,2, 3 on a grid on board.
From the starting dot show the clockwise rule of going up-right, across-right, down-right then across-left.
Emphasise the use of distances between dots NOT the number of dots. The shape is roof-shaped with no lines sticking out or short of meeting at the starting dot.

Does any set of four numbers make this shape using the rule of drawing clockwise? Demonstrate a `non-roof' set of four numbers, e.g. 2, 3, 1, 5
Pair and small group work
Recognising all ideas including partial generalisation. Give out Worksheet 1. Which sets of four numbers produce a roof-shape? Challenge pupils to clarify and record their initial thoughts in any way they like, then to improve or add to them.
Sifting and combining ideas. Give pupils counter-examples that prompt them to improve their ideas. Challenge pupils who complete the task early to think why the rules they found always work.
Whole class sharing and discussion
Call on a few pupils to tell their suggestions, starting with the simplest ideas. Ensure all ideas are recognised (even erroneous) and summarised on the board using their own words as far as possible. You may ask some groups to explain how they spotted the patterns and how they checked them.
Necessary but not sufficient idea, when one rule is needed to make a roof-shape but is not enough to guarantee that the shape made will be a roof A typical list includes: 'the 4th number must be biggest 1st and 3rd numbers must be the same', '4th number is bigger than the 2nd by 2', '1 st and 2nd make up the 4th.

Then go over the listed ideas and involve the class in deciding which ideas are true for all cases, and which are only for some, which are the same but worded differently, and which are linked in some way. Pupils should recognise that a rule may be true but not enough on its own. You could present counter-examples, e.g.2,4,2, 3 has the 1st and 3rd the same, but is not a roof-shape; 1,4, 1, 10 has the 4th number biggest but is not a roof-shape.

End the discussion by finding the fewest rules needed to produce a roof-shape. Now challenge the class to test the rules.
Whole class introduction to variables
A generalisation in mathematics must apply to all cases.

Start with a blank board with only the two rules on it:

  • The 1st and 3rd numbers must be the same.
  • The 4th number must be the sum of the 1st and 2nd (or 2nd and 3rd) numbers.
  • Working with the rules, internalising they apply to ANY number Ask the class to suggest any two small numbers for the 1st and 2nd sides. What should the 3rd and 4th numbers be to make a roof-shape? Ask them to check that they do indeed do so. Ask for a pair of larger numbers, then even larger numbers, then contribute pairs such as 9000 and 2, or 3 and 666, so they can apply the rule easily, and imagine the roof-shape made with them. Now move to placing symbols such as a star and box to represent the 1st and 2nd numbers. Now ask the class what you should write for the 3rd and 4th numbers. Discuss the addition of symbols (e.g. *+ ■) as an expression, whose value is dependent on the values of the symbols.
    Moving to symbolising the general number.
    This is a key developmental step rather than an item of knowledge.Dependent and independent variables
    Bring out that the values, or the 3rd and 4th numbers, are dependent on the values chosen for the 1st and 2nd. Introduce the use of letters to represent any number. Pupils may prefer using Introduce the use of letters natural language as well as to represent any number. Pupils may prefer using 'c' for the 3rd or 4th number, which is then equated to 'a' or 'a + b '.
    Some pupils may recognise at this stage the value of algebra.
    Ask questions such as Why is algebra useful? and encourage pupils to share ideas in their own words.


    Thinking Mathematics Lessons Copyright © by Michael Shayer and Mundher Adhami. All Rights Reserved.

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