Lesson 1 Algebra: Roofs – isometric trapezia

# Episode 1 Isometric grid and four-sided shapes (10-20 mins)

Reasoning Resources
Random dots, isometric dotty grid, projector or visualiser, isometric dotty paper
Whole class preparation
Accepting intuitive ideas from all.

• On the board or projector show a set of random dots and another set of isometric dots, orientated so that there is a horizontal line in them. Ask the class to come up with one or two-word descriptions of each set. What do you notice about each of these pictures of dots?
• Making distinctions and giving descriptions of regularity. Agreeing terms.

• You may end up with:/Disorganised:/lrregular:All-over the-shop, Scattered Messy: etc for one set 'Regular Equal-distance/Equal-space:Triangles,`Organised,'Ordered: 'Lines', etc. for the other.
• Gradual focusing on the feature of equal intervals in three dimensions. Agreeing terms.
• Focus on the ideas of/regularity' and 'equal spacing; drawing more descriptions including that the dots make lines'. Move to the recognition that the dots form only three imaginary straight lines: one 'slanted going up right, or down left'; another 'going up left, or down right; and the horizontal line 'across left to right or right to left.
• Pair and small group work
Following a rule for drawing different quadrilaterals on the grid.

• Give out to each pair of pupils 6-8 pieces of isometric paper in small sizes, e.g. ΒΌ or 1/8 of normal A4 page.
What four-sided shapes with straight sides can you make on the grid using the lines of dots? Pupils should draw large or small shapes as different as possible, with thick lines so that they can show others later. (There may be some dexterity problems here that you could solve through helping, pairing or grouping, rather than making neat drawings an issue.)
• Classification by type.
• Ask the pairs (or pairs of pairs) to classify the shapes into sets. Pose the challenge of the smallest number of sets, and what names or descriptions to give the sets.
• Whole class sharing and discussion: finer differences and using number
Rhombus to parallelogram relationship is the same as the square to rectangle. The four shapes could be looked at together

Pupils should quickly agree on the three sets of shapes: trapezia (alternatively described as skirts, roofs, boats). The parallelograms, and rhombi (diamonds).
For some pupils turning the rhombus (or square) changes its name A discussion about having two sets only, with rhombus defined as a special case of the parallelogram, should naturally follow, with pupils verbalising their reasoning using the notion of parallel lines as the main feature that dominates over different lengths of pairs of sides. You may pose the question, is there something similar here about the square and rectangle?
End the episode by shifting the class attention to using numbers for the lengths of sides.
Pupils should see the lengths measured in spaces between dots rather than the dots
themselves.They should write the lengths on their shapes,then generalise for each shape. So they describe the parallelogram in terms of two pairs of parallel sides of equal lengths, and note how more complex the trapezium is. Some pupils may appreciate that there are more complex trapezia where there is no symmetry.

The next episode concerns the trapezium with the longer side at the bottom (roof) drawn in the order shown (left) using a set of four numbers,3,1,3,4.