FOR TRIAL ONLY. Tangram. Multiplicative Reasoning: Lesson 9A. Summary. Outline of the lesson

Transcription

1 Tangram This tangram consists of three pieces. We want a larger version of the tangram where the 4 cm length becomes a 7 cm length. 4 cm Work in a group of three. Choose one piece each. Draw the larger version of your piece on 1 cm squared paper. Carefully cut it out. Check that the 3 new pieces again fit together. Summary In this lesson students enlarge a 3-piece tangram puzzle. Initially they enlarge just one piece each - the key to the lesson is the considerable surprise that occurs in cases where the resulting pieces don t fit together. Students are asked to map 4 cm onto 7 cm, which involves the quite demanding scale factor 1.7, and which is likely to lead some students to adopt the inappropriate addition strategy +3. They are then asked to enlarge the tangram again, using the scale factor 1.. This is a less demanding task and should allow students to draw on (and develop) some of their geometric knowledge about enlargement. Outline of the lesson 1. Ask students to solve the task in groups of three. Explain the above task and ask students to solve it in groups of three. Provide students with 1 cm squared paper and scissors; make sure each student enlarges just one piece and cuts out the result. 2. Discuss students solutions. Check / briefly discuss students solutions. [If students have used the addition strategy, by adding 3 cm to various lengths, the new pieces won t fit.] 3. Simplify the task. Display this shape. This is a 6 cm square tangram again, but now with 4 pieces (2 of which are squares). Draw a larger version that forms a 9 cm square. Use 1 cm squared paper but don t cut out the pieces. Discuss students drawings. What clues can we use to decide whether the result is right or wrong? 4 cm 4. Revisit the original task. Ask students to repeat the original 4 7 enlargement. Draw a larger version but don t cut out the pieces. What size square does the enlarged tangram form? Evaluate the drawings. What clues can we use...? Why does the +3 addition strategy not work? Discuss the scale factor. If I know a length on the tangram, how can I work out the enlarged length? page 1 Note: These materials are the subject of ongoing research and are made available on request to teachers as draft trial materials only. Please send feedback to Jeremy.Hodgen@kcl.ac.uk or Dietmar.Kuchemann@kcl.ac.uk

2 Overview Mathematical ideas As with Expanded house (Lesson MR-7A) and Stamps (the Lesson 9 starter), this lesson involves geometric enlargement. We start with a fairly demanding scale factor, 1.7, of the sort we have met before, and test students understanding with a task which drammatically highlights the error of using the addition strategy. We go on to use geometric properties of shapes to further bring out the uniform, multiplicative nature of enlargement. Files are also provided (see pages 4 and ) for investigating enlargements by dragging, either in this lesson or later, and for considering the direction and magnitude of the drag (which readily links to the use of centres of enlargement for constructing images). There is also an opportunity to use Cartesian coordinates to represent and analyse enlargements. Students mathematical experiences Students may discover some of the following the addition strategy does not preserve an object s shape an enlargement is uniform all squares within a shape remain as squares after an enlargement slopes (and angles) stay the same all distances are enlarged by the same factor. Key questions Why does adding a constant amount (3 cm) not work? How can you tell whether the drawing is an enlargement? Assessment and feedback The Starter activity should give you an indication of what use students make of geometric properties for judging shape and of what language they use. Thus you might want to focus attention on features like the slope of the slanting line in the tangram, or the fact that the rectangular shape consists of two squares (and that therefore the enlarged version will also consist of two squares). And you might want to use an activity to improve students mathematical language - for example, asking students in pairs to craft an explanation as to why two shapes are or are not similar. A camera (eg a visualiser) can be useful here for displaying the shapes under discussion. Adapting the lesson The Stamps Starter activity gives students the opportunity to think about similarity and enlargement qualitatively (eg, Stamp A is too tall for its width ; In stamp D the circle has become egg-shaped ). Such thinking is important in this lesson, though students initial focus is likely to be numerical. If students struggled with the Starter, you might want to start the lesson with a 4 6 enlargement rather than 4 7. Stages 2 and 3 could then procede as outlined while it might be worth considering a 2 or 1.2 enlargement rather than 1.7 in Stage 4. For other students it is worth going through the four given stages of the lesson, since it is useful to be able to explain why the addition strategy does not work, and to be able to elucidate geometric properties of enlargement, even when students can perform challenging enlargements successfully. For such students there might be time to explore enlargement in terms of dragging and/or by using coordinates and a centre of enlargement, as discussed on pages 4 and. Or this could be pursued at a later date. page 2

3 Outline of the lesson (annotated) 1. Ask students to solve the task in groups of three. Explain the above task and ask students to solve it in groups of three. Provide students with 1 cm squared paper and scissors; make sure each student enlarges just one piece and cuts out the result. 2. Discuss students solutions. Check / briefly discuss students solutions. [If students have used the addition strategy, by adding 3 cm to various lengths, the new pieces won t fit.] It is essential that each student works on their single piece, and cuts out the result. Errors will be far less obvious if students simply enlarge the tangram as a whole (as will be done in Stages 3 and 4 of the lesson). 3. Simplify the task. Display this shape. This is a 6 cm square tangram again, but now with 4 pieces (2 of which are squares). Draw a larger version that forms a 9 cm square. Use 1 cm squared paper but don t cut out the pieces. Discuss students drawings. What clues can we use to decide whether the result is right or wrong? The two square-shapes will remain square-shaped. The right-hand shape has the same-size base as the squareshapes and the enlarged shape will have the same-size base as the enlarged square-shapes. The slanting line has a slope of 1 across, 2 up (ie a gradient of 2), and will continue to have this slope. Any 1 cm line becomes a 1. cm line. The original tangram can be split into nine 2 cm squares (or thirty six 1 cm squares); the enlarged tangram can be split into nine 3 cm squares (or thirty six 1. cm squares). 4. Revisit the original task. Ask students to repeat the original 4 7 enlargement. Draw a larger version but don t cut out the pieces. What size square does the enlarged tangram form? Evaluate the drawings. What clues can we use...? Why does the +3 addition strategy not work? Discuss the scale factor. If I know a length on the tangram, how can I work out the enlarged length? page 3 The addition strategy increases all lengths by the same amount, regardless of how long they are. Rectangles become more like squares. Adding 3 cm to the heights of the original 3 shapes would increase the left hand edge of the tangram by 6 cm while the right-hand edge would only increases by 3 cm.

4 Background The tangram task This lesson is based on a tangram task developed by Guy and Nadine Brousseau in France in the 1970s. Our tangram is simpler than theirs, but in Stage 1 of the lesson we have adopted their idea of asking students to enlarge pieces of a tangram separately, and then to fit the resulting pieces together. This means that students errors, in particular those resulting from the addition strategy, will show up very drammatically, through the pieces not fitting together. Of course, such a demonstration does not of itself offer a way of resolving such errors. Stage 1 of the lesson also involves a quite demanding scale factor ( 1.7) and thus provides quite a stringent as well as drammatic test of how well students have come to understand that the addition strategy does not preserve proportions in geometric shapes. Later stages of the lesson encourage students to use their geometric knowledge to consolidate or enhance their understanding of the uniform, ie multiplicative, nature of enlargement. Enlarging from a centre The related file tangram-results.pdf (below) shows a drawing of the original tangram (in green) and enlargements produced by dragging, using the same scale factors as in the lesson, namely 1. and 1.7. The file can be displayed on a screen or printed out as a worksheet. For the 1. enlargement we have selected a point A (at the centre of the rectangular tangram piece) and drawn arrows to shown how its image, A, is the result of dragging. Students could investigate the images of other points in this way: What can be said about the position, direction and length of the drag-arrows? The file also shows numbered axes (chosen, for the sake of simplicity, with the fixed corner, or invariant point, of the tangram at the origin). We can thus write down the coordinates of points and their images, eg A (2, 1) A (3, 1.). So students could investigate this: What is the relation between the coordinates of a point and its image? 1 1 A Aʹ 0 0 tangram-results.pdf page 4

5 Using the accompanying Word files Enlarging by dragging We have provided three Word files that can be projected on a screen and which allow the tangram to be enlarged by dragging. It is worth trying these for yourself, to get a sense of what they do and to help you decide whether to use any of them in class. In the file tangram-drag-constrain.docx (below left and right) the aspect ratio of the tangram is preserved regardless of the direction of drag, and so this file can be used to provide a simple demonstration of what various enlargements of the tangram will look like tangram-drag-constrain.docx (initial state) tangram-drag-constrain.docx (after random dragging) In contrast to this, the aspect ratio is not constrained in the file tangram-drag-free.docx (below, left), so the shape of the expanded tangram depends on the direction of the drag. A third file, tangram-drag+freepieces.docx (below, right), is like the first in that the aspect ratio of the tangram is constrained so that one can easily produce enlarged versions; however, it also shows the three pieces of the tangram separated out and able to be expanded freely. Here it can be both challenging and illuminating to try to reproduce an enlarged version of the tangram by dragging the individual shapes separately tangram-drag-free.docx (after random dragging) tangram-drag+freepieces.docx (after extensive dragging) page