Fraction Fascination

5061 /www/nrich/html/content/id/5061/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <div style="text-align: center;"><mdo:image width="232" height="241" src="frac1.gif" alt="square divided into four triangles"></mdo:image></div> I drew this picture by drawing a line from the top right corner of a square to the midpoint of each of the opposite sides. Then I joined these two midpoints with another line. Can you see four triangles in the square? What fraction of the area of the square is each of these triangles? Then I drew another picture: <mdo:image width="500" height="500" src="frac2.gif" alt="new design"></mdo:image> How is this made using the first square? What is the shape that has been created in the middle of this larger square? What fraction of the total area of the large square does this shape take up? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> We had several good solutions to this problem. They came from Cong who goes to St Peter's RC Primary, Aberdeen; Terence from Brumby Engineering College and Ben at Kingsbarns Primary. Cong coloured the first image: <mdo:image height="208" width="208" src="cong1.gif" alt="first image coloured"></mdo:image> He said: <div>The green and dark blue shape is $\frac{1}{4}$ of the square's area (each).</div> <div>The blue shape is $\frac{1}{8}$ of the square's area.</div> <div>From these three shapes you can work out the red shape's area which is $\frac{3}{8}$.</div> Cong then sent in a new version of the large design: <mdo:image height="408" width="410" src="cong2.gif" alt="large coloured design"></mdo:image> He goes on to say: <div>This shape is made from combining 4 small squares together to get this big shape.</div> <div>The shape in the middle of the large square is a rhombus.</div> <div>The big blue shape is $\frac{1}{4}$ of the large square's area because it is $\frac{1}{4}$ of the small square and because the large square is four times as big as the small square, which means that it will be the same fraction.</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Fraction Fascination</h2> <div style="text-align: center;"><mdo:image alt="square divided into four triangles" height="241" src="frac1.gif" width="232"></mdo:image></div> I drew this picture by drawing a line from the top right corner of a square to the midpoint of each of the opposite sides. Then I joined these two midpoints with another line. Can you see four triangles in the square? What fraction of the area of the square is each of these triangles? Then I drew another picture: <mdo:image alt="new design" height="500" src="frac2.gif" width="500"></mdo:image> How is this made using the first square? What is the shape that has been created in the middle of this larger square? What fraction of the total area of the large square does this shape take up?</div> <h3>Why do this problem?</h3> <a href="http://nrich.maths.org/public/viewer.php?obj_id=1813&part=index">This problem</a> is useful for those pupils who are coming to terms with spatial representation of fractions where area is concerned rather than just length. Pupils' visualisations vary a great deal and this may prove very difficult for some and yet readily accessible to others. <h3>Possible approach</h3> <div>Pupils can be given the first shape printed out on large sheets and allowed to do some cutting up and then the pupils being brougt together to discuss what their thoughts and finding are. The discussion could lead a teacher to gain some ideas as to how their pupils are grasping the ideas surrounding area and fractions. Decisions coud then be made about whether to go further to look at the later questions or to pursue the first question much further. <h3>Key questions</h3> <div>Tell me about your folding/cutting. What do you think about the area/size of this shape? What have you noticed so far? </div> <h3>Possible extension</h3> You can open out this activity by extending thoughts and ideas. The original triangle could be looked at and ideas for changing it explored. So you may come up with ideas like these new ones; <div style="text-align: center;"><mdo:image src="3New%20TrianglesFF.jpg"></mdo:image></div> These came about by making the corners of the triangles a third of the way along rather than half as in the first, original one. You could usefully ask the pupils what they notice about the four areas in each of these three examples. Triangles can be formed in different ways of course so opening the door to ideas such as: <div style="text-align: center;"><mdo:image src="DiffTriFF.jpg"></mdo:image></div> Now we have only 3 areas to explore, but what can the pupils say about them? [The point on the left hand side is 1/4 of the way down.] They could explore many more examples like this and compare the three areas and triangles you create. Then there is the second part of the question. Asking pupils if they could do something else with the original shape to produce a tiling effect can lead to all kinds of ideas. One that I saw was; <div style="text-align: center;"><mdo:image src="New2ndPartFF.jpg"></mdo:image></div> Again questions about areas can be explored. Alternately you could right away present the same idea using the common A4 size of paper. <div style="text-align: center;"><mdo:image src="landscape.jpg"></mdo:image></div> So it's really a matter of changing the original question slightly and getting the pupils to say what they see and what ideas could be explored. <h3>Possible support</h3> Some pupils may need to have an adult with the to help in sustaining concentration.</div></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> It might help to split the picture into four like this: <mdo:image width="232" height="241" src="frachint.gif" alt="image split in quarters"></mdo:image> You could also label the triangles with a letter or number to help you identify each one. How many of the first square are needed to make the second picture? You could try printing off some of the squares and cutting them out. Can you fit them together to make the larger pattern? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><mdo:image alt="" height="241" src="fracsol.gif" width="232"></mdo:image> A is one quarter C is one quarter D is one eighth Therefore B is three eighths New shape made from four of above, rotated by 90. Central shape is a rhombus. Square above now is quarter of overall shape. C is therefore a quarter of a quarter of the total shape which is a sixteenth, but there are four of them. Therefore rhombus is a quarter of area of whole large square. <h5>These are the original Teachers' notes</h5> This problem requires children to split up the square into smaller pieces in such a way as to help them work out the fraction of space occupied by each triangle. Some pupils might be able to visualise these fractions, others will find it helpful to draw sketches, or annotate the picture in the problem. It would be a good idea simply to talk through the problem with your pupils, just enough to be sure they understand the task, and then to leave them some time to think about how they will tackle it. Ask everyone to think on their own for a few minutes, but then give them the opportunity to talk to a partner about how they might approach the solution. Then, invite the class to share their ideas together. (Think, pair, share.) Once some suggestions have been made, children can work on the solution in pairs. Although this problem at first appears very challenging at this level, it quickly becomes more manageable. The less input you as a teacher have at the beginning of the session in terms of directly steering pupils' ideas, the more satisfaction they will feel at having solved it. (As long as you are there to support them along the way!) You can open out this activity by extending thoughts and ideas. The original triangle could be looked at and ideas for changing it explored. So you may come up with ideas like these two new ones; <mdo:image src="3New%20TrianglesFF.jpg"></mdo:image> These came about by making the corners of the triangles a third of the way along rather than half as in the first, original one. You could usefully ask the pupils what they notice about the four areas in each of these three examples. Triangles can be formed in different ways of course so opening the door to ideas such as: <mdo:image src="DiffTriFF.jpg"></mdo:image> Now we have only 3 areas to explore, but what can the pupils say about them? [The point on the left hand side is 1/4 of the way down.] They could explore many more examples like this and compare the three areas and triangles you create. Then there is the second part of the question. Asking pupils if they could do something else with the original shape to produce a tiling effect can lead to all kinds of ideas. One that I saw was; <mdo:image src="New2ndPartFF.jpg"></mdo:image> Again questions about areas can be explored. Alternately you could right away present the same idea using the common A4 size of paper. <mdo:image src="OriginalFFonA4.jpg"></mdo:image> So it's really a matter of changing the original question slightly and getting the pupils to say what they see and what ideas could be explored.</mdoxml> 2 5 0 1 0 0 0 Fraction Fascination This problem challenges you to work out what fraction of the whole area of these pictures is taken up by various shapes. Fractions, Decimals, Percentages, Ratio and Proportion Fractions 2D Geometry, Shape and Space Mixed triangles 2D Geometry, Shape and Space Squares