Sol's Serviettes
Use the areas of triangles and squares to find unknown lengths
Devise and use problem solving strategies to explore situations mathematically (be systematic, draw a diagram).
This problem involves working with triangles and squares to track down a required area. The problem can be solved by using a scale drawing. However, this can only give an approximate answer to the question. Students should be encouraged to do it as accurately as they can. One way to do this is to notice how the isosceles triangle can be dissected to fit into the squares that we have in the diagram.
It’s worth noting that, generally speaking, scale diagrams can only give approximations to the ‘real’ thing. So it is almost always necessary to do something more sophisticated to get an accurate answer. On the other hand, scale diagrams may give clues to the exact answer (see Rings and Diamonds) and there may be some situations where scale diagrams give sufficient accuracy for more sophisticated methods not to be worthwhile. However, as a general rule, only use scale diagrams when you have tried everything else.
This is another example of breaking up an area to enable particular information to be found. (See also Bill’s Badge.)
The Problem
Sol was out at a boring birthday party for one of his father’s old friends. He had a nice red serviette folded into an isosceles triangle. There was also two square flower vases on the table. Although they were different sizes they both fitted on the serviette - one one way and then the other the other way.
He noted that the bigger vase A had a side length of 21cm. What was the side length of the smaller vase?
Teaching sequence
- Introduce the problem by brainstorming about triangles or ask the students to tell you what it is that makes a triangle a triangle.
- Pose the problem to the students.
- Ask the students for their initial thoughts about how they might proceed with the problem.
- Let the students work on the problem in pairs or small groups.
- If the students are stuck suggest that they construct the triangle or make a scale drawing of it. Hopefully in constructing the triangle they will note the relationship between the area of the square and the area of the triangle.
- Ask the students to write their solution so that it could be shared with others in the class.
- Share reports. Discuss the similarities and differences between the approaches used.
Solution
Because the serviette is isosceles, the larger square is ½ of the area of the triangle. Hence the area of the triangle is 2 x (21)2 = 882 (cm2).
Now the other diagram can be redrawn as below.
Hence the triangle is made up of all 9 small triangles and the smaller square is made up of 4 small triangles. So the area of the smaller square is
4/9 x 882 = 392 cm2.
This gives the side of the smaller square as √392 = 19.80 cm.
| Attachment | Size |
|---|---|
| SolsServiettes.pdf | 41.14 KB |
| SolsServiettesMaori.pdf | 55.26 KB |
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