That Takes the Biscuit

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Purpose

This is a level 3-4 activity from the Figure It Out series.

A PDF of the student activity is included.

Student Activity

    

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Required Resource Materials

square grid paper

scissors (optional)

Activity

This activity invites students to generalise how to find the areas of parallelograms and triangles from the area of the surrounding rectangle. Multiplicative strategies are useful for this activity but are not essential. The reasoning required here is mostly spatial, although questions 6 and 7 invite some use of numbers.

By cutting and pasting, it is possible to change any rectangle to a parallelogram of the same area and vice versa. In general, it is done in this way:

Similarly, it is possible by cutting and pasting to show that a triangle has half the area of its surrounding rectangle, and likewise, that a rectangle has twice the area of any triangle it surrounds, as long as the triangle and rectangle share 2 vertices (have the same base). Diagrammatically, the cutting and pasting looks like this:

Questions 1, 2, and 3 require the students to generalise these properties through spatial visualisation. The students’ ability to directly compare the areas of rectangles and parallelograms (see question 2) will indicate their understanding of the attribute of area. In other words, do they recognise that shapes can be compared by area? For teaching ideas and student progressions in area, you could refer to pages 3–5 and 11–13 of Book 9: Teaching Number through Measurement, Geometry, Algebra, and Statistics in the Numeracy Project series.

Question 2a deals with two rectangles rather than one rectangle and one parallelogram, and they don’t have the same height or base, so the principle stated above does not apply. The shapes do, however, have the same area, and this can be shown by cutting and rearranging. In general, different-shaped rectangles don’t have the same area. Challenge your students to discover what  there is about the relationship between these two rectangles that means that they do have the samearea. (Poochpower is the height of Barking Mad and 1 1/2 times its length. Mathematically, this relationship can be represented like this: 2/3 x 3/2 = 1.)

Note that most of this activity is about direct comparison of area rather than the use of a unit to measure it. Don’t assume that your students have a unit concept for area simply because they can directly compare. Frequently, students do not understand that squares are used as a unit because they tessellate. They will often want to use non-tessellating shapes like beans or counters when the shape to be measured has curved sides. They may not be aware that in measuring area, the square units are located in rows and columns so the amount can be calculated easily with multiplication.

This situation becomes more complex when the shape is made up of whole and part square units, as with triangles and parallelograms. When the students answer questions 4–7, encourage them to apply the area principles they have established for rectangles and parallelograms. This may help them to establish how square units can be partitioned and recombined. For example, in question 5b, extend the activity to include unit measurement by drawing diagrams such as the following:

Look for the students to establish spatially that parts of squares can be recombined to form whole units. This will help them to understand questions 6 and 7 better. In question 6, look for the students to begin with “parent” rectangles of 48 and 24 squares. From these rectangles, they can make an infinite number of correct parallelograms and triangles. For example, consider question 6a:

An 8 x 6 rectangle has an area of 48 squares, so these parallelograms will also have that area.

Similarly for question 7, look for your students to reason spatially about how to create hexagons and octagons with the same area as a rectangle with an area of 40 squares. This rectangle could be 5 x 8, 4 x 10, 2 x 20, 1 x 40, or 2 1/2 x 16. The question doesn’t specify a regular hexagon or octagon, so the side lengths don’t have to be all the same. In the case of the hexagon, the students will need to reason how they can transform a rectangle of area 40 into a hexagon with the same area.

With the octagon, the students will need to consider how a rectangle with an area greater than 40 squares can be reduced to form an octagon with an area of 40 squares. This diagram shows one solution:

Another strategy could be to make 1/4 of an octagon with an area of 10 squares and then use symmetry to make a complete octagon with an area of 40 squares. The diagram below shows one such solution:

Extension


The students could investigate how to find the areas of other shapes. The trapezium is a good example. A trapezium is a quadrilateral with only 1 pair of parallel sides. Challenge the students to find the connection between the area of a trapezium and the area of rectangles of a related size.

The students could also use their knowledge of triangles and angles to create regular hexagons and octagons:


Answers to Activity


1. The two different-shaped biscuits are the same size because their areas match (assuming, based on Brock’s comment, that they are the same thickness).

2. a.–c. All the biscuits have the same area.


3. Answers will vary. If the widths and heights of the rectangles and parallelograms match, then the areas will be the same.


However, rectangles and parallelograms can have the same area even if these measurements don’t match.
4. a. Half the area
b. Explanations will vary. The diagrams below show the matching areas:

c. With a rectangle. It is easier to see how a rectangle can be cut to make 2 triangles of the same size.
5. a.–b. Both biscuits are the same size as the Barking Mad biscuits. The diagrams below show the matching areas:

6. Shapes will vary. Some examples are:
a.

b.

7. a. Hexagon examples will vary. One possible hexagon is:

b. Rotation may vary, but the octagon with 40 squares will look like this:

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