You Can Count on Squares!

Getting Started

In this first session, the students investigate rectangles on dot paper but set in the context of examining the rectangles that two students (Jess and Hannah) have made.  Allow your class to work in pairs as this helps to stimulate discussion, and also allows students to clarify their ideas and any problems.

1. Pose the following problem:
   Jess and Hannah explore the different rectangular shapes that can be made using 24 squares. Hannah uses dot paper and draws these rectangular shapes. show copymaster 1.
   (Note that the students may need to explore the dot paper in order to construct a square of 1 square unit, where a unit is the shortest distance between 2 dots)
2. Are shapes 2 and 3 the same?
   Jess thinks they are. Explain her reasoning.
3. Has Hannah has drawn all of the different rectangular shapes?
   If not, complete Hannah’s picture using your dot paper
4. Now tell your class that Jess has drawn all the possible rectangles with 24 squares. (Show Copymster 2.)
   Jess looks at her shapes and notices a relationship between the number of columns and rows and number of squares in her rectangles. She says that she can find a shortcut way of counting all the squares in the shapes that have been made.
   What method has Jess discovered?
5. Ask the class to use the dot paper to draw all the different rectangular shapes that have an area of 36 square units?  (Note: Some students may realise that they do not have to draw all the shapes to identify which ones will have an area of 36.)
6. With the students construct a table which records the different rectangular shapes.
   

   Rows	Columns	Area
   1	36	36
   2	18
   3	12
   4
   5
   6
   9
   12
   18
   36

7. Complete the table and explain how their system of recording indicates all the different shapes that can be made that have an area of 36 square units.
8. Ask the students to use a table to find all the different rectangles that have an area of 60 square units.
9. From the work that you have done so far, write a rule that would allow you to quickly find the area (in square units) of any labelled rectangular shape.
   What is the area of this rectangle? (See Copymster 3)
10. Give students the following problem:
    A certain rectangular shape has an area of 60 square units, what might the length of its sides be?

Exploring

This section is divided into two parts. In Session A the class discovers the relation between the area of a parallelogram and the area of a rectangle; in Session B, the area of any triangle.

Session A

1. Ask the students to use the dot paper to make a square that has no dots in the middle, and only 4 on the boundary. Tell the students that each side is said to have unit length and that the area is one square unit.
   If the distance between each dot was 1 cm, what would you call the square? (1 square centimetre)
   If the distance between each dot was 1 m, what would you call the square? (1 square metre)
   You might like to show the actual size of the latter square using 4 one-metre rulers.
   (The aim of this task is to allow students to develop techniques that will help them to analyse the mathematical situations that they will meet in the rest of the unit.)
2. Make a rectangle using cardboard strips and split pins.
   
              
3.  
4. Show by pushing the edges a little that you get parallelograms.
   What do we call shapes like this? (Parallelogram.)
   Which is bigger, the area of the rectangle we started with or the area of the parallelogram?
   
         
5. Jess and Hannah wondered how they could find the area of a parallelogram.
   They reasoned that, if they could make these sorts of shapes into rectangles, then they could use what they already knew to solve their problem.
   Is there a rectangle with the same area?
   What rectangles can you get from a parallelogram?
   Why do the two shapes have the same area?

   Show why this is the case. This relies on the fact that the two end triangles in the diagram have the same area (triangles AXD, BYC). This is easily shown by drawing a diagram and cutting out the triangles. One will fit over the top of the other. This shows that the parallelogram ABCD has the same area as the rectangle ABXY.

   
    

   So we need to know the base length and the height to find the area of a parallelogram. Then
                                                       area = base x height.

6. Draw some other parallelograms.
   What rectangle has the same area as this one? This one?
   How would you label your diagrams to indicate a quick way of finding the area for each of your shapes?
   What are the areas of these parallelograms? (Copymaster 4)
   (It will probably be necessary to measure the height of the parallelograms.)
7. Get the students to write a short paragraph using pictures, illustrating how the areas of parallelograms are related to areas of rectangles. Their work should include a formula (in words will do) for the area. (Area = base x height.)

Session B

1. Jess and Hannah were a little worried about the triangles AXD and BYC. It would be nice if we had a way of finding their area.  So they decided to investigate a way of finding the area of right triangles, right-angled triangles – triangles with a right angle in them. They decided to look first at the right triangle below.
   

   

2. How would you try to do this?
   Can you make a rectangle out of it?
   Can you make a rectangle with two copies of it?
   Try putting two together. (See the diagram below.)
   

   

   So what is the area of the right-angled triangle?

3. What is the area of this right-angled triangle?
   
4. Get the students to make up their own right triangles and find their areas.
   Can you find a formula for the area of a right triangle? (Half base times height.)
   Then let a few of the students report what they have found.
5. For more able students you might continue this to find the area of any triangle. By putting two triangles together you’ll get a parallelogram. The area of the triangle is half the area of the parallelogram.
6. Get the students to write a short paragraph using pictures, to illustrate how to find the area of a right triangle. Their work should include a formula (in words will do) for the area. (Area = half base x height.)

Reflecting

Here the students use the formulae that they have obtained earlier to solve some problems.

Problem 1

Get the students to choose one of the shapes below, draw it on dot paper and find its area. The students can their own dimensions for the shapes. This can be done in at least two ways. By cutting it up into rectangles and by counting square on the pot paper.

Problem 2

You have been given 36 metres of string to ‘rope off’ a rectangular area that can be made in to a vegetable garden. What is the largest rectangular garden you can make?
Justify your choice, using diagrams and or tables of information.