MATHEMATICS

Grade 4

SPACE AND SHAPE, PATTERNS, DATA HANDLING

Module 14

INVESTIGATE AND APPROXIMATE THE AREA OF POLYGONS

Activity 1:

To investigate and approximate the area of polygons (using square grids and tiling) in order to develop an understanding of square units [LO 4.8]

1. Square blocks covered by your hand.

1.1 On the grid paper below, carefully place your hand with the fingers spread out. Trace around your hand with your pencil, stopping at the wrist. Lift your hand. You will see a beautiful outline of your hand. We want to find out how many square blocks your hand covers.

1.2 Put a dot in each full block as you count it, and write down the total number of full blocks covered by your hand in the table on the next page. Now look for places where half a block is covered. Two half blocks will make a whole block, so put a dot in each and count them as one whole block. Write down the total. Now combine those less than a half with those more than a half to make more wholes. Write down that total. Now add the totals. That should give you an approximate idea of how many blocks are covered by your hand.

Square blocks covered by my hand.

1.3 Now colour in the shape of your hand on the paper.

2. Count the square blocks covered by the following shape in the same way. First count the whole blocks. Then combine bits to make whole blocks. (Put dots in the blocks as you count them, if it helps.)

Table 3

Total number of blocks covered: …………………. square blocks.

3. Count the blocks covered by the following polygons:

Figure 1

3. 1 _________ square blocks

Figure 2

3.2 _______________square blocks

Measure the square blocks with your ruler. They are 1 cm long and 1 cm wide, so instead of calling them “square blocks”, we can call them SQUARE CENTIMETRES.

4. Now find the number of square centimetres covered by each of the following polygons:

Figure 3

4.1_____________ square cm

Figure 4

4.2_______________ square cm

5. Let’s pretend that you have made a doll’s house for a younger cousin. You have covered the floor of the bathroom with paper on which you have drawn 1 cm square blocks. There is a bath mat on the floor as shown below. How many of the tiles are covered by the bath mat?

Figure 5

6.1 Draw a diagram to show what the mat looks like and label the length and the width.

6.2 Calculate how many square metres of floor are covered by the mat. Write down your calculation and the answer. Remember to write “square metres” with the answer.

6.3 Now draw blocks on your diagram so that it is four blocks long and three blocks wide. Check your answer for 6.2.

7. Dad uses 36 square tiles to tile the floor of a square braai area. He starts to tile the floor by placing 6 tiles next to each other along the edge of this floor.

7.1 How many rows of six tiles each will he have when he has finished?

7.2 Draw a diagram to show what it looks like.

8.

8.2 Make a diagram to show what it would look like.

9. Dad uses 736 tiles to tile a rectangular stoep. There are 23 tiles across the width of the stoep. How many tiles are there down the length of this stoep? Write down your calculation and answer.

10. Little 1 cm square tiles that look like Tiger’s eye semi-precious stones are used to cover a work surface in the kitchen. There are 75 of these tiles along the length of this surface, and 54 tiles across its width. How many tiles are there altogether?

Activity 2:

To investigate and extend numeric and geometric patterns not limited to sequences involving constant difference or ratio [LO 2.1]

1.1

Figure 6

1.2

Figure 7

Figure 8

1.3

Figure 9

Figure 10

Figure 11

1.4

1. Look at the following shapes (which can be made with cuisenaire rods or toothpicks) and write down your answers:

There is a very obvious pattern. Predict what 1.4 will be.

That pattern is so easy because each time the pattern changes in the same way.

Spotting a pattern can save us time and energy when we calculate answers.

2. Now look at the multiples of 9 again. We have already noticed one pattern. Maybe you noticed another? 9; 18; 27; 36; …..

Add the digits that make each multiple: 0 + 9 =____; 1 + 8 =____; 2 + 7 =____; 3 + 6 = ____

Is this true of all the multiples of 9? Try a few more. It can be useful if you are not sure of a multiple. Some learners are not sure if 54 or 56 is a multiple of 9. Which is it? ____

John knows seventy something is a multiple of 9. Help him: seventy-___ All John has to do is to say: 7 + ___ = 9; the multiple is 72.

3. Complete the table by discovering and using the pattern:

3.1

3.2 This information could also be given in a flow diagram. Please complete the flow diagram by looking at 4.1 again:

Figure 12

3.3 Write in words what was done: The input number was____________

4.

4.1 Complete the table and then describe to your friends what was done:

4.2 Put the same information from 5.1 in a flow diagram:

Figure 13

4.3 Write down in words what was done: The input number was ______

5.

5.1 Now find the “recipe” and complete this table:

5.3 Now find the “recipe” to complete this table:

5.5 Why cannot this table in 6.3 be written in a flow diagram? Discuss this with your friends and then write down your answer.

6. Other patterns involving numbers:

6.1 Add: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

One can just add them in that order, or one can look for patterns. Just as we paired the numbers on opposite sides of the dice, so let us pair opposite numbers here: the first and the last and so on. It becomes:

1 + 10 and 2 + 9 and 3 + 8 and 4 + 7 and 5 + 6. What do you notice about the totals?

_______ This can be shortened to: 5 × 11. Explain this to a friend. Where does 5 × 11 come from?

6.2 Add all the numbers from 1 to 20 inclusive. Look for a pattern and a short way. Write down what you did and your answer on the dotted line below. Then check your answer the long way. You may use a calculator to do so.

7. Another interesting pattern can be seen in the answers if you add:

8. More patterns with shapes. The following pattern may be made with toothpicks, one for each straight line.

Figure 14

Hint: Maybe you could look at the toothpicks needed for six triangles and use them to calculate how many toothpicks are needed for 17 triangles, or you could think that you know the general pattern and just apply it to find out how many toothpicks are needed for 17 triangles. Your discussion is important, so the answers are not being given to you.The same applies to the 25 triangles.

8.4 Write down how you calculated the answers for

9. Complete the table:

TEST YOUR PROGRESS

1. Do the following show tessellation? Write “yes” or “no” for each one.

Table 10

Using the trapezium and diamond ________

Using just the trapezium ________

Using circles _______

2. Write down one way in which the sides of a trapezium differ from the sides of a

parallelogram .

3. Why is the triangle used in the building of the framework of the roofs of houses?

4. On a floor there are 10 tiles in a row and there are 17 rows of tiles. How many tiles are there altogether?

5. Dad uses 135 tiles to tile a stoep. He places 9 tiles across the width of the stoep. How many tiles are there in the length of the stoep?

6. Make a diagram to show what a square tiled area would look like if 16 square tiles were used to cover it. Use your ruler to draw in the tiles.

7. Complete the table:

8. Complete this table:

9. Thirty squares are made with toothpicks as shown in the diagram (one toothpick for each straight line). How many toothpicks are needed?

Assessment

Memorandum

ACTIVITY 1area of polygons

1.1 to 1.3 Practical own work and recording of it.

2. 4 whole blocks and 5 and a bit blocks = about 9 blocks

3.1 6

3.2 12

8.1 Yes

8.2 Drawing

9. 23 x ? =736

32 tiles

10. 75 x 54 = 4 050 tiles!

ACTIVITY 2 Patterns

1.4

4 shapes

2. Multiples of 9 – the digits forming each multiple of 9 add up to 9, so 54 is a multiple of 9; 72 is a multiple of 9. This is useful for checking answers.

Output numbers: 7; 14; 21; 28; 35; 42; 49; 56; 63; 70

4.1

4.2 Flow diagram:

Input numbers: 1; 2; 3; 4; 7; 8; 9; 10; 20; 50

Operators: x 5 + 2

Output numbers: 7; 12; 17; 22; 37; 42; 47; 52; 102; 252

4.3 multiplied by 5 and 2 was added to the answer.

5.1

5.2 multiplied by 2 and 1 was added to the answer

5.3

5.3 multiplied by 4 and 1 was subtracted from the answer.

6.2 1 + 20; 2 + 19; 3 + 18; 4 + 17; 5 + 16; 6 + 15; 7 + 14; 8 + 13; 9 + 12; 10 + 11

10 x 21 = 210

7. 55; 155; 255; 355; 455 etc.

Own

8.1 2

8.2

8.4 (a) 17 x 2 + 1

(b) 25 x 2 + 1

9.