Investigate and approximate the area of polygons

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] You know that when we tessellate, a flat space is completely covered without any overlapping and without spaces being left. 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. Whole Blocks Half blocks made into whole blocks Other bits made into whole blocks Total number of 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.)

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

3. Count the blocks covered by the following polygons:

3. 1 _________ square blocks

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:

4.1_____________ square cm

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?

Explain to a friend how you calculated your answer. Write down how you calculated your answer. Also write down your answer. Remember to write “square cm” with your answer. 6. In the family room there is a mat that is 4 m long and 3 m wide. 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. Can one make a square tiled area with 25 square tiles? 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

1.2

1.3

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 Input 1 2 3 4 5 6 7 10 13 Output 7 14 21 28

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

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: 1 2 3 4 7 8 9 10 20 50 7 12 17 22

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

4.3 Write down in words what was done: The input number was ______ 5. 5.1 Now find the “recipe” and complete this table: 1 2 3 4 5 6 9 11 12 20 3 5 7 9 11

Write down in words what was done: The input number was ________ 5.3 Now find the “recipe” to complete this table: In 1 2 3 4 5 6 7 10 14 Out 3 7 11 15

Write down in words what was done: The input number was _______ 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: 1 to 10 inclusive = _____ 11 to 20 inclusive = _____ 21 to 30 inclusive = _____ 31 to 40 inclusive = _____ 41 to 50 inclusive, and so on to 100. Write down the answers, study them and try to explain why this pattern occurs. 8. More patterns with shapes. The following pattern may be made with toothpicks, one for each straight line.

Pattern: Each time we add one triangle, we need ______more toothpicks. We could put our information into a table. Please complete it. Number of triangles 1 2 3 4 5 6 17 25 Number of toothpicks

How are the last two answers calculated? There are at least two different ways (without a calculator) and it is important that you should discuss these with your friends. 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 17 triangles 25 triangles 9. Complete the table: In 1 2 3 4 5 6 10 20 50 Out 8 15 22 29 36

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

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: 1 2 3 4 5 6 10 12 20 4 7 10 13 16

8. Complete this table: 1 2 3 4 7 8 1 4 9 16 100

9. Thirty squares are made with toothpicks as shown in the diagram (one toothpick for each straight line). How many toothpicks are needed? Find a pattern and write down the missing numbers: 5; 13; 21; 29; ___;___

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 about 8 square cm 7 square cm counted the length on the long side and the number of tiles on the short side. 5 x 2 = 10 tiles or 10 square cm Drawing 4m x 3m = 12 square metres Drawing 6 rows Drawing 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. Missing output numbers: 35; 42; 49; 70; 91 Flow diagram: input numbers: 1; 2; 3; 4; 5; 6; 7; 8; 9; 10 Operator: x 7 Output numbers: 7; 14; 21; 28; 35; 42; 49; 56; 63; 70 multiplied by 7 4.1 1 2 3 4 7 8 9 10 20 50 7 12 17 22 37 42 47 52 102 252

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 1 2 3 4 5 6 9 11 12 20 3 5 7 9 11 13 19 23 25 41

5.2 multiplied by 2 and 1 was added to the answer 5.3 In 1 2 3 4 5 6 7 10 14 Out 3 7 11 15 19 23 27 39 55

5.3 multiplied by 4 and 1 was subtracted from the answer. It can! x 4 – 1 Own 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 Triangles 1 2 3 4 5 6 17 25 Tooth-picks 3 5 7 9 11 13 35 51

Discussion 8.4 (a) 17 x 2 + 1 (b) 25 x 2 + 1 9. In 1 2 3 4 5 6 10 20 50 Out 8 15 22 29 36 43 71 141 351

TEST YOUR PROGRESS 1.1 Yes 1.2 Yes 1.3 No 2. Only 1 pair of opposite sides are parallel; they are not equal in length. 3. It is a rigid shape. 4. 170 tiles 5. 15 tiles 6. Diagram 4 by 4 7. 1 2 3 4 5 6 10 12 20 4 7 10 13 16 19 31 37 61

8. 1 2 3 4 7 8 10 1 4 9 16 49 64 100

10. 5; 13; 21; 29; 37; 45