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    This module is included inLens: Siyavula: Mathematics (Gr. 7-9)
    By: SiyavulaAs a part of collection: "Mathematics Grade 9"

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Space and shape

Module by: Siyavula Uploaders. E-mail the author

MATHEMATICS

Grade 9

ALGEBRA AND GEOMETRY

Module 7

SPACE AND SHAPE

Activity 1:

To understand the structure of some regular right prisms

[LO 3.3, 3.4]

A. Building containers

You will be given a sheet of shapes. You will need a ruler that you can measure with, a pair of scissors and glue or sticky tape. Colouring pens will also be helpful. Do the following with these shapes:

  1. Carefully measure all the lines and write down your measurements ( you should be able to measure to the nearest half–millimetre). You must also do your best to measure the radius (or diameter) of the circle. If you have a protractor available, find out where the 90°–angles are.
  2. Using these measurements, calculate the areas of the different shapes, and add the parts together to find out the total area of each of the four shapes. Set your work out very clearly so that anybody can understand what you have done. Use the proper names for the shapes you describe.

For example, for the last figure you could say:

Total area = small rectangle + small rectangle + large rectangle

= (l × b) + (l × b) + (l × b)

and so on . . . (Remember to use appropriate units.)

  1. Very carefully cut out the given shapes. You can colour these shapes to make it easier to see which the top and base are, and which the sides (the sides are striped). Now fold them and use tape, or glue and paper strips, to make four boxes. Keep the sides with the dotted lines on the outside.
  2. Write down what the Total Surface Area (TSA) of each shape is. (You have already calculated the answer!)
  3. Work in groups of two or three to try to find out how many 1cm × 1cm blocks will fit into each box. This is called the volume of the box. If you can find a method or a formula that will work with each of the four shapes, write that down carefully.
  4. At the end of this exercise, you should have two formulas.

B. Right prisms

  • Each of the four boxes is a right prism. These are shapes with a top and base that are exactly the same size and shape, and sides that go up straight at right angles to the base. Look around to see whether you can discover shapes with these characteristics.
  • We name right prisms according to the shape of the base, e.g. square prism, rectangular prism, triangular prism and circular prism (cylinder).
  • Are these two shapes right prisms? Describe the shape of the base of each, and confirm whether the sides go straight up at right angles to the base.
Figure 1
Figure 1 (Picture 9.png)
  • What kind of work did you do in this section? Score yourself in this table.
Table 1
Did i work Excellent Adequately Not well enough
well with my team?      
according to instructions?      
carefully?      
accurately?      
neatly?      

C. Formulas

  • To calculate the total surface area (TSA) and volume (V) of any right prism we use the following general formulas: (Please note that H refers to the prism height.)

TSA = 2 × base area + sides area and V = base area × prism height

Here are some important examples. These are the cut–out prisms you made into boxes. Please note how each part of the calculation is done separately and then put into the formula at the end.

  1. Square prism:

TSA = 2 base area + sides area = (2 × s2) + (H × base perimeter)

Figure 2
Figure 2 (graphics1.png)

s = 28mm

Step 1: Determine what the base is

and sketch it with its dimensions.

Step 2: Calculate the base area.

Base area = s2 = 282 = 784 mm2

Step 3: Calculate the base perimeter.

Base perimeter = 4 × s = 112mm

Step 4: Write down the height of the prism.

H = 52mm

Step 5: Calculate the TSA and V.

V = 784 × 52 = 40 768 mm3 ≈ 40,7 cm3

TSA = (2 × 784) + (52 × 112) = 7 392 mm2 ≈ 73,9 cm2

  1. Rectangular prism:

TSA = 2 base area + sides area = 2 (l × b) + (H × base perimeter)

Figure 3
Figure 3 (graphics2.png)

1 = 41mm; b = 14mm

Step 1: Determine what the base is and sketch it with its dimensions.

Step 2: Calculate the base area.

Base area = l × b = 41 × 14 = 574 mm2

Step 3: Calculate the base perimeter.

Base perimeter = 2 (14 + 41) = 110mm

Step 4: Write down the height of the prism.

H = 54mm

Step 5: Calculate the TSA and V.

V = 574 × 54 = 30 996 mm3 ≈ 31 cm3

TSA = (2 × 574) + (54 × 110) = 7 088 mm2 ≈ 70,1 cm2

  1. Cylinder:

Figure 4
Figure 4 (graphics3.png)
TSA = 2 base area + sides area = 2 (π r2) + (H × base perimeter)

r = 17,5mm

Step 1: Determine what the base is and sketch it with its dimensions.

Step 2: Calculate the base area.

Base area = π r2 = 3,14159 × (17,5)2 ≈ 962,1mm2

Step 3: Calculate the base perimeter.

Base perimeter = 2 π r = 109,956mm

Step 4: Write down the height of the prism.

H = 60,5mm

Step 5: Calculate the TSA and V.

V = 962,1 × 60,5 ≈ 58 207,8 mm3 ≈ 58 cm3

TSA = (2 × 962,1) + (60.5 × 109,956) ≈ 8 576,55 mm2 ≈ 85,8 cm2

  1. Triangular prism:

TSA = 2 base area + sides area = 2 (½ × b × h) + (H × base perimeter)

Figure 5
Figure 5 (graphics4.png)

b = 43,5mm; h = 31,5mm

hypotenuse = 53,7mm (Pyth.)

Step 1: Determine what the base is and sketch it with its dimensions.

Step 2: Calculate the base area.

Base area = ½ b × h = 685,125 ≈ 685,1mm2

Step 3: Calculate the base perimeter.

Base perimeter = b + h + hypotenuse ≈ 128,7mm

Step 4: Write down the height of the prism.

H = 60,5mm

Step 5: Calculate the TSA and V.

V = 685,1 × 60,5 ≈ 41 450,1 mm3 ≈ 41 cm3

TSA = (2 × 685,1) + (60.5 × 128,7) ≈ 9 157,06 mm2 ≈ 91,6 cm2

Exercise:

Calculate the total surface area and the volume of each of the following three prisms.

Figure 6
Figure 6 (Picture 2.png)

Assignment to be done in pairs:

  • Help Granny solve her problem. She has cooked a pot of peach jam. The jam is 2 cm from the top rim of her cooking pot which has a diameter of 24 cm and is 21 cm high.
  • She has some pretty jam jars which she wants to fill to about ½ cm from the top.
  • She has two types of jam jar. The brown kind has a square base (8 cm × 8 cm) and is 12 cm high, and the yellow kind has a base of 6,5 cm × 11,5 cm and is 11 cm high. There are eleven of each kind.
  • Her problem is that she wants to use only one type of jar for the peach jam. This means that she does not want to start filling one kind of jar and then find that she has jam left over when she has used up all eleven jars.
  • Your job is to find out for her whether she has enough jars of one type to fit her jam into, and to tell her which kind to use.

Activity 2

To become acquired with various two- and three-dimensional figures

[LO 3.1, 3.5]

A. Two-dimensional figures

These are figures that can be drawn on flat paper. Therefore they are called plane figures. Of course there are limitlessly many such figures.

Polygons are closed figures with three or more straight sides. If all the sides are the same length, and all the internal angles are equal, we call them regular polygons. Triangles are three-sided polygons, and an equilateral triangle is a regular three-sided polygon. A square is a regular four-sided polygon. Pentagons have five sides, hexagons have six sides and heptagons have seven. Make a list of as many of these special names as you can find.

Here are several closed plane figures. Decorate them and write the name of each polygon on the shape.

Figure 7
Figure 7 (Picture 3.png)

B. Investigation

Choose four polygons from the group above, all regular, but with four different numbers of sides. Now measure the sizes of the internal angles of each. Try to find out whether it is possible to make a formula to tell you how large the angles are, and what they add up to.

The following table will be helpful. As you can see, there are infinitely many polygons.

Table 2
No. of sides a = internal angle size b = 360 – a c = b – 180 Total of a Total of c
Three       3×a = 3×c =
Four       4×a = 4×c =
Five       5×a = 5×c =
Six       6×a = 6×c =
Seven       7×a = 7×c =
Twelve       12×a = 12×c =
  • The characteristics in the table above are needed when you have to decide how to tile a floor (say) with regular polygons so that they don’t overlap and don’t leave gaps. Some of these polygons will work alone, and some can or must be combined.
  • Design and draw a repeating tiling pattern of your own, using only regular polygons and colour it so that the pattern shows clearly.

C. Three-dimensional closed figures

  • If these shapes have sides made up of polygons, then we call them polyhedra. A regular polyhedron has faces that are congruent regular polygons, with internal angles the same shape and size.
  • In contrast to the polygons, there are only five regular polyhedra. They have been known since the time of Plato and the Greek mathematicians; this is why they are known as the five Platonic Solids.

D. Project

Research the five Platonic Solids, finding their names and properties, and other interesting deductions and facts about them. Make an attractive poster or models of these solids showing the facts associated with each. Below are pictures of the five solids.

Figure 8
Figure 8 (Picture 4.png)

Assessment

Table 3
LO 3
Space and Shape (Geometry)The learner will be able to describe and represent characteristics and relationships between two-dimensional shapes and three-dimensional objects in a variety of orientations and positions.
We know this when the learner :
3.1 recognises, visualises and names geometric figures and solids in natural and cultural forms and geometric settings, including:3.1.1 regular and irregular polygons and polyhedra;3.1.2 spheres;3.1.3 cylinders;3.2 in contexts that include those that may be used to build awareness of social, cultural and environmental issues, describes the interrelationships of the properties of geometric figures and solids with justification, including:3.2.1 congruence and straight line geometry;3.3 uses geometry of straight lines and triangles to solve problems and to justify relationships in geometric figures;3.4 draws and/or constructs geometric figures and makes models of solids in order to investigate and compare their properties and model situations in the environment;
3.5 uses transformations, congruence and similarity to investigate, describe and justify (alone and/or as a member of a group or team) properties of geometric figures and solids, including tests for similarity and congruence of triangles.

Memorandum

Discussion

  • This guide includes two pages of figures for constructing simple right prisms. Photocopy enough for the learners to make at least two of the figures. It would be best if the copies could be made on very light card (or heavy paper). If they are asked to colour some of the parts (e.g. the base and top) it might make it easier to explain some of the more difficult formulae.
  • The two formulae for right prisms are, in general:
  • Total Surface Area = double the base area + height of prism × perimeter of base
  • Volume = base area × height of prism
  • Ensure that learners are clear on the units (squared or cubed) appropriate to each formula.
  • Another difficulty that learners might encounter is that the word height is used in calculating the area of triangles as well as being one of the dimensions of right prisms. A useful trick is to use h for the triangle case and H for the prism case.
  • Breaking down the steps required for the calculations is a useful method for learners who get confused by the components in the formula. Of course, very competent learners will substitute values straight into the formula. This is an effective system, and should be encouraged where appropriate.

Solutions – exercise:

Rectangular prism: TBO = 412 cm2 Vol = 480 cm3

Triangular prism: TBO = 307,71 cm2 Vol = 360 cm3

Cylinder: TBO = 402,12 cm2 Vol = 603,19 cm3

Granny’s Jam Pot: Vol = 8 595,40 cm2

11 Square–based jars: Vol = 8 096 cm2

11 Rectangle–based jars: Vol =8 633,63 cm2

So, granny must use the rectangular–based jars if she wants to fit all the jam in!

Figure 9
Figure 9 (Picture 18.png)

3 = triangle; 4 = tetragon; 5 = pentagon; 6 = hexagon; 7 = heptagon; 8 = octagon; * = not polygon

Table 4
No of sides a = internal angle size b = 360° – a c = b – 180° Total of a Total of c
Three 60° 300° 120° 3×a = 180° 3×c = 360°
Four 90° 270° 90° 4×a = 360° 4×c = 360°
Five 108° 252° 72° 5×a = 540° 5×c = 360°
Six 120° 240° 60° 6×a = 720° 6×c = 360°
Seven 308,57° 51,43° –128,57° 7×a = 2160° 7×c = –360°
Twelve 330° 30° –150° 12×a = 3960° 12×c = –360°

TEST 1

1. Explain how you would recognise a right prism.

2. Explain how you could find the base of a right prism.

3. Calculate the total surface area and the volume of each of the following three prisms. Give your answers accurate to two decimal places.

Figure 10
Figure 10 (Picture 19.png)

TEST 1 – Memorandum

1. Essential points in the explanation: three-dimensional; top and base congruent plane shapes; side(s) at right angles to base.

2. Any reasonable explanation, e.g. if the chosen side is placed at the bottom, the description of a right prism fits what you see.

3. Rectangular right prism: TBO = 1 939,68 cm2 Volume = 5 769,72 cm3

Triangular right prism: TBO = 1 507,74 mm2 Volume = 2 312 mm3

Cylinder: TBO = 8 022,37 m2 Volume = 41 593,67 m3

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