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Grade 7 Mathematics Learner’s Book

INNODEMS

Section 3.2 Length

Length in focus.
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Consider a tailor cutting out and doing your school unifom. He has to take your measurements using a tape measure labelled \(\textbf {A} \) alongside, record them down then take the fabric to make your uniform , measure then cut the fabric ready to sew it to come up with a well fitting uniform for you.
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length

Subsection 3.2.1 conversion of units of length

Activity 3.2.1.

\({\color{magenta}text bf{Converting units of length.}}\)
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Washika wants to go up the stairs. The staircases are labelled centimetres \(\text{(cm) }\text{,}\) decimetres \(\text{ (dm) }\text{,}\) metres \(\text{ (m) }\text{,}\) Decametres \(\text{ (Dm) }\) and Hectometres \(\text{ (Hm) }\) Every step he makes is equivalent to ten times the previous step from the picture above.
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  1. How many decimetres are there in a metre?
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  2. How many decimetres are there in a Decametre?
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  3. How many metres are there in a Hectometre?
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length conversion
\(\text bf{ Conversion of length}.\)
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  • \(\displaystyle {\color{Magenta} \text {To perfom the operations involving length, measurements must be of the same units. }}\)
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  1. Collect a number of sticks and cut them according to the exact height of the staircase in your environment.
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  2. Measure the sticks to uniform length in my case it’s \(10\text{ cm }\) long.
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    I have \(7\) what about you?
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  3. We will also measure the width of our staircases the same way we did to the length and we shall get the total length and the total width.
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  4. Fill in the table below to test mastery of skills learnt from the activity...
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Table 3.2.1.
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______mm = 1cm
______cm = 1dm
______dm = 1m
Table 3.2.2. Learning point.
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10mm = 1cm
10cm = 1dm
10dm = 1m
10m = 1Dm
10Dm = 1Hm
10Hm = 1km
\({\color{Magenta} \text {The standard unit for measuring length internationally is Metres.}}\)
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a. We multiply a unit by \(10\) to convert it to the next in descending order. \({\color{Magenta} \text{ Km } \xrightarrow[] {\times 10} \text{ Hm } \xrightarrow[] {\times 10} \text{ Dm } \xrightarrow[]{\times 10} \text{ m } \xrightarrow[]{\times 10} \text{ dm } \xrightarrow[]{\times 10}\text{ cm }}\)
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b. We divide a unit by \(10\) to convert it to the next in ascending order. \({\color{Magenta} \text{ Km } \xleftarrow[]{\div10} \text{ Hm } \xleftarrow[]{\div10} \text{ Dm } \xleftarrow[]{\div 10}\text{ m } \xleftarrow[]{\div 10}\text{ dm } \xleftarrow[]{\div 10}\text{ cm }}\)
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Interact with the exrcise below. \({\color{blue}\text{Drag the blue box across to your right and back}}\) and observe conversion of different units of measuring length from \({\color{blue}\text{Km to mm.}}\)> You can also click on the dropown arow written ⟦\(mathbf{metres}\)⟧ and change the units as you prefer. The \(\mathbf{\text {randomizing tab} can also help you generate a new number.}\) \({\color{magenta}\text{Enjoy as you play around with the mathematical figures!}}\)
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Figure 3.2.3. Converting various units of measuring length.
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Example 3.2.4.

Moraa walks from home to school taking \(40\text{ m } \text{.}\) Convert Moraa’s distance in \(\text{km}\text{.}\)
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Solution.
Since,
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\(1000 \text{ m } =1\text{ km }\)
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\begin{align*} 40 \text{ m }=\amp \frac{40 \cancel {m}}{1\,000\cancel{m}} \times 1\text{ km }\\ =\amp 0.04 \text{ km } \end{align*}
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Moraa walks \(0.04 \text{ km }\) from home to school.
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Example 3.2.5.

The path from the gate to the office in Myra school is \(40 \text{m} 450 \text{cm}\text{.}\) What is the distance in centimetres?
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Solution.
Distance is \(40 \, \text{m} \, 450\, \text{cm}\text{.}\)
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The conversion is,
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\(100\,\text{cm}= 1\,\text{m}\)
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Therefore,
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\begin{align*} 40 \, \text{m} \, 450\, \text{cm}=\amp (40 \times 100)\,\text{cm}+ 450\,\text{cm}\\ =\amp4\,000\,\text{cm} +450\,\text{cm} \\ = \amp 4\,450 \text{cm} \end{align*}
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The path from the gate to the school is \(4\,450 \text{cm} \text{.}\)
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Subsection 3.2.2 Addition and Subraction of units of length

Activity 3.2.2.

  • Work in groups using a 30cm ruler.
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  • Measure the width of your classroom window.
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  • Measure the distance from door to the furthest corner of your classsroom.
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  • What do you notice about the width of the windows and the door?
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  • They are smaller than the wall measurement.
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  • Take the total length of all walls and subtract the width of windows and classroom door and note down.
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(Measuring length)
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Example 3.2.6.

Work out\(8\text{ dm }\) \(5\text{ cm }\) +\(7\text{ dm }\) \(6\text{ cm }\)
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Hint.
Allign the same units vertically.
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Solution.
Dm cm
+ 8 5
7 6
16 1
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Example 3.2.7.

Add \(9\text{ Hm }\) \(5\text{ Dm }\) \(4\text{ m }\) \(4\text{ dm }\) to \(1\text{ Hm }\) \(6\text{ Dm }\) \(5\text{ m }\) \(6\text{ dm }\)
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Hm Dm m dm
1 6 5 6
+ 9 5 4 4
11 2 0 0
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Activity 3.2.3.

  • Measure and cut off a string of length \(1\text{ m }\text{.}\)
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  • From the string cut off a string of length \(60\text{ cm }\text{.}\)
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Example 3.2.8.

Subtract \(3\text{ m }\) \(15\text{ cm }\) from \(5\text{ dm }\) \(25\text{ cm }\)
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Hint.
Allign the same units vertically starting from the second then the first one.
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Solution.
m cm
5 25
- 3 15
2 10
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Example 3.2.9.

work out \(6 { Hm } \) \(5 { Dm }\) \(3\text{ m }\) \(4 \text{ dm }\)- \(4\text{ Hm }\) \(6 \text{ Dm }\) \(5 \text{ m }\) \(3 { cm }\)
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Hm Dm m dm cm
6 5 3 4 3
- 4 6 2 5 3
1 9 0 9 0
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Subsection 3.2.3 Multiplication involving units of length

Activity 3.2.4.

  1. Take the number of steps in our staircase and multiply by the height and you will have the total height without measuring the like in the case of my continuos arrow labelled \(\text {h}\)
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  2. Measure their total length and width.
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Example 3.2.10.

Work out \(\, 8\text{ Dm }\) \(9\text{ m } \times3\)
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Solution.
Dm m
8 9
\(\times\) 3
26 7
  • \(\rightarrow\) \(9 \times3 = 27\text{,}\) write \(7\text{ m } \) regroup \(20 \text{ m }\) to \(2\text{ Dm }.\)
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  • \(\rightarrow\) \(8\times3 =24 +2 = 26\)
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Example 3.2.11.

Calculate:
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Hm Dm m
6 8 5
\(\times\) 5
Solution.
Hm Dm m
6 8 5
\(\times\) 5
34 2 5
  • \(\rightarrow\) \(5 \times5 = 25\text{,}\) write \(5 \text{ m } \) regroup \(20 \text{ m }\) to \(2 \) add it to \(8 \times5 = 40\) get \(42 \text{ Dm },\) \(2 \text{ Dm }\) to \(4 \text{ Hm }\)
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  • \(\rightarrow\) Add \(4 \text{ Hm }\) to \(6\times5 = 30 +4 \) to get \(34 \text{ Hm }\)
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Activity 3.2.5.

  • Prepare a string of a certain length.
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  • Fold and cut it into pieces of equal length.
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  • Measure the length of each piece of string.
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  • Relate the length of each string to that of the original string.
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Prepare
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Example 3.2.12.

Work out \(13 \text{ dm } \) \(2 \text{ m } \div 6\)
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Solution.
2Dm 2m
6 13Dm 2m
12Dm
\(1\times10\) 10
+2
12
-12
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  • Divide \(13\text{ Dm }\) by \(6\) to get \(2\text{ Dm }\) remainder \(1\text{ Dm }\text{.}\)
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  • Convert \(1\text{ Dm }\) to \(\text{ m }\) to get \(10\text{ m }\text{.}\)
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  • Add \(2\text{ m }\) to \(10\text{ m }\) to get \(12\text{ m }\text{.}\)
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  • Divide \(12\text{ m }\) by \(6\) to get \(2\text{ m }\text{.}\)
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\(13 \text{ dm } \) \(2 \text{ m } \div 6 = 2 \text{ Dm } 2 \text{ m }\)
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Example 3.2.13.

Calculate \(37\text{ Hm } 9 \text{ Dm } 2\text{ m } \div 12 \)
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Solution.
3Hm 1Dm 6m
12Hm 37Dm 9m 2m
-36 \(\downarrow\)
\(1\times10\) 10 \(\downarrow\)
+9 \(\downarrow\)
19
-12
\(7\times10\) 70
+2
72
-72
00
\begin{equation*} 37\text{ Hm } 9 \text{ Dm } 2\text{ m } \div 12 = 3\text{ Hm } 1 \text{ Dm } 6\text{ m } \end{equation*}
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Subsection 3.2.4 Perimeter of plane figures

Activity 3.2.6.

We need paper cut-outs and a ruler.
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  1. Make paper cut-outs of different plane figures.
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  2. Make paper cut-outs of different plane combined shapes.
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  3. Measure the distance round each shape using a piece of string and a ruler.
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  4. Compare the answers in step 2 and 3 and note the difference.
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Perimeter is the distance round a plane figure. To work out perimeter, you add the lengths of the sides of a given plane figure.
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Example 3.2.14.

Work out the perimeter of the given plane figures.
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Solution.
  1. Perimeter of a square.
    \begin{align*} =\amp l+l+l+l \\ =\amp 4l\\ =\amp 4 \times 10cm \\ =\amp \textbf {40cm} \end{align*}
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  2. Perimeter of a rectangle.
    \begin{align*} =\amp l+w+l+w \\ =\amp 2 \times (l + w) \\ =\amp 2 \times (40cm +8cm) \\ =\amp 2 \times 48cm \\ =\amp \textbf {96cm} \end{align*}
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Develop the formula for a perimeter of a rectangle in this activity. Drag th
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Figure 3.2.15. Calculating perimeter.
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Example 3.2.16.

Work out the perimeter of the figure alongside
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Solution.
We add all the sides to get the perimeter.
\begin{align*} \text{perimeter}=\amp4\text {cm} + 3 \text {cm} + 7\text {cm }+ 7\text {cm} + 3\text {cm} + 4\text {cm} + 15\text {cm}\\ =\amp \textbf {43cm} \end{align*}
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Exploration 3.2.7.

Extended activity
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Ombui made a paper cut out of right-angled triangle like the one shown on the right. He then made three similar cut outs then joined them as shown below to form a combined cut-out. Work out the perimeter of the combined cut
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We can as well make different shapes and find their perimeters. Explore on the folowing shapes and count the number of sides they have in the video below. https://www.youtube.com/watch?v=WTeqUejf3D0
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Figure 3.2.17. Perimeter of different plane figures.
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Subsection 3.2.5 Circumference of circles

Activity 3.2.8.

  1. We need a ruler, a string, a set square, a tapemeasure and a circular object.
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  2. Measure the circumference of each circular object using a tapemeasure as shown alongside, Or a string if you don’t have a tapemeasure and record your answer.
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  3. Use a ruler to measure the length of the string that is equal to the string that is equal to the circumference.
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  4. Measure the diameter of each circular object as shown below. The distance \(d\) is the diameter of each object.
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5.Draw and fill in the table like the one shown below.
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Table 3.2.18.
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Number of Item(s) Object Circumference (C) Diameter (d) Circumference \(\div\) diameter \((\frac{C}{d})\)
1
2
3
4
6. What do you notice about the values of \((\frac{C}{d})\text{?}\)
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Connect.
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  • The circumference is the distance round a circle.
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  • When you divide the circumference by the diameter of any circular object, the number you get is approximately \(3.14\) correct to two decimal places.
    \begin{equation*} \frac{Circumference}{diameter} = 3.14 \end{equation*}
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    Circumference \(= 3.14 \times\text {diameter}\)
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  • The number \(3.14\) is called \(\textbf {pi}\) and is represented by the symbol \(\pi\text{.}\) As a fraction, \(\pi = \frac {22}{7}\text{.}\) Therefore, circumference \(= \pi d\)
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  • Two radii are equal to the diameter. Hence \(d = 2r\)
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    Circumference
    \begin{align*} = \amp \pi \times d\\ = \amp \pi \times 2r\\ = \amp 2\pi r \end{align*}
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Example 3.2.19.

The diameter of a circle is \(21\, cm\text{.}\) Find its circumference.
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Solution.
Circumference \(= \pi D\) where \(D\) is the diameter of a circle.
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Therefore,
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\begin{align*} \text{Circumference} =\amp (\frac{22}{7} \times 21)cm \\ =\amp 66\, cm \end{align*}
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The circumference\(= {\color{blue} 66\,cm}\)
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Figure 3.2.20. Perimeter of different plane figures.
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Example 3.2.21.

Work out the circumference.
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Solution.
\begin{align*} \text{Circumference }=\amp \pi D \\ =\amp \frac{22}{7} \times 14\,cm\\ =\amp \frac{308}{7}\,cm \\ =\amp 44\, cm \end{align*}
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