SUBJECT: Number theory
TOPIC : Maps and Scales
LESSON :Calculating actual distances
on the earth’s surface from distances on maps and vice versa
PREVIOUS KNOWLEDGE: Students
can solve problems involving ratios and proportions as well as convert from one
unit of measurement to another. Students can identify a map.
OBJECTIVES: By the end of
the lesson, students should b
i)
Able to calculate distances on maps given the scale,
corresponding actual distances on the earth’s surface and vice versa.
ii)
ii) Able
to calculate scales if distances on map and earth’s surface are given.
iii)
iii) Able to accurately convert from one unit of
measurement to the other.
TEACHING AIDS: Maps with
scales and routes demarcated in it
TEACHING METHODS: Demonstration, Group discussion,
Questioning, and Assignment methods
GRADE LEVEL: 7
DURATION: 45 minutes
DATE: 27/11/2019
INTRODUCTION
When a map is drawn to scale of
say 1:50,000, it means 1cm on the map represents 50,000cm on the earth’s surface
or on land. 1:50,000 is a ratio showing the distance o map to distance on land.
This ratio can be reversed to have it as 50,000:1 that is distance on land to
distance on map.
PRESENTATION
If the ratio 1:50,000 means 1cm on map is 50,000cm on land. When this
ratio is reversed it is 50,000:1 or written as
50,000
1 is called a linear scale factor denoted by.
Length on land
= K or distance on land = k,
Length on map distance on map
Using the linear scale factor above, it follows that 50,000
=K, cross multiplying by the denominator
1
Example 1
A map is drawn to a scale of 1:50,000.Calculate the length of a
road which appears as 3cm on the map.
Solution
Scale, 1:50,000
Length on map=3cm.
Length on land=X
Linear Scale Factor (K) = 50,000
Length on land = k
Length on map
X =
50,000
3cm
X x 3cm = 50,000 x 3cm
3cm
X=150,000cm.
Therefore length of road which appears to be 3 cm on maps
actually 150,000cm on land.
Example 2.
A map is drawn to a scale of 1:100,000.Calculate the distance
between two towns A and B which appear to be 12.3cm apart on a map.
Solution
Scale 1:100,000
Distance on map=12.3cm
Distance on land=X
Linear scale factor (K) =100,000
Distance on land = k
Distance on map
X = 100,000
12.3
X x 12.3cm = 1000,000x 12.3 cm
12.3cm
X=1,230,000cm
Therefore distance between the two towns which appear to be
12.3cm on map is 1,230,000 cm on land.
Example 3.
A map is drawn to a scale of 1:30,000. Calculate the length of a
lake on the map which is 10cm long on land.
Solution
Scale 1:30,000
Length on map=Y
Length on Land=10cm
Linear Scale Factor (K) =130,000
Therefore the length of lake on the map is 0.000077cm on the
map.
Distance on land=
k
Distance on map
10cm= 130,000
=
k
Y
10 cm x Y = 130,000 x Y
Y
10cm= 130,000 Y
130,000Y = 10cm
130,000Y
= 10cm
130,000 130,000
Y = 0.000077cm
Note. From the statement distance
on land=k; we write, distance on land= K x distance on map
distance on map
Example4.
If the scale of a map is 1:10,000. What will be the length of a road on a map which is 50.000cm long?
If the scale of a map is 1:10,000. What will be the length of a road on a map which is 50.000cm long?
Solution
Scale 1:10,000
Length map= Y
Length on land =50,000cm
Linear Scale Factor (K) =10,000
Distance on land= K
Distance on map
50,000cm = 10,000
Y
50,000cm x Y = 10,000 x Y
Y
50,000cm=10,000Y
10,000Y=50,000cm
10,000 Y = 50,000 CM
10,000 10,000
Y= 5cm
Therefore, length of road on map=5cm.
Example 5.
A map is drawn to a scale of 1:20,000.Calculate;
a)
the distance between two villages in km
which appear to be 2cm apart on the map
b)
The length on the map of a lake which is
10km long.
Solution
a)
Scale 1:20,000
Distance between the
villages on map=2cm
Distance between the
villages on land=X
Linear scale (K) =
120,000
Distance on land
=K
Distance on map
X=20,000
2cm
X x
2cm= 20,000x2cm
2cm
X =
40,000cm
Distance on land in cm
is 40,000cm. This needs to be converted to kilometers (km)
100cm= 1m
40,000cm=T
40,000cm
= T
1000 cm 1m
400m x 1m = T
T=400m.
1000m=1Km
400m=p
400 m
= p
1000 m
1km
400m=
p
10000m 1km
P= 400x 1km
1000
P=0.4km
Therefore,
distance on land in kilometers is 0.4km
b)
Scale 1:20,000
Length of lake on
land=10km= (10
) cm =1,000,000cm
Length of lake on map=Y
Linear scale factor (K)
=20,000
Length of lake on land=
k
Length of lake on map
1,000,000cm=
20,000
y
1,000,000cm=20,000Y
20,000Y=1,000,000cm
20000y=
1,000,000
20,000 20,000cm
Y=50cm
The lake is 50cm long
on the map.
Example 6
On a map of scale 1cm
to 5m, the distance between two palm trees is 14cm. Calculate the distance
between these trees on the ground.
Solution
Scale 1cm to 5m means
1cm: 500cm giving 1:500
Distance on the
ground=X
Distance on the map
=14cm
Linear Scale factor (K)
= 500
X=
500
14cm
`X =14cm
X=7000cm
Distance between the
palm trees on the ground is 7000cm apart.
EVALUATION
1)
On a map of scale 1:35000 the distance
between two telephone poles is 26cm. calculate the distance between the two
poles on land.
2)
If the scale of a map is 1:10,000, what
will be the length on the map of a road which is 100m?
3)
Find the length of a road represented by
i)
21.7cm on map
ii)
O.75cm on map
When
the scale is 1cm to 5m
4)
Find the length on a map represented by
i)
7.2cm on land
ii)
28.6cm on land
When the scale is 1cm to 10m.
SUMMARY
Ø
For any calculation to be done, make sure
that the scale has the same units on numbers involved and write this out as a
ratio in the form 1:n
Ø
Write down the linear scale factor K
obtained from the ratio dimension on land: dimension on map or plan.
Ø
Use the formula; Length on land=K
Length on map or plan.
Ø
Substitute the values in the formula and
solve as requested.
Ø
Take not of the units of measurement in
use and make sure you convert them to one.
CONCLUSION
1)
If a scale is 1:10,000. What length will
60cm on the map represent on land:
i)
in centimeters(cm)
ii)
in meters(m)
iii)
in kilometers(km)
2)
The distance from Bamenda to Ndop is 32km.
How far apart will they be on a map of scale 1:5000?
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