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Dividing a real distance by the scale to get the line — and choosing which scale a map should be drawn at.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
You will work out how long a line should be on a map for a given real distance, and choose a sensible scale for a map's job.
You can turn a measured line into a real distance by multiplying. This lesson is the same scale sentence used the other way around, and the only new thing is which of the two numbers you start with. You will divide, which undoes multiplying.
| Term | What it means |
|---|---|
| Draw to scale | To make every line on a map stand for the right distance. |
| Close-up scale | A scale giving many centimeters to a little ground. |
| Wide scale | A scale fitting a lot of ground into a few centimeters. |
| Choose a scale | To decide which kind a map's job needs. |
The scale sentence has not changed: 1 cm stands for 2 km. Last lesson you had the line and wanted the distance, so you multiplied. This time you have the distance and want the line, so you divide:
real distance ÷ what one centimeter stands for = length of the line
Ten kilometers at 1 cm to 2 km is $10 \div 2 = 5$ cm.
There is a check you can do without redoing the sum. The line on the map is always the smaller number, because the map is smaller than the place. If your answer is bigger than the real distance, you multiplied when you should have divided.
Another way: story
Think of folding a long piece of string to fit in a box. However you do it, what goes in the box is shorter than the string. A map is the world in a box.
Another way: example
The same 10 km drawn at three different scales: 5 cm at 1 cm to 2 km, 10 cm at 1 cm to 1 km, 2 cm at 1 cm to 5 km. Same journey, three lengths of line, because the scale, not the journey, decides how long the line is.
A route must meet access rules as well as a distance limit. Suppose an invented museum map uses 1 cm for 10 meters. The 2 cm shortcut has stairs, the 4 cm path has a ramp, and the 7 cm path is step-free too. A visitor needs a step-free journey of at most 50 meters. First remove the stairs route. Then compare 40 and 70 meters with 50. Choose the 40-meter path. The shortest line on the page was not usable for this visitor. Check current access notes: a scale tells length, not whether a gate is locked.
Here is a 12 km road drawn on maps at different scales.
| Scale | Divide | Line on the map |
|---|---|---|
| 1 cm to 1 km | 12 ÷ 1 | 12 cm |
| 1 cm to 2 km | 12 ÷ 2 | 6 cm |
| 1 cm to 3 km | 12 ÷ 3 | 4 cm |
| 1 cm to 4 km | 12 ÷ 4 | 3 cm |
The more kilometers each centimeter carries, the shorter the line. A wide scale fits a long road into a small space.
Going from the map to the ground, you multiplied: 6 cm times 2 km is 12 km. Going back, you divide: 12 km divided by 2 km is 6 cm. You end up where you started.
That gives a check for every answer. Multiply your line by the scale number; you should get back the real distance.
A map is always smaller than the place it shows, so a line on the map is always a smaller number than the distance it stands for, as long as the scale is more than 1 km to the centimeter.
If you work out that a 10 km road needs a line of 20 cm, stop: the line came out bigger than the road. You multiplied instead of dividing. This check takes a second and catches the most common mistake.
Before dividing, both numbers must be in the same unit. If a path is 3 km long and the scale is 1 cm to 500 m, change the path to meters first: 3 km is 3000 m. Then 3000 divided by 500 is 6 cm.
Dividing 3 by 500 without changing units gives a tiny number, and nobody could draw a line that short. Matching units is part of the answer, not tidiness.
When you make a map, you choose its scale. A close-up scale, like 1 cm to 50 m, draws a small place big, with room for gates, paths and benches. A wide scale, like 1 cm to 50 km, fits a whole state on one sheet, but a town becomes a dot.
No scale does everything. A map of a park cannot show the whole country, and a map of the country cannot show the park's gate. Choosing a scale means choosing what to give up.
Mapmakers often start from the page. If a state is 400 km wide and the paper has room for a 20 cm map, then each centimeter must carry 400 divided by 20, which is 20 km. The scale is 1 cm to 20 km.
Then every other distance on the map is divided by 20. A 60 km road becomes a 3 cm line, and a 5 km lake becomes a line only a quarter of a centimeter long.
Checking an answer. Both checks should pass. If the line is bigger than the distance, you multiplied by mistake.
Dividing is allowed because it asks how many equal lots of the scale distance fit into the real distance, and each lot is one centimeter of line.
Changing kilometers to meters is allowed because it changes only how the distance is written, not the distance itself. Checking by multiplying is allowed because multiplying undoes dividing.
The most common slip is multiplying out of habit. Another is dividing with mixed units, kilometers by meters.
A third is putting the real distance on a ruler instead of the line's length. A fourth is believing one scale is always best, when every scale is chosen for a job.
A fourth-grade class in Ohio is making a wall map of their town for the school library. The town is about 6 kilometers across, and the poster paper has room for a map 60 centimeters wide. Each centimeter must carry 6 divided by 60 of a kilometer, which is a tenth of a kilometer, or 100 meters. So the scale is 1 cm to 100 m.
Now every distance they know on the ground has to be divided by 100 meters. Main Street is 1500 m long, so it gets a line of 15 cm. The school's front field is 200 m across, so it gets 2 cm. The class checks each line with the size check: every line is a smaller number than its distance.
They also argue about what to leave out. At 1 cm to 100 m the houses are too small to draw one by one, so they color whole blocks instead. Choosing the scale decided what the map could show.
The U.S. Geological Survey, called USGS, has drawn maps of the whole country for more than a hundred years. Its famous topographic maps cover the United States in small squares, drawn at a scale where 1 inch stands for 2000 feet, about 4 centimeters to the kilometer.
At that close-up scale a single map sheet covers only about 7 miles by 9 miles, so it takes more than 50,000 sheets to cover the lower 48 states. But each sheet shows every road, stream and building, and even the shape of hills. Hikers, firefighters and engineers depend on that detail.
For a view of a whole state, USGS draws wider-scale maps where one inch stands for many miles. Those fit on one sheet but leave out the small streams and buildings. Neither kind is better. Mapmakers divide the ground by the scale they choose, and choose the scale by the job.
The commonest slip is multiplying out of habit, straight after a lesson spent multiplying. The check catches it every time: the line is smaller than the distance. An answer of 20 cm for a 10 km road on a map at 1 cm to 2 km is wrong before you look at the arithmetic.
The second is dividing with mixed units. If the distance is in kilometers and the scale is in meters, one of them has to change first. Dividing 3 by 500 gives an answer far too small, and it will not look obviously wrong on the page.
The third is expecting one right scale to exist. It does not. A scale is chosen for a job, and every scale gives up something: either the ground it cannot fit on the sheet, or the detail it cannot draw big enough to see. A good mapmaker names the job first and picks the scale second, never the other way around, and then checks that the one thing the reader most needs is big enough to find.
A road is 10 km. The map is 1 cm to 2 km. Write the scale first.
$1 \text{ cm} = 2 \text{ km}$
What each centimeter carries.
Divide the distance by the scale.
$10 \div 2 = 5$
Lots of 2 km.
Give the line with its unit.
$5 \text{ cm}$
A length of paper.
Check the size.
$5 < 10$
The line is smaller.
A path is 3 km long and the scale is 1 cm to 500 m. Spot the problem.
$\text{kilometers and meters}$
Different units.
Change the path to meters.
$3 \text{ km} = 3000 \text{ m}$
Before dividing.
Divide by the scale.
$3000 \div 500 = 6$
Lots of 500 m.
Give the line.
$6 \text{ cm}$
A drawable length.
Check by multiplying back.
$6 \times 500 = 3000 \text{ m}$
The distance again.
A road is $12$ km. Draw it on a map at $1$ cm to $2$ km.
$12 \div 2 = 6 \text{ cm}$
Divide.
Draw it on a map at $1$ cm to $4$ km.
$12 \div 4 = 3 \text{ cm}$
Divide again.
Compare the lines.
$6 - 3 = 3 \text{ cm}$
The first is longer.
Say why the lines differ.
$\text{each cm carries less on the first}$
A closer-up scale.
Check both sizes.
$6 < 12, \; 3 < 12$
Both smaller than the road.
Check by multiplying back.
$6 \times 2 = 12, \; 3 \times 4 = 12$
The same road.
Write the scale.
$1 \text{ cm} = 4 \text{ km}$
What each centimeter carries.
Divide the distance.
$12 \div 4 = 3 \text{ cm}$
Lots of 4 km.
Check the size.
You are drawing one map to help visitors find the gate of a small park. Which scale should you draw it at?
Complete the worked solution: a road is $40$ km long. One map is drawn at $1$ cm to $4$ km, and another at $1$ cm to $5$ km. How long is the road's line on each map, and how much longer is it on the first?
Draw the road on the first map.
$\text{distance} \div \text{first scale} =$ a
Centimeters of line.
Draw it on the second map.
$\text{distance} \div \text{second scale} =$ b
Centimeters of line.
Compare the two lines.
$\text{first line} - \text{second line} =$ d
Same road, different lines.
Check the sizes.
$\text{both lines} < \text{the distance}$
A map is smaller than the place.
Match each job to the map scale that suits it.
| a map where 1 cm stands for 50 km | a map where 1 cm stands for 20 km | a plan where 1 cm stands for 50 m | |
|---|---|---|---|
| seeing where the mountains of a country are | |||
| planning a drive from one city to another | |||
| finding the gate into the park |
A map is being drawn at $1$ cm to $2$ km. On the ground, the school is $6$ km from the market. How long should the line between them be on the map?
Answer: unit: cm / mm
The same $8$ km between the pond and the playground is now being drawn on a different map, at $1$ cm to $1$ km. How long is the line on **this** map?
Answer: unit: cm / mm
At $1$ cm to $5$ km, you are drawing the $10$ km from the bridge to the mill. Below is a ruler in centimeters, from $0$ to $12$. Put the marker where your line should end.
0 |——————————| 12
Mark the position with a cross, then write the value:
A footpath is $4$ km long. You are drawing it on a plan where $1$ cm stands for $500$ m: meters this time, not kilometers. How long should the line be, in centimeters?
Answer: unit: cm / mm
A new visitor map uses 1 cm for 15 meters. A wheelchair user needs a step-free route no longer than 60 meters. The only routes are: A, 2 cm with stairs; B, 3 cm, step-free; C, 6 cm, step-free. What is the ground length, in meters, of the route meeting BOTH requirements? Enter a value and its ground-distance unit: m or cm. If using cm, convert the ground distance (1 m = 100 cm); do not report the paper length.
Answer: unit: m / cm
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
At $1$ cm to $2$ km, you are drawing the $8$ km from the pond to the playground. Below is a ruler in centimeters, from $0$ to $12$. Put the marker where your line should end.
0 |——————————| 12
Mark the position with a cross, then write the value:
You can turn a real distance into a length on a map. Tell somebody the check that catches a multiplication where a division was needed.
21. Your turn: a map is 1 cm to 4 km. How long a line do you need for 12 km?, step 3
$3 < 12$
The line is smaller.