From the map to the ground · printable packet

Measure the line, multiply by what one centimeter stands for, and the paper becomes a real distance.

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.

1. What you will learn

You will turn a length measured on a map into a real distance, using the map's scale, and give the answer with its unit.

2. What you already do

You can read a scale bar and say what one step of it is worth. A written scale, 1 cm stands for 2 km, is the same fact in a sentence instead of a picture, and from here on it is used as a multiplication rather than counted along.

3. Words for this lesson

4. One multiplication, every time

Term	What it means
Scale	What one centimeter on a map stands for on the ground.
On the map	Measured on the paper, in centimeters.
On the ground	The real distance, in meters or kilometers.
Leg	One straight part of a route.

A map's scale is a sentence: 1 cm stands for 2 km. It says what one centimeter of paper is worth on the ground, and it is the same everywhere on the sheet.

A line 3 cm long on a map at 1 cm to 2 km is $3 \times 2 = 6$ km. The 3 was never a distance; it was a length of paper. The multiplication is what turns it into one.

Another way: story

It is like a recipe that serves one: if one person needs 2 potatoes, then 3 people need $3 \times 2$. Nobody would answer 3 potatoes, and answering 3 km is exactly the same slip.

Another way: steps

Write the scale down first: 1 cm = 2 km. Then write the measurement: 3 cm. Then multiply. Doing it in that order makes the mistake almost impossible, because the number you measured never sits on its own next to a kilometer sign.

5. Straight across or along a route

A straight line gives the separation of two places on a local plan. Travel distance follows a usable route, including bends and crossings. In an invented park, two gates are 3 cm apart across a fenced pond. A path around it is 5 cm and another is 7 cm. With 1 cm representing 10 meters, the shorter permitted walk is 50 meters; the straight separation is 30 meters. The detour adds 20 meters. Add every route segment before converting, or convert all segments and then add. Both give the same total. Do not treat a line across water as an available footpath.

6. One scale, many lines

The table is a multiplication table for the number 2. A map at 1 cm to 5 km would use the table for 5 instead.

7. Measuring the line

To measure a straight line on a map, put a ruler's zero at one end and read the number at the other. Make sure you start at zero, not at the ruler's end, which on many rulers is a little way before the zero mark.

If the route bends, measure each straight leg separately and add the lengths, or lay a string along the route and then measure the string.

8. Routes with more than one leg

A route with two legs can be done two ways. Convert each leg and add: at 1 cm to 2 km, legs of 3 cm and 4 cm are 6 km and 8 km, which make 14 km.

Or add the centimeters first, 3 plus 4 is 7 cm, and multiply once: 7 times 2 is 14 km. Both ways give the same answer, which makes a good check.

9. Scales in meters

On the map	Multiply	On the ground
1 cm	1 × 2	2 km
3 cm	3 × 2	6 km
5 cm	5 × 2	10 km
7 cm	7 × 2	14 km

Maps of small places, like a town or a park, use scales in meters: 1 cm stands for 250 m. The rule is exactly the same. A street 4 cm long on that plan is 4 times 250, which is 1000 meters, or 1 kilometer.

Always read the unit in the scale sentence and give it with your answer. 1000 m and 1000 km are very different trips.

10. Scales in inches and miles

Many American maps give their scale in inches and miles: 1 inch stands for 10 miles. Nothing changes except the units. A road 3 inches long on that map is 3 times 10, which is 30 miles.

Some maps give both scales, one metric and one American. Whichever you use, measure the map in the unit the scale sentence starts with, and answer in the unit it ends with.

11. Checking with sense

A good check is to ask whether the answer makes sense. A walk across a park should be hundreds of meters, not hundreds of kilometers. A drive between two cities might be tens or hundreds of kilometers, not three.

If your answer is the same number you measured, you probably forgot to multiply, unless the scale really is 1 cm to 1 km.

12. Why the measured number is not the answer

The number you read off the ruler is a length of paper. It tells you how much of the map the line covers, and nothing yet about the ground. Only the scale sentence can turn it into a distance, because only the scale says what each centimeter of paper is worth.

That is why the same ruler reading can mean very different journeys. A line four centimeters long is eight kilometers on a map at one centimeter to two, twenty kilometers on a map at one centimeter to five, and one kilometer on a town plan at one centimeter to two hundred fifty meters. The paper is the same in every case; the ground is not. So before you write any answer, look back at the scale sentence and ask whether you have used it. If your answer is the same number you read off the ruler, and the scale is not one to one, you have not used it yet. Go back, write the scale down, and multiply before you answer.

13. The method, step by step, and how to check it

Checking an answer. Divide your answer by the scale number. You should get back the length you measured.

14. Why each step is allowed

Multiplying is allowed because every centimeter on the map stands for the same distance, so a line of several centimeters is several equal groups of that distance.

Adding legs, in either order, is allowed because distance simply adds up along a route. Dividing to check is allowed because division undoes multiplication.

15. Common slips

The most common slip is answering with the measured number, forgetting to multiply. Another is dividing when you should multiply.

A third is dropping the unit or giving the wrong one. A fourth is measuring from the end of the ruler instead of from its zero mark.

16. In the world: a state road map

A family in Colorado is planning a drive from Denver to Colorado Springs. The state road map says 1 inch stands for 10 miles. With a ruler, the highway between the two cities measures about 7 inches on the map, following its gentle curves.

Seven inches times ten miles an inch is 70 miles, which is close to the real distance along Interstate 25. At highway speed, that is a drive of a little more than an hour, so the family knows when to leave to arrive for lunch.

If they had answered 7, they would have planned a drive of a few minutes. The scale sentence is printed in a corner of every state road map, usually next to the scale bar, so that anyone with a ruler can turn inches of paper into miles of road. Many maps also print a small table of distances between big cities, which makes a good check on a measurement.

17. In the world: a city's bike plan

Cities across the United States are adding bike lanes, and planners draw them on large maps of the city. A planner's map might say 1 cm stands for 250 m. A new lane drawn along three straight streets measures 4 cm, 6 cm and 2 cm on the map.

Added up, that is 12 cm of map. Twelve times 250 meters is 3000 meters, or 3 kilometers of new bike lane. The planner can also do it street by street: 1000 m, 1500 m and 500 m, which also add to 3000 m.

The number matters, because paint, signs and curbs are priced by the meter. A city council deciding whether to pay for the lane wants to know its real length, not its length on the paper. A planner who forgot to multiply would ask for money to build a bike lane 12 meters long.

18. Where this goes wrong

The big one is answering with the number you measured. A 3 cm line becomes 3 km, and the multiplication has been skipped entirely. The giveaway is that the answer is the same number as the measurement; if that happens, check whether you multiplied.

The second is multiplying by the wrong thing: at 1 cm to 2 km, a 3 cm line is $3 \times 2$, not $3 \div 2$. Dividing is next lesson's job, and it is the other direction.

The third is dropping the unit. 6 is not an answer; 6 km and 6 m are different by a factor of a thousand, and the scale sentence says which one you are in.

19. A line of 3 cm at 1 cm to 2 km

20. A walk in two parts

21. A town street in meters

22. Your turn: a map is 1 cm to 4 km. How far is a line of 3 cm?

23. Guided practice

A different map: $1$ cm stands for $5$ km. Match each length measured on it to the distance it stands for.

24. Guided practice

Complete the worked solution: on a map, $1$ cm stands for $2$ km. A bike route has two straight legs, measured at $6$ cm and $5$ cm on the map. How long is each leg on the ground, and the whole route?

25. Guided practice

On this map, $1$ cm stands for $2$ km. The straight line from the market to the bridge measures $5$ cm. How far apart are they really?

26. Guided practice

Still $1$ cm to $5$ km. A walker goes from the harbor to the lighthouse, which is $3$ cm on the map, and then another $2$ cm on to a third place. How far have they walked altogether?

27. Practice

At $1$ cm to $2$ km, the line from the pond to the playground measures $4$ cm. Work out the real distance and put the marker on it. The scale below runs from $0$ to $20$ km, marked every kilometer.

28. Practice

A map is drawn at $1$ cm to $2$ km. Three lines have been measured on it. Fill in what each one is on the ground, in kilometers.

29. Somewhere new

A town plan says: $1$ cm stands for $250$ m. Not kilometers this time: meters. A street measures $7$ cm on the plan. How long is it, in meters?

30. Somewhere new

On an invented walking map, 1 cm represents 38 meters. School and library are 4 cm apart in a straight line across a pond. Walking is allowed only on two paths around it: route A measures 6 cm, and route B measures 8 cm. How many meters LONGER than the straight-line distance is the shortest allowed walking route? 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.

31. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

32. Test question

A map is drawn at $1$ cm to $2$ km. Three lines have been measured on it. Fill in what each one is on the ground, in kilometers.

33. What you can do now

You can turn map centimeters into real distance. Tell somebody how you can spot, just by looking at your answer, that you forgot to multiply.

Working for the steps left to you

	On the map (cm)	On the ground (km)
the first line	2
the second line	7
the third line	9

	On the map (cm)	On the ground (km)
the first line	3
the second line	6
the third line	8

Kilometers of ground.

	10 km	15 km	30 km
a line 2 cm long
a line 3 cm long
a line 6 cm long