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Reading a scale bar

The marked line that says what distances on a map are worth — and why it is not the ruler it looks like.

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 work out what one step of a scale bar stands for, and read a distance off the bar without mistaking it for a ruler.

2. What you already do

You can read a number line and count along it, and you know that a grid square on Barrow's map stands for a set distance on the ground. You can multiply and divide small numbers. A scale bar is the same idea printed on the map itself, so that any map can be measured and not only one with squares on it.

3. Words for this lesson

4. The bar that turns ink into ground

A scale bar is a short line printed on a map, cut into equal steps, with numbers along it saying what distance each step stands for on the ground.

Reading it is two questions, always in this order. First: what is one step worth? Look at the numbers printed on the bar and see what they go up by, or, if only the whole bar is labeled, divide that by the number of steps. Second: how many steps? Multiply, and that is the distance.

Never skip the first question. A bar marked in tens and a bar marked in ones look almost identical, and the same three steps along them are thirty kilometers and three kilometers.

5. Estimate before calculating

If a path is measured as 4.2 cm and the scale says 1 cm represents 20 meters, first estimate with 4 cm: 4 times 20 gives about 80 meters. The measured length is a little greater, so the unrounded result should be a little greater too. An estimate is useful for catching an answer that is ten times too large. State the unit and the rounding choice. A thick printed line or a rough ruler reading limits precision; report about 80 meters rather than pretending the drawing gives every centimeter on the ground.

6. Reading different bars

Here are some scale bars and what one step of each is worth.

Two of the bars stand for the same 4 km and are cut up differently. The look of a bar tells you nothing; only its numbers do.

7. Kilometers and miles

The maps in this course use kilometers, the unit used in science and by most of the world. Most American road maps use miles instead, and many print two scale bars, one in each unit, one above the other.

The two questions do not change with the unit. What is one step worth? How many steps? Just be sure to read the right bar, and to say the unit in your answer: 6 km and 6 miles are different distances.

8. Counting gaps, not marks

A scale bar's numbers sit at marks, but the distance is in the gaps between them. From 0 to the first mark is one gap. From 0 to the third mark is three gaps.

Some bars have small marks between the numbered ones. If three small marks sit between 0 and 4 km, they cut that stretch into four gaps, so each small gap is 1 km. Count the gaps, and the small marks will not fool you.

9. Two questions, in order

First divide, then multiply. If a bar of 5 equal steps stands for 20 km, one step is 20 divided by 5, which is 4 km. A road that takes 3 steps is then 3 times 4, which is 12 km.

Doing it the other way around does not work. You cannot multiply by one step's worth until you know it, and you cannot know it without dividing the whole bar.

10. Measuring with the bar

To measure a distance on a map, lay the edge of a strip of paper along the route, mark both ends on the strip, and then lay the strip beside the scale bar. Count how many steps fit between your two marks.

If the route bends, mark the paper at each bend and turn it, like walking the route in small straight pieces. The strip adds up the pieces for you.

11. Why maps need a bar

A map drawn without a scale bar can show where things are and which way, but not how far. Two maps of the same town can be printed at very different sizes, so a ruler alone cannot tell you anything about the real distance.

The bar also stays true when a map is copied bigger or smaller, because it shrinks or grows along with everything else on the page. A note like 1 cm stands for 2 km does not, which is why many mapmakers prefer a bar.

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

1. Find the bar on the map, not the north arrow.
2. Read its unit: kilometers, miles, meters.
3. Work out one step: read the printed numbers, or divide the whole bar by its steps.
4. Count the steps a distance takes, gaps not marks.
5. Multiply steps by one step's worth, and give the unit.

Checking an answer. A distance that takes fewer steps than the whole bar must be less than the whole bar's distance.

13. Why each step is allowed

Dividing the whole bar by its steps is allowed because the steps are equal, so they share the whole distance evenly. Multiplying steps by one step's worth is allowed for the same reason: every step adds the same amount.

Counting gaps rather than marks is required because a distance is the space between two points, and each gap is one step of space.

14. Common slips

The most common slip is reading the bar as a ruler: giving centimeters of paper as the answer instead of kilometers of ground. Another is counting marks as though each were worth one, when the printed numbers go up by more.

A third is counting marks instead of gaps. A fourth is reading the miles bar when the question asks for kilometers.

15. In the world: planning a hike

A family planning a day hike in a national park looks at the park's trail map. In the corner is a scale bar cut into 4 equal steps, with the whole bar marked 2 miles, so one step is 2 divided by 4, which is half a mile.

They lay a strip of paper along the trail from the parking lot to the waterfall, bending it at each turn, and find the trail takes 6 steps of the bar. That is 6 times half a mile, which is 3 miles each way, or 6 miles there and back.

Hikers walk about 2 miles an hour on an easy trail, so the family plans about 3 hours of walking, plus time to rest and eat lunch by the falls. Park rangers say one of the most common problems they see is hikers who guessed the distance from how short the line looked on the map. The scale bar is what turns that short line into a real walk.

16. In the world: a school field trip

A third-grade class is planning a field trip to a science museum. The teacher shows a map of the city with a scale bar cut into 5 equal steps, the whole bar standing for 10 kilometers. One step is 10 divided by 5, which is 2 kilometers.

The class measures the route from the school to the museum and finds it is 4 steps long. That is 4 times 2, which is 8 kilometers, about 5 miles. The bus company charges by distance, so the number matters for the trip's cost.

One student measured the route with a ruler, found it was 8 centimeters long, and said the trip was 8. The class noticed that the number happened to match, but only by luck: on a map with a different scale bar, the same 8 centimeters could be 16 or 40 kilometers. Only the bar can say what the centimeters are worth.

17. Where this goes wrong

A scale bar looks like a small ruler and it is not one. The numbers printed along it are distances on the ground. Measuring 3 cm along the bar does not mean 3 of anything; it means whatever the bar says 3 cm of it is worth.

The sign of it is an answer in the wrong units: a child who says three when the bar is marked in kilometers has given a length of paper, not a distance on the ground.

The second mistake is counting marks instead of reading them. If the numbers on the bar go up in twos, the third mark is 6 and not 3. Read the printed numbers before you count anything.

The third is counting lines instead of gaps. Three small marks between two numbers cut that gap into four parts, not three.

18. A bar marked every 2 km

1. The printed numbers go 0, 2, 4, 6. Find what one mark is worth.

   $2 \text{ km}$

   Ask this first.

2. Find the third mark after zero.

   $3 \text{ gaps along}$

   Count gaps from zero.

3. Multiply the gaps by their worth.

   $3 \times 2 = 6 \text{ km}$

   Never just count.

4. Check against the printed number.

   $\text{the third mark says } 6$

   They agree.

19. Only the whole bar labeled

1. The bar is cut into 4 equal steps and the whole thing stands for 8 km. Record both.

   $4 \text{ steps}, 8 \text{ km}$

   No number on each step.

2. Divide to find one step.

   $8 \div 4 = 2 \text{ km}$

   Equal steps share it.

3. A path takes two steps. Multiply.

   $2 \times 2 = 4 \text{ km}$

   Steps times one step.

4. Check against the whole bar.

   $4 < 8$

   Fewer steps, less distance.

5. Give the unit.

   $4 \text{ km}$

   Not 4 steps.

20. A road measured along a bar

1. A bar of $5$ equal steps stands for $20$ km. Find one step.

   $20 \div 5 = 4 \text{ km}$

   Divide first.

2. A road on the map is as long as $3$ steps. Multiply.

   $3 \times 4 = 12 \text{ km}$

   Then multiply.

3. Check it is less than the whole bar.

   $12 < 20$

   Three steps of five.

4. Check by adding step by step.

   $4 + 4 + 4 = 12$

   The same answer.

5. Say what the count alone was.

   $3 \text{ steps}$

   Not a distance.

6. Give the answer with its unit.

   $12 \text{ km}$

   Kilometers of ground.

21. Your turn: a bar of 5 equal steps stands for 20 km. What is one step worth?

1. Record the whole bar and its steps.

   $20 \text{ km}, 5 \text{ steps}$

   What the bar says.

2. Divide the whole by the steps.

   $20 \div 5 = 4$

   Equal steps share it.

3. Your turn: work this step out. Its working is at the end of the packet.

   Give the answer with its unit.

22. Guided practice

Three maps, three scale bars. What is one step worth on each?

23. Guided practice

Complete the worked solution: a map's scale bar is cut into $4$ equal steps, and the whole bar stands for $12$ km. A road on the map is as long as $2$ steps of the bar. What is one step worth, and how long is the road on the ground?

1. Find what one step is worth.

   $\text{whole bar} \div \text{steps} =$ _s

   Kilometers per step.

2. Find the road's real length.

   $\text{steps along} \times \text{one step} =$ _d

   Kilometers on the ground.

3. Check the units.

   $\text{kilometers, not steps}$

   A count of steps is not a distance.

4. Check the size.

   $\text{road} < \text{whole bar}$

   It took fewer steps than the bar has.

24. Guided practice

A map's scale bar runs from $0$ at its left end to $7$ km at its right end, with a mark every $1$ km. Put the marker on $2$ km.

0 |——————————| 7

Mark the position with a cross, then write the value:

25. Guided practice

This scale bar is marked every $5$ km, and it runs from $0$ to $25$ km. Put the marker $2$ marks after zero.

0 |——————————| 25

Mark the position with a cross, then write the value:

26. Practice

Two small marks are printed beside this park plan, outside the park itself. One of them answers *which way*; the other answers *how far*. Click the one that answers how far.

This task has no paper form; do it on a device.

27. Practice

A scale bar is cut into $2$ equal steps, and the whole bar stands for $8$ km. How much ground does one step stand for?

Answer: unit: m / km

28. Somewhere new

Here is the trap this whole lesson is about. A scale bar is $4$ cm long on the paper and stands for $12$ km on the ground. You measured a road and it came to $2$ cm along the bar. Put the marker at the distance that road really is.

0 |——————————| 12

Mark the position with a cross, then write the value:

29. Somewhere new

A visitor measures a straight path on an invented park plan. The scale says 1 cm represents 28 meters. The path is just over 4 cm and is measured as 21/5 cm. Round that map length to the nearest whole centimeter to estimate the ground distance. Give the estimate in meters. 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

30. Lesson test

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

31. Test question

A scale bar is cut into $2$ equal steps, and the whole bar stands for $14$ km. How much ground does one step stand for?

Answer: unit: m / km

32. What you can do now

You can read a scale bar. Tell somebody the first question you ask of one, and why measuring three centimeters along it does not mean three of anything.

Working for the steps left to you

21. Your turn: a bar of 5 equal steps stands for 20 km. What is one step worth?, step 3