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Reading a scale

Numbered marks, smaller marks between them, and what one small mark is worth.

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 mark on a scale is worth, and read a value from it.

2. What you already do

You can already count along a number line and land on the right number. A scale is a number line printed on something — a ruler, a jug, a thermometer — and the only new thing is that some of the marks have no number printed on them.

You can also share a number equally. If 100 is shared between 4, each share is 25. Reading a scale uses exactly that: the printed numbers give an amount, and the small marks share it out.

3. Words for this lesson

4. Three scales, one question

Here are three instruments. Each has numbers printed on it and smaller marks between them, and on each one a pointer sits three small marks past the last number.

Write down what you think each pointer reads, before you read on. Three marks past 2 on the ruler; three marks past 100 on the scale; three marks past 100 on the jug.

Now the question this lesson exists for: did you use the same number three times? If you counted three and added three every time, you have said the ruler reads 5 cm and the jug holds 103 mL. Check the jug against the numbers printed on it: three small marks cannot cross a gap of 100 mL three milliliters at a time.

So one small mark is not worth one, and it is not worth the same on all three. What is it worth? You have everything you need to find out — the printed numbers and how many spaces fill the gap between them — and the next part names what you did.

5. What one mark is worth

A scale has numbered marks and smaller ones between them. To read it, find two numbers you can read, count the small spaces between them, and divide. That tells you what one small mark is worth. Only then count along to the pointer.

If ten small spaces fill the gap from 2 to 3, each one is worth $1 \div 10 = 0.1$, and three marks past 2 reads 2.3. If the numbers go up in tens, each mark is worth 10 and the third mark reads 30, not 3. If four spaces fill a gap of 100, each is worth 25, and three marks past 100 reads 175.

The rule has two parts, and the order matters. First find what one mark is worth. Then count. A learner who counts first and thinks later nearly always treats each mark as one, and every reading comes out wrong in the same way.

6. Spaces, not lines

The most common slip in reading a scale is counting the lines when you should count the spaces. Look at a gap between 0 and 100 with three small lines inside it:

Three lines cut the gap into four spaces, just as three cuts across a sandwich make four pieces. So each space is $100 \div 4 = 25$, not $100 \div 3$.

A quick way to be sure: count the spaces by putting your finger in each one, from the first printed number to the next. Or remember that there is always one more space than there are small lines. Nine small lines make ten spaces, which is why a ruler with nine little lines between each centimeter is marked in tenths.

7. Reading when the pointer is between marks

Sometimes the pointer does not sit exactly on a mark. Then you choose the nearest mark. If a ruler is marked in millimeters and the end of a leaf is a little past the 4th millimeter mark after 6 cm, you read 6.4 cm. If it is almost at the 5th, you read 6.5 cm.

Scientists call this reading to the nearest mark. It means every reading has a small wobble in it, never more than half a mark. That is fine: it is how all measuring works, even with the best instruments. What is not fine is a reading that is out by a whole number because the marks were counted wrong. A good reading can be a little off; it cannot be off by a factor of ten.

8. Looking straight at the scale

Where you put your eye matters. If you look at a measuring jug from above, the water level seems lower than it is; from below, it seems higher. This is called parallax. To avoid it, bend down so your eye is level with the water, and read the scale straight on.

On a ruler, put the zero mark exactly at the end of the thing you measure, not the end of the ruler. Many rulers have a little blank piece before the zero, and starting there adds a few millimeters to every reading. On a kitchen scale, check that it reads zero before you put anything on it. These three habits — eye level, start at zero, check the zero — are what careful measurers do every single time. Scientists in a laboratory follow the same three habits with instruments that cost thousands of dollars.

9. Checking a reading

Every reading should pass two checks.

1. Is it between the two numbers? If the pointer is between 100 and 200, the reading must be more than 100 and less than 200. A reading of 103 passes this check but fails the next one.
2. Is it a whole number of marks past the lower number? On a scale marked in 25s, readings are 100, 125, 150 and 175. A reading of 103 cannot be right, because no mark sits there. The pointer can sit between two marks, but then the reading is the nearest mark, and 103 is not near any of them.

If a reading fails either check, go back to step two of the method and find what one mark is worth again. Nearly every wrong reading is fixed there.

10. Scales that go up and scales that go round

Not every scale is a straight line. A kitchen scale often has a round dial, and a clock has numbers in a circle. The method does not change: find two printed numbers, count the spaces between them, divide, then count along.

On a clock face, the numbers 12, 1, 2 and so on are printed, and between each pair there are five spaces. For the minute hand, the numbers stand for 0, 5, 10 and so on minutes, so each small mark is worth $5 \div 5 = 1$ minute. A minute hand two marks past the 3 reads 17 minutes past the hour. A clock is a scale you have been reading for years, and it follows exactly the same rule.

11. In the world: a car's fuel gauge

Every car in the United States has a fuel gauge on the dashboard. It is a scale with only two letters printed on it: E for empty and F for full. Between them are three small marks, which cut the scale into four spaces. So each mark is one quarter of a tank.

Suppose a car's tank holds 16 gallons when full. One quarter is $16 \div 4 = 4$ gallons. If the needle points at the second mark from E, the tank holds $2 \times 4 = 8$ gallons. If a trip needs 10 gallons, the driver knows to stop for gas first.

A driver who thought each mark was one gallon would think the car had only 2 gallons, and would worry for nothing. A driver who counted the three lines as three spaces would think each mark was about 5 gallons and might run out on the highway. Counting the spaces and finding one mark's worth is what makes the gauge useful.

12. In the world: a doctor's thermometer

When you feel sick, a nurse may take your temperature. In the United States, body temperature is usually measured in degrees Fahrenheit, and a normal temperature is about 98.6. A glass thermometer has numbers printed every whole degree — 97, 98, 99, 100 — and between them are small marks.

On many of these thermometers there are four small lines between each pair of numbers, making five spaces. Each mark is $1 \div 5 = 0.2$ of a degree. If the liquid ends three marks past 98, the reading is $98 + 3 \times 0.2 = 98.6$ degrees, which is normal. If it ends two marks past 100, the reading is 100.4 degrees, which doctors call a fever.

Reading the scale correctly matters here: a nurse who treated each mark as one whole degree would read 101 instead of 98.6, and might send a healthy child home from school.

13. One mark is not always one

Almost every wrong reading comes from assuming one mark means one. On a kitchen scale a mark might be 25 g; on a jug it might be 50 mL; on a ruler it is a tenth of a centimeter. A learner who counts three marks past 100 and says 103 has not read the scale at all, only counted it.

The other trap is counting marks instead of gaps. Three small marks between two numbers split the gap into four parts, not three: count the spaces, not the lines.

Both mistakes are caught by the same habit. Before you count anything, say out loud what one mark is worth, with its unit. If you cannot say it, you are not ready to read the scale yet.

14. A ruler between 2 cm and 3 cm

1. Find the two printed numbers.

   $2\ \text{cm and } 3\ \text{cm}$

   The pointer sits between them.

2. Find the gap between them.

   $3 - 2 = 1\ \text{cm}$

   That is the amount the small marks share.

3. Count the spaces in the gap.

   $10 \text{ spaces}$

   Nine small lines make ten spaces.

4. Find one mark's worth.

   $1 \div 10 = 0.1\ \text{cm}$

   Ten equal spaces make tenths.

5. Count along and add.

   $2 + 3 \times 0.1 = 2.3\ \text{cm}$

   Three marks past 2 is three tenths more.

15. A kitchen scale with a twist

1. Find the two printed numbers.

   $100\ \text{g and } 200\ \text{g}$

   The pointer is between them.

2. Find the gap between them.

   $200 - 100 = 100\ \text{g}$

   The small marks share one hundred grams.

3. Count the small lines.

   $3 \text{ small lines}$

   This is the twist: lines are not spaces.

4. Turn the lines into spaces.

   $3 + 1 = 4 \text{ spaces}$

   Always one more space than lines.

5. Find one mark's worth.

   $100 \div 4 = 25\ \text{g}$

   Each space is a quarter of the gap.

6. Count along and add.

   $100 + 3 \times 25 = 175\ \text{g}$

   Three marks past 100 grams.

16. A thermometer read in two places

1. Find the printed numbers near the top.

   $20 \text{ and } 30 \text{ degrees}$

   The liquid ends between them.

2. Find the gap between them.

   $30 - 20 = 10 \text{ degrees}$

   The small marks share ten degrees.

3. Count the spaces in the gap.

   $5 \text{ spaces}$

   Four small lines make five spaces.

4. Find one mark's worth.

   $10 \div 5 = 2 \text{ degrees}$

   Each space is two degrees, not one.

5. Read the morning temperature.

   $20 + 1 \times 2 = 22 \text{ degrees}$

   The liquid was one mark past 20.

6. Read the afternoon temperature.

   $20 + 4 \times 2 = 28 \text{ degrees}$

   Later it was four marks past 20.

7. Find how much it warmed up.

   $28 - 22 = 6 \text{ degrees}$

   Three marks of two degrees each.

17. Your turn: five spaces fill the gap from 4 to 5. The pointer is two marks past 4.

1. Find the gap between them.

   $5 - 4 = 1$

   The spaces share one whole.

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

   Find one mark's worth.

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

   Count along and add.

18. Guided practice

The scale runs from 0 to $9$ and every mark is one step. Where does $4$ sit?

0 |——————————| 9

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

19. Guided practice

Complete the worked solution: a kitchen scale is numbered every $100$ g, with $3$ small marks between the numbers. The pointer is $2$ small marks past $400$ g. What does it read?

1. Count the spaces between numbers.

   $3 \text{ marks make}$ _a spaces

   There is always one more space than there are marks.

2. Find one mark's worth.

   $100\ \text{g} \div \text{spaces} =$ _b g

   The hundred grams are shared equally between the spaces.

3. Count along and add.

   $400 + 2 \times \text{one mark} =$ _c g

   Each mark past the number adds one mark's worth.

20. Guided practice

Match each scale description to the value of one small mark.

21. Guided practice

A scale runs from $0$ to $100$ and is marked every ten. Where is $60$?

0 |——————————| 100

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

22. Practice

Between $3$ and $4$ there are ten small spaces. The pointer is $4$ marks past $3$. What does it read?

3 |——————————| 4

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

23. Practice

Three instruments, each with a pointer past a printed number. The ruler has ten spaces to each centimeter. The kitchen scale is numbered every $100$ g with four spaces in each gap. The jug is numbered every $100$ mL with two spaces in each gap. The ruler reads $3.8$ cm, the scale reads $75$ g and the jug $50$ mL. Fill in what one small mark is worth on each, and how many marks each pointer is past the number below it.

24. Practice

A bathroom scale is numbered every $10$ kg, with four small marks between the numbers. The pointer stops $1$ small marks past $60$. What is the reading?

Answer: unit: g / kg

25. Somewhere new

A measuring jug in a kitchen is numbered every $100$ mL, with three small marks in each gap. A cook pours water until it stands $1$ small marks above the $100$ mL line. How much water is in the jug?

100 |——————————| 200

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

26. Lesson test

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

27. Test question

Three instruments, each with a pointer past a printed number. The ruler has ten spaces to each centimeter. The kitchen scale is numbered every $100$ g with four spaces in each gap. The jug is numbered every $100$ mL with two spaces in each gap. The ruler reads $7.4$ cm, the scale reads $350$ g and the jug $350$ mL. Fill in what one small mark is worth on each, and how many marks each pointer is past the number below it.

28. What you can do now

You can read a scale. Tell someone how you work out what one small mark is worth before reading anything.

Working for the steps left to you

17. Your turn: five spaces fill the gap from 4 to 5. The pointer is two marks past 4., step 2

17. Your turn: five spaces fill the gap from 4 to 5. The pointer is two marks past 4., step 3