Decimals on a number line

When you reach for it

At the school frog-jumping contest, each frog makes three hops in a row, and a judge measures how far it got from the start line. A long tape lies on the grass with a big mark at every whole meter: 0, 1, 2, 3, and so on. Aditya’s frog lands past the 3-meter mark but before the 4-meter mark. The judge writes down 3.7 meters. Where on the tape is that?

You reach for a number line whenever a decimal needs a place: to see how big it is, to see what it sits between, or to see which mark it is closer to. In tenths and hundredths you met the line from 0 to 1 cut into ten steps of 0.1. This lesson moves the same idea to any gap between two whole numbers, and then cuts a tenth into ten again to place hundredths.

One tool from earlier comes along. On any number line, the ticks step by a fixed amount, and you find it by dividing the gap between two labels by the number of equal parts between them; that is number lines with different scales.

The procedure

The lesson is taught in two parts, one a day.

Part 1: tenths between two whole numbers

Take 3.7. The digit before the point says 3 ones. The digit after it says 7 tenths. So

That tells you two things at once. First, 3.7 is more than 3, because it is 3 and a bit more. Second, it is less than 4, because the bit more is only 7 tenths, and it takes 10 tenths to make another whole. So 3.7 sits in the gap between 3 and 4.

The gap from 3 to 4 is one whole. Cut it into ten equal parts and each part is one tenth, 0.1. The ticks are then 3.1, 3.2, 3.3 and so on, each one tenth more than the tick before. Count seven parts to the right of 3 and you are at 3.7.

One tenth past 3.9 is not a new decimal inside the gap: ten tenths past 3 make one more whole, so the last tick is labeled 4. That is why the gap holds exactly nine ticks between its ends: 3.1 through 3.9.

Reading a marked point runs the same way backward. Most number lines do not label every tick. Find the two labels on either side of the point, count the equal parts between them, and work out what one part is worth. Then count parts from the left label to the point. In the next figure the labels are 4 and 5 and there are ten parts, so each is worth 0.1.

A number line does not have to start at 0. The contest tape starts at 0, but a line drawn only from 4 to 5 is still a number line; every point on it keeps its value.

Part 2: hundredths between two tenths

Now take 5.83, the length of Citlali’s frog’s jump. It has 5 ones, 8 tenths and 3 hundredths:

The 5 ones put it between 5 and 6. The 8 tenths put it past 5.8. The 3 hundredths are less than one more tenth, since a tenth is 10 hundredths, so it has not reached 5.9. So 5.83 sits in the gap between the tenths 5.8 and 5.9.

To see where, zoom in on that one gap. It is one tenth wide. Cut it into ten equal parts and each part is one hundredth, 0.01. Writing the ends as hundredths helps: 5.8 = 5.80 and 5.9 = 5.90, so the ticks are 5.81, 5.82, 5.83 and so on. Count three parts past 5.80.

The zoomed line starts at 5.80, not at 0. That is normal. A line only needs to show the part you care about, as long as its parts are equal and you know what one part is worth.

Closer to which? A number line also shows which of two marks a point is nearer. Look at the halfway point of the gap. Between 3 and 4 it is 3.5, five tenths from each end. A point left of halfway is closer to the left end; a point right of it is closer to the right end. So 3.7, at seven tenths past 3 and only three tenths short of 4, is closer to 4.

The same works one size smaller. Between 5.8 and 5.9 the halfway point is 5.85. The point 5.83 is three hundredths past 5.8 and seven hundredths short of 5.9, so it is closer to 5.8. As for whole numbers, 5.83 is 0.83 past 5 but only 0.17 short of 6, so it is closer to 6. Choosing the closer mark is the heart of rounding, which is taught in rounding to any place and rounding decimals.

Worked examples

Examples 1, 2 and 3 go with Part 1. Examples 4, 5, 6 and 7 go with Part 2.

Boundaries

• The parts must be equal. Counting ticks gives a value only when the gap is cut into equal parts. If the parts are unequal, the ticks do not step by a fixed amount and counting them says nothing.
• Ten parts is a choice of line, not a rule. A line from 3 to 4 with five parts steps by 0.2, not 0.1. Always work out what one part is worth from the labels before counting; see number lines with different scales.
• Exactly halfway is closer to neither. 3.5 is five tenths from 3 and five tenths from 4, and 5.85 is five hundredths from each of 5.8 and 5.9. Which way to go from a halfway point is a separate agreement made in rounding, not a fact about distance.
• This lesson stops at hundredths. Each hundredth can be cut into ten again to place Thousandths: reading, writing and expanded form; the zoom works the same at every step.
• This lesson stays at 0 and to its right. Every decimal here is a distance along the tape, so none is less than 0.

Where this goes next

Once decimals have places on a line, their order is the order of the points: the one farther right is larger. Using that, and the place values, to say which of two decimals is bigger is comparing and ordering decimals. A jump forward along the line is an addition, which you have met in adding tenths and hundredths, and counting hundredths across a whole number, as in Example 7, is the same move. The number line is also where Thousandths: reading, writing and expanded form get their places, by one more zoom.

Exercises

Exercises 1 to 8 go with Part 1, and 9 to 17 with Part 2. The reasoning group draws on both.

Part 1: tenths between two whole numbers

1. Between which two whole numbers is each decimal, and how many tenths past the smaller one is it? (a) 6.2, (b) 1.9.
2. A number line from 8 to 9 is cut into ten equal parts. What is each part worth? At which tick after 8 is 8.5?
3. Read points A, B and C on the line in the figure for Exercise 3, above.
4. Aditya’s frog lands 4 tenths past the 2-meter mark. Write the jump as a decimal.
5. Which decimal is one tenth short of 5? Which is one tenth past 5?
6. A tape runs from 0 to 2 meters with a tick at every tenth. How many tenth-parts does it have in all? At which tick after 0 is 1.3?
7. Hemi draws a line labeled 7 and 8 at its ends, with nine ticks between. A dot sits on the third tick after 7. What number is it?
8. How many tenths apart are 0.8 and 1.4 on a number line?

Part 2: hundredths between two tenths

9. Between which two tenths is each decimal? (a) 7.46, (b) 2.91.
10. A line from 1.5 to 1.6 is cut into ten equal parts. What is each part worth? What number is at the fourth tick after 1.5?
11. Read points P and Q on the line in the figure for Exercise 11, above.
12. Citlali’s frog lands 5 hundredths past the 4.7-meter mark. Write the jump as a decimal.
13. Which whole number is each decimal closer to? (a) 8.6, (b) 3.21.
14. Which tenth is each decimal closer to? (a) 1.47, (b) 6.02.
15. A line from 0.3 to 0.4 is cut into ten equal parts. What number is at the ninth tick after 0.3? How many hundredths does the whole line span?
16. Hemi’s frog jumps 2.98 meters. How many hundredths short of the 3-meter mark is it, and which whole number is it closer to?
17. What decimal sits exactly halfway between 7.2 and 7.3?

Reasoning

18. Aditya puts 3.08 on the eighth tick of a line from 3 to 4 cut into tenths. Which number is really at that tick, and where does 3.08 belong? Explain.
19. Explain why 4.5 is just as close to 4 as it is to 5, and why every decimal between 4 and 4.5 is closer to 4.
20. Hemi says there is no number between 2.6 and 2.7, because they are next to each other on the line. Explain why she is wrong, and name two numbers between them.
21. Explain how the digits of a decimal such as 9.14 tell you, without drawing anything, which gap between whole numbers it is in and which smaller gap between tenths.