A Peek Beyond the Point · Lesson 7 of 11
Locating and Comparing Decimals
“Find decimal positions on number lines and compare quantities by place value.”
• Locate tenths, hundredths, and thousandths on number lines. • Explain when adding zeros preserves a decimal’s value. • Determine interval sizes from the labelled endpoints of a scale. • Compare and order decimals using corresponding places.
Locating and Comparing Decimals
A decimal gives a distance from zero on a number line. The number 1.4 contains one whole unit and four tenths, so it lies between 1 and 2. Divide that unit interval into ten equal intervals and move four intervals beyond 1. The point you reach is 1.4.
Hundredths require a finer view. The interval from 1 to 1.1 is one-tenth of a unit. Divide that interval into ten equal smaller intervals and each new interval is one-hundredth of a unit. The internal divisions are 1.01, 1.02, and so on up to 1.09, followed by the endpoint 1.10, which equals 1.1.
Problem
Where is 1.04 on a number line?
- 1.It contains 1 whole unit, 0 tenths, and 4 hundredths.
- 2.Choose the interval from 1.00 to 1.10 and split it into ten equal parts.
- 3.Count four small intervals from 1.00. The resulting point is 1.04.
- 4.It is less than 1.1 because four hundredths are fewer than ten hundredths.
There is Zero Dilemma!
Does writing 0.2 as 0.20 change its value? Place value answers the question. In 0.2 there are two tenths. In 0.20 there are still two tenths, with zero extra hundredths. In 0.200 there are zero extra hundredths and zero extra thousandths. All three forms name the same quantity.
Zeros inserted between the point and a nonzero digit have a different effect. The 2 in 0.02 represents two hundredths, and the 2 in 0.002 represents two thousandths. Their positions have changed. A zero before the whole-number part, as in 04.50, does not change that whole-number part: 04.50 and 4.50 are equal.
| Forms | Relationship | Reason |
|---|---|---|
| 0.2, 0.20, 0.200 | Equal | Each contains two tenths |
| 0.2, 0.02, 0.002 | Different | The 2 occupies different places |
| 4.5, 4.50, 04.50 | Equal | All contain four units and five tenths |
| 4.05, 4.050 | Equal | Both contain four units and five hundredths |
Problem
Which values are equal among 4.5, 4.05, 0.405, 4.050, 4.50, 4.005, and 04.50?
- 1.Write enough fractional zeros to compare corresponding places: 4.500, 4.050, 0.405, 4.050, 4.500, 4.005, and 4.500.
- 2.The forms 4.5, 4.50, and 04.50 are equal.
- 3.The forms 4.05 and 4.050 are equal.
- 4.The values 0.405 and 4.005 differ from these groups and from each other.
A zero can be removed from the end of fractional digits without changing value, but removing an internal placeholder can change the value. Also, removing the final zero from the whole number 20 would change it to 2. State exactly where a zero is placed before applying a rule.
Magnifying the Number Line
A long decimal can be located by repeatedly zooming into the interval containing it. For 4.185, first select the unit interval from 4 to 5. Within it, select 4.1 to 4.2. Within that tenth, select 4.18 to 4.19. Dividing the last interval into ten equal parts produces thousandth intervals.
Problem
Locate 0.407 using successive subdivisions.
- 1.Start in the unit interval from 0 to 1.
- 2.Select the tenth interval from 0.4 to 0.5.
- 3.Because the hundredths digit is 0, select the first hundredth interval, from 0.40 to 0.41.
- 4.Divide this interval into ten equal parts. Seven intervals after 0.400 gives 0.407. It lies below 0.410.
Check the Size of Each Interval
Ten equal intervals do not always represent tenths of one unit. Their size depends on the distance between the labelled endpoints. If 5 to 10 is split into ten equal intervals, the full distance is five units, so each interval is half a unit. Always find the interval size before identifying an unlabelled point.
Problem
A line from 4.3 to 4.8 is divided into ten equal intervals. What is the value at the third interval after 4.3?
- 1.The endpoint difference is 4.8 − 4.3 = 0.5 unit.
- 2.Ten equal intervals share that distance, so each interval is 0.05 unit.
- 3.Three intervals cover 0.15 unit.
- 4.Add this to the starting value: 4.3 + 0.15 = 4.45.
Comparing Corresponding Places
Compare whole-number parts first. If those parts are equal, compare tenths, then hundredths, then thousandths as needed. Stop at the first differing place. The number with the larger digit there is larger because the following smaller places cannot make up an entire unit of that place.
Problem
Which is larger: 6.456 or 6.465?
- 1.Both have 6 whole units, so their whole-number parts do not decide the comparison.
- 2.Both have 4 tenths. Compare hundredths next: 5 hundredths versus 6 hundredths.
- 3.The second number has more at this first differing place, so 6.465 > 6.456.
- 4.The extra thousandths in the first number cannot bridge the gap: 6.456 is below 6.460, whereas 6.465 is above it.
The number of written digits does not decide size. For example, 2.5 = 2.500, which is greater than 2.05 = 2.050. Expressing both to the same number of fractional places can make a comparison clearer, but the added zeros do not change either quantity.
Check Your Understanding
Use the ideas from this lesson to choose an answer. Explain your choice to yourself before opening the explanations below.
Quiz
Which forms all represent the same quantity?
Which is greater?
A line from 5 to 10 has ten equal intervals. What is each interval?
In which interval does 4.185 lie?
Does writing more fractional digits necessarily make a positive decimal larger?
Each form in the third choice contains 4 units and 5 tenths.
Practice Problems
- Locate 9.876 by naming its unit, tenth, and hundredth intervals.
- Order 0.2, 0.02, and 0.002 from smallest to largest.
- Arrange 11.01, 1.011, 1.101, 11.10, and 1.01 in descending order.
- Arrange 2.567, 2.675, 2.768, 2.499, and 2.698 in descending order.
- Order 33.13 m, 33.31 m, 33.133 m, 33.331 m, and 33.313 m from longest to shortest.
- A number line from 8 to 8.1 has ten equal intervals. Name its sixth division after 8.
- Arrange 4.678 g, 4.595 g, 4.600 g, 4.656 g, and 4.666 g in descending order.
It lies in 9 to 10, then 9.8 to 9.9, then 9.87 to 9.88. Divide that last interval into ten equal parts and take the sixth interval after 9.870.
Key Takeaways
• Decimals identify positions and distances on a number line. • Find a scale’s interval size from its labelled endpoints. • Successive subdivisions locate tenths, hundredths, and thousandths. • Appending fractional zeros preserves value. • Internal zeros can be essential placeholders. • Compare whole-number parts first, then corresponding fractional places. • More written digits do not necessarily mean a larger quantity.