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Math Lesson 2.2.3 - Rounding through Intervals and Segments Explained
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Welcome to our Math lesson on Rounding through Intervals and Segments Explained, this is the third lesson of our suite of math lessons covering the topic of Upper and Lower Bounds. Intervals and Segments, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Rounding through Intervals and Segments Explained
Intervals and segments can be used to explain rounding. More specifically, we use the concept of half-segment to indicate the possible values of a number which, when rounded, provides the same value. In this case, the number included in the lower bound is included in the set while, after the upper bound, there is always an "uncovered" part which is dictated by the precision of numbers involved.
For example, if a number is 30 when rounded to the nearest ten, we have taken the lower bound as 25 and the higher bound as 34. But when the rounded number is 40, this includes values from 35 to 44. It is clear that the part from 34 to 35 remains uncovered as long as we don't increase the precision of numbers to include non-whole values in the set.
To avoid these issues, we introduce the concept of a "half-segment" to represent the possible range of a rounding. We say "all numbers included in the half-segment [25, 34) are 30 when rounded to the nearest ten, this includes non-whole numbers". In this way, we no longer need to specify the set of numbers involved; all number sets have the same structure of representation when rounding to the nearest ten. The same can be said for other types of rounding (to the nearest hundred, thousand, unit, tenths, thousandths, etc.).
Example 8
Show, using symbols and number line, the set of numbers that are:
- 70 when rounded to the nearest ten.
- 400 when rounded to the nearest hundred.
- 9000 when rounded to the nearest thousand.
- 14 when rounded to the nearest unit.
- 5.6 when rounded to the nearest tenth.
Solution 8
- The numbers that are 70 when rounded to the nearest ten extend from 65 (including it) to 75 (without including it). Hence, when represented by symbols, this set is written as [65, 75). When shown in a number line, this set gives
- The numbers that are 400 when rounded to the nearest hundred extend from 350 (including it) to 450 (without including it). Hence, when represented by symbols, this set is written as [350, 450). When shown in a number line, this set gives
- The numbers that are 9000 when rounded to the nearest thousand extend from 8500 (including it) to 9500 (without including it). Hence, when represented by symbols, this set is written as [8500, 9500). When shown in a number line, this set gives
- The numbers that are 14 when rounded to the nearest unit extend from 13.5 (including it) to 14.5 (without including it). Hence, when represented by symbols, this set is written as [13.5, 14.5). When shown in a number line, this set gives
- The numbers that are 5.6 when rounded to the nearest tenth extend from 5.55 (including it) to 5.65 (without including it). Hence, when represented by symbols, this set is written as [5.55, 5.65). When shown in a number line, this set gives
Remark! 5.60 and 5.6 represent the same number; the extra zero simply provides a higher precision in measurement.
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