Chapter 2 · Operations with rational numbers
Word Problems and Error Analysis with Rational Numbers
Turn stories about temperature, water levels and depth into sums and differences of rational numbers. Then find and fix the first false line in a worked solution.
Stories with two opposite directions (up and down, above and below, gain and loss) need numbers with signs. In this lesson you work with rational numbers: numbers you can write as a fraction , where and are integers () and is not . So , , and are all rational numbers, and you already know how to add and subtract them.
You use that skill in two ways here. First, you turn a story into an expression (numbers joined by operations) and find its value. Second, you read a finished solution, find the first line that is not true, and fix it. Both jobs depend on one thing: knowing what each sign and each operation means.
In a story, a signed number (a number written with its sign, such as or ) tells you two things: how far and which way from a reference point called zero. Set up the story before you compute:
- Choose the zero, the positive direction and the unit. For example: zero is sea level, up is positive, and the unit is meters.
- Write each amount as a signed number. With that choice, m below sea level is .
- Write each rise or fall as an operation. With up as positive, a climb of m adds and a drop of m subtracts .
- Find the value, then check that its sign and size fit the story.
Keep the same zero, direction and unit for the whole problem.
| Story | Zero, positive direction, unit | Expression and value | What the value means |
|---|---|---|---|
A fish is m below the surface and swims up m. | The surface; up is positive; meters | The fish is m below the surface. | |
Your game score is points and you lose points. | A score of ; gaining is positive; points | Your score is points below . | |
You walk m forward and then m back. | Your starting spot; forward is positive; meters | You are back at your starting spot. |
Key concept
Each sign and operation has a meaning you can check
For any rational numbers and :
- Adding. is the number that is units from . It is in the positive direction (toward greater numbers) when is positive, and in the negative direction (toward lesser numbers) when is negative. When , . Here is the absolute value of : its distance from .
- Subtracting. means . The number is the additive inverse, or opposite, of : the number that gives . For example, the additive inverse of is , because .
These facts let you turn a story into an expression, find its value, and test every line of a solution.
Worked example
A temperature that rises and falls
At 6 a.m. the temperature is °C below zero. By noon it rises °C. By evening it falls °C. What is the evening temperature?
Set up the story. Let zero be 0 °C, let warmer be the positive direction, and measure in degrees Celsius. Then 3.5 °C below zero is −3.5. With warmer as positive, the rise adds 8.2 and the fall subtracts 6.4.
The expression has only addition and subtraction, so work from left to right and add the rise first. Adding the positive number 8.2 moves 8.2 units in the positive direction from −3.5. The first 3.5 units reach 0, because −3.5 and 3.5 are opposites. That leaves 8.2 − 3.5 = 4.7 units above 0.
Subtract the fall. Subtracting 6.4 means adding its additive inverse, −6.4, so move 6.4 units in the negative direction from 4.7. The first 4.7 units reach 0. That leaves 6.4 − 4.7 = 1.7 units below 0.
Check the sign and size against the story. A rise of 8.2 and a fall of 6.4 together make an overall rise of 8.2 − 6.4 = 1.8. Starting 3.5 below zero and rising 1.8 does not reach zero, so the answer must be negative. It must also be 1.8 greater than −3.5.
Answer
The evening temperature is °C, which is °C below zero.
Some stories ask for a change: how far, and in which direction, a quantity moved from a start value to an end value. The change is the number you add to the start value to get the end value, so you find it by subtracting the start value from the end value. Both values must use the same zero and the same unit.
The sign of the change gives the direction. A positive change means the end value is greater than the start value. A negative change means the end value is less than the start value. A change of means the two values are equal.
Worked example
A change that crosses zero
On Monday the water in a reservoir is ft below its normal level. On Friday it is ft above its normal level. What is the change in the water level from Monday to Friday?
Set up the story. Let zero be the normal level, let above be the positive direction, and measure in feet. Monday's level is the start value. It is below normal, so it is negative: −2½. Friday's level is the end value. It is above normal, so it is positive: 1¾. The change is the end value minus the start value.
Rewrite the subtraction as adding the additive inverse, which keeps the value the same. The additive inverse of −2½ is 2½, because the two numbers sum to 0.
Write both numbers as fourths so the parts are the same size. 1¾ is 4 fourths and 3 fourths, so 7 fourths. 2½ is 5 halves, and each half is 2 fourths, so 10 fourths. Then add the numerators, which count the fourths: 7 + 10 = 17. Last, 17 fourths is 16 fourths and 1 fourth, which is 4¼.
Check. Adding the change to the start value must give the end value. The start value, −2½, is 10 fourths below 0. Adding 17 fourths uses 10 of them to reach 0 and leaves 7 fourths, which is 1¾. This fits the story: the water rises 2½ ft to reach normal, then 1¾ ft more.
Answer
The change is ft: the water level rose ft from Monday to Friday.
Error analysis means reading a finished solution, finding the first line that is not true, explaining why it is not true, and correcting the work. It works on your own solutions as well as on someone else's.
Read each line as a claim. The first line claims to match the story. Every later line claims to have the same value as the line before it. Test the claims in order. The first claim that fails is where the error starts. Lines after it can use correct arithmetic and still give a wrong value, because they build on the false line.
The table shows four tests. The first one checks the setup. The other three come from the facts in the key idea, so they work for any rational numbers.
| Test | A line that fails | Why it fails | Corrected line |
|---|---|---|---|
Does the first line match the story? | A fish m below the surface swims up m: | With the surface as zero and up as positive, m below is , not . | |
Does the line rewrite subtraction as adding the additive inverse? | It changes subtraction to addition but keeps . The additive inverse of is . | ||
Is the sum exactly units from , in the direction of ? | Adding the positive number moves toward greater numbers, but is less than . | ||
Does a result of come from adding two opposites? | It adds and . They are equal, not opposites, so their sum is not . |
Worked example
Find and fix the error
A diver starts m below the surface. She descends m and then rises m. To find her final position, Kai writes three lines, each meant to equal the line before it. Line 1: . Line 2: . Line 3: . Find the first line that is not true, explain the error, and correct the work.
Test line 1 against the story. Let zero be the surface, let up be the positive direction, and measure in meters. The start is −4.5. With up as positive, the descent subtracts 2.75 and the rise adds 3.5. Line 1 matches the story, so it is correct.
Rewrite line 1 so that every operation is addition. Subtracting 2.75 means adding its additive inverse, −2.75. Nothing else changes, so 3.5 is still added.
Rewrite Kai's line 2 the same way and compare. Subtracting 6.25 means adding −6.25, and −6.25 is the sum of −2.75 and −3.5. So line 2 adds −3.5 where line 1 adds 3.5. The two lines do not have the same value, so line 2 is the first line that is not true.
Explain the error. In line 1 the minus sign applies only to 2.75, the number right after it. Kai added 2.75 + 3.5 = 6.25 first and subtracted the total. That turns the rise of 3.5 m into a descent. Line 3 follows from line 2 with correct arithmetic, but it builds on a false line, so its value is wrong too.
Correct the work from line 1. Adding −2.75 moves 2.75 units in the negative direction from −4.5, which gives −7.25. Adding 3.5 then moves 3.5 units in the positive direction. That is not enough to reach 0, and 7.25 − 3.5 = 3.75 units remain below it.
Check with a different grouping. Every operation is now addition, so you may regroup the numbers being added without changing the sum (the associative property of addition). The descent and the rise together make a change of −2.75 + 3.5 = 0.75, so the diver ends 0.75 m above where she started.
Answer
Line 2 is the first line that is not true: Kai subtracted instead of adding it. The diver's final position is m, which is m below the surface.
Takeaway. Choose the zero, the positive direction and the unit, and a story about rational numbers becomes an expression. For any rational numbers and , subtracting means adding its additive inverse , and is units from in the direction of . Use the same two facts to check work: test each line in order, find the first one that is not true, explain why, and correct it. Then ask whether the sign and size of the answer fit the story.