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Win Elements · Grade 7 Mathematics · Notes

Adding Signed Fractions and Decimals

Skill 1.03 CA CCSSM 7.NS.1b About 10 minutes

The walk has not changed. The only new job is measuring how far the jump goes.

Here is the good news, and it is bigger news than it sounds: nothing you learned about adding on the number line changes when the numbers become fractions or decimals.

Start at the first number. Jump the second number. Its size is how far, its sign is which way. That is still the entire instruction.

What is new is only this: when the jump is 12 or 0.9, you have to do a little work to figure out how far that actually is before you can take it. That is the whole lesson.

1. The walk is the same walk

Look at this and say the two instructions out loud: start at −34, jump 12 to the right.

Figure 1. Same picture as last lesson, smaller tick marks. The line is now cut into quarters, so you can land between the whole numbers.

You did not need a rule for that. You needed to know one thing: how many tick marks is a jump of one-half?

2. The one new job: measuring the jump

On a line cut into quarters, a jump of 12 is two ticks, because one-half is two-quarters. Same distance, different name.

Figure 2. The gold line shows it: 1/2 and 2/4 are the same place. Renaming a fraction does not move it. It just describes it with the tick marks you are using.

What a common denominator is really for Finding a common denominator does not change either number. It only puts both of them in the same units, so you can say how far the jump is in tick marks you can count.

That is why you cannot add 34 and 12 directly. It is the same reason you cannot add 3 feet and 1 yard without converting first. The amounts are real; the units just do not match yet.

3. Predict before you compute

Before touching a denominator, answer one question: which way does the answer end up?

You are standing at −34, three-quarters of the way left of zero. Your jump is only one-half long, pointing right. A short jump cannot carry you all the way back across zero.

Figure 3. Compare the two distances before you compute. The leftward one is longer, so you finish left of zero. You now know the sign of your answer before doing any fraction work.

This thirty-second habit is what catches the mistake in section 6 before it costs you anything.

4. Now do the measuring

Worked example

Find −34 + 12.

Predict. Start three-quarters left, jump one-half right. The start is farther than the jump, so the answer is negative and small.

Match the units. Quarters and halves do not match. Rename the half as two quarters:

−34 + 24

Take the walk. Start three ticks left of zero. Jump two ticks right. You land one tick left of zero:

−34 + 24 = −14

Check the prediction. Negative and small. It matches, so the fraction work did not wander off.

Notice the denominator never moved. Four stayed four the whole way through, because the tick marks did not change size mid-walk. Only the count of ticks changed: three left, then two right, leaves one left.

5. Decimals are the same walk

A decimal is already a fraction wearing different clothes. 0.9 means nine tenths, so on a line cut into tenths, a jump of 0.9 is nine ticks. Try −2.4 + 0.9.

Figure 4. −2.4 + 0.9 = −1.5. Nine small ticks to the right. The prediction holds: the jump was too short to reach zero.

With decimals the units usually already match, which is why decimals often feel easier. Line up the decimal points and every digit is sitting over one of the same size.

6. When both numbers are negative

Same as last lesson: you start left and keep going left. −1.5 + (−0.75) starts at one and a half left of zero and jumps another three-quarters left.

Figure 5. −1.5 + (−0.75) = −2.25. Both numbers negative, so the answer is negative and farther from zero than either one. Exactly as before.

7. The mistake that costs the most points

Treating the sign and the fraction as two separate problems

This is the one to watch for, and it is sneaky because half of the work is usually right. A student splits −34 + 12 into two jobs: do the fractions, then deal with the sign. They compute 34 + 12 = 54, remember there was a minus somewhere, and write −54.

The fraction arithmetic in that is flawless. The answer is still wrong, because the minus sign was never a label to attach at the end — it was an instruction about which direction to walk.

Figure 6. The wrong walk sends both pieces left. But only one of them was pointing left — the other had a plus sign on it the whole time.

Look at the two landings. One is farther from zero than where you started; the other is closer. A rightward jump has to bring you closer to zero when you start on the left. That is the tell.

The check: do the prediction from section 3 first, and write down the sign and a rough size before you touch a denominator. If your computed answer disagrees with the prediction, trust the prediction and find the slip.

8. A real situation

Worked example

A diver is 4.5 meters below the surface. She rises 1.75 meters to look at a reef. Where is she now?

Figure 7. Rising is the positive direction, so the jump points up. It is not long enough to reach the surface, so she is still below it.

Predict. She starts 4.5 below. A rise of 1.75 is much shorter than 4.5, so she is still underwater — the answer is negative, somewhere near 3 below.

Compute. Start at −4.5, jump 1.75 up:

−4.5 + 1.75 = −2.75

Two and three-quarter meters below the surface. Negative, near 3. The prediction holds.

9. A look ahead: subtraction is about to disappear

Coming up next

What would you do with 12 − 34? It is a subtraction, and you have only been adding.

Watch what happens if you read the minus as a direction instead of an operation: start at one-half, jump three-quarters to the left. That is a walk you already know how to take.

Figure 8. 1/2 − 3/4 lands on −1/4 — the same place as the very first example on these notes. Two different-looking problems, one identical walk.

That is the next skill, and it is less a new topic than a relabeling. Every subtraction is an addition whose jump points the other way.

Words to know

common denominator: A shared tick size for two fractions, so their distances can be counted in the same units.
equivalent fractions: Different names for the same place on the number line. 12 and 24 are the same point.
rational number: Any number you can write as one integer over another. Fractions, decimals and whole numbers are all rational.
estimate: A rough answer found on purpose before the exact one, used to catch mistakes.

What to carry away

The walk is unchanged: start at the first number, jump the second.
A common denominator does not change either number. It only makes the jump countable.
Renaming a fraction never moves it on the number line.
Predict the sign and rough size before computing, then check your answer against it.
The sign is not a label you attach at the end. It is the direction you were walking the whole time.

Check your understanding

You are ready to move on when you can answer these out loud, in your own words, without looking back:

What is a common denominator actually for? Answer without using the word "add."
Why does renaming 12 as 24 not change the answer?
Before computing −34 + 12, how do you know the answer is negative?
In Figure 6, what did the wrong walk do with the plus sign?
How is Figure 8 the same walk as Figure 1, even though one is a subtraction?