015i - Efficient Strategies for Adding and Subtracting Rational Numbers

Lesson 15 — Efficient Strategies for Adding and Subtracting Rational Numbers

Strand: Number | Descriptor: AC9M8N04 | Duration: 45 minutes

Learning Intentions

• To add and subtract integers, fractions and decimals efficiently, choosing a strategy suited to the numbers involved.
• To understand subtraction of a negative as equivalent to addition, and use this to simplify calculations.

Success Criteria

I can:

1. Rewrite “subtracting a negative” as “adding a positive”, and vice versa.
2. Choose an efficient mental strategy (compensation, number line jumps, common denominators) for a given pair of rational numbers.
3. Add and subtract fractions and decimals with unlike denominators or place values.
4. Justify why a chosen strategy is more efficient than direct column working for a given example.

Warmup

(5 minutes — number line jumps, mini whiteboards)

Using a number line, find each result. Sketch the jump(s) you make.

Discussion: Ask students to describe question 3 in words before solving (“take away a debt of 6” or similar). Reveal that and land on the same point. This is today’s central idea.

Activities

Activity 1 — Explicit Instruction: Subtracting a Negative (10 min)

I do / We do / You do.

I do: Model the equivalence directly using temperature as a context.

The temperature is . It then drops by a further below what it would be if we removed a drop that already happened — instead, consider: if “subtracting ” means undoing a 6-degree drop, that is the same as adding 6 degrees.

Rule: Subtracting a negative is the same as adding its positive. Adding a negative is the same as subtracting its positive.

We do: Rewrite and evaluate together: , , , .

You do: Rewrite and evaluate: , , , .

Activity 2 — Guided Practice: Efficient Strategies for Fractions and Decimals (10 min)

I do: Compare two approaches to (common denominators already shared — straightforward) versus (denominators differ — find the lowest common denominator first).

For decimals, model compensation: adjust one number to a “nice” value, then correct.

We do: Choose and apply an efficient strategy together: , (compensation: think ), .

You do: Evaluate, and briefly note which strategy you used: , , , .

Activity 3 — Inquiry: is there Always a “best” Strategy? (14 min)

Pairs, then whole-class share.

Give students a mixed set of 6 addition/subtraction problems (some integers, some fractions, some decimals, some “nice” numbers, some awkward). In pairs, students solve each and rate, from 1 (very efficient) to 3 (clunky), how well a compensation or number-line strategy worked compared to a standard column/common-denominator method.

Example set: , , , , , .

Which problems were faster with a “clever trick” (compensation, rounding, spotting a shared structure)? Which were faster with the standard method? Is there a pattern to when each strategy wins?

Socratic scaffolding for pairs who need support:

Prompt	Purpose
Understand: what makes a strategy “efficient”?	Fewer steps, less written working, or less chance of error — not necessarily “the one you were taught first”.
Look at . What do you notice about the numbers?	Both are close to — a compensation opportunity.
Devise a plan using that observation	.
Now look at . Does the same trick apply?	No — these are fractions with unlike denominators; compensation to a “nice” decimal doesn’t help here, but converting to twelfths does: .
Carry out the plan for all six problems	Complete the full set, strategy by strategy.
Looking back — generalise	Compensation shines when numbers are close to a round number; common denominators shine for “awkward” fractions; number lines help visualise small integer jumps. The “best” strategy depends on the shape of the numbers, not a fixed rule.

Finish with a whole-class share: each pair nominates their favourite “trick” from the set and explains why it worked.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Evaluate .
2. Evaluate .
3. Use compensation to evaluate efficiently. Show your adjustment.
4. Reasoning. Explain why is not equal to , even though is.

Answers: 1. ; 2. ; 3. ; 4. Subtracting means adding , so ; only adding a number and its opposite gives zero, not subtracting.

Common Misconceptions

Misconception	How to pre-empt it
Treating “subtract a negative” the same as “subtract a positive”.	Contrast with side by side every time the rule is introduced.
Applying compensation incorrectly (adjusting both numbers in the same direction).	Model explicitly: what is added to one number must be subtracted from the other to keep the total unchanged.
Finding a common denominator but forgetting to convert the numerator too.	Insist the multiplication is shown on both numerator and denominator, as in .
Believing a negative fraction like means “the whole fraction is somehow different from “.	Reinforce are all the same value; the negative can sit anywhere without changing the result.
Assuming decimals with different numbers of decimal places cannot be compensated.	Show using compensation () to demonstrate the strategy still works cleanly.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Evaluate .

Answer

E2 (Kangaroo style). The sum of two numbers is . One of the numbers is . What is the other number?

Answer

.

E3 (Challenge). Without a calculator, find .

Answer

Each bracket equals , and there are brackets: .

E4 (Investigation). A number and its opposite (negative) are both marked on a number line. Their distance apart is . If the number is positive, find both numbers.

Answer

If the number is , its opposite is , and the distance between them is , so . The numbers are and .

Homework

1. Evaluate: (a) (b) (c) (d) .
2. Use compensation to evaluate efficiently, showing your adjustment: (a) (b) (c) .
3. Evaluate: (a) (b) (c) .
4. A diver is at m relative to sea level. She rises m, then descends m. What is her new depth? Express your working as a sum of integers/decimals.
5. Reasoning. A student claims ” is always smaller than .” Investigate using values where is negative, and explain whether the claim is always true.
6. Challenge. Three rational numbers have a sum of . Two of them are and . Find the third, giving your answer as a fraction in simplest form.

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) (b) (c) . Q3 — (a) (b) (c) . Q4 — m, i.e. m below sea level. Q5 — False in general: if is negative, becomes plus a positive (larger), while becomes plus a negative (smaller) — so is actually larger than when , the opposite of the claim. Q6 — .