067i - Efficient Calculation Strategies

Lesson 67 — Efficient Calculation Strategies Across Fractions, Decimals and Percentages

Strand: Number | Descriptor: AC9M7N06 | Duration: 45 minutes

Learning Intentions

• To choose the most efficient form — fraction, decimal or percentage — for a given calculation.
• To use equivalence to sidestep awkward arithmetic.

Success Criteria

I can:

1. Recognise when converting form makes a calculation easier.
2. Use fraction equivalents of common percentages and decimals.
3. Use commutativity and factorisation to simplify calculations.
4. Justify my choice of strategy.

Warmup

(6 minutes — same answer, different routes; pairs)

Three students calculate of 48$:

• Ana: by long multiplication.
• Ben: , cancelling.
• Cara: “a quarter of is .”

1. Do all three get the same answer?
2. Rank the methods by speed.
3. What did Cara know that made her fastest?

Answers: 1. Yes, 1225% = \tfrac14$, and quartering is a single mental step.

Today’s theme: the answer doesn’t depend on the form — but the effort does.

Activities

Activity 1 — Explicit Instruction: Choosing the Best Form (14 min)

The decision guide:

Situation	Best form	Why
Percentage is a friendly fraction (, , , , , )	Fraction	One-step mental division
Awkward percentage (, )	Decimal	Calculator- and algorithm-friendly
Repeated halving or doubling available	Fraction	Halve/double chains are fast
Money	Decimal	Matches cents
Exact thirds, sixths, sevenths	Fraction	Decimals recur
Comparing quantities	Percentage or decimal	Common scale

I do — three showcases.

Showcase 1: of 64$.

(Versus by hand — slow.)

Showcase 2: .

(Quarter it, triple it.)

Showcase 3: of 45.60\tfrac1345.603$ cleanly:

(Fractions of money work fine when the division is exact.)

We do — choose a form and calculate:

1. of
3. of 56$
4. of
5. of 8.60$

(Answers: ; ; 211\tfrac12 \times 18 = 2710% \div 2 = 0.86 \div 2 = $0.43$.)

Activity 2 — Clever Moves (12 min)

Move 1 — swap the percentage. of equals of (Lesson 66 E5).

Move 2 — factor and reorder. Multiplication is commutative and associative:

Move 3 — compensate. : think .

Move 4 — halve and double. .

You do — find the slick route:

1. of
4. of
8. of 200$

(Answers: 1. of ; 2. ; 3. ; 4. of ; 5. ; 6. ; 7. ; 8. 90$.)

Discussion of Q8: “of ” means doubling the percentage number — because of is . Similarly “of ” means halving it. Percentages of , , and all have instant routes.

Activity 3 — Inquiry: the Strategy Tournament (12 min)

Pairs, then whole class.

Each pair receives the same six calculations. For each:

1. Find the answer any way you like.
2. Write down your route in one line.
3. As a class, collect the different routes for each item and vote for the most elegant.

The six: (a) of 320.6 \times 35\dfrac{5}{6}4285%$401.25 \times 4.830%$15 + 70%$15$

Socratic scaffolding (deploy per item as needed):

Prompt	Purpose
For (a): what fraction is ?	— so .
For (d): is the direct route or the complement faster?	: . Both fine.
For (e): can you turn into a fraction?	.
For (f): must you calculate both parts?	No — of 15 = $15$. The whole is one step.
Looking back	The most elegant route often avoids calculation altogether.

Answers: (a) 122135$346$15$.

Checks for Understanding

(5 minutes — exit ticket)

1. Calculate using a fraction shortcut: (a) of 9612.5%$72$.
2. Calculate cleverly: .
3. Calculate: of .
4. Calculate of 45 + 40%$45$ in one step.
5. Reasoning. Explain why of is the same as , using the swap move.

Answers: 1. (a) 24$9(0.25 \times 8) \times 13 = 2650%2 = 1100%$45 = $4514%50 = 50%14 = 7$.

Common Misconceptions

Misconception	How to pre-empt it
Believing there is one “correct” method per question.	The tournament makes multiple valid routes visible and celebrated.
Always reaching for the calculator algorithm.	Time a benchmark race: mental of versus written .
Misremembering fraction–percentage pairs (, not ).	Keep the Lesson 56 conversion table displayed.
Applying the swap move to division or subtraction.	It relies on commutativity of multiplication only.
Compensating the wrong way: .	You over-counted by one lot of , so subtract. Estimate to check.
Treating elegance as optional.	Efficient routes reduce errors as well as time — fewer steps, fewer slips.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Calculate without long multiplication.

Answer

E2 (AMC Junior style). Which is largest: of , of , or ?

Answer

All three are equal — of of .

E3 (Challenge). Calculate of of mentally.

Answer

Faster by cancelling: , , leaving .

E4 (Challenge). A shop takes off, and a member’s card takes a further off the reduced price. Express the total effect as a single fraction of the original price.

Answer

The customer pays () of the original.

E5 (Challenge). Without a calculator, decide which is bigger: of 8165%$84$?

Answer

. of . The second, narrowly.

Homework

1. Use a fraction shortcut: (a) of 8.4025%$6875%$12012.5%$9620%$45$.
2. Use a clever reorder: (a) (b) (c) (d) .
3. Use the swap move: (a) of (b) of (c) of (d) of .
4. Use compensation: (a) (b) (c) .
5. Calculate the total in one step: of 80 + 55%$80$.
6. Choose your own best route and name it: (a) of 480.75 \times 44\dfrac{2}{3}$9.6095%$60$.
7. Reasoning. Explain why of equals of , and evaluate both.
8. Reasoning. A student calculates as . Diagnose and correct the error.
9. Challenge. Calculate mentally, explaining your grouping.

Answers: Q1 — (a) 4.20$17$90$12$92317404534.518170 - 0.7 = 69.340 + 0.4 = 40.430 - 1.5 = 28.5$80\tfrac38 \times 48 = $18\tfrac34 \times 44 = 339.60 \div 3 \times 2 = $6.4060 - 5% = 60 - 3 = $57\tfrac{16 \times 75}{100} = 1298 \times 0.5 = 100 \times 0.5 - 2 \times 0.5 = 50 - 1 = 49(0.2 \times 800) = 160\times 0.25 = 40\times 0.5 = 20$.