Lesson 84 — Choosing Efficient Calculation Strategies; Digital Tools
Strand: Number | Descriptor: AC9M8N05 | Duration: 45 minutes
Learning Intentions
- To choose efficient strategies for calculating percentages, including benchmark percentages and decimal multipliers.
- To use digital tools (calculator, spreadsheet) appropriately for complex or repeated percentage calculations.
Success Criteria
I can:
- Calculate common “benchmark” percentages (1%, 10%, 5%, 25%, 50%) mentally and combine them to find others.
- Convert a percentage change into a single decimal multiplier (e.g. a 15% increase = ×1.15).
- Justify when a mental strategy, written strategy, or digital tool is most efficient for a given problem.
- Use a calculator or spreadsheet formula to compute percentage changes accurately.
Warmup
(5 minutes — mental relay, mini whiteboards)
Using $360 as the amount, find each of the following as fast as you can, building each answer from the one before:
- 10% of $360
- 1% of $360 (from Q1)
- 5% of $360 (from Q1)
- 20% of $360 (from Q1)
Answers: 1.
Discussion: Which answers did you build directly from 10%, rather than starting again from scratch?
Activities
Activity 1 — Explicit Instruction: Benchmark Percentages and Decimal Multipliers (12 min)
I do — benchmarking: Find 17% of
I do — decimal multipliers: An increase of 8% is the same as multiplying by
We do: Find 23% of
You do: Find each using whichever benchmark strategy is fastest: (a) 15% of
Activity 2 — Digital Tools for Repeated or Complex Calculations (12 min)
Discuss: benchmark strategies are fast for single calculations with “nice” numbers, but become inefficient for repeated calculations (e.g. many items) or compounding situations (a percentage applied repeatedly over time).
I do: Show a spreadsheet-style table where a formula such as =A2*1.15 is applied down a column to add 15% GST-equivalent to a list of prices — one formula, many results.
| Item | Price | Price incl. 15% |
|---|---|---|
| A | $40 | =A2*1.15 → $46.00 |
| B | $65 | =A3*1.15 → $74.75 |
| C | $128 | =A4*1.15 → $147.20 |
We do: Model a compounding scenario by hand for just 3 steps, to show why a digital tool becomes valuable beyond that: a town of
Note the growth is not linear — each year’s increase is larger in dollar terms than the last, because it is calculated on a growing base. Calculating this by hand for, say, 20 years would be slow and error-prone — exactly the kind of repeated, compounding task suited to a spreadsheet or calculator’s memory/power functions.
You do: Decide, for each scenario, whether mental benchmarking, written working, or a digital tool would be most efficient, and justify your choice: (a) finding 10% of
Activity 3 — Applied Task: Choosing the Right Tool (10 min)
Pairs, then whole-class share.
A canteen manager needs to: (i) quickly estimate a 15% tip on a
2,400 invoice, as a quick sanity check before paying it. For each, decide the most efficient strategy and calculate the result.
Socratic scaffolding:
| Prompt | Purpose |
|---|---|
| Understand: what does each task actually require? | (i) one quick estimate, (ii) many repeated identical calculations, (iii) one calculation with a “nearly whole” percentage. |
| Devise a plan for (i) | Use benchmarks: |
| Devise a plan for (ii) | 60 repeated calculations of the same type strongly suggest a spreadsheet formula (=price*1.07), not manual repetition. |
| Devise a plan for (iii) | Notice |
| Look back | Which task would have taken longest without choosing an efficient strategy? (ii), by far — reinforcing why digital tools matter most for scale, not for single “nice” numbers. |
Checks for Understanding
(6 minutes — exit ticket, collected)
- Use benchmarks to find 23% of
. - Write the decimal multiplier for (a) a 12% increase (b) a 7% decrease.
- A café applies a 6% price rise to all 45 items on its menu. Would a mental strategy or a digital tool be more efficient? Justify your answer.
- Find 98% of
using an efficient shortcut, and explain your method.
Answers: 1.
Common Misconceptions
| Misconception | How to pre-empt it |
|---|---|
| A decrease of 8% corresponds to a multiplier of | Contrast |
| Digital tools are always the fastest method, even for a single simple calculation. | Model Activity 3(i): a quick mental benchmark beats opening a calculator app for one “nice” number. |
| A calculator removes the need to estimate first or check the answer’s reasonableness. | Require an estimate (e.g. “roughly a fifth of the amount”) before any calculator use, and compare. |
| Percentages close to 100% (e.g. 99%, 98%) must be calculated the “long way”, by direct multiplication. | Reinforce the “distance from 100%” shortcut used in Activity 3(iii) and the CFU. |
| Compounding growth can be estimated by simply multiplying one year’s increase by the number of years. | Show the town-growth table: each year’s dollar increase is larger than the last, so simple multiplication underestimates the true total. |
Enrichment — Competition-Style Problems
E1 (Kangaroo style). Using benchmarks only (10%, 5%, 1%), find 47% of
Answer
E2 (AMC Junior style). A price increases by 10%, then the new price increases by a further 10%. What single percentage increase on the original price is this equivalent to?
Answer
Combined multiplier:
This is a single 21% increase, not 20% — an important contrast with simply adding the percentages.
E3 (Challenge). A quantity is repeatedly increased by 5% each step. Using the multiplier method, find (to the nearest whole number) how many steps are needed before the quantity has more than doubled, without computing every step individually — estimate using the fact that
Answer
Since
This shows the power of the multiplier method combined with a digital tool: computing
Homework
- Use benchmarks to find: (a) 13% of
(b) 24% of (c) 45% of . - Write the decimal multiplier for: (a) a 15% increase (b) a 15% decrease (c) a 2.5% increase.
- A landlord raises rent on 12 identical properties by 3.5% each. Explain, in a sentence, why a spreadsheet formula would be more efficient here than mental benchmarking.
- Find 97% of
using an efficient shortcut. - Reasoning. Explain why
of a number is always greater than the number itself, but of a number is always less, using multiplier language. - Challenge. A savings account balance grows by exactly 2% each year. Using the fact that
, find the approximate balance after 10 years if the starting balance was , and state whether simply adding “10 lots of 2% of ” would give the same answer. Explain the difference.
Answers: 1. (a)