086i - Interpreting Percentage Modelling Solutions

Lesson 86 — Interpreting and Communicating Modelling Solutions

Strand: Number | Descriptor: AC9M8N05 | Duration: 45 minutes

Learning Intentions

• To interpret the numerical result of a percentage calculation in terms of the original real-world context.
• To communicate a modelling solution clearly, including appropriate rounding and units.

Success Criteria

I can:

1. State the meaning of a calculated percentage result in a full sentence tied to the context.
2. Round and present financial answers appropriately (e.g. to the nearest cent).
3. Check whether an answer actually addresses the question that was asked.
4. Identify when an answer is unreasonable and explain why.

Warmup

(5 minutes — mini whiteboards)

For each pre-solved calculation, write the missing “answer sentence” that a customer or client would actually want to read.

1. A 128 \times 0.75 = 96$.
2. A wage of /week, increased 4%: .
3. Simple interest: .

Sample answers: 1. “The item now costs 884.” 3. “The investment earns $120 in interest over 2 years.”

Discussion: A bare number is not a complete answer — the modelling cycle’s Interpret stage means restating the number in the language of the original question.

Activities

Activity 1 — Explicit Instruction: from Number to Sentence (12 min)

I do: Take the raw result from a discount-and-GST calculation (a item discounted 20%, then 10% GST added) and build a complete interpretive statement:

“The customer pays **348 - 278.40 \times \dots278.40348$ does not. Model the correct comparison:

“The customer pays 76.56 compared to the full GST-inclusive price.” This shows why interpreting correctly sometimes requires solving an extra comparison step, not just restating the first number found.

We do: Interpret this result in a full sentence: a investment at simple interest earns over 3 years — what should the client be told, including the final balance?

You do: Convert each raw answer into a complete, correctly rounded interpretive sentence:

1. (a discounted price, in dollars).
2. (a profit margin, as a proportion).
3. (a population after growth, in people).

Activity 2 — Checking Reasonableness and Spotting Errors (12 min)

Students act as “auditors” of pre-worked solutions.

Present 4 worked solutions, each containing a plausible-looking but incorrect final answer. For each, students decide: is this reasonable? If not, why, and roughly what should it be?

1. “A 72.” (Error: used instead of — a decrease was calculated as an increase. Correct: $48.)
2. “GST of 10% is added to a 45.10.” (Error: added . Correct: $49.50.)
3. “Simple interest on at 5% p.a. for 3 years: .” (Error: forgot to divide by 100. Correct: $300.)
4. “A profit margin of on cost price , profit , reported as a 25% margin.” (Error: , not 25%.)

Activity 3 — Applied Inquiry Task: Reviewing a Naive Estimate (10 min)

Pairs, then whole-class share.

A shopper estimates that a 20% discount on a 68 off the original ticket price. Check the shopper’s estimate against the full calculation, interpret any difference, and suggest what the shopper’s estimate left out.

Socratic scaffolding:

Prompt	Purpose
Understand: what did the shopper actually estimate?	20% of 68 — but this ignores that GST also applies to both the original and discounted prices.
Devise a plan	Calculate the true final price with GST, and the true “no discount” price with GST, then find the real saving.
Carry out the plan	With discount: . Without discount: . Real saving: .
Interpret	The shopper’s estimate of 74.80, because GST is charged as a percentage of a larger “no discount” reference price too, not simply added on top of the naive $68 estimate.
Look back	Is a ~\dfrac{74.80-68}{74.80}\times100%\approx9%$ — worth knowing before relying on a quick mental estimate for a large purchase.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. A calculation gives a raw answer of for the price of a discounted item. Write a complete interpretive sentence.
2. A worked solution claims a 50.10$. Identify the error and state the correct price.
3. Round to the nearest cent, and explain why this rounding matters in a financial context.
4. Reasoning. A calculation intended to find “how much was saved” instead reports the new discounted price. Explain why this does not answer the question, even though the arithmetic may be correct.

Answers: 1. “The item costs 5050.1055.00214.39$ — money amounts are conventionally reported to the nearest cent (2 decimal places), since fractions of a cent cannot be paid. 4. The discounted price and the amount saved are two different quantities; reporting the new price does not tell the reader how much less they are paying than before, which was what was asked.

Common Misconceptions

Misconception	How to pre-empt it
Rounding partway through a multi-step calculation does not affect the final answer.	Show a worked example where rounding an intermediate result shifts the final answer by several cents; insist rounding happens only at the final step.
A number on its own is a complete answer.	Require every Checks for Understanding and homework answer to include a full interpretive sentence, not just a value.
”Percentage saved” and “dollar amount saved” mean the same thing and can be used interchangeably.	Contrast the two directly using the Activity 3 example — $74.80 saved is roughly a 20% saving on the GST-inclusive price, not simply “20%” restated.
If the working looks correct, the final answer must be correct.	Model the Activity 2 “auditor” habit: always sanity-check an answer’s size against the context, independent of re-checking the steps.
Reporting an unrounded calculator display (e.g. ) is acceptable in a financial answer.	Require rounding to the nearest cent in every financial context, and discuss why banks and shops never display more than 2 decimal places.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A calculation finds that a discount saves a shopper , which is stated to be “15% of the original price”. What was the original price?

Answer

The original price was $250.

E2 (Kangaroo style). Two claims are made about the same 20, a 25% reduction.” Claim B says “the new price is $60, an 80% reduction.” Which claim, if any, is mathematically consistent?

Answer

Claim A: saving 20$80\dfrac{20}{80}\times100%=25%$ — consistent.

Claim B: a new price of from an original of is a reduction of , not — not consistent (Claim B has confused the remaining percentage, , with the reduction percentage).

E3 (Challenge). A worked solution reports a final answer of "" for a profit margin. Explain what has most likely gone wrong in the interpretation stage, and suggest what a sensible corrected answer might look like.

Answer

A margin of over is almost certainly the result of forgetting to convert a decimal proportion to a percentage correctly (e.g. reporting directly as "" instead of recognising it should already have been only if profit were 10.5 times the cost — implausibly large for most retail contexts). A sensible corrected process is to re-check: profit margin , confirm the profit and cost values used, and expect a typical retail margin in the tens of percent, not the thousands — this is the “look back” reasonableness check.

Homework

1. Write a complete interpretive sentence for each raw answer: (a) (a discounted price) (b) (a new wage, in dollars per week) (c) (a profit margin).
2. A worked solution claims “a 30.25$.” Identify the likely error and state the correct price.
3. Round each to the nearest cent: (a) (b) (c) .
4. A shopper says a discount on a 30$.” Check this mentally and state whether it is a reasonable estimate.
5. Reasoning. Explain why a modelling solution should always be checked against the original question before being reported, using an example where the “obviously correct” number is not actually what was asked for.
6. Challenge. A report states that a company’s revenue “increased by 150% and then decreased by 60%, an overall drop of 90%.” Check this claim using multipliers and state the true net percentage change.

Answers: 1. (a) “The item costs 312.” (c) “The profit margin is 18.5% of the cost price.” 2. of is , so the correct price is , not — likely a decimal-point slip. 3. (a) (b) (c) . 4. of , so “302.50\times0.40=1.00150-60=90$) with the correct multiplier method.