054i - Interpreting Linear Model Solutions

Lesson 54 — Interpreting Solutions in Terms of the Original Situation

Strand: Algebra | Descriptor: AC9M8A03 | Duration: 45 minutes

Learning Intentions

• To translate an algebraic solution back into the real-world situation, using correct units and a full sentence.
• To decide whether a solution should be rounded up, rounded down, or rejected, based on what the variable represents.

Success Criteria

I can:

1. State a solved value with correct units in a sentence that answers the original question.
2. Decide whether to round a solution up or down, based on the real-world meaning of the variable.
3. Recognise when a solution lies outside the sensible domain of a model, and explain why.
4. Interpret a break-even solution as a threshold, stating which option is preferable on each side of it.

Warmup

(5 minutes — quick-fire, mini whiteboards)

Here are three raw algebraic answers, stripped of context. For each, decide: can it be used exactly as calculated, or is something suspicious about it?

1. Solving a “how many buses are needed” problem gives .
2. Solving a “how many weeks until she can afford it” problem gives .
3. Solving a phone-plan break-even problem gives .

Discussion: Q1 cannot mean buses — a bus is a whole, indivisible unit. Q2’s negative answer suggests the target was already affordable before the model’s starting point, or that the model’s assumptions don’t hold there. Q3 is a threshold, not a count, so the decimal itself carries meaning and shouldn’t simply be rounded away. Today’s big idea: solving an equation (Lesson 53) is only half the job — Stage 4 of the modelling cycle, Interpret, is where the raw number becomes a real answer.

Activities

Activity 1 — Explicit Instruction: Rounding with Purpose (12 min)

I do: A group of friends has 250$27$10$ booking fee.

Talk aloud: ” counts tickets — a whole-number quantity. The budget of 25089$ would exceed the budget.” Check leftover funds:

We do: Together, a painter needs to cover a wall. One tin covers .

Discuss: here the requirement is a minimum — the whole wall must be covered, so partial coverage is not acceptable. Round up to tins, even though is closer to .

You do: For each, decide whether to round up or down, and state the final answer with units.

1. A school needs to seat students on buses that each hold . Solving gives .
2. A customer has 40$6.50n = 6.15$.
3. items are packed into boxes holding each. Solving gives .

Activity 2 — Explicit Instruction: Writing the Interpretive Sentence and Checking the Domain (12 min)

I do: Recall the rideshare break-even from Lesson 53: km. Write the full interpretation:

“For trips shorter than about km, Company A is cheaper. For trips longer than about km, Company B is cheaper.”

Note: a break-even value is a threshold, not a count of objects — rounding to one decimal place preserves useful precision here, unlike Activity 1’s ticket and tin problems.

We do: Together, a gym membership model is solved for a target balance and gives . Recall from Lesson 51 that the domain of this model is . Interpret together: ” is not a valid number of weeks. It falls outside the domain of the model — the target balance would have needed to be reached before membership even began, which the model does not describe.”

You do: Interpret each solved value in a full sentence, with correct units and rounding.

1. Solving the printing-shop model gives .
2. Solving the taxi model for gives km.
3. Solving a savings model for a target balance gives .

Activity 3 — Inquiry Task: the Printing Shop Budget (11 min)

Pairs, then whole-class share.

A printing shop charges a 412$50$ budget. Determine the maximum number of pages they can print without exceeding their budget, and state exactly how much of the budget remains unspent.

Socratic scaffolding:

Prompt	Purpose
Understand: are we finding an exact decimal, or a maximum whole number?	The real answer must be a whole number of pages — cannot be fractional.
Devise a plan	Solve exactly first, then decide how to round.
Carry out the plan	.
Look back — is 50$ a maximum or a minimum?	A maximum (a budget ceiling). Rounding up to pages would exceed it.
So how should be rounded?	Down, to pages.
Carry out the final step	Find the actual cost of pages, then subtract from 50$ to find the leftover.

Answer: The customer can print pages, leaving cents unspent.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. A hall seats groups of at each table. Solving for the number of tables needed for guests gives . State the answer, with reasoning, in a full sentence.
2. Solving a savings model for a 400w = 15.5$ weeks. State the answer, with reasoning, in a full sentence.
3. A break-even calculation gives km. Write a one-sentence interpretation naming which option is cheaper on each side of this value.
4. Reasoning. Explain why the ticket problem in Activity 1 required rounding down, while the paint-tin problem required rounding up, even though both involved whole units.

Answers: 1. Round up to tables — tables would only seat guests, leaving without a seat; 2. Round up to weeks — after weeks the target has not yet been reached; 3. “For trips shorter than km [Option A] is cheaper; for trips longer than km [Option B] is cheaper.”; 4. The ticket budget was a maximum that could not be exceeded (round down to stay within it), while the wall area was a minimum that had to be fully covered (round up to meet it) — the direction of rounding depends on whether the constraint is a ceiling or a floor.

Common Misconceptions

Misconception	How to pre-empt it
Always rounding to the nearest whole number, regardless of context.	Ask explicitly: “is this a ceiling (maximum) or a floor (minimum)?” before rounding.
Reporting the raw decimal with no units and no sentence.	Require every final answer to be a full sentence naming the quantity and its units.
Rounding up when the constraint is a maximum budget.	Check the rounded answer against the original constraint: does it still fit?
Treating an out-of-domain solution (e.g. negative time) as “no answer” rather than explaining what it means.	Model the phrase “this falls outside the domain because…” every time an unusual solution appears.
Over-rounding a threshold value (e.g. break-even distance) to a whole number, losing the precision needed to compare options near the boundary.	Discuss what level of precision the purpose of the answer actually requires.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Solving a seating problem gives for the number of minibuses needed, each holding students, for a group of . How many students are in the minibus that is not full, and how many empty seats does it have?

Answer

Round up to minibuses. The first carry students, leaving in the final bus — empty seats.

E2 (Kangaroo style). A model gives the break-even point between two shipping companies as kg. Company X is cheaper below this mass. A customer has a kg parcel and a kg parcel. Which company should they use for each?

Answer

The kg parcel is below the break-even mass, so Company X is cheaper. The kg parcel is above it, so Company Y is cheaper.

E3 (Challenge). Solving a savings-target equation gives weeks. Explain what this means about the target amount, given the model with domain .

Answer

Since is outside the domain , the target balance was already smaller than the starting balance of 80w=0$.

Homework

1. Solving a bus-seating problem (seats of ) for students gives . State the number of buses needed, with reasoning.
2. Solving a savings-target problem gives weeks. State the answer, with reasoning, in a full sentence.
3. A break-even calculation between two market stalls gives items. Write a one-sentence interpretation of which stall is cheaper below and above this value.
4. Solving a model for a delivery cost gives kg. Explain why this solution should be rejected.
5. Reasoning. A rectangular garden bed needs edging. Solving for the number of m edging boards required gives . Explain why this must be rounded up, and state how much excess edging (in metres) is left over from the final board.
6. Challenge. A charity needs to raise at least 1500P = 45c - 120P = 1500c = 36c = 35.43635.435$.

Answers: 1. Round up to buses — buses only seat ; 2. Round up to weeks — after weeks the target is not yet reached; 3. “For fewer than items the first stall is cheaper; for more than items the second stall is cheaper.”; 4. Mass cannot be negative, so falls outside the sensible domain of the model; 5. Round up to boards, since boards ( m) would not fully edge the bed; boards provide m, leaving m excess (accepting reasonable rounding in the check); 6. 150035.436$ guarantees the target is met or exceeded.