Lesson 7: Paying for a Purchase Over Time

Prepared by Georgia Wang

Year level: Year 10 (middle ability set)Duration: 60 minutesFocal concept/topic: Paying for a Purchase Over Time

Lesson aim:

Calculate complete payment-plan costs and distinguish total cost from payment manageability.

Learning intention:

We are learning to calculate and compare the complete cost of purchases paid over time.

Success criteria:

□ I can calculate a percentage deposit and the number and total of repayments.
□ I can calculate total paid and additional cost above the cash price.
□ I can justify a comparison using complete totals.
□ I can distinguish fitting stated payment limits from genuine affordability.

Syllabus outcomes and content:

NSW Education Standards Authority (NESA, 2022).

MA5-FIN-C-01: solves financial problems involving simple interest, earning money and spending money
MAO-WM-01: develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly
Calculate the cost of buying items on terms, by paying an initial deposit and making regular repayments

Working Mathematically indicators:

Organise deposits and repeated payments
Justify unequal-duration comparisons
Identify limits of a conclusion based on incomplete information

ICT:

Calculator use supports arithmetic only.

LIT:

Explicit financial vocabulary
Annotate offer terms
“Plan __ costs $__ in total, compared with $__; therefore …”
Silent writing and partner rehearsal before sharing

Prerequisite knowledge:

Percentages of amounts
Addition and multiplication of money
12 months in a year
Calculator use

Students have studied earning money and simple interest in the proposed sequence, but this lesson does not require a loan-interest formula.

Resources:

Calculators
Mini-whiteboards/A–D cards
Board
Printed Tasks 7A–7F below
Labelled calculation rows
Payment timeline
Observation checklist
Exit ticket

Labels used in this plan: WM Working Mathematically LIT Literacy ICT Technology AfL Assessment for learning AoL Assessment of learning

Key teacher question → Expected: suggested student response. Self-designed WM indicators are in italics.

Teaching sequence

Steps & timing	Teaching & learning activities
ORIENTATION0–6 minContext and retrievalAfL	Teacher: Present a fictional $900 purchase. Reveal LI/SC and inquiry: “Can a smaller monthly repayment cost more overall?” Use Task 7A, including the Lesson 6 simple-interest bridge. Allocate 2 minutes to the interest check, 2 minutes to deposit/payment retrieval and 2 minutes to purpose and feedback. Students: Independently calculate 20% of $900 and payments per year. AfL Simultaneous responses; refresh percentage conversion or months only where needed.
ENHANCING6–18 minExplicit modelWMLIT	Teacher: Model Task 7B with separate deposit, repayments and total rows. Draw: deposit now → 12 monthly payments. Students: Annotate calculations and explain why the deposit is counted once. WM LIT Ask “Is repayment total the same as total paid?” Expected: no; add the deposit. Key question: What does 20% represent in the $900 plan, and why do we add it once? Expected: It is a $180 deposit. It is one initial payment, followed by the 12 separate repayments.
ENHANCING18–21 minHinge checkAfL	Teacher: Display Task 7C; collect individual A–D answers simultaneously. AfL Identify structural errors separately from arithmetic errors. At least 80% correct: proceed with targeted prompts. Below 80%: use next segment to remodel dominant error. The threshold is a pacing rule, not proof of mastery. Key question: Which answer includes both the deposit and all 12 monthly payments? Expected: A: $200 + 12 × $90 = $1,280. B omits the deposit; C counts only 11 payments; D counts only one.
ENHANCING21–26 minResponsive teachingAfL	Teacher: Reteach the dominant hinge error on a timeline, or use Task 7D as a partially worked transfer if secure. Students: All answer Task 7D independently to check transfer. AfL Check one deposit, correct payment count and inclusion of both amounts. Key question: What does the $840 in Task 7D represent, and what is missing from it? Expected: It is the repayment total. Add the $80 deposit to obtain $920.
ENHANCING26–43 minMain comparisonWMLITAfL	Teacher: Give Task 7E. Students calculate both plans individually before partner comparison. Then reveal buyer limits. WM LIT Require three separate conclusions: lower cost; fitting stated limits; what remains unknown. AfL Observe complete calculation, comparison and interpretation using I/P/N checklist. Key question: How can Plan B have the smaller monthly payment but the higher total cost? Expected: B lasts 24 months: $120 + 24 × $65 = $1,680, compared with A at $1,500.
ENHANCING43–48 minAdvertising promptLITWM	Teacher: Display “Laptop from $65 a month”. Students: Identify missing cash price, deposit, payment count/frequency, total and possible charges. LIT WM Explain why the monthly figure alone is insufficient. Key question: What information is missing from “Laptop from $65 a month”? Expected: Cash price, deposit, payment frequency and count, total, and any fees or conditions.
SYNTHESIZING48–55 minConsolidation	Students: Rehearse an evidence-based explanation with a partner; revise an anonymous claim: “B is cheaper because $65 is less than $105.” Expected: B has a smaller monthly payment but costs $180 more overall. Teacher: Revisit success criteria and clarify cost versus affordability. Key question: Does fitting $150 upfront and $80 per month prove that Plan B is affordable? Expected: No. It fits the stated limits, but income, essential expenses, other commitments and unexpected costs are unknown.
SYNTHESIZING55–60 minIndependent exitAoLAfL	Students: Complete Task 7F without partner help. AoL End-of-lesson individual mastery record against the success criteria. Mark the exit task out of 4 using the criteria below; this records attainment after instruction. AfL Collect calculations and explanations. Next lesson, respond to deposit, payment-count or comparison errors separately. Contingency: Shorten public sharing if needed; retain independent exit evidence.

Worked solutions

7A Retrieval: Find 20% of $900; how many monthly payments are in one year? Answers: 0.20 × 900 = $180; 12 payments. Lesson 6 bridge: $900 at 5% p.a. simple interest for one year gives I = 900 × 0.05 × 1 = $45 and $945 altogether. Ask: “Is this $45 calculated in the same way as the $180 additional purchase cost?” Expected: no; the simple-interest task uses a stated rate and time, while the purchase-plan cost comes from deposit plus repayments minus cash price. No interest rate is given in the purchase plan.

7B Model: Cash price $900; 20% deposit followed by $75 monthly for 12 months; no other charges. Deposit = $180. Repayments = 12 × $75 = $900. Total paid = $180 + $900 = $1,080. Additional cost = $1,080 − $900 = $180.

7C Hinge: Cash price $1,000; 20% deposit followed by $90 monthly for one year. A $1,280; B $1,080; C $1,190; D $290. Correct A: $200 + 12 × $90 = $1,280. B omits deposit; C uses 11 payments; D uses one payment.

7D Fresh check: Cash price $800; 10% deposit then $70 monthly for 12 months. Deposit $80; repayments $840; total $920; additional cost $120. Ask: “What must be added to $840?” Expected: the one-off $80 deposit.

7E Comparison: Cash price $1,200. Plan A: 20% deposit then $105 monthly for 12 months. Plan B: 10% deposit then $65 monthly for 24 months. Neither has other charges.

Quantity	Plan A	Plan B
Deposit	$240	$120
Repayment total	$1,260	$1,560
Total paid	$1,500	$1,680
Additional cost	$300	$480

Buyer statement: “I can pay no more than $150 upfront and $80 per month.” Questions: Which costs less overall? Which fits both limits? Does this prove affordability? Answers: A costs $180 less; only B fits both limits; no — income, essential expenses, other commitments and unexpected costs are unknown. Delaying or not buying is defensible. Extension: Find B’s monthly payment for equal total cost: 120 + 24m = 1,500, so m = $57.50.

7F Exit: Tablet cash price $600. X: 10% deposit plus $55 monthly for 12 months. Y: $120 deposit plus $95 monthly for 6 months. Calculate X’s deposit and total; compare. X: $60 + $660 = $720. Y: $120 + $570 = $690. Y costs $30 less despite its larger monthly repayment.

Exit assessment criteria

Task 7F: deposit $60 (1); correct payment count and repayment total $660 (1); total $720 (1); Y is $30 cheaper using both totals (1). AoL records four criteria separately. Use the same evidence formatively to decide next-lesson feedback. A correct total without an explanation does not demonstrate the reasoning criterion.

Adjustments and anticipated misconceptions

Barrier / misconception	Specific support and retained demand
Percentage confusion	Use 10% as one tenth, then build 20%; retain the same deposit calculation.
Omitting deposit / miscounting payments	Provide separately labelled rows and a timeline; require payment count before multiplication.
Language / participation barriers	Offer vocabulary, sentence frame and partner rehearsal; every student still writes their own comparison.
Early finisher	Use the equal-total monthly-payment extension; require equation and interpretation.

Observation checklist: Record Complete calculation / Evidence-based comparison / Careful interpretation for each student as I (independent), P (prompted) or N (not yet). Partner observations show process; the individual exit is the stronger transfer evidence.

Vocabulary

Term	Student-friendly definition
Cash price	Price paid for the item outright.
Deposit	Initial amount paid before the later repayments.
Instalment / repayment	One scheduled payment.
Duration	How long the payment plan lasts.
Repayment total	Sum of the scheduled repayments, excluding the initial deposit in these offers.
Total paid	Deposit plus all repayments and any separately stated charges.
Additional cost	Total paid minus cash price.
Affordability	Whether a person can meet payments while covering other needs and commitments.

Sources acknowledged in this lesson

NSW Education Standards Authority. (2022). Mathematics K–10 syllabus: Stage 5, Financial mathematics A. https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-k-10-2022/content/stage-5/facb3aa952

Syllabus content and outcome descriptions above reproduce NESA wording; italicised lesson WM indicators are teacher-designed applications.

Adapted from Georgia Wang’s Assignment 1, Appendix 1: Paying for a Purchase Over Time. Original numerical model, hinge, main comparison and exit ticket are retained. Explicit curriculum, literacy and assessment labels, vocabulary definitions and the transfer-check solution have been added.

All offer terms are fictional classroom data, not current financial-product or legal information. Each lesson totals 60 minutes. Support should respond to observed student work, and all learners retain the core calculation and reasoning goals.