Lesson 9: Short-term Small Loans: Comparing Borrowing Costs

Prepared by Georgia Wang

Year level: Year 10 (middle ability set)Duration: 60 minutesFocal concept/topic: Short-term Small Loans: Comparing Borrowing Costs

Lesson aim:

Calculate and compare borrowing costs for small short-term loans and assess independent spending reasoning.

Learning intention:

We are learning to compare loan costs using the amount borrowed, stated charges and repayment schedule.

Success criteria:

□ I can calculate a percentage fee based on the original amount borrowed.
□ I can calculate total repaid and borrowing cost without double counting.
□ I can compare equal borrowing amounts and identify schedule constraints.
□ I can justify a decision with calculations and acknowledge missing information.

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
Examine the principles behind short-term loans involving small dollar amounts and compare borrowing costs associated with using these products

Working Mathematically indicators:

Select an appropriate fee model
Distinguish principal from borrowing cost
Compare like amounts and explain a duration-dependent result

ICT:

Build a comparison model and change loan duration while holding principal and fee basis constant. Check outputs against hand arithmetic and interpret the crossover.

LIT:

Read “original amount borrowed”, “per month” and “in addition”
Explain “Total repaid is __; borrowing cost is __”

Prerequisite knowledge:

Lessons 7–8 total-cost comparisons and conditional charges
Percentages
Basic spreadsheet formulas

No reducing-balance or compound-interest calculation is assumed.

Resources:

Tasks 9A–9F
Calculators
Spreadsheet devices/paper fallback
Mini-whiteboards
Individual assessment sheet and marking guide

Teacher model: Spending_Money_Teacher_Spreadsheet.xlsx, L9 Loans sheet. A completed formula-and-values snapshot follows below. Students construct their own model before seeing the teacher version.

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
ORIENTATION0–8 minRetrievalAfL	Teacher: Address a Lesson 8 charge error and introduce LI/SC. Task 9A distinguishes total repaid and cost. Students: Respond independently. AfL If students say $540 is borrowing cost, use borrowed/returned/extra rows. Key question: If you borrow $500 and return $540, what is the borrowing cost? Expected: $40, not $540. Cost is total repaid minus principal.
ENHANCING8–20 minExplicit modelWMLIT	Teacher: Model Task 9B. Identify exactly what the percentage applies to; draw the monthly schedule. Students: Complete labelled principal, fees, total and cost rows. WM LIT Explain why the same principal is counted once. Key question: Which amount is the base for the 2% monthly fee, and why does it stay $12? Expected: The stated original $600 principal. This fictional fee is not based on a reducing balance.
ENHANCING20–25 minHinge and responseAfL	Task 9C: simultaneous individual response. AfL Reteach fee base or omitted principal when needed; use fresh check before the comparison. Key question: Why is $456 the correct total in the hinge? Expected: $400 principal + $40 establishment + 2 × $8 monthly fees = $456.
ENHANCING25–38 minICT comparisonICTWMAfL	Students: Calculate Task 9D by hand, then build formulas and test durations 2, 3 and 4 months. Swap checking/entry roles. ICT WM Predict whether the cheaper offer changes before changing inputs. AfL Teacher checks original-principal fee base and month count. Key question: Why can the cheaper loan change when the term changes? Expected: L has a larger establishment fee but a smaller monthly fee. M is cheaper at 2 months ($910 versus $912); L at 3 months ($928 versus $940).
ENHANCING38–40 minInterpretationLITWM	Use Task 9E. Students: Identify lower cost separately from the payment ceiling; explain missing information and the possibility of delaying. LIT WM Use numerical evidence rather than naming one universally “best” loan. Key question: Can we say L is genuinely affordable because its repayments fit $310? Expected: No. Its schedule fits, but other commitments and income reliability are not provided.
SYNTHESIZING40–55 minIndependent integrated assessmentAoL	AoL Task 9F, completed individually with calculator; no peer/spreadsheet help. Teacher: Collect as evidence of spending content and mathematical reasoning. Include Q4 to connect earning money, simple interest and spending.
SYNTHESIZING55–60 minReview and next steps	After collecting work, students self-check against success criteria; teacher revisits principal / total / cost distinctions. AfL next steps: Feedback groups for percentage-base, charge-count or interpretation errors. Contingency: Shorten the preceding discussion to preserve 15-minute assessment. Key question: How do earnings, simple interest and spending calculations combine in the closing case? Expected: The matured savings are $420; weekly surplus is $30, so six weeks reaches $600. The loan costs $684. Waiting avoids $84 of borrowing cost if the purchase can wait.

Worked solutions

9A: Borrow $500 and repay $540 altogether. Borrowing cost = $540 − $500 = $40. Total repaid is $540.

9B: Fictional loan of $600 for two months. Establishment fee 10% of original principal; monthly fee 2% of original principal for each month; no other charges or interest. Fees are included in two equal monthly repayments; no upfront fee. Establishment $60; monthly fee $12; two months $24; borrowing cost $84; total $684; each repayment $342. These fee rates are invented teaching inputs, not actual or permitted market rates.

9C Hinge: Borrow $400 for two months; establishment $40; monthly fee $8; fees included in repayments. Total repaid? A $48, B $56, C $448, D $456. Correct D = 400+40+2×8. A uses one month and omits principal; B omits principal; C uses one month. Fresh check: borrow $300, establishment $30 and $6 per month for two months → $342 total and $42 cost.

9D: Both fictional offers lend $800 for three months, with fees included in three monthly repayments. L: establishment fee 10% of original principal, plus 2% of original principal per month. M: $50 establishment plus $30 per month. No other charges or interest. Neither fee uses a reducing balance.

Quantity	L	M
Establishment	$80	$50
Monthly fee	$16	$30
Fees over 3 months	$48	$90
Borrowing cost	$128	$140
Total repaid	$928	$940
Repayment schedule	$309.33, $309.33, $309.34	$313.33, $313.33, $313.34

Teacher spreadsheet inputs: B4 principal $800; B5 months 3. L total = B4+B4*10%+B4*2%*B5; M total = B4+50+30*B5; cost = total−B4. Two months: L $912, M $910; three: L $928, M $940; four: L $944, M $970. Round earlier payments to cents and adjust the final payment to preserve the total. Students build their own table and may use different cell locations; paper fallback evaluates the same expressions.

9E: Buyer has a stated ceiling of $310 per month. At three months, L’s three payments fit, M’s do not. L also costs $12 less. This does not prove affordability: other expenses, income reliability and contractual eligibility are unknown. Ask why changing duration can change the cheaper offer: L has a larger establishment fee but a smaller monthly fee. Extension: 880+16m=850+30m → m=30/14≈2.14 months; for whole-month terms M is cheaper at 1–2 months, L from 3 months.

Task 9F independent assessment and marking guide 17 marks

Q1 (4 marks): Cash price $750. Deposit 20%, then $60 monthly for 12 months. Calculate deposit, repayment total, total paid and additional cost. Answers: $150; $720; $870; $120. Award one mark per correct quantity with identifiable working; follow through a single earlier arithmetic error when the later process is correct.

Q2 (3 marks): $240 item; four $60 payments including today; $2 account charge at each date; one additional $8 fee for exactly one late payment. Find on-time total and specified late total; explain whether interest-free necessarily means no charges. Answers: $248 (1); $256 (1); no — the stated fees add cost (1).

Q3 (5 marks): Borrow $500 for two months. A: $40 establishment and $10 per month. B: establishment 8% of original principal and $15 per month. All fees included in two equal repayments; no other charges or interest. Find each total and borrowing cost; recommend the lower-cost offer using the difference. Answers: A total $560, cost $60; B establishment $40, total $570, cost $70. A costs $10 less. Award A total (1), A cost (1), B total (1), B cost (1), numerical justification (1). Do not require a claim of genuine affordability.

Task 9F Q4 integrated closing case

Q4 (5 marks). A fictional buyer earns $120 per week after deductions and has $90 per week in essential expenses. They deposited $400 one year ago at 5% p.a. simple interest; that account matures today. A needed item costs $600 cash. The Task 9B loan provides $600 and requires $684 total repayment. Calculate the weekly saving, interest earned and savings available today. How many whole weeks of saving are needed to pay cash? Explain the financial trade-off between waiting and using the loan. Assume the cash price stays $600 and no further interest is earned during the waiting period.

Answers: surplus = 120 − 90 = $30 per week (1); I = 400 × 0.05 × 1 = $20 (1); matured savings = $420 (1); gap = $180, so 180 ÷ 30 = 6 weeks (1); waiting avoids $84 of borrowing cost, but delays the purchase six weeks (1). Choosing the loan requires additional evidence of repayment capacity; the information does not guarantee affordability. Allow follow-through for an earlier arithmetic slip where the later method is correct.

Suggested assessment allocation: Q1 3 minutes, Q2 3 minutes, Q3 4 minutes, Q4 5 minutes. Total 15 minutes, within 40–55 minutes. Retain the final 5-minute synthesis.

Interpretation of evidence: Review success criteria by item, not only score. Q1 tests deposit/payment coordination; Q2 tests charge interpretation; Q3 tests loan comparison. Provide feedback on structural errors before more complex tasks. These are teacher-designed assessment criteria, not school reporting grades. Q4 additionally checks the link among earning, accumulated savings and a spending choice.

Completed teacher spreadsheet example

File: Spending_Money_Teacher_Spreadsheet.xlsx, L9 Loans. B4 = principal 800; B5 = months 3. Offer L total = B4+B4*10%+B4*2%*B5; offer M total = B4+50+30*B5. Borrowing cost = total minus B4. Outputs at 3 months: L $928 / $128 cost; M $940 / $140 cost. A duration table shows 2 months: $912 versus $910; 3 months: $928 versus $940; 4 months: $944 versus $970. Two earlier payments round to cents and the final payment adjusts to preserve the total.

Adjustments and anticipated misconceptions

Barrier / misconception	Specific support and retained demand
Uses monthly repayment as percentage base	Colour-code original principal and circle the fee basis; keep the same comparison.
Double-counts included fees	Write total = principal + all charges, then split that total into repayments; do not add charges again.
Confuses loan cost with total	Use original / returned / extra labels and require subtraction.
Language / executive-function needs	Chunk offer cards; provide calculation rows and read directions aloud without supplying answers. Retain individual reasoning.
High prior attainment	Use crossover equation and explain why the whole-month decision differs from the fractional equality.

Vocabulary

Term	Student-friendly definition
Principal	Original amount borrowed in these offers.
Establishment fee	One-off charge for setting up the loan.
Fee basis	The amount on which a percentage fee is calculated.
Loan term	Time over which the loan is repaid.
Total repaid	All payments returned to the lender, including stated charges.
Borrowing cost	Total repaid minus the amount borrowed.
Repayment ceiling	A stated maximum payment; it does not by itself establish affordability.

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.

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.