Curriculum-Design Prototype · Cross-System Mathematics
One Learner, Three Systems
Designing an accessible proportional-reasoning pathway for the United States, England, and New Zealand
How should a mathematics pathway change — and not change — when the learner stays the same but the education system changes? One fictional learner. One proportional-reasoning pathway. Three systems, verified at the source.
All screens are illustrative design prototypes — not product screenshots
Australia included as a verified comparison
South Korea marks the localization boundary
LearnerAlex, 13, autistic with ADHD (fictional composite)
StrandProportional reasoning, lower secondary
SystemsUS, England, New Zealand (+ Australia verified; South Korea boundary)
Deliverables12-skill pathway, 7 item specs, evidence routing, adult views
StatusIllustrative curriculum-design prototype
Short on time? Review the 12-skill pathway, Items 3, 5, and 6, the evidence-routing example, and the three-system adaptation.
1 · The design problem
Proportional reasoning is often presented as separate topics: ratios, unit rates, fractions, percentages, scale, speed, best value, graphs, and linear relationships. Mathematically, these depend on a connected understanding of multiplicative relationships.
A learner may complete isolated calculations without understanding the relationship. Another learner may understand the relationship but struggle to demonstrate it because of mathematical language, working-memory demands, written organization, or an inaccessible response method. The design problem therefore has two parts:
- Distinguish a mathematical misconception from a language, fluency, memory, attention, sensory, motor, or interface barrier.
- Preserve a coherent mathematical progression while adapting curriculum placement, terminology, contexts, and assessment expectations for each system.
This is a curriculum-design prototype. It does not claim that the systems are interchangeable, that one sequence should be used unchanged in every country, or that a fictional learner profile produces measured outcomes.
United StatesCalifornia's Common Core-aligned Grade 6–7 ratio and proportion standards as a concrete reference — not a universal map for every state.
EnglandThe National Curriculum for England at Key Stage 3. "England," not "the UK," because education is devolved and my teaching experience was in England.
New ZealandThe official Mathematics and Statistics Phase 3 and Phase 4 content now published through Tāhūrangi, plus my own NZ classroom experience.
AustraliaA focused comparison using Australian Curriculum Version 9.0 codes and wording from ACARA's official curriculum workbook. State or territory implementation would still need to be checked.
South KoreaNot independently curriculum-mapped. I taught in South Korea for a year (English, not mathematics), and even that classroom familiarity is not expertise in its curriculum, mathematical language, classroom norms, or assessment system.
2 · Learner profile: Alex
Alex is a fictional composite learner aged 13. Alex is autistic and has ADHD. The profile is not intended to represent all autistic learners, all learners with ADHD, or all neurodivergent learners.
Strengths
- Recognizes numerical and visual patterns quickly
- Understands multiplication as scaling
- Notices structural similarities and generalizes
- Explains ideas clearly in conversation
- Engages deeply when a task has visible purpose
- Develops valid methods that differ from the taught method
- Connects visual patterns with early algebraic ideas
Current barriers
- Inconsistent fact retrieval under pressure
- Part-to-part vs part-to-whole confusion
- "How many more" vs "how many times as many" in unfamiliar language
- Losing intermediate values or units in multistep work
- Effortful handwritten organization
- Reduced accuracy when reading, input, and calculation demands stack
- Difficulty resuming interrupted work
- Disengagement without a visible stopping point
The response is not to assign easier mathematics by default. The goal is to identify which demands are intrinsic to proportional reasoning and which can be reduced, externalized, or presented differently.
Evidence-to-design analysis (hypotheses to test, not diagnoses)
| Evidence | Working interpretation | Design response |
|---|---|---|
| Explains multiplicative patterns orally but produces incomplete written solutions | Conceptual reasoning may exceed written-output fluency | Allow structured or oral explanation when extended writing is not the construct |
| Answers familiar ratio problems correctly but uses additive reasoning in unfamiliar word problems | May recognize a problem type without identifying the relationship | Teach additive-vs-multiplicative discrimination before formal proportion solving |
| Confuses "increase by 20%" with "increase to 20%" | Language may be interfering with an accessible calculation | Teach the phrases in contrast before multistep work |
| Drops units after correctly finding a unit rate | Working-memory or organizational demand masking understanding | Keep quantity labels and units persistent |
| Solves correctly using an unexpected method | The response is mathematically valid even off the anticipated route | Define accepted equivalences and method-independent success criteria |
| Loses progress after interruption | Resume behavior creates avoidable executive-function demand | Preserve entered work, representations, and exact place |
End-of-pathway objective
By the end of the pathway, Alex will recognize, represent, and explain proportional relationships using ratios, unit rates, percentages, tables, graphs, diagrams, and equations — and distinguish them from additive or otherwise non-proportional relationships. Evidence of success includes identifying quantities and units, distinguishing difference from scale, interpreting part-to-part and part-to-whole ratios, generating equivalent ratios, finding and interpreting unit rates, connecting representations, explaining why a relationship is or is not proportional, diagnosing the first invalid step in a solution, and applying the idea in an unfamiliar context.
3 · Curriculum alignment and version control
A cross-system map is only useful when each alignment is tied to a named source, version, and date. This prototype separates the shared mathematical progression from official curriculum wording, likely local placement, and product-facing skill structure.
✓ US — codes verified
✓ England — statutory strand verified
✓ NZ — policy + wording verified
✓ Australia — v9.0 codes verified
◌ South Korea — no alignment claimed
| System | Placement of the ratio & proportion pathway |
|---|---|
| United States | Grade 6–7: ratio and rate reasoning (6.RP.1–3), proportional relationships (7.RP.1–3) |
| England | Key Stage 3: ratio, proportion, and rates of change (notation, part:part / part:whole, direct proportion, compound units, gradients) |
| New Zealand | Phase 3 core (Years 7–8), Phase 4 extension into rates, percentage change, and gradient (Years 9–10) |
| Australia (v9.0) | Year 7 ratios, Year 8 rate comparison and ratio-rate modelling |
United States and England — full verified wording and sources
United States — codes verifiedGrade 6: 6.RP.1 (ratio concepts and language), 6.RP.2 (unit rates), 6.RP.3 (ratio and rate problems via tables, diagrams, double number lines, equations, unit pricing, constant speed, percentages, measurement conversion). Grade 7: 7.RP.1 (unit rates with fractional quantities), 7.RP.2 (recognize and represent proportional relationships), 7.RP.3 (multistep ratio and percentage problems). Source: California CCSS Mathematics (PDF)
England — statutory strand verifiedKey Stage 3: ratio notation and simplest form; part-to-part and part-to-whole division; one quantity as a fraction of another; ratio ↔ fractions ↔ linear functions; scale factors, diagrams, and maps; percentage change; direct proportion graphically and algebraically; compound units (speed, unit pricing, density); gradients and intercepts of linear graphs. Source: DfE KS3 programme of study (PDF)
New Zealand — policy status and wording verified
The official Phase 3 (Years 7–8) page states this content became statement of official policy for all English-medium state and state-integrated schools from 1 January 2026.
Phase 3 and Phase 4 wording, quoted exactly
Phase 3, quoted exactly: "A fraction can describe a proportional relationship between two amounts." · "Ratios can be used to describe proportional relationships and unequal division of a whole." Related Phase 3 content covers percentages that proportionally increase or decrease a quantity, part-to-part and part-to-whole division, financial applications with NZ currency conventions, and map scale as a ratio.
Phase 4 (Years 9–10), quoted exactly: "A rate proportionally compares two quantities that have different units of measure." · "Finding equivalent ratios and rates by scaling up or down." Phase 4 extends into rates, percentage change, financial mathematics, and gradient as the steepness of a linear graph.
Design implication: anchor the core ratio pathway in Phase 3 and use Phase 4 as the extension into formal rates, percentage change, gradient, and linear relationships. Sources: Phase 3 · Phase 4 (checked 11 July 2026).
Australia — Version 9.0 codes and wording verified against ACARA's curriculum workbook
Version 9.0 places explicit ratio work in Year 7, rate comparison and ratio-rate modelling in Year 8, and links the pathway to modelling, financial contexts, graphs, and linear relations. The ten verified codes and their official wording:
Ten Version 9.0 codes with official content descriptions
| Code | Content description (official wording) |
|---|---|
| AC9M7N08 | recognise, represent and solve problems involving ratios |
| AC9M7M06 | use mathematical modelling to solve practical problems involving ratios; formulate problems, interpret and communicate solutions in terms of the situation, justifying choices made about the representation |
| AC9M7N09 | use mathematical modelling to solve practical problems, involving rational numbers and percentages, including financial contexts; formulate problems, choosing representations and efficient calculation strategies, using digital tools as appropriate; interpret and communicate solutions in terms of the situation, justifying choices made about the representation |
| AC9M7A04 | describe relationships between variables represented in graphs of functions from authentic data |
| AC9M7A05 | generate tables of values from visually growing patterns or the rule of a function; describe and plot these relationships on the Cartesian plane |
| AC9M8M05 | recognise and use rates to solve problems involving the comparison of 2 related quantities of different units of measure |
| AC9M8M07 | use mathematical modelling to solve practical problems involving ratios and rates, including financial contexts; formulate problems; interpret and communicate solutions in terms of the situation, reviewing the appropriateness of the model |
| AC9M8N05 | use mathematical modelling to solve practical problems involving rational numbers and percentages, including financial contexts; formulate problems, choosing efficient calculation strategies and using digital tools where appropriate; interpret and communicate solutions in terms of the situation, reviewing the appropriateness of the model |
| AC9M8A02 | graph linear relations on the Cartesian plane using digital tools where appropriate; solve linear equations and one-variable inequalities using graphical and algebraic techniques; verify solutions by substitution |
| AC9M8A03 | use mathematical modelling to solve applied problems involving linear relations, including financial contexts; formulate problems with linear functions, choosing a representation; interpret and communicate solutions in terms of the situation, reviewing the appropriateness of the model |
Wording verified against the official ACARA curriculum workbook (checked 11 July 2026).
Shared conclusion. The shared mathematical spine is real, but the systems do not necessarily introduce each concept in the same year, use the same terminology, weight every representation equally, expect identical written methods, connect ratio to algebra at the same point, use the same consumer contexts, or assess the same content in the same way.
4 · Prerequisite dependency map
Five strands feed the pathway. The diagnostic does not require flawless mastery of every prerequisite — it determines which supports should be embedded and which prerequisites need direct instruction before the learner continues.
Number
Whole-number × and ÷
↓
Multiplicative comparison
↓
Scaling up or down
↓
Equivalent ratios
↓
Unit rates
↓
Proportional relationships
↓
Tables, graphs, equations
↓
Direct proportion & linear transfer
Fractions
Fraction equivalence
↓
Part-to-whole interpretation
↓
Fractions ↔ decimals ↔ %
↓
Percentage of a quantity
↓
% increase and decrease
↓
Consumer & financial applications
Language
"More than" vs "times as many"
↓
"For every" and "per"
↓
Ratio order & rate direction
↓
Interpreting equations, graphs, answers
Representation
Read a two-column table
↓
Plot & interpret ordered pairs
↓
Axes and scale
↓
Diagram ↔ table ↔ graph ↔ equation ↔ words
Organization
Identify the two quantities
↓
Retain units
↓
Record intermediate results
↓
Calculated value vs what was asked
5 · The 12-skill pathway
Phase 1 · Establish the relationship
1Additive vs multiplicative comparison2Quantities, units, direction of comparison3Part-to-part & part-to-whole ratios
Phase 2 · Build equivalence and unit-rate meaning
4Generate equivalent ratios5Spot non-equivalent ratios6Find & interpret a unit rate
Phase 3 · Connect representations
7Ratios ↔ fractions ↔ decimals ↔ %8Ratio tables9Graph & interpret points10Express with an equation
Phase 4 · Discriminate, apply, transfer
11Proportional vs additive vs fixed-start linear12Diagnose errors, compare methods, localized transfer
Item template. Every item below follows one 15-part specification: pathway position, mathematical objective, curriculum connection, prerequisites, construct boundary, learner-facing prompt, correct response, accepted alternatives, response interaction, likely incorrect responses (with reasoning, evidence, next action, and competing explanations), student-facing feedback, accessibility review, localization review, data and revision criteria, and connection to the next skill. The screens below show the learner-facing view first; the design spec sits underneath each one.
6 · Item specifications 1–7
Item 1 · Additive and multiplicative comparison try it
Mina has 4 square tiles. Arun has 12 triangular tiles.
Two students compare the quantities:
Statement 1: “Arun has 8 more tiles than Mina.”
Statement 2: “Arun has 3 times as many tiles as Mina.”
Statement 2: “Arun has 3 times as many tiles as Mina.”
✓ Both statements are true. "8 more" tells the difference (12 − 4 = 8). "3 times as many" tells the scale factor (12 = 3 × 4). These answer different questions about the same two quantities.
Statement 1 is true — but check statement 2 as well. How many 4s make 12?
Statement 2 is true — and what is 12 − 4?
Compare the numbers two ways: what is 12 − 4, and how many 4s make 12?
Illustrative prototype — not a product screenshot
Design spec — objective, diagnosis, feedback, accessibility, localization
Objective & position
Distinguish additive difference from multiplicative comparison. Skill 1; prepares ratio interpretation. Curriculum: US 6.RP.1 · England KS3 multiplicative relationships · NZ Phase 3 proportional reasoning · Australia AC9M7N08.
Construct boundary
Measures interpretation of two valid comparison types — not reading stamina, color recognition, or calculation speed.
Likely incorrect responses
| Response | Likely thinking |
|---|---|
| A only | Recognizes additive difference but may not recognize scale or understand "times as many." |
| B only | Recognizes scale but may assume only one comparison can be valid. |
| D | May not understand the comparison language, may attend to shapes rather than quantities, or may be disengaged. |
Accessibility & localization
Shapes labeled, not distinguished by color alone; quantities remain visible while options are read; keyboard-selectable; text-to-speech available; no time limit or drag-only action. Localization: verify the local instructional phrase for multiplicative comparison — translation must preserve the difference/scale distinction.
Next-skill rule
Correct with explanation → ratio direction. Correct without explanation → one confirmation item. A only → short multiplicative-comparison lesson. D or inconsistent → language-supported prerequisite check.
Item 2 · Part-to-part and part-to-whole
A box contains 6 square tiles and 4 triangular tiles.
1. The ratio of square tiles to triangular tiles:
6 : 4
2. The fraction of all the tiles that are square:
6 / 10
“Squares to triangles” compares 6 with 4: 6:4 = 3:2. “The fraction that are square” compares the 6 squares with all 10 tiles: 6/10 = 3/5.
Illustrative prototype — responses shown completed
Design spec — objective, diagnosis, feedback, accessibility, localization
Objective & position
Distinguish a part-to-part ratio from a part-to-whole fraction. Skill 3. Curriculum: US 6.RP.1 · England KS3 ratio division · NZ Phase 3 part:part and part:whole · Australia AC9M7N08.
Accepted alternatives
6:4 or 3:2 if simplest form is requested; 6/10 or 3/5. Unsimplified forms are correct when the objective is identifying the comparison — the item must explicitly say "in simplest form" before rejecting them.
Likely incorrect responses
| Response | Likely thinking |
|---|---|
| 6:10 | Part-to-whole used instead of part-to-part. |
| 4:6 | Order reversed. |
| 6/4 | Part compared with the other part rather than the total. |
| 3:5 for squares:triangles | Simplified numbers without preserving the original relationship. |
Accessibility & localization
Shapes named in text; screen-reader text gives each group and the total; direct entry and structured selection available; the two parts can display separately to reduce density; wording remains visible during response. Localization: check "counters" vs "tiles"; "simplest form" vs "lowest terms"; when colon notation is introduced.
Next-skill rule
Both correct → equivalent ratios. Part-to-whole confusion → targeted comparison sort. Order reversal → ratio-language practice. Correct but unsimplified → continue unless simplification is the objective.
Item 3 · Unit rate and best value Recommended read
A store sells the same rice in two packages.
Package A: 3 kg for $12.60 · Package B: 5 kg for $20.50
Package A: 3 kg for $12.60 · Package B: 5 kg for $20.50
Which package has the lower price per kilogram? Show or explain how you know.
12.60 ÷ 3 = 4.20dollars per kilogram
20.50 ÷ 5 = 4.10dollars per kilogram
Package B has the lower unit price: $4.10 per kilogram, compared with $4.20.
Illustrative prototype — responses shown completed
Design spec — objective, diagnosis, feedback, accessibility, localization
Objective & position
Calculate and interpret a unit price, then compare two offers. Skill 6. Curriculum: US 6.RP.2–3 · England KS3 unit pricing · NZ Phase 4 rates and financial mathematics · Australia AC9M8M05, AC9M8M07.
Accepted methods
Division; equivalent-ratio table; scaling to a common mass; a correct mental strategy with explanation. A calculator may be used if decimal division is not the intended construct.
Likely incorrect responses
| Response | Likely thinking |
|---|---|
| Package A because $12.60 is less | Compares total prices rather than equivalent quantities. |
| 3 ÷ 12.60 and 5 ÷ 20.50 | Reverses the requested rate. |
| Package B with no evidence | May be correct by guess or superficial comparison. |
| Correct values without units | Calculation understood; interpretation incomplete. |
Feedback (total-price comparison): The packages contain different amounts, so the total price is not a fair comparison. Find the cost for the same amount, such as one kilogram.
Feedback (reversed division): The question asks for dollars per kilogram. Divide dollars by kilograms.
Accessibility & localization
Structured list and table; units stay attached; calculator availability follows the construct; direct entry, selection, or oral explanation; one optional hint at a time; no drag-only comparison. Localization: US — metric or realistic customary values without adding conversion difficulty; England — pounds sterling, price per kg or 100 g; NZ & Australia — local dollars, metric, GST-inclusive displayed prices and unit-pricing conventions.
Next-skill rule
Correct rates and interpretation → representation connection. Correct comparison without unit rates → one explicit-rate item. Total-price comparison → unit-rate instruction. Reversed division → unit-direction scaffold.
Item 4 · Connecting tables, graphs, and equations
A cyclist rides at a steady 12 kilometers per hour.
Time t (hours): 0, 1, 2, 3 · Distance d (kilometers): 0, 12, 24, 36
Time t (hours): 0, 1, 2, 3 · Distance d (kilometers): 0, 12, 24, 36
Which equation connects distance d and time t — and what does the point (2, 24) mean here?
d = 12t· After 2 hours the cyclist has traveled 24 kilometers.
Illustrative prototype — responses shown completed
Design spec — objective, diagnosis, feedback, accessibility, localization
Objective & position
Recognize the same proportional relationship in a context, table, graph, and equation. Skills 8–10. Curriculum: US 7.RP.2 · England KS3 direct proportion and linear functions · NZ Phase 4 gradient and y = mx + c · Australia AC9M7A04, AC9M7A05, AC9M8A02.
Options as delivered
A. d = 12 + t B. d = 12t (key) C. d = t − 12 D. d = 12t + 12 (shuffled at delivery)
Likely incorrect responses
| Response | Likely thinking |
|---|---|
| d = 12 + t | Additive reasoning. |
| d = 12t + 12 | Treats the rate as a starting value. |
| Correct equation, weak point interpretation | Symbolic pattern recognized without contextual meaning. |
| "24 divided by 2 is 12" only | Calculation understood; quantities and units not interpreted. |
Accessibility & localization
Table, graph, equation, and context available as structured information; axis labels, units, and points exposed to screen readers; graph access not hover-only; keyboard users move among points; oral or structured interpretation accepted. Localization: US may foreground "constant of proportionality" and "slope"; England and NZ "direct proportion," "rate of change," "gradient" — terminology must match the phase without changing the relationship.
Next-skill rule
All representations connected → proportional-vs-linear discrimination. Equation correct but interpretation weak → contextual point practice. Additive equation → return to multiplicative comparison. Starting-value equation → fixed-intercept contrast item.
Item 5 · First-error analysis in a percentage solution try it Recommended read
A jacket originally costs $80. The price is reduced by 25%. A student writes the four steps below.
Click the first incorrect step.
✓ Step 3 is the first error. The first two steps correctly find the discount: 25% of $80 is $20. Because the price is reduced, subtract: 80 − 20 = 60. (Direct method also accepted: 0.75 × 80 = 60.)
That step is correct so far — the discount really is $20. Keep looking for the first step that goes wrong.
This conclusion is wrong, but it follows from an earlier mistake — find the first step where the working goes wrong.
Illustrative prototype — not a product screenshot
Design spec — objective, diagnosis, feedback, accessibility, localization
Objective & position
Identify the first incorrect step in a multistep percentage solution and correct it. Skill 12. Curriculum: US 7.RP.3 · England KS3 percentage change · NZ Phase 3–4 proportional increases and decreases · Australia AC9M7N09, AC9M8N05.
Likely incorrect responses
| Response | Likely thinking |
|---|---|
| Step 2 | May not understand percentage as multiplication by a decimal, or questions the calculation itself. |
| Step 4 | Recognizes the conclusion is wrong but does not locate the first invalid operation. |
| Step 3 without a valid correction | May notice a mismatch without understanding the decrease. |
Accessibility & localization
Each step separately selectable and announced with its number; accessible math notation; color not the only indication of selection; correction by equation, short explanation, or both; reduced-density mode can show one line at a time while preserving earlier lines. Localization: local currency and price conventions; verify "sale price" vs "reduced price"; whether "decrease by" vs "decrease to" needs explicit contrast.
Next-skill rule
First error found and corrected → transfer task. Step 4 → additional first-error item. Step 2 → percentage-of-quantity prerequisite. Misreads "reduced by" → language contrast lesson.
Item 6 · Proportional or linear? try it Recommended read
Two bike-rental companies charge by the hour.
Company A: 0 h → $0 · 1 h → $6 · 2 h → $12 · 3 h → $18
Company B: 0 h → $4 · 1 h → $10 · 2 h → $16 · 3 h → $22
Company A: 0 h → $0 · 1 h → $6 · 2 h → $12 · 3 h → $18
Company B: 0 h → $4 · 1 h → $10 · 2 h → $16 · 3 h → $22
Which company's total cost is proportional to the number of hours rented?
✓ Company A charges $6 per hour with no starting charge: C = 6h. Company B also climbs $6 per hour but starts with a $4 fixed charge: C = 6h + 4 — linear, but not proportional. A proportional relationship must have output 0 when the input is 0.
Both companies do increase by $6 for each additional hour — but look at the 0-hours row. A proportional relationship must have output 0 when the input is 0.
Look at Company B's 0-hours row: it charges $4 before any time passes. What does Company A charge at 0 hours?
Test each table: does doubling the hours double the cost? Try it for Company A.
Illustrative prototype — not a product screenshot
Design spec — objective, diagnosis, feedback, accessibility, localization
Objective & position
Determine whether a relationship is proportional and explain the role of the zero input and fixed charge. Skill 11. Curriculum: US 7.RP.2 · England KS3 direct proportion, gradients, intercepts · NZ Phase 4 y = mx + c and gradient · Australia AC9M8A02, AC9M8A03.
Likely incorrect responses
| Response | Likely thinking |
|---|---|
| C. Both | Notices the constant $6-per-hour change but not yet that proportionality also requires output 0 at input 0. |
| B only | Attending to larger values, treating the fixed charge as a "stronger" relationship, or picking the more complex pattern. |
| D. Neither | May not know how to test proportionality from a table, may expect corresponding values to be equal, or may not connect the zero row with the equation and graph. Deliberately distinct from C: "both increase by $6" supports C, not D. |
Follow-up prompt
"A graph of Company B is still a straight line. Why does that not make the relationship proportional?" Expected: a straight line shows constant rate of change; a proportional relationship must also pass through the origin — Company B begins at (0, 4), not (0, 0).
Accessibility & localization
Both tables visible together with clear labels; companies identified by text, not color; accessible math markup; no drag-only interaction; a graph can be offered without being mandatory. Localization: "bicycle rental" vs "bike hire"; local currency and plausible pricing; the ordinary term for the starting charge — initial fee, fixed charge, unlock fee, or call-out fee.
Next-skill rule
Correct with origin/fixed-charge explanation → localized transfer task. C → targeted proportional-vs-linear comparison. D → table-to-equation and origin prerequisite. Correct without explanation → graph or equation confirmation item.
Item 7 · Scaling a recipe without changing the relationship try it
A rice dish for 6 people uses 450 g of rice and 750 mL of water. The same recipe needs to serve 9 people.
How much rice and water are needed? Show or explain how you scaled both quantities.
Hint: the servings go from 6 to 9, so multiply both ingredients by 9/6 = 3/2 — the amounts are still yours to find.
Illustrative prototype — responses shown completed
Design spec — objective, diagnosis, feedback, accessibility, localization
Objective & position
Scale two related quantities by the same factor and explain why the ratio remains equivalent. Skill 12 transfer task. Curriculum: US 6.RP.3 · England KS3 ratio and scale · NZ Phase 3–4 scaling equivalent ratios · Australia AC9M7M06, AC9M8M07.
Accepted methods
Unit-rate method (75 g and 125 mL per person, then × 9); scale-factor method (× 3/2); ratio table (6 → 3 → 9 servings); equivalent decomposition (halve the 6-serving amounts to get 3 servings, then add: 6 + 3 = 9). Any mathematically valid method is accepted if both quantities are scaled consistently.
Likely incorrect responses
| Response | Likely thinking |
|---|---|
| 453 g and 753 mL | Adds 3 because the number of people increased by 3 — additive reasoning. |
| 675 g and 1,050 mL | Scales the quantities using different relationships. |
| 675 g and 1,125 liters | Correct numbers, incorrect unit interpretation. |
| 300 g and 500 mL | Reverses the scale factor (× 2/3) and makes the recipe smaller. |
Feedback (additive scaling): Adding 3 to each ingredient does not preserve the recipe. The servings change from 6 to 9, so multiply both ingredients by the same scale factor, 9/6.
Feedback (reversed scaling): The recipe is becoming larger, so the scale factor must be greater than 1. Use 9/6, not 6/9.
Accessibility & localization
Quantities in a list and structured table; units always visible; ratio table, equations, or oral explanation accepted; calculator allowed when the construct is scaling; optional support can reveal the scale factor without the final amounts; no decorative food imagery or drag-only actions. Localization: England/NZ/Australia keep metric with natural recipe language; a US version may keep metric or use carefully chosen customary quantities without making conversion the main difficulty; a Korean version needs review of mathematical language, familiar quantities, and context — not just names and won amounts.
Next-skill rule
Correct with equivalence explanation → pathway complete or extension. Correct amounts without explanation → one equivalence justification item. Additive scaling → return to Skills 1 and 4. Reversed factor → direction-of-scaling scaffold.
7 · Worked example: response data changes the next recommendation
The pathway uses evidence rather than treating every wrong answer as a request for more practice at the same level. Starting item: Item 3, unit rate and best value.
Alex's first response
Chooses Package A "because $12.60 is less than $20.50." Evidence: compares total prices rather than equal quantities. It does not yet show whether Alex can calculate a unit rate when the required direction is explicit.
System decision
Do not assign another visually similar best-value problem. Route to a short unit-rate direction scaffold: "The question asks for dollars per kilogram. Complete the label before calculating: ___ dollars ÷ ___ kilograms = ___ dollars per kilogram."
Second response
Enters 12.60 ÷ 3 = 4.20 and 20.50 ÷ 5 = 4.10, selects Package B — but omits units from the explanation. Updated evidence: Alex can calculate and compare unit rates when the desired units are explicit. The remaining issue is interpretation and communication, not division or comparison.
Next recommendation
One confirmation item asking Alex to state what a unit rate means in context — labels stay visible, the completed division structure is removed. Do not recommend: a return to basic division, a large set of identical best-value questions, or a harder proportional-equation item before interpretation is confirmed.
Confirmation & pathway update
Alex writes: "Package B costs $4.10 for each kilogram, which is less than $4.20 per kilogram." Mark unit-rate calculation and interpretation demonstrated; move to connecting a rate with a table, graph, and equation.
Why this matters: the sequence distinguishes a learner who doesn't understand fair comparison, one who understands but reverses or can't organize the unit rate, and one who calculates correctly but doesn't interpret the units. Those three states should not produce the same feedback or the same next recommendation.
8 · One item across three systems: unit price comparison
The mathematical objective is stable across all versions: calculate and interpret a unit price, fairly compare two offers, and justify the decision. Two packages contain different quantities; total price alone is insufficient; the answer needs the unit and a contextual interpretation; the main misconceptions remain total-price comparison and reversed rate direction. Everything else is a localization decision:
| Decision | United States | England | New Zealand |
|---|---|---|---|
| Curriculum placement | Grade 6 ratio and rate reasoning, Grade 7 connections to proportional reasoning; state-specific verification outside the California reference | KS3 ratio, proportion and rates of change, incl. unit pricing and compound measures; placement varies across Years 7–9 | Phase 3 proportional relationships and financial contexts; Phase 4 extension into rates, financial maths, gradient |
| Learner-facing language | "Which package has the lower unit price?" / "Which is the better buy?" — test the phrase, don't assume equivalence | "Which packet offers the better value?" / "lower price per kilogram" — verify ordinary classroom wording | "Which packet is better value?" — confirm wording with local teachers and learners |
| Units & quantities | Keep kilograms, or use realistic pounds/ounces with carefully chosen values; never mix systems accidentally | Metric mass; per-100 g comparison may be more realistic for some products but changes the arithmetic | Metric mass and locally plausible package sizes |
| Currency & tax | US dollars; posted grocery prices typically exclude sales tax where applicable — decide whether tax is excluded or explicitly part of the problem | Pounds sterling; displayed prices generally include VAT where it applies — don't introduce tax unless it's the objective | NZ dollars; retail prices are ordinarily GST-inclusive — compare displayed totals without adding GST again |
| Likely terminology | Math, unit rate, unit price, dollars per pound/kg, slope (later) | Maths, unit price, rate, direct proportion, gradient (later), simplest form | Maths, unit rate or rate, NZ dollars, GST, gradient (later) |
| Assessment note | Calculator access depends on whether decimal division or unit-rate reasoning is the construct | Expected written methods and calculator use differ by school and assessment context | Match the active curriculum phase; avoid NCEA-specific language for a Years 7–10 pathway (NCEA is New Zealand’s senior-secondary qualification) |
Localization is not a currency-symbol swap. Each version requires decisions about placement, units, realistic quantities, tax assumptions, ordinary language, expected method, calculator access, and where the concept connects to algebra.
9 · The same evidence, three adults
Alex distinguishes additive from multiplicative comparisons in visual examples but often reverts to additive reasoning on unfamiliar text problems. Can calculate a unit rate when quantities are clearly labeled; may reverse the division when "per" is embedded in dense language.
Recommended: the mathematical-language contrast activity and two guided unit-rate examples before proportional-equation tasks.
Begin with one pair of quantities: ask Alex to describe both "how many more" and "how many times as many." For unit rates, have Alex label the requested unit before calculating: "I need dollars per kilogram, so dollars are divided by kilograms."
Focus: direction of comparison and of the rate — not re-teaching calculation.
Alex understands that quantities can be compared in more than one way. The next step is recognizing which comparison a question asks for. Helpful prompts: "Are we finding how many more, or how many times as many?" · "What does per mean here?" · "Which unit should the answer use?"
Framing: no diagnostic jargon; a short, targeted teaching need — not general mathematical inability.
Illustrative prototypes — how one result could be reported to three audiences
10 · Accessibility framework used across items
For every item, five decisions are documented: the barrier (what could prevent access or distort the evidence), the construct (what the item measures), permissible flexibility (what can change without weakening the construct), the non-negotiable demand (what the learner must still do), and the implementation requirement (what the interface and assistive-technology support must provide).
- If dragging is difficult, allow keyboard or structured selection when movement is not the construct.
- If writing is effortful, allow a structured or oral explanation when handwriting is not being assessed.
- If reading load masks ratio knowledge, simplify syntax without changing the quantities or relationship.
- If a graph is the construct, provide accessible graph navigation and structured descriptions rather than replacing it with a table.
- If an interruption causes work loss, preserve the exact state rather than interpreting restart errors as a lack of understanding.
- If continuous progress movement distracts the learner, test alternative presentation timing while keeping progress information available.
11 · South Korea: the localization boundary
I taught in South Korea for a year — English, not mathematics — and even that classroom familiarity is not expertise in its curriculum or mathematical language. Before adapting this pathway for South Korean learners, local collaboration would need to verify:
Localization checklist · what local collaboration verifies
- Current national mathematics curriculum and intended school year
- Korean mathematical terminology and word-problem syntax
- Conventional textbook representations
- Expected written-solution formats
- Calculator expectations
- How classroom content relates to assessment preparation
- Whether translation preserves the mathematical demand
- Whether examples feel locally ordinary, not externally themed
An English-language problem involving Korean names and won amounts is not, by itself, a localized Korean learning experience.
12 · Validation and evaluation plan
Because Alex is fictional, this case study claims no measured learner outcomes. Before publication as a product pathway, validate:
CurriculumCorrect source, version, date, phase, year, grade, key stage, and code · prerequisite assumptions and local sequence · state, territory, or school implementation differences.
Mathematical languageTerms teachers and learners actually use · natural translated syntax · absence of avoidable ambiguity.
Item qualityObjective–evidence alignment · diagnostic value of distractors · acceptance of valid equivalent forms and methods · feedback accuracy · comparable difficulty across local versions.
AccessibilityKeyboard-only and screen-reader completion · accessible equation and graph interaction · whether flexibility preserves the construct · resume behavior · session length and progress presentation.
Pilot evidence to examine: correct-response and distractor rates; reversed-quantity and unit-omission patterns; performance before and after feedback; transfer across representations; use of optional supports; errors caused by input mechanics; learner explanations; educator and family interpretation; local reviewer feedback.
13 · Source and version register
- United States: California Department of Education, California Common Core State Standards: Mathematics — Grade 6 and Grade 7 Ratios and Proportional Relationships.
- England: Department for Education, National Curriculum in England: Mathematics Programmes of Study, Key Stage 3.
- New Zealand: Ministry of Education — Mathematics and Statistics Phase 3 (Years 7–8) and Phase 4 (Years 9–10), via Tāhūrangi.
- Australia: ACARA, Australian Curriculum Version 9.0 — official curriculum downloads and workbook, checked for AC9M7N08, AC9M7M06, AC9M7N09, AC9M7A04, AC9M7A05, AC9M8M05, AC9M8M07, AC9M8N05, AC9M8A02, and AC9M8A03.
- IXL public product-context references (skill organization only — not curriculum authority): US Grade 6 · US Grade 7 · NZ Year 8 · NZ Year 9 · Australia Year 7.
All curriculum wording checked against the sources above on 11 July 2026. The fictional learner profile is a design tool, not research evidence.
Curriculum-design prototype by Christopher Welch. All screens are illustrative prototypes in this site's own design system — not screenshots of any product. Designed with accessibility in mind: keyboard-operable controls, semantic headings, visible focus, text equivalents for figures, and reduced-motion support. Formal assistive-technology testing remains part of publication QA. Single self-contained page, no libraries. ← All samples
© 2026 Christopher Welch · All rights reserved. Everything in this portfolio — the case studies, item specifications, interactives, course materials, text, and code — is Christopher’s original work, published so prospective employers and collaborators can evaluate it. No license is granted by this site or its source repository being public. Hiring conversations are always welcome; reproduction or adaptation without written permission is not.