Subject
Mathematics — Pāngarau (Number and Algebra)
Pāngarau / Mathematics · Ratios and Proportional Reasoning · Years 9–10
Recipe Scaling Mathematics
He kai kei aku ringa · There is food at the end of my hands — ratios, proportions, and cost calculations through hāngī and rewena bread preparation.
Free class-ready resource, premium progression path
This handout is ready to print and use. For cost-optimisation extensions, dietary ratio problems, or a full cooking mathematics unit, Te Wānanga can generate adapted tasks.
- Generate budget-constraint optimisation problems.
- Add exponential growth rewena fermentation tasks.
- Save your proportional reasoning unit in My Kete.
Ngā Whāinga Akoranga · Learning Intentions
- We are learning to use ratios and scale factors to increase recipe quantities proportionally.
- We are learning to simplify ratios to their lowest terms.
- We are learning to calculate costs using rates and proportional reasoning.
- We are learning to apply exponential growth thinking to fermentation ratios.
Paearu Angitu · Success Criteria
- I can calculate a scale factor and apply it to each ingredient in a recipe.
- I can simplify a ratio such as 2.5 : 1.8 to its simplest form.
- I can calculate the total cost of ingredients for a given number of servings.
- I can describe how a rewena starter doubles and calculate its volume after several days.
Hononga Marautanga · Curriculum Alignment
NZC Mathematics — Number and Algebra
- Ratios and rates: simplifying, applying, comparing
- Proportional reasoning and multiplicative thinking
- Percentage and cost calculations
- Exponential growth patterns (extension)
Te Ao Māori integration
- Hāngī as a context for large-scale proportional planning
- Rewena bread (paraoa) — fermentation as mathematics
- Manaakitanga — feeding the community with precision and generosity
- Kai Māori as a site of practical mathematical knowledge
Horopaki · Context — He kai kei aku ringa
The whakataukī "He kai kei aku ringa" (There is food at the end of my hands) speaks to self-sufficiency, manaakitanga, and the power of skilled labour. A hāngī for a hui requires precise calculation: too little food dishonours guests; too much wastes precious resources. Every head cook at a traditional hāngī is applying multiplicative reasoning in real time. Rewena bread uses a living fermentation starter that grows predictably — a natural exponential function baked into every loaf.
Te Rēhita Tūāpohihi · Base Recipe (Serves 10)
This is the standard hāngī recipe for 10 people. You will scale it to feed larger groups.
| Ingredient / Kai | Amount (10 people) | Unit | Notes |
|---|---|---|---|
| Pork (poaka) | 2.5 | kg | Shoulder or belly cut |
| Chicken (heihei) | 1.8 | kg | Legs and thighs |
| Kūmara | 1.2 | kg | Red or orange variety |
| Potato (rīwai) | 1.5 | kg | Agria or similar |
| Pumpkin (paukena) | 800 | g | Buttercup preferred |
| Cabbage (kāpeti) | 0.5 | head | Cut into wedges |
Mahi 1 · Activity 1: Scaling Table
Complete the table by calculating the amount of each ingredient needed for 20, 50, and 200 people. Show your scale factor for each column.
Scale factor = desired servings ÷ base servings (10)
| Ingredient | 10 people (base) |
Scale factor for 20: | Scale factor for 50: | Scale factor for 200: |
|---|---|---|---|---|
| Pork | 2.5 kg | |||
| Chicken | 1.8 kg | |||
| Kūmara | 1.2 kg | |||
| Potato | 1.5 kg | |||
| Pumpkin | 800 g | |||
| Cabbage | 0.5 head |
Check: convert the pumpkin amount for 200 people from grams to kilograms. Write your answer here:
Mahi 2 · Activity 2: Ratio Simplification
The hāngī recipe contains several ingredients in ratio relationships. Simplify each ratio to its lowest terms. Round decimals to one decimal place before simplifying, or use fractions.
| Ratio description | Original ratio | Simplified ratio | Show your working |
|---|---|---|---|
| Pork : Chicken | 2.5 : 1.8 | ||
| Kūmara : Potato | 1.2 : 1.5 | ||
| Pork : Kūmara : Pumpkin | 2.5 : 1.2 : 0.8 | ||
| Meat total : Vegetable total (pork+chicken : kūmara+potato+pumpkin) |
Explain in words what the pork : chicken ratio means for a cook planning a hāngī.
Mahi 3 · Activity 3: Cost Calculation
Scenario: Hāngī for 50 people
Use your scaled amounts for 50 people from Activity 1. Calculate the total cost of each ingredient using the prices below.
- Pork: $12.00 per kg
- Chicken: $9.50 per kg
- Kūmara: $4.00 per kg
- Potato: $2.50 per kg
- Pumpkin: $3.00 per kg
- Cabbage: $2.80 per head
| Ingredient | Amount for 50 people | Price per unit | Total cost |
|---|---|---|---|
| Pork | $12.00/kg | ||
| Chicken | $9.50/kg | ||
| Kūmara | $4.00/kg | ||
| Potato | $2.50/kg | ||
| Pumpkin | $3.00/kg | ||
| Cabbage | $2.80/head | ||
| TOTAL |
a) What is the cost per person for 50 people? Round to the nearest cent.
b) If pork cost rises by 15%, what is the new total cost for 50 people? How much more per person does this add?
Mahi 4 · Activity 4: Rewena Bread — Fermentation Ratios
He aha te rewena?
Rewena paraoa (Māori potato bread) uses a naturally fermented starter called a "buggy". The starter is kept alive by feeding it flour and water. The mathematical behaviour of the starter is predictable and follows a clear doubling pattern.
Starter ratio: 1 cup flour : 1 cup water (approximately 250 mL each)
Growth rule: Under warm room conditions (~20°C), the starter approximately doubles in volume every 24 hours for the first three days after feeding.
a) Complete the table showing starter volume over three days, starting with 1 cup (250 mL) of starter.
| Day / Rā | Volume at start of day | Doubling calculation | Volume at end of day |
|---|---|---|---|
| Day 0 (start) | 250 mL | — | 250 mL |
| Day 1 | |||
| Day 2 | |||
| Day 3 |
b) Write a formula for the volume of starter after n days, starting with 250 mL.
V(n) = 250 ×
c) Each loaf of rewena requires ½ cup (125 mL) of starter. After 3 days, how many loaves could you make from your starter? Show your working.
d) The flour-to-water ratio in the starter is 1 : 1. If you want to make the starter richer by using a 2 : 1 flour-to-water ratio, and you need 500 mL total volume, how much flour and how much water do you need?
Whakaaro Hohonu · Extension: Feeding 200 People
The big challenge
A hui is expected to bring 200 people to the marae. You are the head cook.
- Scale the base recipe to 200 people (use your table from Activity 1).
- Calculate total ingredient costs using the prices in Activity 3.
- Your budget is $1,500. Are you over or under budget? By how much?
- If you are over budget, which ingredient(s) would you reduce and by what percentage? Explain your reasoning in terms of manaakitanga — which ingredients are most important to maintain?
Aronga Mātauranga Māori
In te ao Māori, the preparation and distribution of kai is a mathematical act of manaakitanga. Every hāngī ever cooked required proportional reasoning — calculating how much food would feed the people honourably without waste. The rewena starter is a living fermentation system: a natural exponential function that Māori bakers have managed for generations without formal algebra, but with a precise understanding of growth, ratio, and timing. When ākonga scale a recipe or calculate the doubling of starter, they are participating in a tradition of practical mathematical knowledge that is embedded in cultural practice rather than abstracted from it. This is the essence of mātauranga Māori in mathematics: knowledge that is purposeful, communal, and alive.
Ngā Rauemi Tautoko · Support Materials
Resources already provided:
- Base recipe table: hāngī for 10 (pork, chicken, kūmara, potato, pumpkin, cabbage)
- Scaling table: student fills in quantities for 20, 50, and 200 people
- Ratio simplification activity with four ratio pairs
- Cost calculation scenario: hāngī for 50 people with price list and totals table
- Rewena bread fermentation: doubling table, formula, and loaf calculation
- Extension: 200-person budget challenge with manaakitanga reasoning prompt
Aronga Rerekē · Differentiated Pathways
Entry-level support
Provide a calculator and a completed scale factor worked example (×2 scale shown step by step). Focus on scaling to 20 people only. For ratio simplification, offer counters or a GCF table. Skip the fermentation formula; complete the doubling table with a prompt showing Day 0 → Day 1 worked.
On-level
Complete all four activities. Use the scaling table for all three quantities. Simplify ratios to lowest terms showing clear working. Calculate total costs and explain the pork price-rise impact. Write the fermentation formula and calculate loaves from 3 days of growth.
Extension / Whakaaro Hohonu
Complete the 200-person budget challenge. Research bulk buying: if pork is 12% cheaper per kg when ordering over 10 kg, how does this change the budget? Model the rewena fermentation as V(t) = 250 × 2^(t/d) and find d if the starter only doubles every 36 hours at 15°C. Compare exponential growth rates at 15°C, 20°C, and 25°C.
📋 Teacher Planning Snapshot
Ngā Whāinga Ako — Learning Intentions
Students will engage with this resource to build pāngarau (mathematical) understanding — developing number sense, pattern recognition, and mathematical reasoning through hands-on, culturally grounded activities that connect to tamariki's world.
Ngā Paearu Angitū — Success Criteria
- ✅ Students can explain their mathematical thinking using words, objects, drawings, or symbols.
- ✅ Students can apply the number or pattern concept in this resource to a real or everyday context.
Differentiation & Inclusion
Scaffold support: Use concrete materials (blocks, counters, fingers) for entry-level engagement before progressing to abstract representations. Offer extension challenges asking students to generalise a pattern, write their own word problem, or explain their strategy to a partner.
ELL / ESOL: Mathematical language is a discipline-specific barrier — pre-teach key terms (e.g., equals, more than, fewer, pattern, factor) using visual representations. Allow students to demonstrate mathematical understanding non-verbally or through drawing. Pair with a bilingual buddy where possible.
Inclusion: Embed choice in how students engage — oral, written, or diagrammatic responses are all valid. Neurodiverse learners benefit from short, chunked task sequences with immediate feedback loops. Avoid timed drills in favour of exploratory tasks that reward curiosity. Make the maths classroom a safe place to be wrong and try again.
Mātauranga Māori lens: Pāngarau is a living tradition in Te Ao Māori — from the geometric precision of tukutuku and kōwhaiwhai patterns to the navigational mathematics of waka hourua, and the seasonal calculations embedded in maramataka. Framing early number sense within these contexts shows tamariki that mathematics is a human, culturally rich endeavour — not a foreign import. Encourage students to see counting, measuring, and patterning as acts of knowing their world.
Prior knowledge: Designed for early learners. No prior formal mathematics knowledge required. Teachers should assess current number knowledge before selecting appropriate entry points.
Curriculum alignment
- Number and Algebra — Number Strategies: Use a range of counting, grouping, and equal-sharing strategies with whole numbers and fractions.
- Number and Algebra — Patterns and Relationships: Generalise that the next counting number gives the result of adding one object to a set and that counting the number of objects in a set tells how many.