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.
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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
Curriculum alignment for this handout has not yet been verified against the live curriculum statements. A generated placeholder that stood here was removed on 2026-08-29 because it matched no real statement.
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.