Best for
Students who are confident with Phase 2 core strategies and ready to develop flexible multiplicative thinking, including strategy selection and justification.
Pāngarau / Mathematics · Years 4–6 · Multiplicative Thinking
Numeracy Phase 2 — Challenge Extension
Multiplicative strategies, area models, doubling/halving, and fact families for ākonga working at or beyond Phase 2 expectations. Every problem draws on Hamilton Zoo fencing projects, habitat maps, and conservation data from real NZ endangered species.
Free challenge resource, premium adaptation path
This handout provides the full multiplicative strategy sequence with scaffolded area model diagrams and conservation contexts. Te Wānanga can rebuild this for different number ranges, local contexts, or bilingual delivery.
- Generate harder multi-step versions using three-digit multipliers.
- Swap in local conservation data — your region's pest-control or planting counts.
- Save adapted copies in My Kete for group rotations.
Kaiako planning snapshot
- Use length: 50–70 minutes or across two sessions.
- Grouping: Small group launch with teacher, then independent or paired challenge problems.
- Prep: Print one handout per student. Grid paper helpful for area models. Conservation data visible on screen optional.
- Teaching move: "Which strategy would you choose and why?" — push for reasoning before calculation.
- Extension: Students who finish early can create their own conservation data problem and solution key.
Resources already provided
- Printed handout (one per student)
- Pencils or pens for write-on spaces
- Fact family scaffold with write-on boxes
- Doubling/halving worked examples and table
- Area model grid scaffold
- "Make a tidy group" strategy worked example
- Six multi-step challenge problems (conservation data)
- Learning intentions and success criteria
- Mātauranga Māori curriculum alignment notes
If the lesson covers multiplicative strategies, area models, or fact families, everything you need is on this page. No additional preparation required.
Ngā Whāinga Akoranga / Learning Intentions
- We are learning to use multiplicative strategies including doubling/halving and area models.
- We are learning to select and justify the most efficient strategy for a given problem.
- We are learning to recognise the inverse relationship between multiplication and division.
Paearu Angitu / Success Criteria
- I can use an area model to solve 2-digit × 1-digit problems.
- I can apply doubling/halving to make multiplication easier.
- I can explain why I chose a particular strategy.
Strategy 1: Fact Families (× and ÷)
Hamilton Zoo context: The keeper team packs supply kits. Each kit holds 6 items. If 7 kits are packed, how many items in total? If 42 items need to be packed into kits of 6, how many kits are needed?
How it works
Every multiplication fact has a fact family — four related equations using the same three numbers. Knowing one means knowing all four.
Example fact family for 6, 7, and 42:
6 × 7 = 42
7 × 6 = 42
42 ÷ 6 = 7
42 ÷ 7 = 6
Complete the fact families below:
| Three numbers | × fact 1 | × fact 2 | ÷ fact 1 | ÷ fact 2 |
|---|---|---|---|---|
| 8, 9, 72 | ||||
| 4, 7, 28 | ||||
| 5, 12, 60 |
Strategy 2: Doubling & Halving
Hamilton Zoo context: The zoo is building new enclosures. Each fence panel is 4 m wide and they need 36 panels for the giraffe paddock. What is the total fencing length?
How it works
Double one factor, halve the other. The product stays the same. Use this to turn a tricky multiplication into an easier one.
Example: 4 × 36
Double 4 → 8; halve 36 → 18
8 × 18 = still tricky, so double/halve again:
Double 8 → 16; halve 18 → 9
16 × 9 = 144 ✓ (or think: 16 × 9 = 16 × 10 − 16 = 160 − 16 = 144)
Total fencing: 144 m
Practice — use doubling/halving to simplify, then calculate:
| Original | After D/H step 1 | After D/H step 2 (if needed) | Answer |
|---|---|---|---|
| 6 × 24 | |||
| 8 × 35 | |||
| 14 × 15 | |||
| 4 × 125 |
Strategy 3: Area Model
Hamilton Zoo context: The habitat planning team overlays a map grid to estimate the area of a new kiwi habitat. The rectangle measures 34 m × 6 m.
How it works
Partition the larger factor by place value. Calculate each partial product, then add them together.
Example: 34 × 6
Partition 34 into 30 + 4.
30 × 6 = 180
4 × 6 = 24
Total: 180 + 24 = 204
Area model diagram for 34 × 6:
30
4
6
6 × 30 = 180
6 × 4 = 24
Total area: 180 + 24 = 204 m²
Draw your own area model and calculate:
| Problem | Partition the larger number | Partial products | Total |
|---|---|---|---|
| 47 × 8 | 40 + 7 | ||
| 63 × 7 | |||
| 85 × 9 |
Draw the area model for 63 × 7 in the space below:
Draw your area model rectangle here, labelled with partial products.
Strategy 4: Make a Tidy Group
Hamilton Zoo context: A keeper shift lasts 9 hours and there are 28 keepers rostered this week. How many keeper-hours is that in total?
How it works
Adjust one factor to the nearest 10, multiply, then compensate.
Example: 28 × 9
Round 28 up to 30 (added 2 extra groups).
30 × 9 = 270
Remove the 2 extra groups: 270 − (2 × 9) = 270 − 18 = 252
Another example: 19 × 7
Round 19 up to 20. 20 × 7 = 140. Remove 1 group of 7: 140 − 7 = 133.
Practice:
| Problem | Tidy group I use | Tidy calculation | Compensation | Answer |
|---|---|---|---|---|
| 29 × 6 | 30 × 6 | 180 | − 1 × 6 | |
| 38 × 4 | ||||
| 49 × 8 |
Challenge Problems — NZ Conservation Data
For each problem, state your strategy and explain why you chose it. Show all working. Use the back of this page if you need more space.
Problem 1 — Kākāpō recovery
As of a recent census, there are 247 kākāpō in the Aotearoa recovery programme. Each bird receives on average 8 health checks per year. How many health checks does the team carry out across the whole population each year?
Strategy chosen: ___________________ Why: ________________________
Show working here.
Answer: _______________
Problem 2 — Kiwi egg incubation
A kiwi incubation centre hatches kiwi eggs over a season. Each of their 6 incubators holds 39 eggs. How many eggs in total are in incubation? Use doubling/halving OR area model and show both — then decide which was more efficient.
Show both strategies here.
Which was more efficient and why?
Answer: _______________
Problem 3 — Tuatara population estimate
A sanctuary island has 9 monitored zones. Each zone holds approximately 47 tuatara. Estimate the total population and then calculate the exact total. How close was your estimate?
My estimate: _______________ Strategy used: ___________________
Show exact calculation here.
Exact total: _______________ Difference from estimate: _______________
Problem 4 — Fencing the new kiwi zone
Hamilton Zoo's new kiwi zone needs fencing on three sides (the fourth side is a wall). Two sides are each 38 m long. The third side is 26 m long. Fencing costs $9 per metre. What is the total cost?
Hint: find the total metres first, then calculate the cost.
Step 1 — total metres:
Working here.
Step 2 — total cost:
Working here.
Answer: $_______________
Problem 5 — Keeper shift handover
At a shift handover, the outgoing keeper reports they completed 7 rounds of all 68 enclosure checks during the week. The incoming keeper needs to complete 4 more full rounds before the end of month. How many total enclosure checks will have been done across both keepers by month end?
Show working here — you may need two calculations.
Answer: _______________
Strategy explained:
Problem 6 — Map grid overlay (multi-step)
A habitat map uses a grid where each square = 5 m × 5 m. The tuatara reserve covers a rectangle of 9 squares wide by 14 squares long. What is the actual area of the reserve in square metres? Show your method and check your answer using a different strategy.
Hint: find the number of grid squares first, then find the actual area.
Show both methods here.
Area of reserve: _______________ m²
Curriculum integration / Te Mātaiaho alignment
These Phase 2 challenge strategies align with the Pāngarau / Mathematics learning area of Te Mātaiaho. Multiplicative strategies, inverse operations, and area models are core progressions for Years 4–6. The NZ conservation contexts integrate with the Science and Social Sciences strands through population mathematics and environmental stewardship.
Hononga Marautanga · Curriculum Alignment
Pāngarau / Mathematics
Years 4–6: Use a range of multiplicative strategies with whole numbers; understand the inverse relationship between multiplication and division; use area models to represent multiplication; select and justify efficient strategies; apply multiplicative reasoning in multi-step problems.
Science / Social Sciences — Conservation context
Level 3–4: Apply mathematical reasoning to environmental contexts; interpret conservation population data; understand how quantitative analysis supports decision-making in conservation settings; connect multiplicative thinking to real-world environmental stewardship.
Aronga Mātauranga Māori
This resource sits within a kaupapa that recognises mātauranga Māori as a living knowledge system with its own frameworks, values, and ways of understanding the world. In te ao Māori, whakapapa describes not only genealogy but the multiplicative branching of relationships — one ancestor becomes many descendants, one kākāpō population grows through compounding care. This multiplicative structure mirrors the area model: break apart, understand each part, then recombine. The concept of kaitiakitanga grounds every conservation problem in this handout — accurate multiplication is not merely an academic exercise but a tool for protecting endangered species. The population of kākāpō, kiwi, and tuatara in these problems reflects real recovery efforts where kaitiaki (guardians) must count carefully to make good decisions. Whanaungatanga — the value of relationships — extends here to peer learning: discussing strategy choice with a partner, explaining your reasoning, and checking each other's thinking are all expressions of whanaungatanga in the learning space. Te Aranga design principle taiao (natural environment) grounds all number work in the living world.
Tuhia ōu whakaaro · Write Your Thoughts
Which strategy was most efficient for you today? Was there a problem where you changed strategies partway through? Why?
Ngā Rauemi Tautoko · Support Materials
This handout is designed to be used alongside the broader unit resources available at Te Kete Ako handouts library. The Core Strategies handout and Strategy Passport are companion resources for this phase. All resources are provided — no additional preparation is required to use this handout in your classroom.