Scaling Rational Quantities

Overview

This is lesson 4 of 10 in Rational Number Knowledge Practices. Students use multiplicative relationships to scale recipes, distances and quantities while preserving ratios, connecting fractions, decimals, measurement and everyday contexts.

Learning intentions

• WALT recognise multiplicative relationships in proportional situations.
• WALT identify and use a scale factor to enlarge or reduce quantities.
• WALT scale recipes, distances and measures using fractions and decimals.
• WALT explain why additive thinking may not preserve a ratio.

Success criteria

• I can identify the scale factor.
• I can calculate a scaled quantity using multiplication or division.
• I can show my thinking with a ratio table, equation or diagram.
• I can explain what remains constant and check whether my answer makes sense.

Curriculum links

• Number: use multiplication, division, fractions and decimals to solve problems involving proportional relationships.
• Measurement: apply metric units and interpret quantities such as capacity, length and distance.
• Algebraic thinking: identify and describe multiplicative relationships and use symbols or tables to represent them.
• Mathematical practices: reason, communicate, represent, make connections and use mathematical ideas to solve authentic problems.

Lesson structure (60 minutes)

1. 0–7 minutes – Hook and retrieval

Open with the hook and retrieval slides. Display a recipe for four people: 2 kūmara, 1 cup spinach, ½ cup dressing and 250 mL water. Ask: “How could we make enough for eight people?” Students think, pair and share, then briefly revisit multiplication, division, equivalent fractions and decimals.

Prompt: “What is changing? What must stay the same?” Record both sensible and unsensible strategies without correcting immediately.

2. 7–17 minutes – Explicit teaching and modelling

Use the modelling slides to model scaling the recipe from four to eight people. Explain that the scale factor is 2, so every quantity is multiplied by 2:

• 2 kūmara × 2 = 4 kūmara
• 1 cup × 2 = 2 cups
• ½ cup × 2 = 1 cup
• 250 mL × 2 = 500 mL

Contrast this with “adding 2 to every amount”. Discuss why adding 2 does not preserve the relationship. Emphasise the unit rate: each person receives the same amount.

3. 17–28 minutes – Guided ratio tables

Display the guided practice slides and complete a ratio table together. Use a context such as enlarging a harakeke garden plot or preparing class kai. For example, if 3 people need 600 mL of water, calculate the amount for 6 and 9 people.

Ask regularly: “What is being kept constant?” “What is the scale factor?” “Could you use division to find the amount for one person?” Students explain their strategy to a partner before answers are shared.

4. 28–43 minutes – Paired application

Distribute the scaling and proportional thinking worksheet to pairs. Students choose either recipe cards or map distances of local places. They complete ratio tables and record an equation and a diagram for each problem.

Include whole-number scale factors first, such as ×2, ×3 and ×4. Encourage students to use measuring cups, rulers or visual bar models where helpful. Circulate and ask: “Does your answer make sense?” “How do you know the ratio is still equivalent?”

5. 43–51 minutes – Compare and discuss strategies

Use the strategy comparison slides to show two possible solutions to one problem: additive thinking and multiplicative thinking. Pairs decide which preserves the ratio and justify their choice using precise language such as proportion, multiplicative, scale factor, per and unit rate.

Invite several students to explain different representations. Highlight that equivalent fractions and decimals can describe the same scaled quantity.

6. 51–57 minutes – Challenge and consolidation

Students who finish early solve a fractional scale-factor problem: make three-quarters of a recipe. For example, calculate three-quarters of 2 cups, ½ cup and 250 mL. Other students complete one final worksheet question with teacher support.

Use the challenge and reflection slides for a quick class reflection: “When would we multiply? When might we divide?”

7. 57–60 minutes – Exit ticket

Finish with the exit ticket question: “250 mL is scaled by 1.5. Calculate the new quantity and explain your thinking.” Students must show an equation or diagram and state what the scale factor means. Collect responses to plan the next lesson.

Resources

• the scaling and proportional thinking slide deck
• the scaling and proportional thinking worksheet
• Recipe cards, including a kūmara salad example
• Measuring cups, jugs and spoons
• Maps of familiar local places
• Rulers or string for measuring map distances
• Ratio tables displayed on the board
• Calculators for checking, not replacing, reasoning
• Vocabulary display: proportion, ratio, scale factor, multiplicative, quantity, per, unit rate

Assessment

• Observe pair discussion and ratio-table completion, noting whether students use multiplication rather than repeated addition.
• Check equations, diagrams and explanations for preserved ratios and accurate units.
• Use the exit ticket to identify students needing further support with decimal scale factors, particularly 1.5.

Differentiation

• Begin with whole-number scale factors and concrete measuring tools; provide partially completed ratio tables and sentence stems: “The scale factor is…”, “I multiplied by…”, “This keeps the ratio because…”.
• Support learners with a worked example, visual bar models, partner talk and opportunities to choose recipe, garden or map contexts. Do not assume cultural knowledge; invite students to use familiar foods, places or community contexts.
• For EAL and students needing additional support, pre-teach vocabulary with pictures, read questions aloud and allow drawing, oral explanation or home-language discussion before recording.
• Extend confident learners with fractional or decimal scale factors, such as ¾ or 1.5, and ask them to create a scaling problem for another pair, including a convincing explanation of why additive thinking fails.