--- title: "Checking a model" subtitle: "Number Foundations · Modelling · Lesson 30" --- # Checking a model **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A map app says the drive takes 3 hours. It takes 3 and a half. The maths was right; the model was too simple. Finding out why is the last step of modelling. :::teacher-note **In one sentence:** Students compare simple models with real outcomes (taxi fares, trip times, exchange rates), name the assumption that failed, and refine the model with a starting charge, stops or fees (AC9M8M07, node 2). **Students should be able to say:** I can name an assumption a model makes. I can explain how an assumption makes a model's answer differ from real life. I can refine a model and say when a simple one is good enough. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict :::keep-together **A model says a 5 km taxi ride costs 5 × $2.10 = $10.50. The real fare is $14.70. Why?** ☐ 5 × 2.10 is not 10.50 ☐ The model leaves out a starting charge ☐ Taxi fares are random ☐ The taxi went faster than expected My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** The model leaves out a starting charge (a flag fall) of $4.20: 4.20 + 10.50 = 14.70. The arithmetic was fine. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Assumptions :::keep-together ### The trip Here is the routine. 1. $\text{model: water bill} = \$2.50 \times \text{kL}$ — A simple model: 20 kL costs \$50. 2. $\text{real bill: } \$80$ — Compare with what happened. 3. $\text{assumption: no fixed charge}$ — Name what the model took to be true. 4. $\$30 + \$2.50 \times 20 = \$80$ — Refine: add the \$30 service charge. **A model says a 300 km trip takes 300 ÷ 100 = 3 hours. What does it assume?** ☐ The road is straight ☐ The car travels at exactly 100 km/h the whole way, with no stops ☐ The car uses no petrol ☐ Nothing: it is exact ::: :::teacher-note **Answer:** The car travels at exactly 100 km/h the whole way, with no stops Steady speed, no stops, no traffic. ::: :::keep-together ### Find the mistake **Sam's model and the real trip disagree. Click the line where his review goes wrong.** ``` 1 Sam's model: time = 300 km ÷ 100 km/h = 3 hours. 2 The real trip took 3.5 hours. 3 So the calculation 300 ÷ 100 must be wrong. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: 300 ÷ 100 = 3 is right. The assumption (a steady 100 km/h with no stops) did not hold. Refine the model: add the stops, or use a lower average speed. ::: :::keep-together ### Refine it **How could you refine the trip model?** ☐ Divide 300 by 100 again ☐ Use 3.14 instead of 3 ☐ Add the time for a stop, or use a lower average speed ☐ Make the trip shorter ::: :::teacher-note **Answer:** Add the time for a stop, or use a lower average speed Add what the model left out: a 30-minute stop makes it 3.5 hours. ::: :::typical ## Refine and judge **Backwards.** **Exchange fees.** **Add the stops.** **Good enough.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Checking models :::keep-together ### The phone plan **A model for a phone plan is cost = $0.10 per MB used. The real bill was $20 more than the model said. Jo says: 'The model's sum must be wrong.' What is more likely?** ☐ The model leaves out a fixed $20 monthly fee ☐ Jo is right: the sum is wrong ☐ The phone company made a mistake ☐ The price per MB changed ::: :::teacher-note **Answer:** The model leaves out a fixed $20 monthly fee A gap of the same amount every time suggests a fixed charge: cost = 20 + 0.10 × MB. ::: :::keep-together ### A taxi ride **With the refined taxi model ($4.20 plus $2.10 per km), what does an 8 km ride cost, in dollars?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 21 4.20 + 2.10 × 8 = 4.20 + 16.80 = $21. ::: :::keep-together ### Refined phone model **A refined phone model is cost = 20 + 0.10 × MB. What is the cost for 350 MB, in dollars?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 55 20 + 0.10 × 350 = 20 + 35 = $55. ::: :::keep-together ### Review: speed **Review: a car goes 270 km in 3 hours. What is its average speed in km/h?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 90 270 ÷ 3 = 90 km/h. ::: ## Make: a plumber's quote Then name one assumption in the plumber's model, and say how a real bill might differ. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A model says 4 kg of apples at $3.50 per kg cost $14. The shop charges $12.60 for 4 kg. Name a possible assumption that failed.** {{lines-3}} ::: :::teacher-note **Answer:** A fixed price per kg: the shop may give a discount for buying more (a bulk price). :::