You will learn
- Write a rule for a situation.
- Use a formula.
- Use a table.
- Test a model.
Version 0.2 · expanded edition · built from cited public sources Content is based on the official frameworks and sources listed under Sources and standards. No official endorsement is claimed. Written in the style of common primary school assessments. No exam result is promised.
Student lesson
- A model is a maths description of a situation.
- A formula uses letters for the rule.
- Fixed cost plus rate times amount.
- Test the model with real values.
Words to know
A maths description.
Rule with letters.
Cost that stays the same.
Cost per unit.
1. Discover: taxi fare
A taxi charges 5 to start plus 2 per kilometre. How much for 10 km?
2. Understand: modelling
- Model: cost = 5 + 2k for k kilometres.
- For 10 km: 5 + 2 × 10 = 25.
- For 3 km: 5 + 6 = 11.
- Table: 1 km → 7, 2 km → 9, 3 km → 11: it grows by 2 each km.
Idea 1: Rule
A taxi charges a fixed 5 plus 2 per km: C = 5 + 2k.
Idea 2: Use it
4 km: 5 + 2 × 4 = 13. 0 km: 5. 10 km: 25.
Idea 3: Backwards
Cost 15: 15 − 5 = 10, then 10 ÷ 2 = 5 km.
Why a model? One formula answers every distance.
3. Worked examples
Example 1 (I show you): Cost
C = 5 + 2k, k = 4
- 2 × 4 = 8.
- 5 + 8.
Answer: 13
Example 2 (We do it together): Fixed
Cost for 0 km
- Only the fixed charge.
Answer: 5
Example 3 (You try): Distance
Cost 15
Show the steps and answer
- 15 − 5 = 10.
- 10 ÷ 2.
Answer: 5 km
Did you know?
The fixed charge is the cost when the amount is zero.
Why this matters
Models predict costs, times and amounts before they happen.
4. Explore: model
You need paper.
- Find the fixed part.
- Find the rate.
- Write the formula.
- Test with numbers.
Think about it: what did you notice?
5 + 2k.
5. Practise
Say it in a sentence
Use these starters when you talk about the lesson. Say them out loud.
- The fixed charge is ___.
- The rate is ___ per ___.
- The formula is C = ___.
- For ___, the cost is ___.
Guided practice
Question 1. Cost for 10 km?
25.
Question 2. Cost for 3 km?
11.
Question 3. Growth per km?
2.
Independent practice
Level 1: Start
- Cost for 4 km
- Cost for 0 km
- Kilometres for cost 15
- Fixed charge
- Rate per km
Level 2: Grow
- A phone hire costs 3 plus 4 per day. Cost for 5 days
- Same hire: cost for 0 days
- Same hire: rate per day
Level 3: Stretch
- Same hire: days for a cost of 19
- C = 5 + 2k. A journey costs 41. How far?
- Ben says the cost for 10 km is 2 × 10 = 20. What is wrong?
Answers are in the answer key below. Try first, then check.
6. Apply: everyday mathematics
A taxi is 5 plus 2 per km. Cost for 10 km?
Show a sample answer
25.
Another one. A gym charges 10 joining fee plus 6 per visit. Write a formula and find the cost of 8 visits.
Show a sample answer
C = 10 + 6v; 58
7. Common mistakes
Fix: Start fee is fixed.
Fix: Add it to the rate part.
Fix: Rate changes; fixed cost does not.
8. Lesson summary and revision card
- I can write a rule for a situation.
- I can use a formula.
- I can use a table.
- I can test a model.
9. Check and master
Answer all 14 questions. You will see feedback after each one.
Your result appears here after you answer every question.
10. Key takeaways
- Fixed plus rate.
- Test the formula.
- Use a table.
11. Next lesson
Next lesson: Explaining solutions. Keep going with the next idea in this course.
Exam practice
These questions are written in the style of common school and international primary assessments. Try them on your own, without looking back. Then check your answers in the marking guide. The marks are shown in brackets.
- A taxi charges a fixed 5 plus 2 per km: C = 5 + 2k. Find the cost for 4 km. [2 marks]
- Find the cost for 0 km. [1 mark]
- A journey costs 15. How many km? [3 marks]
- State the fixed charge and the rate per km. [2 marks]
- Ali says the cost for 10 km is 2 × 10 = 20. Is he right? Explain. [3 marks]
Total: 11 marks.
Marking guide and model answers
- 5 + 2 × 4 [1]; 13 [1]
- Only the fixed charge: 5 [1]
- 15 − 5 = 10 [1]; 10 ÷ 2 [1]; 5 km [1]
- Fixed charge 5 [1]; rate 2 per km [1]
- No [1]. He forgot the fixed charge [1]. C = 5 + 2 × 10 = 25 [1].
Score guide: 10 to 11 = secure. 7 to 9 = nearly there, revise the parts you missed. Under 7 = work through the lesson again with a grown-up.
Printable worksheet
- Cost for 4 km
- Cost for 0 km
- Km for cost 15
- Fixed charge
- Rate per km
- Cost for 4 km (C = 5 + 2k)
- Cost for 0 km
- Kilometres for cost 15
- Fixed charge
- Rate per km
Homework
Model the cost of a call with a fixed fee.
Answer key and marking scheme
| Question | Expected response | Marks |
|---|---|---|
| Cost for 4 km | 13 | 2 |
| Cost for 0 km | 5 | 2 |
| Km for cost 15 | 5 | 2 |
| Fixed charge | 5 | 2 |
| Rate per km | 2 | 2 |
| Total | 10 |
Practice answers: Worksheet: 1 13; 2 5; 3 5; 4 5; 5 2.
Quiz answers: 1 A, 2 B, 3 C, 4 A, 5 B, 6 A, 7 B, 8 C, 9 A, 10 B, 11 C, 12 A, Challenge 1 C, Challenge 2 C.
Extra worksheet answers (items 6 onward): 6 13; 7 5; 8 5; 9 5; 10 2.
Independent practice answers. Level 1: 1 13; 2 5; 3 5 km; 4 5; 5 2. Level 2: 1 23; 2 3; 3 4. Level 3: 1 4; 2 18 km; 3 He forgot the fixed charge: 5 + 20 = 25..
Mastery and remediation
| Objective | Bronze (Developing) | Silver (Proficient) | Gold (Advanced) |
|---|---|---|---|
| O1 Write a rule for a situation | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
| O2 Use a formula | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
| O3 Use a table | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
| O4 Test a model | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
Courseversity badge thresholds (overall quiz score): Gold = 90–100%; Silver = 80–89.9%; Bronze = 70–79.9%. A badge is never a full record of progress.
If a learner struggles
- Find the misconception: Charging 2 for every km including the start. Start fee is fixed.
- Explain another way: use real objects first, then pictures.
- Show a simpler example: Patterns.
- Guided practice: work through one example together and let the learner try the next one.
- Reassess: use the quiz again or the worksheet.
- Move on when the learner shows Silver-level evidence independently.
For parents and guardians
In this lesson your child writes and uses simple maths models. You do not need to be good at mathematics to help.
- Look at taxi or phone plans together.
- Ask: what is fixed? what changes?
- Test the rule with real numbers.
- Ask “how do you know?” after each answer, and listen.
- Praise effort, not only right answers. Mistakes are how we learn.
- Ten to fifteen minutes a day is better than one long session.
Teacher package
1–5. Objectives, prerequisites, materials, preparation, duration
Lesson objectives. Learners will: write a rule for a situation; use a formula; use a table; test a model.
Prerequisite knowledge. Lessons 2 and 3.
Materials. Counters (stones, seeds or bottle tops), paper, pencils, board or chalkboard. Low-resource options: use stones, seeds, sticks, bottle tops or drawings in the sand or on the board instead of printed materials.
Preparation. Gather the counters and draw the example pictures on the board before the lesson.
Suggested duration. 45–60 minutes. It can be split into two shorter sessions.
6–9. Teaching procedure, explanation notes, guided and independent activities
Introduction (5 min). Start with the Discover situation and let learners suggest ideas before you teach.
Explanation (10 min). Teach the Understand section with real objects first, then pictures, then symbols and words.
Guided activity (10 min). You need paper. Learners follow the Explore steps in pairs.
Independent activity (10 min). Learners complete the independent practice and the worksheet.
Assessment and reflection (5 min). Ask each learner to say one thing they learned and how they know it.
Key questions. What is fixed? What changes with each unit?
Teacher explanation notes. Use the concrete, pictorial, abstract order: real objects first, then pictures, then symbols and words. Return to real objects whenever a learner needs support.
10–12. Differentiation, formative assessment, common misconceptions
Support. Use real objects and work with a partner; say each step aloud together. Extension. Ask learners to make up their own example and explain it to a partner.
Formative assessment. Listen to learner explanations, ask “how do you know?”, and check the independent work.
Misconceptions. Charging 2 for every km including the start. Start fee is fixed.
13–15. Assessment questions, answer keys, marking scheme
Use the quiz and worksheet on this page. The 11-mark exam practice scheme, the 10-mark scheme and all answers are in the answer sections above. Re-check every answer and the marking scheme before publication.
16–19. Remediation, enrichment, homework, teacher reflection
Remediation. Patterns. Enrichment. Invite learners to find or create three more examples at home or in the classroom and explain them. Homework. See above.
Teacher reflection. After teaching, record: which concepts learners understood; which misconceptions occurred; which activities worked; which learners need more practice; and what to improve next time.
Sources and standards
This lesson is based on the sources below, and no official endorsement is claimed. Reference starting points named in the Courseversity standard: Common Core State Standards for Mathematics, Cambridge Primary and the IB Primary Years Programme. Where a standard code is named, it points to the official framework.
Official standards for this topic area. Common Core State Standards for Mathematics (official site): Standards for Mathematical Practice. These pages list the standards that cover this topic area; read them alongside the lesson.
Research used to shape this lesson. Common Core 6.EE.C.9 and MP4; Cambridge Primary Mathematics Stage 6 (using formulas). These sources informed the lesson.