Modelling with linear relations: a week of ready-to-teach maths

Five lessons for Year 8 Algebra. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus rule, table and graph triad cards, linear-or-not sort cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet. The graphs are printed on the worksheets.
3. Print the two cut-out sheets, one copy per pair, and cut them before Day 3: the triad cards are for Day 3, the sort cards for Day 5.
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

Many real plans have a fixed start and a steady rate: $20 to join and $5 for each item, $8 plus $4 an hour. Such a plan is a linear relation, C = 20 + 5n, where c = 20 is the start value and m = 5 the rate. Its graph is a straight line that meets the vertical axis at the start value and rises 5 for every step across. The same plan can be shown as a rule, a table or a graph, and each is best for something: the rule for exact answers, the table for patterns, the graph for comparing. A model answers questions both ways: substitute to predict a cost, or solve to find how many items a budget buys. Then review: a linear model assumes the rate never changes, and real life has bulk discounts, caps and limits.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

Letters and units

Each rule names its letters: C for cost, B for a balance, h for a height, n for the number of items, hours or weeks. Rates carry units such as dollars per hour. Spreadsheets or graphing tools can make the tables and graphs if your class has them; every graph in this pack is also printed, so nothing depends on them.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 formulates rules from situations; Day 2 reads them back from graphs; Day 3 connects rule, table and graph and compares two plans; Day 4 uses a model to answer questions in both directions; and Day 5 interprets, checks and reviews. It builds on the linear equations and graphs unit, and on percentage modelling in Year 8 Number. It prepares gradient and intercept in Year 9 and the comparison of linear and non-linear models in Year 10.

Materials for the week (one trip)

• From the classroom: rulers, pencils, coloured pencils, scissors, this pack printed.
• Cut-out sheet 1, one per pair: six rules, six tables and six graphs to match (Day 3).
• Cut-out sheet 2, one per pair: 16 linear-or-not sort cards (Day 5).
• Calculators for Days 4 and 5; a spreadsheet if available. No other equipment to buy.

Words for the word wall

model, linear relation, rule, start value, rate, per, fixed cost, variable cost, table of values, graph, intercept, steepness, predict, solve, substitute, interpret, review, assumption, break-even, linear, not linear

Dear families

This week in maths, Year 8 is modelling real plans with straight-line rules: a gym with a joining fee and a weekly charge, a taxi fare, a savings plan. Students write the rule, draw its graph, use it to predict and to plan, and ask whether the model is fair to the situation.

Try this at home

• Find a price with a fixed part and a per-unit part: a taxi, a hire, a delivery fee. Write its rule.
• Ask: a plan costs $20 to join and $5 a month. What does a year cost? ($80.)
• Ask: with $50, how many months of that plan can you afford? (6 months: 20 + 5 × 6 = 50.)
• Compare two plans you know. After how many months is the cheaper one no longer cheaper?

My models this week

Fill one row a day. Find a real plan, write its rule, and use it once.

Printed from the free seegongsik Modelling with linear relations teaching pack · seegongsik.com/au/y8/algebra/AC9M8A03/pack

From a real plan to a straight line

A fixed start plus a steady rate makes a linear relation. Students find the start and the rate in real plans, write rules and tables, and decide which situations are not linear at all.

We are learning to

• identify the start value and the rate in a situation, with their units,
• write a linear rule such as C = 20 + 5n and make a table from it,
• decide whether a situation can be modelled by a linear relation, and say why.

Success criteria

• I can turn “$20 to join and $5 an item” into C = 20 + 5n and explain each number.
• I can explain why a doubling pattern is not linear.

You need

The worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Swapping the start and the rate: C = 5 + 20n for $20 to join and $5 an item.
• Forgetting the start: C = 5n leaves out the joining fee.
• Calling any growing pattern linear: linear means the same amount added each step.

Answers

• Part A: C = 35 + 12n (start 35, rate 12 dollars per week); C = 8 + 4n (start 8, rate 4 dollars per hour); C = 60 + 90n (start 60, rate 90 dollars per hour); B = 150 + 25n (start 150, rate 25 dollars per month); h = 30 − 2n (start 30, rate -2 cm per hour).
• Part B: C = 35 + 12n: 35, 47, 59, 71; C = 8 + 4n: 8, 12, 16, 20.
• Part C: 1. linear (fixed start, same amount per km); 2. linear (start 0, $15 per month); 3. not linear (it multiplies, not adds, each hour); 4. not linear (the rate changes at 10); 5. linear (same amount lost each minute).
• Part D: the candle reaches 0 after 15 hours; the rule only makes sense from n = 0 to 15.

From a real plan to a straight line

A. Start, rate, rule

B. Tables

Make a table for plans 1 and 2 from n = 0 to n = 3.

C. Linear or not?

Circle yes or no, and give a reason.

D. Stretch: when does the model stop?

For the candle (plan 5), when does the height reach 0? For which values of n does the rule make sense?

Reading m and c from the graph

The start value is where the line meets the vertical axis; the rate is how much it rises (or falls) for each step across. Students read both from graphs, write the rules and explain them in context, with units.

We are learning to

• read the start value c where a line meets the vertical axis,
• find the rate m from the rise over one step, or over several steps,
• write the rule and explain m and c in the situation, with units.

Success criteria

• I can write the rule of a graph that starts at 40 and rises 5 per step.
• I can explain what the 2.2 in a taxi rule means, with its units.

You need

The worksheet and a ruler, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Reading c where the line meets the horizontal axis.
• Taking the rate as the height of the line rather than the rise per step.
• Using the axis scale wrongly: one grid square may be 10 or 20 dollars.

Answers

• Part A: A: c = 10, m = 15, C = 10 + 15n; B: c = 40, m = 5, C = 40 + 5n; C: c = 90, m = -15, C = 90 − 15n.
• Part B: graph A costs $10 before any hours and $15 each hour; graph C starts with $90 credit and loses $15 a week.
• Part C: 3.6 is the flag fall, $3.60 charged before moving; 2.2 is $2.20 for each kilometre; 10 km costs 3.6 + 2.2 × 10 = $25.60.
• Part D: graph C reaches 0 after 6 weeks (90 ÷ 15), where the line meets the horizontal axis.

Reading m and c from the graph

A. Read the graphs

For each graph, mark the start and draw a rise-and-step triangle. Then write c, m and the rule (C for the amount, n across).

B. What do they mean?

For graphs A and C, write a sentence saying what the start value and the rate mean in the situation, with units.

C. Taxi rule

A taxi fare is F = 3.6 + 2.2d, where d is the distance in kilometres. What do 3.6 and 2.2 mean? What would a 10 km trip cost?

D. Stretch: when does it run out?

Use graph C and its rule to find when the credit reaches $0. Where does that show on the graph?

One relation, three representations

A rule, a table and a graph can describe the same relation, and each is good for something. Students move between them, find a rule from a table, and compare two plans where their lines cross.

We are learning to

• make a table and a graph from a rule, and a rule from a table or a graph,
• spot a linear table by its equal differences,
• compare two plans with a table and a graph and find where they cost the same.

Success criteria

• I can write the rule for a table that goes 5, 9, 13, 17 and explain each number in it.
• I can say when Plan A becomes cheaper than Plan B, using a graph or a table.

You need

The worksheet, a ruler and two coloured pencils; the triad cards (cut-out sheet 1), one set per pair.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the triads; start the second with Part A.

Watch for these ideas

• Taking the first table value as the rate.
• Finding the rate from a table whose n goes up in 2s without dividing by 2.
• Thinking the plan with the lower start is always cheaper.

Answers

• Part A: C = 12 + 3n: 12, 15, 18, 21, 24, 27; the graph starts at 12 and rises 3 per step.
• Part B: 7, 13, 19, 25, 31 gives C = 7 + 6n; 50, 40, 30, 20, 10 gives C = 50 − 5n (the second table steps in 2s, so the rate is half the difference).
• Part C: the graph starts at 12 and rises 3 each step, so the rule is C = 12 + 3n, the same as Part A.
• Part D: C = 30 + 10n and C = 16n cost the same at n = 5 ($80); below 5 Plan B is cheaper, above it Plan A.

One relation, three representations

A. From a rule

Complete the table for C = 12 + 3n, then plot the points and rule the line.

B. From a table

Check the differences, then write the rule.

D. Stretch: two plans

Plan A: C = 30 + 10n. Plan B: C = 16n. Make a table for n = 0 to 8 for both. When do they cost the same? Which plan would you choose for 3 months, and for 8?

Using the model to answer a question

A model answers two kinds of question. Substitute to predict the total for a given input; solve to find the input that gives a total. Students do both for a hire plan and a savings goal, then find a break-even point.

We are learning to

• predict a total by substituting into a linear rule,
• find the input that gives a total by solving a linear equation,
• answer in a sentence with units, and round in the way the situation needs.

Success criteria

• I can use C = 20 + 5n to find the cost of 6 items and the items for $45.
• I can explain why a savings answer of 17.9 weeks becomes 18 weeks.

You need

The worksheet, one per student; calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with Part C.

Watch for these ideas

• Substituting the total in place of n.
• Forgetting to subtract the start before dividing by the rate.
• Rounding to the nearest when the situation needs rounding up.

Answers

• Part A: 2 hours: 25 + 18 × 2 = $61; 5 hours: 25 + 18 × 5 = $115; 8 hours: 25 + 18 × 8 = $169.
• Part B: $79: 18n = 54, n = 3 hours; $133: 18n = 108, n = 6 hours; $205: 18n = 180, n = 10 hours.
• Part C: B = 120 + 35n; 120 + 35n = 750 gives n = 18, so 18 weeks.
• Part D: 30 + 10n = 16n gives 6n = 30, n = 5; both cost $80.

Using the model to answer a question

A trailer costs $25 to book and $18 an hour. Rule: C = 25 + 18n, where n is the number of hours.

A. Predict the cost

B. How many hours?

Write an equation and solve it.

C. A savings goal

Jun has $120 saved and adds $35 a week towards a $750 bike. Write a rule for the balance B after n weeks, then find when Jun can buy the bike. Answer in a sentence.

D. Stretch: break-even

Plan A: C = 30 + 10n. Plan B: C = 16n. Solve to find when they cost the same, and check by substituting.

Interpreting and reviewing the answer

A model’s answer means something only in its situation. Students state answers in context, check them, decide whether a linear model fits a table of data, sort situations by whether they are linear, and play the quiz.

We are learning to

• state an answer as a sentence with units that answers the original question,
• check an answer by substituting it into the model,
• review a model’s assumptions and decide whether a linear model fits.

Success criteria

• I can turn “n = 5” into a sentence a customer would understand.
• I can test a table for equal differences and say whether a linear model fits.

You need

The worksheet; the sort cards (cut-out sheet 2), one set per pair; the quiz on the unit page.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with Part C and the quiz.

Watch for these ideas

• Giving a bare number with no unit or meaning.
• Checking a growing table only once instead of every difference.
• Trusting a model far outside the values it was built on.

Answers

• Part A: "Jun can buy the bike after 18 weeks." "Hiring the trailer for 10 hours costs $205." "The two plans cost the same, $80, at 5 months."
• Part B: check each answer by substituting it back into the rule from Day 4.
• Part C: Taxi fares: equal differences, linear; Phone photos stored: differences grow, not linear; Water in a tank (L): equal differences, linear.
• Part D: for example C = 20 + 5n for n up to 10, then $4 an item after that; or a cap such as “never more than $60”.

Interpreting and reviewing the answer

A. Say it in context

Rewrite each bare answer as a full sentence that answers the original question, with units.

B. Check it

Choose two of your Day 4 answers. Substitute each back into its rule and show it works.

C. Does a linear model fit?

Write the differences under each table. Circle linear or not linear.

D. Stretch: a better model

The board’s plan (C = 20 + 5n) has no bulk discount. Write a model for a shop that charges $4 an item after the first 10. Where does it stop being one straight line?

Rule, table and graph cards

For Day 3, one set per pair: six rules

Six tables

Six graphs

Teacher note: the triads are C = 10 + 5n (10, 15, 20, 25); C = 8n (0, 8, 16, 24); C = 20 + 2n (20, 22, 24, 26); C = 5 + 10n (5, 15, 25, 35); C = 30 − 5n (30, 25, 20, 15); C = 15 (15, 15, 15, 15).

Linear or not? Sort cards

For Day 5, one set per pair: 16 situations

Sort into two piles: linear (the same amount added or taken each step) and not linear.

Teacher note: linear 9 cards, not linear 7 (the price halves every year; area of a square of side s; free for the first 3 hours, then $5 an hour; a tree growing faster each year; interest added to interest each month; a ball dropped: distance fallen each second; a phone plan capped at $50).

What we know: Modelling with linear relations

Work on your own. You may use a calculator. Show your working.

1. A school band hire costs $40 plus $25 an hour. Write a rule for the cost C of n hours. ______________________
2. A plant doubles in height every week. Can a linear rule model it? Why? ______________________
3. Write the rule for this table: n = 0, 1, 2, 3 gives C = 9, 13, 17, 21. ______________________
4. Use C = 6 + 3n to find C when n = 7. ______________________
5. Use C = 6 + 3n to find n when C = 42. ______________________
6. Plan X: C = 50 + 20n. Plan Y: C = 20 + 26n. When do they cost the same? ______________________
7. A dog walker charges $45 for the first visit and $15 for each visit after. Is C = 45 + 15n right for n visits? Explain. ______________________
8. A phone has 200 minutes of credit and uses 15 a day: M = 200 − 15d. For which days does the rule make sense? ______________________

Answers and marking guide

Answers with working

1. C = 40 + 25n.
2. No: doubling multiplies each week; a linear rule adds the same amount each week.
3. Differences of 4, start 9: C = 9 + 4n.
4. C = 6 + 3 × 7 = 27.
5. 3n = 36, n = 12.
6. 50 + 20n = 20 + 26n gives 6n = 30, n = 5; both cost $150.
7. Not quite: the first visit is $45 and every later one $15, so n visits cost 45 + 15(n − 1) = 30 + 15n.
8. From d = 0 to d = 13.333333333333334: after 13.333333333333334 days the credit is 0 and cannot go negative.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and states answers in context.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: formulating, reading graphs, representations, using models and reviewing.