Relationships between variables in graphs: a week of ready-to-teach maths

Five lessons for Year 7 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 situation cards, graph and story 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.
3. Print the two cut-out sheets, one copy per pair: situation cards (Day 2) and graph and story cards (Day 4).
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

A graph shows how one variable changes with another. Read the axis titles and scales first, then read across and down to match values. The shape tells the story: a rising line means both variables increase together, a falling line means one decreases as the other increases, and a level line means nothing is changing. Steepness shows how fast: on a distance-time graph, a steeper line is a faster speed, and a flat part is a stop. Describing a relationship means saying all of this in words, with numbers from the graph.

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.

How we describe a graph

Name both variables and use the axis titles: “as time increases, the temperature rises”. Say how fast with numbers read off the graph: “about 2 °C every hour”. Point to changes: “it levels off after 1 pm”. An estimate between plotted points is fine if it is called an estimate. Calculators are not needed this week.

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 reads values off a graph; Day 2 describes its shape; Day 3 adds steepness and rate; Day 4 tells and draws journey stories; and Day 5 describes a real data set from start to finish. It builds on the Year 6 line graphs and on formulas (AC9M7A01), and it leads into tables of values and the Cartesian plane (AC9M7A05).

Materials for the week (one trip)

• From the classroom: pencils, rulers, scissors, small envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 12 situation cards (Day 2).
• Cut-out sheet 2, one per pair: 6 graph cards and 6 story cards (Day 4).
• No calculators and no other equipment to buy.

Words for the word wall

graph, axis, scale, title, variable, value, read off, estimate, rising, falling, level, increase, decrease, constant, steep, gentle, rate, per hour, distance-time graph, stop, trend

Dear families

This week in maths, Year 7 is reading graphs of real data and describing what they show: when something rises or falls, how fast it changes, and what a flat part means.

Try this at home

• Look at a weather app’s graph for today. When is it warmest? When does it change fastest?
• Describe a trip to school as a graph of distance from home: where would the line be flat?
• Ask: what would a graph of a bath’s water level look like while it fills, then drains?
• Ask: on a graph of distance against time, what does a steeper line mean? (Going faster.)

Graphs this week

Fill one row a day with a graph you saw: what it showed and one thing it told you.

Printed from the free seegongsik Relationships between variables in graphs teaching pack · seegongsik.com/au/y7/algebra/AC9M7A04/pack

Reading a graph

A graph shows how one variable changes with another. Students read the titles and scales first, then read values across and down a class’s temperature graph, estimate between points, and describe the day in words.

We are learning to

• read the axis titles and scales before the line,
• read a value off a graph, across and down, with its unit,
• estimate between plotted points and call it an estimate.

Success criteria

• I can read a value off a graph and say its unit.
• I can estimate a value between two plotted points.

You need

The worksheet and a ruler. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after across and down; start the second with Part A.

Watch for these ideas

• Reading the wrong axis: 16 °C is a temperature, so start on the vertical axis.
• Counting grid lines instead of reading the scale. Here each line is 2 °C.
• Treating a value between readings as exact. Call it an estimate.

Answers

• Part A: 9 am 12 °C, 1 pm 19 °C, 3 pm 17 °C.
• Part B: 16 °C at 11 am; the highest was 19 °C at 1 pm and 2 pm; from 7 am to 11 am it rose 10 degrees.
• Part C, for example: it rose all morning, quickly from 8 am to 9 am, peaked at 19 °C at 1 pm and 2 pm, then fell to 17 °C by 3 pm.
• Part D: about 13 °C (an estimate); from 8 am to 9 am, up 4 degrees, the steepest part of the line.

Reading a graph

A class recorded the temperature in the school yard every hour, from 7 am to 3 pm.

A. Read the temperature

B. Read the time and the change

1. When was the temperature 16 °C? ________
2. What was the highest temperature, and when? ________________
3. How much did the temperature rise from 7 am to 11 am? ________

C. Describe the day

Describe how the temperature changed from 7 am to 3 pm, in two sentences with numbers from the graph.

D. Stretch

Estimate the temperature at half past 9. ________   In which hour did it rise fastest? How can you tell from the graph? ________________

Rising, falling and level

The shape of a graph describes the relationship: rising means both variables increase together, falling means one decreases as the other increases, and level means one stays the same. Students sort situations by shape, sketch graphs, and describe a tank draining.

We are learning to

• describe a graph as rising, falling or level, or a mix,
• sketch a graph for a situation,
• describe a relationship in a sentence that names both variables.

Success criteria

• I can sort situations into rising, falling and level, and explain one.
• I can describe a graph with a sentence beginning “As time increases, …”.

You need

The situation cards from cut-out sheet 1 (one set per pair), the worksheet. No calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Describing the line, not the situation: say “the water decreases”, not just “it goes down”.
• Reading a graph as a picture, such as a hill shape for a walk up a hill.
• Missing a change partway, such as rising and then levelling off.

Answers

• Part A: 1 rises steadily; 2 stays level; 3 falls steadily; 4 rises quickly, then levels off.
• Part B: 1 falling; 2 rising; 3 rises, then falls back to the ground; 4 level.
• Part C: as time increases, the water decreases, by about 40 L each hour (between 36 and 44); about 100 L were left between 2 and 3 hours, about 2.5 hours; empty after 5 hours.
• Part D: rises quickly, then more slowly, then levels off as an adult. Any rising-then-falling story, such as a ball thrown up.
• Cards: rising 1, 4, 6, 11; falling 2, 7, 9, 12; level 3, 5, 8, 10.

Rising, falling and level

A. Describe each graph in words

B. Sketch a graph for each situation

C. A tank draining

1. Describe the relationship in a sentence that names both variables. ________________________________
2. About how much water drains each hour? ________
3. When was there about 100 litres left? ________

D. Stretch

Sketch the height of a person from birth to age 30, then describe it. Then sketch a graph that rises and then falls, and write a story for it.

Steepness and rate

Steepness shows how fast one variable changes compared with the other: on a distance-time graph, the steeper line is the faster speed. A rate, such as 60 km an hour, is how much the line rises for each step across.

We are learning to

• compare the steepness of two lines and say what it means,
• read a rate, such as kilometres per hour or dollars per week, off a graph,
• find where a graph is steepest.

Success criteria

• I can say which of two lines is faster and give both speeds.
• I can read a rate off a graph from the change between two points.

You need

The worksheet and a ruler. No calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Thinking the higher line is always faster. Compare steepness, not height.
• Reading a rate from one point when the line does not start at zero. Use the change between two points.
• Calling a flat line slow. On a distance-time graph, flat means stopped.

Answers

• Part A: the cyclist’s line is steeper, so the cyclist is faster; after 2 hours 10 km and 30 km; after 3 hours they are 30 km apart.
• Part B: A saves $10 and B $15 a week; at week 2, A has $40 and B $30; they are equal at week 4 ($60); at week 6, B has more ($90 against $80).
• Part C: the second part is steeper; 60 km/h then 90 km/h; 240 km in all.
• Part D: 300 ÷ 4 = 75 km/h. A flat line means the distance is not changing: stopped.

Steepness and rate

A. A walker and a cyclist

1. Whose line is steeper? Who is faster? ________________
2. How far has each gone after 2 hours? ________________
3. How far apart are they after 3 hours? ________

B. Two savers

How much does each save a week? Who has more at week 2? When do they have the same? Who has more at week 6? ________________________________

C. A trip at two speeds

A car travels 60 km in the first hour, then reaches 240 km after 3 hours. Which part of its graph is steeper? What are the two speeds? ________________________________

D. Stretch

A line on a distance-time graph goes from (0, 0) to (4, 300), in hours and kilometres. What is the speed? What would a flat line mean? ________________

Graph stories

A distance-from-home graph tells a journey: rising means moving away, falling means coming back, flat means stopped, and steeper means faster. Students match stories to graphs, read a journey with stops, and sketch graphs of their own.

We are learning to

• say what each part of a journey graph means,
• read times, distances and speeds off a journey graph,
• sketch a graph for a story, and tell a story for a graph.

Success criteria

• I can explain what rising, falling and flat parts of a journey graph mean.
• I can sketch a graph for a story with a stop.

You need

The graph and story cards from cut-out sheet 2 (one set per pair), the worksheet, a ruler.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Reading the graph as a picture of the route: a rising line is not a hill.
• Reading a falling line as slowing down. Falling means heading back home.
• Drawing a stop as a slope. A stop is flat.

Answers

• Part A: 2 km; the first stop lasted 10 minutes; fastest from 30 to 40 minutes (4 km in 10 minutes, 24 km/h), the steepest part; 6 km away for 20 minutes; home at 80 minutes.
• Part B: rise, flat, then a steeper fall; rise steeply, then rise gently (the walk), no flat part.
• Part C, for example: walks to a friend’s, stays, catches a lift further away, then comes straight home.
• Part D: a vertical line would mean being at two distances at the same moment. Home trip: 6 km in 20 minutes, 18 km/h.
• Cards: 1 C, 2 E, 3 F, 4 A, 5 D, 6 B.

Graph stories

A. A journey from home

1. How far from home was the traveller after 20 minutes? ________
2. How long was the first stop? ________
3. Which part of the journey was fastest? How can you tell? ________________________
4. What was the furthest distance from home, and for how long? ________________
5. When did the traveller get home? ________

B. Sketch a graph for each story

C. Tell a story for this graph

D. Stretch

Why can a journey graph never have a vertical part? How fast did the traveller in Part A come home? ________________

Real data

A class measured a seedling’s height every day for 12 days. Students read values, find where it grew fastest, describe the whole shape with numbers, and predict what comes next, with reasons.

We are learning to

• describe a real data graph from start to finish, with numbers,
• find where a graph is steepest and say what that means,
• predict from a graph and explain the prediction.

Success criteria

• I can describe the whole graph in three sentences with numbers.
• I can say where the seedling grew fastest and how the graph shows it.

You need

The worksheet, a ruler, and the unit’s self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 3; start the second with problem 4.

Watch for these ideas

• Finding the tallest day instead of the day of most growth. Look for the steepest part.
• Predicting by continuing the steepest part. The last days have levelled off.
• Describing without numbers. Every sentence can use a value from the graph.

Answers with working

1. day 2 (0.5 cm).
2. 7.5 cm.
3. between day 5 and day 6, 2.5 cm, where the line is steepest.
4. nothing for 2 days, then slow growth, fastest around days 5 to 6, then slower, levelling off at about 13 cm.
5. 11 − 3 = 8 cm.
6. about 13 cm: the last days show almost no growth, so the line has levelled off.
7. 12.5 ÷ 10 = 1.25 cm a day.
8. any sketch that follows the data to day 6, then rises more slowly or levels off, with a reason.

Real data: a growing seedling

A class planted a bean seed and measured the seedling’s height each day at the same time.

Problems (7 and 8 are the stretch)

1. On which day did the seedling first show above the soil?
2. How tall was it on day 6?
3. Between which two days did it grow the most? How can you tell from the graph?
4. Describe the shape of the graph from day 0 to day 12, in two or three sentences with numbers.
5. How much did it grow from day 4 to day 8?
6. Predict its height on day 14, and explain your prediction.
7. Stretch. What was its average growth per day over the first 10 days?
8. Stretch. Sketch the graph you would expect if the class had stopped watering the seedling on day 6, and explain your sketch.

Situation cards

Cut out the 12 cards. Day 2: sort them into three piles by the shape of their graph against time: rising, falling and level.

Graph and story cards

Cut out the 12 cards. Day 4: match each story to the graph of its distance from home.

Stories

Graphs

What we know: Relationships in graphs

The graph shows a bike ride from home. Work on your own, and use numbers from the graph.

1. How far from home was the rider after 20 minutes? ________
2. How long did the rider stop? What does the flat part show? ________________
3. Which part of the ride was faster? How can you tell from the graph? ________________
4. How far did the rider travel in all? ________
5. What was the rider’s speed in the first 20 minutes, in kilometres per hour? ________
6. Describe the whole ride in two sentences with numbers. ________________________________
7. Describe the shape of a graph of a hot drink’s temperature as it cools on a bench. ________________________________
8. On a graph of savings ($) against weeks, a line goes from (0, 20) to (5, 70). Describe the relationship and the weekly rate. ________________

Answers and marking guide

Answers with working

1. 6 km.
2. 10 minutes, from 20 to 30 minutes; the distance did not change, so the rider had stopped.
3. From 30 to 50 minutes: 9 km in 20 minutes, the steeper part.
4. 15 km.
5. 6 km in 20 minutes is 18 km/h.
6. For example: rode 6 km in 20 minutes, stopped for 10, then rode faster, 9 km in 20 minutes (27 km/h).
7. It falls quickly at first, then more slowly, and levels off near room temperature.
8. As the weeks increase, the savings increase steadily, from $20 by $10 a week.

A quick three-level guide

Eight questions, four ideas. A student at standard reads values with units and describes each part of a graph in a sentence that names both variables.

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: reading, shape, steepness, graph stories, and real data.