Skill companion: Representations and Mathematical Models (Year 8)

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Skill companion · Year 8 Science Inquiryseegongsik /au

Representations and Mathematical Models: a skill companion

A small set of reusable sheets that grow one inquiry skill: choosing the representation that suits a purpose, and turning data into a mathematical relationship. Students test whether a relationship is directly proportional, linear but not proportional, or neither, write the rule, use it to predict, and judge what a model shows and leaves out. Print the scaffolds once and slot them into the science lessons you already teach.

AC9S8I04

select and construct appropriate representations, including tables, graphs, models and mathematical relationships, to organise and process data and information

What a skill companion is

Inquiry skills are not a topic of their own. They grow inside the science units a class teaches all year, wherever data is organised or explained: cell sizes, heart rates, the sea floor, rock samples, energy, particles and reactions. So this pack is not a full term of lessons. It is three reusable scaffolds, a map of where they fit, and a short stand-alone lesson for teaching the skill on its own first.

Start here: five minutes

Read the pairing map on the next page: it shows which scaffold fits which science lesson.
Print the rule sheet whenever results come in pairs of numbers, and the representation sheet when students choose how to show something.
Cut out the cards once. They are reused all year, and the blank cards fit any topic.
Open the free interactive unit on your board when you want a worked example of the skill.
Run the one-page mini-lesson first if you want to teach the skill before folding it into a topic.

The step up from Year 7

In Year 7 students chose between a table, a bar chart and a line graph, and drew the graph. In Year 8 they process the data into a relationship: dividing one quantity by the other to test for direct proportion, finding the constant, writing the rule (such as extension = 3 × load) and predicting with it. They tell a proportional line from a linear one that starts above zero, and from a curve, and they judge what a diagram or physical model shows and what it leaves out.

On the board

This pack is the printable half of a free interactive unit. On screen, students switch “Spring extension against load” from “Table” to “Line graph”, compare it with “A stiffer spring, same kind of rule” on the same scale, click the load that breaks the line in “Read the graph: find the load that breaks the line”, and judge conclusions with “Supports” and “Does not support” in “Test conclusions against the spring model”.

seegongsik.com/au/y8/inquiry/AC9S8I04

Aligned to the Australian Curriculum V9 (AC9S8I04). This pack is original material from seegongsik, independently produced and not endorsed by ACARA. Curriculum content descriptors are (c) ACARA, licensed under CC BY 4.0. Free to print and use in class.

Where the skill fitsPairing map

Slot the skill into your science lessons

Every set of results in Year 8 can be represented more than one way. This map shows data in each Year 8 topic, the representation that processes it, the relationship to look for, and which scaffold to reach for.

When you teach	Data or information	Representation	Relationship or rule	Scaffold
Cells: the Basic Units of Life (AC9S8U01)	Image size of cells at different magnifications	A table, then a rule	Real size = image size ÷ magnification	Rule sheet
Cells, Tissues and Organ Systems (AC9S8U02)	Heart rate each minute after exercise	A line graph	A curve: it falls fast, then levels off	The cards
Plate Tectonics (AC9S8U03)	Age of the sea floor at distances from a ridge	A line graph through the origin	Proportional: the slope is the spreading rate	Rule sheet, part C
The Rock Cycle (AC9S8U04)	Mass and volume of rock samples	A line graph, then a rule	Mass = density × volume (proportional)	Rule sheet, then the cards
Kinetic and Potential Energy (AC9S8U05)	Bounce height for each drop height	A line graph with a line of best fit	Close to proportional, below the drop height	Rule sheet, part C
Elements, Compounds and Mixtures (AC9S8U06)	Particles in elements, compounds and mixtures	A diagram or model, and formulas	What each model shows, and leaves out	Representation sheet
Physical and Chemical Changes (AC9S8U07)	Temperature every minute during a reaction	A line graph	Linear while it warms steadily, then a curve as it slows	The mini-lesson, then the representation sheet
Any science topic	Any data or information	The form that suits the purpose	Proportional, linear or neither	Representation sheet first

The three moves, and the picture that backs each one

When you want a worked example on the board, open the interactive unit and use the picture that matches the move.

Switch from a table to a graph to see the shape: “Spring extension against load”.
Read the slope as the number in the rule: “A stiffer spring, same kind of rule”.
Use the expected line to spot a point that misses it: “Read the graph: find the load that breaks the line”.

How the scaffolds build the skill

The rule sheet divides one quantity by the other to test for proportion, writes a rule and predicts with it, then meets a straight line that does not start at zero, and draws a line of best fit. The representation sheet matches jobs to the form that suits them and asks what a particle model shows and leaves out. The cards sort twelve small data tables into proportional, linear but not proportional, and not linear.

Scaffold 1 · Rule sheetOne per student

From a table to a rule

NameClassDate

Two quantities are directly proportional when one is always the same multiple of the other: divide one by the other and you get the same number every time, give or take the wobble of real readings. That number is the constant in the rule.

A. Spring C

Load (N)	0	1	2	3	4	5
Extension (cm)	0	3.1	5.9	9.2	11.8	15.1
Extension ÷ load	–

Is it proportional? How do you know?

The rule: extension =

Predict the extension for 3.5 N:

Can the rule predict 8 N? What might go wrong?

B. A beaker heated steadily

Time (min)	0	1	2	3	4	5
Temperature (°C)	21.0	24.1	26.9	30.2	33.0	35.9

Each minute the temperature rises by about:

Linear? Proportional? Why?

The rule: temperature =

C. A line of best fit

Plot spring C’s points. Lay a ruler so the points sit as evenly as possible on either side, and draw one straight line. It need not pass through every point. Then read two points on your line, far apart, and work out the slope.

Slope =

Scaffold 2 · Representation sheetOne per student

Choose the representation, and know what it leaves out

NameClassDate

Each representation does a different job. A table keeps exact values, a bar chart compares categories, a line graph shows how one measured quantity changes with another, a rule calculates, and a diagram or model shows parts and movement at once.

A. Match the job to the representation

Write the best one: table, bar chart, line graph, rule, diagram or model. Then say why in a few words.

The job	Best representation	Why
Show how much salt dissolves in 100 mL of water at 20, 40, 60 and 80 °C, and read off a value in between
Compare how many grains of quartz, feldspar and mica are in a granite sample
Keep every reading from every trial, exactly as it was measured
Work out the extension of a spring for any load
Show how two plates move apart at a mid-ocean ridge as new rock forms
Show how one student’s heart rate falls, minute by minute, after exercise

B. What a model shows, and what it leaves out

The particle model of a solid, a liquid and a gas, with the same twelve particles in each box.

One thing this model shows well:

A second thing it shows well:

One thing it leaves out, or gets wrong:

Why scientists still use it:

Scaffold 3 · Relationship cards (cut out)Reuse all year

Proportional, linear, or neither?

Cut out the cards. Lay down the three grey heading cards, then sort the tables under them. Check the steps from one reading to the next, and whether the second quantity is zero when the first is.

Directly proportionalA straight line through the origin: the second quantity is always the same multiple of the first.Heading

Linear, not proportionalA straight line that does not pass through the origin: the same step each time, from a start that is not zero.Heading

Not linearThe steps change size: a curve on a graph.Heading

Time cooling (min)
0	5	10	15
Cup temperature (°C)
80	62	50	42

Card A

Load (N)
1	2	3	4
Extension (cm)
3	6	9	12

Card B

Volume of granite (cm³)
10	20	30	40
Mass (g)
27	54	81	108

Card C

Depth in a mine (km)
0	1	2	3
Rock temperature (°C)
18	43	68	93

Card D

Time a candle burns (h)
0	1	2	3
Candle height (cm)
20	17	14	11

Card E

Time (h)
0	1	2	3
Number of bacteria
100	200	400	800

Card F

Mass of sugar (g)
5	10	15	20
Food energy (kJ)
85	170	255	340

Card G

Drop height (cm)
25	50	100	200
Time to fall (s)
0.23	0.32	0.45	0.64

Card H

Time walking at a steady pace (min)
2	4	6	8
Distance (m)
160	320	480	640

Card I

1.5 V cells joined in a row
1	2	3	4
Voltage (V)
1.5	3	4.5	6

Card J

Time heating (min)
0	1	2	3
Water temperature (°C)
20	24	28	32

Card K

Books in a 300 g box
1	2	3	4
Total mass (g)
800	1300	1800	2300

Card L

Teacher note: the piles are not the same size. The answer sheet lists which card is which, and why.

Mini-lesson · Teacher planAbout 30 minutes

Two rules from one stack of cups

Use this stand-alone lesson to teach the skill on its own, before you fold it into a science topic. Nesting cups give two relationships from the same objects. The height of the stack climbs by the same amount for each cup but starts at the height of the first cup, so it is linear but not proportional; the mass of the stack is always the same multiple of the number of cups.

We are learning to

organise measurements in a table and graph them,
test for direct proportion by dividing one quantity by the other,
write a rule and use it to predict, inside and outside the measured range.

Success criteria

I can write a rule for the height of the stack and use it for 25 cups.
I can explain why the height is linear but not proportional, and the mass is proportional.

You need

for each group: ten identical paper or plastic cups that nest inside each other, a ruler, and an electronic balance if you have one,
the rule sheet (scaffold 1), one per student, and the cards from scaffold 3, one set per table.

Lesson flow (about 30 minutes)

5 min	Table or graph? Show the board (see the box). Ask: “The table and the graph hold the same numbers. Which one shows the rule faster?”
10 min	Measure the stack Groups measure the height of stacks of 1 to 10 cups, and the mass if they have a balance, in a table. Ask: “Divide the height by the number of cups. Is it the same each time? Why not?”
10 min	Graph and rule Groups sketch both graphs and write both rules: height = first cup + step × (cups − 1), and mass = one cup × cups. Ask: “Where does each line meet the height or mass axis? What does that tell you?”
5 min	Sort the cards Tables sort the cards under the three heading cards. Ask: “Card L goes up the same step each time. Why is it not proportional?”

Running it shorter? Stop after the rules, and sort the cards in your next science lesson.

On the board

For a worked example, open the unit and switch “Spring extension against load” from “Table” to “Line graph”. Then open “A stiffer spring, same kind of rule” and compare how steep the two lines are on the same scale.

seegongsik.com/au/y8/inquiry/AC9S8I04

Watch for these ideas

Calling every straight line proportional, even one that does not start at zero.
Joining the dots point to point instead of drawing one line of best fit.
Trusting a rule far outside the measured range, such as for 1000 cups.

Make it easier, make it bigger

Easier: give the table with the height divided by the number of cups already worked out for two rows.
Bigger: predict how many cups stack to the height of the classroom door, then say what could make the prediction wrong.

Answers and look-fors

The next sheet has the spring and beaker answers, the representations, the card key and a quick three-level guide. Typical paper cups: a first cup about 9 cm tall and about 0.5 cm more for each extra cup.

Answers · For the teacherModel responses

Answers and look-fors

Rule sheet

A. Extension ÷ load: 3.10, 2.95, 3.07, 2.95, 3.02, all close to 3, so it is directly proportional (and the extension is 0 at 0 N). Rule: extension = 3 × load. At 3.5 N, about 10.5 cm. At 8 N the rule gives 24 cm, but only if the spring stays proportional; a heavy load can stretch it past its limit.

B. The rises are 3.1, 2.8, 3.3, 2.8, 2.9 °C, about 3 °C each minute: linear, but not proportional, because it starts at 21 °C, not 0. Rule: temperature = 21 + 3 × time.

C. A line of best fit runs from (0, 0) to about (5, 15) or (6, 18): slope about 3 cm for each newton, the constant in the rule.

Representation sheet

Job	Best	Why
Show how much salt dissolves in 100 mL of water at 20, 40, 60 and 80 °C, and read off a value in between	line graph	Both quantities are measured numbers, and a line lets you read between the points.
Compare how many grains of quartz, feldspar and mica are in a granite sample	bar chart	The minerals are separate categories; a line joining them would mean nothing.
Keep every reading from every trial, exactly as it was measured	table	Only a table keeps every exact value.
Work out the extension of a spring for any load	rule	A rule, such as extension = 3 × load, calculates any value.
Show how two plates move apart at a mid-ocean ridge as new rock forms	diagram or model	A diagram or model shows the movement and the parts at once.
Show how one student’s heart rate falls, minute by minute, after exercise	line graph	Time is a measured number; the line shows how fast it falls.

B. The model shows well: particles in a solid are close and in rows, in a liquid close but jumbled, in a gas far apart; and the same particles in each state. It leaves out: the particles are always moving; they are not coloured balls; in a real gas the gaps are far bigger than any drawing can show. Scientists use it because it explains melting, diffusion and compressing a gas simply.

Relationship cards

Card	Quantities	Pile	Why
A	Cup temperature (°C) against time cooling (min)	Not linear	Drops of 18, 12 and 8: cooling slows as the cup nears room temperature.
B	Extension (cm) against load (N)	Directly proportional	Always 3 times the load.
C	Mass (g) against volume of granite (cm³)	Directly proportional	Always 2.7 times the volume: the density.
D	Rock temperature (°C) against depth in a mine (km)	Linear, not proportional	Up 25 for each kilometre, from 18 at the surface.
E	Candle height (cm) against time a candle burns (h)	Linear, not proportional	Down 3 each hour, from 20.
F	Number of bacteria against time (h)	Not linear	The number doubles each hour, so the steps grow.
G	Food energy (kJ) against mass of sugar (g)	Directly proportional	Always 17 times the mass.
H	Time to fall (s) against drop height (cm)	Not linear	Twice the height is far less than twice the time.
I	Distance (m) against time walking at a steady pace (min)	Directly proportional	Always 80 times the time.
J	Voltage (V) against 1.5 V cells joined in a row	Directly proportional	Always 1.5 times the number of cells.
K	Water temperature (°C) against time heating (min)	Linear, not proportional	Up 4 each minute, from 20.
L	Total mass (g) against books in a 300 g box	Linear, not proportional	Up 500 for each book, but the empty box adds 300.

A quick three-level guide

Move	Working towards	At standard	Beyond
Choose the representation	uses the same kind of graph for everything	chooses the form that suits the job and says why	says what a model or diagram leaves out, and why it is still useful
Test the relationship	calls any straight line proportional	divides one quantity by the other to test for proportion, and checks the start	tells proportional, linear and curved apart from the steps alone
Write and use a rule	reads values off the table only	writes the rule with its constant and predicts inside the range	draws a line of best fit, reads its slope, and says how far outside the range a rule can be trusted

A student at standard chooses the representation for the job, tests a relationship by dividing, and writes and uses a rule. The skill grows all year, so keep the scaffolds coming back whenever results come in.