Representations and Descriptive Statistics: a skill companion
A small set of reusable sheets that grow one inquiry skill at Year 10 depth: finding the mathematical relationship hidden in a table, redrawing a curve as a straight line to read its gradient, and choosing the summary that a set of repeated results really needs. Print the scaffolds once and slot them into the science lessons you already teach.
What a skill companion is
Inquiry skills are not a topic of their own. They grow inside the units a class teaches all year. 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 gives the representation and the relationship for the data in each Year 10 topic.
- Print the relationship sheet when a class has a table of paired results, and the straighten-and-summarise sheet when a graph curves.
- Cut out the cards once. They are reused all year.
- 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 9
The descriptor is the same in Years 9 and 10; the depth is not. In Year 9 students chose a representation for a purpose and processed data with means and ranges. In Year 10 they test a table for the relationship it follows (a constant ratio, difference or product), write it as an equation, plot a changed variable such as 1 ÷ m so that a curve becomes a line, and read the gradient as a quantity with units.
Slot the skill into your science lessons
Every data set in the year asks which representation shows it best and which summary or relationship processes it. This map answers both for the data in each Year 10 topic.
| When you teach | The data | The representation | Summary or relationship |
|---|---|---|---|
| Heredity: Cell Division and Inheritance (AC9S10U01) | Counts of offspring with each trait | A table of counts and a Punnett square as a model | The ratio of the counts, compared with 3 : 1 |
| Evolution by Natural Selection (AC9S10U02) | Share of each colour over several generations | A line graph of percentage against generation | The trend, and how fast it changes |
| The Big Bang and the Universe (AC9S10U03) | Galaxy distance and speed of moving away | A scatter graph with a line through the origin | v = H × d: the gradient is Hubble’s constant |
| Energy Flow and Climate Change (AC9S10U04) | Yearly mean temperatures over decades | A line graph, with a running mean to show the trend | The change per decade |
| Newton’s Laws of Motion (AC9S10U05) | Acceleration at different masses, same force | a against 1 ÷ m, to make the curve a straight line | a = F ÷ m: the gradient is the force |
| Atomic Structure and the Periodic Table (AC9S10U06) | Atomic radius across a period and down a group | A bar chart in periodic order, or a line graph | The trend, and the exceptions to it |
| Reaction Types and Rates (AC9S10U07) | Mass lost against time at three temperatures | Three curves on one set of axes | The gradient at the start: the initial rate |
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.
- See a relationship by changing the representation: “Acceleration of a fixed-mass trolley at rising force” as a line graph.
- Summarise repeats by their centre and spread: “Reaction time across five repeated runs”.
- Check each claim against the representation: “Test conclusions against the representations”.
Find the relationship hidden in a table
Test each table four ways. If y ÷ x stays the same, y is proportional to x (a line through the origin). If the steps in y stay the same for equal steps in x, the relationship is linear. If x × y stays the same, it is inverse. If y ÷ x² stays the same, y is proportional to the square of x. Then write the equation. The first table is done for you.
1. A 2 kg trolley pulled with rising force
| Force (N) | 2 | 4 | 6 | 8 | 10 |
| Acceleration (m/s²) | 1 | 2 | 3 | 4 | 5 |
Done for you: a ÷ F = 0.5 every time, so a is proportional to F: a = 0.5 × F. The steps are equal too, because a proportional relationship is a linear one through the origin.
2. Water heated at a steady rate
| Time (min) | 0 | 1 | 2 | 3 | 4 |
| Temperature (°C) | 20 | 26 | 32 | 38 | 44 |
3. Trolleys of different mass pulled with 6 N
| Mass (kg) | 0.5 | 1 | 1.5 | 2 | 3 |
| Acceleration (m/s²) | 12 | 6 | 4 | 3 | 2 |
4. A ball dropped from rest
| Time falling (s) | 0.2 | 0.4 | 0.6 | 0.8 |
| Distance fallen (m) | 0.196 | 0.784 | 1.764 | 3.136 |
Make the curve a straight line
A curve is hard to read; a straight line through the origin is easy, and its gradient means something. Trolleys of different mass were pulled with the same force. Plot a against m, then a against 1 ÷ m.
| Mass m (kg) | 0.5 | 1 | 1.5 | 2 | 3 |
| a (m/s²) | 12 | 6 | 4 | 3 | 2 |
| 1 ÷ m (per kg) | 2 |
B. Summarise repeats
Five runs of the same reaction took 22, 21, 24, 20, 23 s. A sixth run, with the tube knocked over and set up again, took 41 s.
Which graph does each equation draw?
Cut out the twelve cards. Six are equations; six are graphs drawn without numbers. Pair each equation with its graph. Then, for each pair, name a Year 10 experiment whose results could look like it.
Teacher note: the answer sheet lists the pairs and an experiment for each.
From a stretched spring to an equation
Each 50 g mass hung on a spring pulls down with a force of about 0.5 N. Pairs add masses one at a time, measure the extension, and record a table. The line graph of extension against force is straight and passes through the origin, so the extension is proportional to the force. Its gradient, in centimetres per newton, turns the table into an equation that predicts the extension for any force in the range.
We are learning to
- organise results in a table with units in the headings,
- choose the graph that shows a relationship and draw a line of best fit,
- read a gradient with its unit and write the relationship as an equation.
Success criteria
- I can say why a line graph suits these results better than a bar chart.
- I can write an equation for my spring and use it to predict an extension I did not measure.
You need
- for each pair: a spring and a clamp stand, six 50 g masses on a hanger, a metre ruler clamped beside the spring, safety glasses, and a soft mat under the masses,
- the relationship and straighten sheets (scaffolds 1 and 2), and the cards from scaffold 3, one set per table. Stop adding masses if the spring stops springing back.
Lesson flow (about 30 minutes)
| 10 min | Measure Pairs read the spring’s length with no mass, then after each mass, and work out the extension each time. Ask: “Why do we record the extension, not just the length? What should the extension be for no mass?” |
| 8 min | Represent Pairs draw extension against force and a line of best fit. Show the board (see the box). Ask: “Is the line straight? Does it pass through the origin? What would that tell us?” |
| 7 min | Equation Pairs find the gradient from two points far apart on their line, with its unit, and write extension = gradient × force. While the class compares springs, tables match the cards. Ask: “Two springs give different gradients. Which is the stiffer spring?” |
| 5 min | Predict Pairs predict the extension for 1.25 N, then check it with a mix of masses, and say how far beyond 3 N they would trust the equation. |
Running it shorter? Stop after the graph, and find the gradient and the equation in your next science lesson.
Watch for these ideas
- Plotting the spring’s length instead of its extension, so the line misses the origin.
- A gradient with no unit, or read from two table rows instead of two points on the line.
- Joining the points dot to dot instead of drawing a line of best fit.
Make it easier, make it bigger
- Easier: give pairs a table with the force column filled in and the graph axes drawn and numbered.
- Bigger: hang two springs end to end, then side by side. Predict each new gradient before measuring it.
Answers and look-fors
The next sheet has the four relationships worked, the straightened graph and its gradient, the summaries, the card pairs, and a quick three-level guide.
Answers and look-fors
Which relationship?
| Table | The test that stays the same | Equation |
|---|---|---|
| A 2 kg trolley pulled with rising force | y ÷ x: 0.5, 0.5, 0.5, 0.5, 0.5 | a = 0.5 × F (a = F ÷ m with m = 2 kg) |
| Water heated at a steady rate | the steps in y: 6, 6, 6, 6 | T = 20 + 6t |
| Trolleys of different mass pulled with 6 N | x × y: 6, 6, 6, 6, 6 | a = 6 ÷ m |
| A ball dropped from rest | y ÷ x²: 4.9, 4.9, 4.9, 4.9 | d = 4.9t² (half of 9.8, the acceleration of falling) |
The second table is linear but not proportional: the steps are equal, but it starts at 20 °C, not zero.
Straighten and summarise
1 ÷ m = 2, 1, 0.667, 0.5, 0.333 per kg. Against m the points fall in a curve; against 1 ÷ m they lie on a straight line through the origin with gradient 6 m/s² per (1/kg), which is 6 N: the force that pulled every trolley, since a = F × (1 ÷ m). Five runs: mean 22, median 22, range 4 s. With the sixth: mean 25.17, median 22.5, range 21 s. The mean and range jump; the median barely moves. Report the five runs and note the sixth separately with its reason (the knocked tube), or report the median.
Equation and graph cards (pairs)
| Cards | Equation | Graph | An experiment like it |
|---|---|---|---|
| G and A | y = 3x | a straight line through the origin | force and the extension of a spring |
| D and J | y = 2x + 5 | a straight line that starts at 5 | temperature of water heated at a steady rate (it does not start at zero) |
| E and L | y = 12 ÷ x | a curve that falls steeply, then levels off | acceleration and mass at a fixed force |
| B and H | y = x² | a curve through the origin that grows steeper | distance fallen and time |
| K and F | y = 10 − 2x | a straight line that falls | length of a candle burning at a steady rate |
| I and C | y = 4 | a flat line: y does not depend on x | swing time of a pendulum and the mass of its bob |
A quick three-level guide
| Move | Working towards | At standard | Beyond |
|---|---|---|---|
| Representation | draws whichever graph comes to mind | chooses a line graph for continuous data and a table with units | changes a variable (such as 1 ÷ m) so a curve becomes a straight line |
| Relationship | describes a trend in words only | tests a table and writes the equation it follows | reads the gradient as a physical quantity with its unit |
| Summary | reports every run, or only the mean | reports a centre and a spread for repeats | chooses the median when an unusual run would mislead, and says why |
In the mini-lesson, extension against force is a straight line through the origin until the spring is overstretched; the gradient, in cm per N, is larger for a softer spring, and two springs end to end double it.