Representations and Descriptive Statistics: a skill companion

A small set of reusable sheets that grow one inquiry skill: choosing the representation that fits the data and the purpose, then processing the data into descriptive statistics and a mathematical relationship that can be calculated with, including where that relationship stops working. 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 science units a class teaches all year, from Body Systems to Chemical Equations. 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

1. Read the pairing map on the next page: it shows a representation and a statistic or relationship for each topic.
2. Print the representation chooser whenever a class has data to show, and the processing sheet when it has data to summarise.
3. Cut out the cards once. They are reused all year.
4. Open the free interactive unit on your board when you want a worked example of the skill.
5. 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 8

In Year 8 students chose a table or a graph to show a pattern. In Year 9 they also process the data: a mean, a median or a range that sums it up (and the choice between them when one value is unusual), and a mathematical relationship, such as y = kx, that turns a trend into a rule. The rule is a model, so it comes with a range where it holds.

Slot the skill into your science lessons

Every data set in the year can be organised and processed. This map shows, for data from each Year 9 topic, a representation that organises it and a statistic or relationship that processes it.

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.

• Match the representation to the purpose: “Energy use of six household appliances”, as a table and as a bar chart.
• Choose the statistic that represents the data: “Standby power of six household devices”, where one device pulls the mean up.
• Turn a trend into a rule, and know its limits: “Read the graph: find the reading that breaks the trend” and the conclusions to judge.

Which representation, for which purpose?

The same data can be shown in many ways. The right representation depends on what you want it to do: keep every value, compare groups, show a change, show whether two things go together, sum the data up, or calculate a new value.

A. Match the purpose

B. My own data

Teacher note: a graph needs a title, both axes named with units, and an even scale. A bar chart separates its bars; a line graph or a scatter plot uses a number line on both axes.

From readings to a rule, and its limits

Processing turns readings into something you can use: a ratio that stays the same becomes an equation, the equation predicts, and the points where it stops fitting show the edge of the model.

A. The readings

Bicarbonate of soda was added to 75 mL of vinegar in an open cup, and the cup was weighed before and after the fizzing. The mass lost is the carbon dioxide that escaped.

B. The rule and its limit

Fill the last column to two decimal places, and rule the line through zero that fits the steady points.

C. Predict

D. Sum it up

Five groups each added 2 g; one group spilled some liquid. Mass lost (g): 0.93, 0.95, 0.94, 0.62, 0.96.

A table, a graph, or a statistic or equation?

Cut out the cards. Lay down the three grey heading cards, then sort the purposes under them. On the blank cards, write a purpose from your own investigation for each heading.

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

One data set, four representations

Use this stand-alone lesson to teach the skill on its own, before you fold it into a science topic. Pairs measure the height of stacks of 1, 2, 4, 6, 8 and 10 identical washers. The same readings become a table, a scatter plot with a straight line through zero, an equation (height = k × number of washers, where k is the thickness of one washer), and descriptive statistics of the class’s values of k.

We are learning to

• construct a table, a graph, a statistic and an equation from the same readings,
• choose the representation that fits a purpose,
• use an equation to predict, and say where it would stop working.

Success criteria

• I can write the equation for my stack and use it to predict the height of 25 washers.
• I can say which representation answers which question about our data.

You need

• for each pair: ten identical metal washers and a ruler marked in millimetres,
• the representation chooser and the processing sheet (scaffolds 1 and 2), one each per student, and the cards from scaffold 3, one set per table.

Lesson flow (about 30 minutes)

Running it shorter? Stop after the graph and the rule, and sum up the class and sort the cards in your next science lesson.

Watch for these ideas

• Joining the points of a scatter plot dot to dot instead of ruling one line of best fit.
• Reporting the mean without looking for a value that pulls it away from the rest.
• Using an equation far outside the range that was tested, as if it had no limits.

Make it easier, make it bigger

• Easier: give the axes already scaled, and a table with a column for height ÷ number of washers.
• Bigger: mix two sizes of washer in one stack and write an equation with two terms.

Answers and look-fors

The next sheet has the representation choices, the worked rule and statistics for the bicarbonate of soda data, the card key, and a quick three-level guide.

Answers and look-fors

Representation chooser

Process the data

Lost ÷ added: 0.46, 0.47, 0.47, 0.47, 0.47, 0.41. The first 5 ratios are steady at about 0.47, so mass lost = 0.47 × mass added, for up to 5 g added. The last point (2.45 g at 6 g) falls below the line because the vinegar is used up: extra bicarbonate of soda has nothing left to react with. For 3.5 g the rule predicts 0.47 × 3.5 ≈ 1.6 g. For 8 g the rule would say 3.76 g, but the vinegar runs out first, so about 2.45 g, the level the points reached. Five groups at 2 g: mean 0.88, median 0.94, range 0.34 g. The median represents a typical group: the spill pulls the mean down and stretches the range. A Beyond answer notes that the reaction makes 0.52 g of carbon dioxide for each gram of bicarbonate of soda, and the cup lost less because some gas stays dissolved in the liquid.

Purpose cards

A quick three-level guide

In the mini-lesson, k is the thickness of one washer, and the height of 25 washers is 25k. The class median of k is the safer summary if one pair measured a bent or odd washer.