Exploring Linear Relationships

Overview

Students use paired numerical data to construct a scatter plot, identify direction and strength of association, and interpret a linear line of best fit in context. The lesson develops prior learning about coordinates, scales, variables and interpreting graphs through a higher-order statistical investigation suitable for Year 10.

Learning intentions

• WALT create an accurate scatter plot from paired data.
• WALT describe the direction, strength and form of a relationship.
• WALT draw and interpret a linear line of best fit.
• WALT use evidence from a graph to make a sensible statement about the data.

Success criteria

• I can identify the explanatory and response variables and label both axes correctly.
• I can choose a sensible scale, plot all points accurately and include a clear title.
• I can describe a relationship using direction, strength, form and any unusual features.
• I can explain what the line of best fit suggests, including a prediction within the data range.

Curriculum links

• Mathematics and Statistics — Mathsteasers: higher-order thinking questions that challenge advanced learners and deepen understanding.
• Mathematics and Statistics — Mathsteasers / Alignment: applying relevant statistical ideas through connected, challenging tasks.
• Mathematics and Statistics — Mathsteasers / Additional resources for advanced learners: extending reasoning, interpretation and explanation.

Lesson structure (41 minutes)

1. 0–5 min · Hook and activate prior knowledge. Teacher opens the hook and comparison slides showing two scatter plots: one with a strong positive association and one with a weak association, then asks, “Which graph would be more useful for making a prediction, and why?” Students discuss with a partner and identify visible differences without using calculators.

2. 5–11 min · Explicit teaching. Teacher uses the key ideas slides to model the essential features of a scatter plot: two quantitative variables, explanatory variable on the horizontal axis, response variable on the vertical axis, appropriate scales, plotted ordered pairs, title and units. Teacher introduces the language “positive, negative or no association”, “strong, moderate or weak”, “linear” and “outlier”. Students annotate a small example and suggest a complete description.

3. 11–16 min · Model the line of best fit. Teacher displays the line-of-best-fit modelling slides and demonstrates drawing a balanced line through the overall trend rather than joining points or forcing the line through the origin. Discuss that approximately equal numbers of points should lie above and below the line. Students estimate two points on the line and explain how the line could be used to make an interpolation.

4. 16–29 min · Paired investigation. Teacher distributes the scatter plot and analysis worksheet to pairs and directs students to use the paired data provided. Students identify the variables, select scales, create the scatter plot, draw a line of best fit and answer questions about direction, strength, linearity, outliers and a prediction within the observed data range. Teacher circulates, checking axis labels, scale intervals and plotting accuracy.

5. 29–36 min · Compare and justify. Teacher uses the discussion and answer-check slides to reveal the same data and invites pairs to compare their lines of best fit. Students justify why slightly different lines may be reasonable, then write a four-part description: direction, strength, form and evidence. Pairs share one prediction and explain whether it is interpolation or extrapolation.

6. 36–41 min · Plenary and assessment. Teacher presents the plenary prompt and asks students to complete an individual exit response: “The relationship is … because … The line of best fit suggests …” Students hand in the response, including one warning about making predictions outside the data range.

Resources

• the complete scatter-plot lesson deck
• the scatter plot and analysis worksheet
• Rulers, pencils and erasers
• Graph paper or squared exercise books
• Calculators, if normally used by the class
• Visualiser or interactive whiteboard
• Highlighters for identifying evidence and unusual points

Assessment

• During modelling, ask students to explain why the explanatory variable belongs on the horizontal axis and why a line of best fit should not simply join points.
• During paired work, check three features: sensible scale, accurately plotted points and a balanced line of best fit. Question students about the meaning of their variables rather than accepting unsupported descriptions.
• Collect the individual exit response to assess whether students can use statistical vocabulary and interpret a prediction in context.

Differentiation

• Support students with a partially completed axis, a reduced data set, colour-coded variable headings and a step-by-step checklist on the worksheet. Provide sentence starters such as “There is a ___ association because …” and “As ___ increases, ___ tends to …”.
• Offer dyslexia-friendly reading options: use a clear sans-serif font, uncluttered spacing, short instructions, bold key words, read instructions aloud and allow students to explain responses orally before writing.
• Pair students strategically and provide a word bank containing positive, negative, no association, strong, weak, linear, outlier, interpolation and extrapolation. Accept labelled diagrams or speech-to-text for students who need an alternative recording method.
• For advanced learners, ask them to compare two possible lines of best fit, explain which is more balanced, estimate the gradient and intercept from their chosen line, and discuss why a strong association does not necessarily prove that one variable causes the other.

Extension

• Students create a new contextual question that could produce a positive or negative linear relationship, identify the variables and sketch the expected scatter plot.
• Students investigate an extrapolation scenario and explain why a prediction beyond the observed data range may be unreliable.