Grouped Shoe Size Histograms

Overview

Today students collect real-world data (shoe sizes), organise it into a grouped frequency table, and use that to construct a histogram. They will compare how different class intervals change the histogram and how that affects interpretation.

Learning intentions

• Students will collect discrete data from a real-world context and record it accurately.
• Students will group data into class intervals and calculate grouped frequencies.
• Students will draw histograms from grouped frequency tables, using frequency density where appropriate.
• Students will interpret histograms and explain how changing class widths affects the histogram’s shape and scale.

Success criteria

• I can create a grouped frequency table from raw data.
• I can calculate frequency for each class interval correctly.
• I can construct a histogram with correct axes and bar widths that match the intervals.
• I can compare two histograms made from different interval choices and explain how the appearance changes.

Curriculum links

• Construct and interpret diagrams for grouped discrete data, including histograms with equal and unequal class intervals (KS4 Statistics).
• Interpret, analyse and compare distributions using univariate empirical distributions (KS4 Statistics).
• Appropriate graphical representation involving grouped data (KS4 Statistics).

Lesson structure (60 minutes)

1. 0–5 min · Hook (real data). Teacher shows a quick prompt: “What shoe size do people in the class have?” and demonstrates how data will be collected. Students write a hypothesis (e.g., “Most people will have sizes around …”) and note one prediction.

2. 5–15 min · Collect data (real-world context). Teacher models safe data collection: students share shoe sizes they know (e.g., from their own label) and the teacher records all values on the board in a tally list. Students convert classmates’ shoe sizes into a list (or tally marks) of discrete values.

3. 15–25 min · Organise into a grouped frequency table. Teacher chooses a first set of class intervals (e.g., shoe size 3–4, 5–6, 7–8, 9–10 depending on the class range) and models how to define boundaries consistently. Students copy the teacher’s table layout, then sort the shoe sizes into intervals and calculate the frequency for each interval.

4. 25–33 min · Frequency and histogram basics (equal intervals). Teacher explains: for a histogram, the x-axis uses class intervals and the bar height represents frequency (or frequency density if class widths differ). Students check their intervals match the bar widths and draft axes on paper.

5. 33–42 min · Construct the histogram. Teacher draws the histogram for the first interval set: bars aligned to each class interval with heights based on the grouped frequencies (and confirms whether frequency density is needed—this first set uses equal widths if possible). Students draw their own histogram carefully, labelling axes (shoe size interval and frequency/frequency density).

6. 42–50 min · Compare interval choices (unequal intervals). Teacher proposes a second grouping with different class widths (e.g., narrower bins in the middle range and wider bins at the ends). Students re-use the same raw shoe size data, create a new grouped frequency table, and prepare to redraw a second histogram.

7. 50–58 min · Interpret and justify changes. Teacher leads a discussion: “Which histogram looks ‘taller’ and why?” and prompts students to explain that changing class width changes the vertical scale (frequency density) and can change the apparent shape. Students compare the two histograms and write 2–3 sentences: one describing what changes visually and one explaining why it happens mathematically.

8. 58–60 min · Quick check (exit). Teacher asks one final question: “If you use wider intervals, what must you consider so the histogram stays fair to compare?” Students answer on paper in one or two sentences.

Resources

• Class list/tally board (or spreadsheet) for shoe sizes
• Paper for grouped frequency tables and histograms
• Rulers and graph paper
• Pre-written interval options on cards (equal-width set and unequal-width set)
• Example histogram image (teacher-made, no links)
• Calculator (optional, for checking totals)

Assessment

• Teacher checks frequencies during table building (spot-checks intervals and counts).
• Teacher checks histogram construction: axis labels, correct bar widths, and correct use of frequency/frequency density.
• Exit question to confirm understanding of why interval width affects histogram interpretation.

Differentiation

• Support: Provide a partially completed grouped frequency table template and interval boundaries (e.g., “3 ≤ size < 5” style) so students focus on counting.
• Support: Sentence starters for the comparison statement: “The histogram looks more/less … because …”.
• Extension: Ask students to compute frequency density for unequal intervals and verify that the total area corresponds to the total frequency.
• SEN/EAL: Keep interval sets small, use colour coding for intervals, and allow verbal responses that the teacher converts into written explanations.

Exit ticket prompt (use in last 2 minutes)

“Your histogram has different-looking bars after changing the class intervals. What has changed, and how do you ensure the histogram is still correctly representing the data?”