Use one relationship and keep asking the same mathematical question: Where are the fixed term and rate in this representation?
A simple context is a fixed hire charge plus a cost per hour. Represent it in a value table, a straight-line graph and an equation. Praktikal can hold all three forms, the pupil responses, live result pattern and feedback in one lesson. It does not convert among the representations automatically.
Choose one notation convention, a meaningful domain and consistent units. Make the context, axes, scale, equation and distractors agree.
The same fixed charge and rate in three representations
Bike hire costs €6 once, then €3 for each hour: C = 6 + 3h , shown for 0 ≤ h ≤ 4.
Value table
h = time · C = total cost
| h (hours) | C (€) |
|---|---|
| 0 | 6 |
| 1 | 9 |
| 2 | 12 |
| 3 | 15 |
| 4 | 18 |
Straight-line graph
C, total cost (€)
h, time (hours)
Equation
The same relationship in symbols
C =6+3h
- Fixed term
- €6 when h = 0
- Rate
- €3 for each extra hour
Shown domain: 0 ≤ h ≤ 4
Establish the quantities and generate values
Name the quantities before writing symbols. For example:
- time hired, in hours;
- total cost, in euros;
- a fixed charge paid once;
- an additional cost for each hour.
In Praktikal, use text and formula content with either a static table and numeric questions or a data-table question. Each pupil completes their own table. Include zero hours and consecutive one-hour intervals so the fixed charge and hourly rate are visible in the values.
Ask pupils to complete a small set of values and explain what changes by the same amount. State the units and make clear which cells they should complete.
- Teacher decision
- if errors are arithmetic, correct them without abandoning the representation link; if pupils cannot explain the first value, return to the fixed charge.
Match the table to a graph and equation
Present several graph and equation choices. Use graph and formula blocks with a pairing, grouping or closed matching question.
Require one reason for the match:
- Which graph value corresponds to the fixed charge?
- How does the hourly rate appear between neighbouring table rows?
- Where are those two quantities in the equation?
Simply placing the three forms together does not mean pupils will understand how they connect. Prompt pupils to identify where the same quantities and relationships appear in each representation.
[1]Map the fixed term and rate explicitly
During live delivery, Praktikal shows answers and progress. Use the response pattern to choose one connection for whole-class discussion.
Temporarily mark the fixed term and rate consistently across the table, graph and equation. Pair colour with text labels, line styles or shapes so the meaning does not depend on colour vision.
Ask pupils to complete the sentence: “The ___ in the context is the ___ in the table, the ___ on the graph and the ___ in the equation.”
The platform can collect the response; the teacher decides which connection needs attention. A correct match does not reveal all of the pupil’s reasoning.
Reject a plausible mismatch
Show an equation or graph with one feature correct and one wrong – for example, the correct hourly rate but an incorrect fixed charge. Ask pupils to select the mismatch and explain precisely where it conflicts with the table, graph or context.
Use a closed item for the selection and a separate short response for the explanation. Automatic scoring applies to the closed component only; review the explanation yourself. Praktikal does not automatically diagnose which connection failed.
- Teacher decision
- revisit the first value if the fixed term is missed; compare equal input intervals if the rate is missed; continue when pupils can use one representation to test another.
Remove the cues and change the starting point
For the transfer check, remove the temporary labels and colours. Start with a new graph or context and ask pupils to construct or match the table and equation.
Use data-table, numeric, formula and matching questions, then review the results. Define how equivalent equation forms will be accepted. Praktikal does not generate the graph or convert between representations automatically.
Review the results, give feedback and decide whether the next lesson should revisit fixed terms, rates, scales, units or domains.
Removing a cue creates a stronger check than repeating the colour match, but one response still does not prove durable transfer.
[1] [2]Source notes
Teaching Strategies for Improving Algebra Knowledge US practice guidance supporting explicit attention to the structure of algebraic representations. It does not test this exact lesson sequence.
Improving Mathematics in Key Stages 2 and 3 EEF guidance on using representations, assessment and mathematical discussion. One response remains limited evidence for the next teaching decision.
Teacher feedback
You checked both the fixed charge and the rate.