Function tables and reading real-data graphs
Evaluating a linear rule, finding the rule from a table of values, plotting the pattern, and reading a rate from a real-world graph
About three to four lessons of 45 to 60 minutes
Every rule has a table, and every table has a graph
A taxi charges a flat $5 fee plus $2 for every kilometre. A phone plan gives you 2 GB of data included, then charges for every extra GB. A garden tap fills a water tank at a steady rate. Every one of these stories is a function: put a number in (kilometres, GB, hours), follow the rule, and a number comes out (the cost, the total data, the litres in the tank).
Once a rule is turned into a table of input-output pairs, the same information can be drawn as a graph, and the shape of that graph tells its own story. A straight line means a constant rate: the same amount is added (or taken away) every single step. This unit connects the rule, the table and the graph so you can move between all three, and read a real rate straight off a description with no equation given at all.
- Taxi fare ruley = 2x + 5, a flat fee plus a rate per kilometre
- Filling a tank at a steady ratethe litres added each hour is the gradient
- A table of x and y valuesthe constant difference between y-values is the rule's rate
- A straight-line graphwhere it crosses the y-axis is the start value
What students will be able to do
Students will evaluate a linear rule for a given input, read and build a table of values, find the rule from a table by identifying the constant difference and the starting value, plot a table of values on the Cartesian plane, and find a rate of change from a description of a real-world graph with no rule given.
- I can substitute a value of x into a rule y = mx + c to find y.
- I can read a table of values and describe how y changes each time x increases by 1.
- I can find the gradient and starting value from a table and write the rule y = mx + c.
- I can plot a table of values on the Cartesian plane.
- I can find a rate of change from a real-world description, without being given the rule.
Standards this unit teaches
- AC9M7A04Australian Curriculum v9 (ACARA)Read graphs of real data
Describe the relationship between variables shown in graphs of functions drawn from real data.
- AC9M7A05Australian Curriculum v9 (ACARA)Patterns, tables and graphs
Generate tables of values from visually growing patterns or the rule of a function, then describe and plot these relationships on the Cartesian plane.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Year 7 linear equations & expressions teaching unitsubstitution into a rule is exactly the skill section 1 uses to evaluate a function
- Grade 6 coordinate plane worksheetsplotting an (x, y) point, the skill section 3 builds on
- Rate in the glossarya related refresher on comparing one quantity's change to another's
Words to teach and display
- Function
- a rule that gives exactly one output for every input
- Input / output
- the value that goes into a rule, and the value the rule gives back
- Table of values
- a list of matching input-output pairs generated from a rule
- Gradient
- how much y changes each time x increases by 1, the constant rate of a linear rule
- y-intercept
- the value of y when x is 0, where the graph crosses the y-axis
- Cartesian plane
- the grid formed by a horizontal x-axis and a vertical y-axis, used to plot (x, y) points
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. What a function machine does: input, rule, output
ConcretePicture a function as a machine: a number goes in, the machine applies a fixed rule, and a number comes out. The rule y = mx + c means: multiply the input x by m, then add c. Every input has exactly one output, no matter how many times you run the machine.
Reading the rule aloud helps: y = 3x + 2 means 'take x, multiply by 3, then add 2'. The order matters, because multiplication happens before the addition, exactly as in the order of operations.
A rule is y = 4x - 3. Find y when x = 5.
- Multiply x by 4: 4 x 5 = 20.
- Subtract 3: 20 - 3 = 17.
Answer: y = 17.
- What would y be if x = 0 in this rule?
- How does y change each time x increases by 1?
2. Finding the rule from a table of values
PictorialA table hides its rule in two places: the constant difference between consecutive y-values is the gradient m, and the y-value when x = 0 is the starting value c. Find both, and the rule y = mx + c is complete.
Always check the difference between EVERY consecutive pair, not just the first one; a table only comes from a linear rule if that difference stays exactly the same all the way along.
A table of values follows a rule y = mx + c: (0, 5), (1, 9), (2, 13), (3, 17). Find the rule.
- Find the differences between consecutive y-values: 9 - 5 = 4, 13 - 9 = 4, 17 - 13 = 4. The difference is always 4, so the gradient m = 4.
- Read off the y-value when x = 0: c = 5.
- Write the rule using m and c: y = 4x + 5.
Answer: y = 4x + 5.
- Why do you need to check more than one difference before trusting the gradient?
- Which row of the table gives you c directly?
3. Plotting a table on the Cartesian plane
PictorialEvery (x, y) pair in a table is a point that can be plotted. When a rule is linear, every one of those points lands on the same straight line, and the steepness of that line is the gradient.
Use the graph above to find the gradient and y-intercept of the line, without looking back at the table.
- Read where the line crosses the y-axis: it crosses at y = 5, so c = 5.
- Pick two neighbouring points, such as (0, 5) and (1, 9): y increases by 4 while x increases by 1.
- The gradient is that rise-over-run: m = 4.
Answer: Gradient m = 4, y-intercept c = 5, matching y = 4x + 5.
- If the line were steeper, would the gradient be bigger or smaller?
- Where would the line cross the y-axis if c were negative instead?
4. Reading a rate of change from a real-world graph description
AbstractSometimes there is no table and no rule, just a description of what a graph shows: a starting amount and an amount reached after some time, with a constant rate in between. Find the rate by dividing the total change by the time it took.
The same move works for any 'starting amount, constant rate, amount after some time' story: subtract to find how much changed, then divide by how long it took to change.
A graph shows that after 5 hours, a water tank holds 340 litres, having started at 90 litres and filling at a constant rate. Find the fill rate in litres per hour.
- Find the total litres added: 340 - 90 = 250 litres.
- Divide by the time it took: 250 / 5 = 50.
Answer: 50 litres per hour.
- Which number in the story is the starting value (c), and which is the rate (m)?
- How would the calculation change if the tank were being drained instead of filled?
Common misconceptions and how to address them
MisconceptionReading c from the first row shown in a table, even when that row's x-value is not 0.
Why it happens: Students assume 'the first row' always means 'the starting value', without checking which x-value it actually has.
How to address it: Underline the row where x = 0 before reading off c. If no such row is shown, work backwards or forwards along the constant difference until you reach x = 0.
MisconceptionSwapping the gradient and the y-intercept when writing the rule from a table, e.g. writing y = 5x + 4 instead of y = 4x + 5.
Why it happens: Both numbers come from the same table, so it is easy to mix up which one multiplies x and which one is added once.
How to address it: Label each number as you find it: 'm is how much for every step', 'c is the start, before any steps'. Say both labels out loud before writing the rule.
MisconceptionAssuming a table is linear from just the first difference, without checking every difference is the same.
Why it happens: One matching difference feels like enough evidence, especially under time pressure.
How to address it: Make checking every consecutive difference a required step, not optional. A table with even one different gap does not have a single constant gradient.
MisconceptionOn a graph, confusing a steep line with a line that has a big y-intercept.
Why it happens: Both 'steepness' and 'starting height' make a line look bigger on the page, so the two ideas blur together.
How to address it: Find them one at a time and in a fixed order: first mark where the line crosses the y-axis (that is c alone), then separately measure the rise over the run between two points (that is m alone).
MisconceptionIn a real-world rate question, dividing the final amount by the time instead of the CHANGE by the time, e.g. computing 340 / 5 instead of (340 - 90) / 5.
Why it happens: The final amount is the number stated last in the sentence, so it feels like the one to use.
How to address it: Always subtract the starting amount from the final amount first, to isolate how much was actually added, before dividing by the time.
Guided practice (with answers)
1. A rule is y = 2x + 7. Find y when x = 6.
Answer: 19, because 2 x 6 = 12, and 12 + 7 = 19.
2. A table shows (0, 3), (1, 6), (2, 9), (3, 12). Find the rule.
Answer: y = 3x + 3, because the difference is always 3 and the value at x = 0 is 3.
3. A rule is y = 5x - 4. Find y when x = 0.
Answer: -4, the y-intercept, because 5 x 0 = 0, and 0 - 4 = -4.
4. A table shows (0, 20), (1, 17), (2, 14). Find the gradient.
Answer: -3, because y decreases by 3 every time x increases by 1.
5. Find the gradient and y-intercept of the line through these points, then write the rule.
Answer: m = 3, c = 2, so y = 3x + 2, because y rises by 3 for every 1 across and the line crosses the y-axis at 2.
6. A tank starts with 40 litres and after 4 hours holds 200 litres, filling at a constant rate. Find the fill rate.
Answer: 40 litres per hour, because (200 - 40) / 4 = 160 / 4 = 40.
Independent practice worksheets
Set the matching ChalkBee worksheet for independent work. The answer keys are computed in code, so they are never wrong.
Differentiation
- Keep every table starting at x = 0 until finding c directly from the table feels automatic.
- Pre-draw a 'difference' row under the table so the constant gap is visible before asking for the rule.
- Limit gradients to positive whole numbers before introducing negative (decreasing) rates.
- Give the plotted points already marked, and ask only for the connecting line and the reading of m and c.
- Use tables and stories with negative or fractional gradients.
- Compare two functions given in different forms (one as a table, one as a real-world description) without either being given as an equation.
- Pose a two-step real-world rate problem, such as a plan with an included amount before the constant rate begins.
- Ask students to invent their own table, equation and real-world story that all describe the same line.
Assessment: exit ticket
A short exit ticket covering evaluating a rule, finding a rule from a table, and reading a real-world rate.
1. A rule is y = 6x - 1. Find y when x = 3.
Answer: 17, because 6 x 3 = 18, and 18 - 1 = 17.
2. A table shows (0, 8), (1, 11), (2, 14). Write the rule.
Answer: y = 3x + 8, because the difference is always 3 and the value at x = 0 is 8.
3. A car park charges a $5 flat fee plus $2 per hour. Show that the total after 4 hours is $13 using the rule y = 2x + 5.
Answer: Substitute x = 4: y = 2(4) + 5 = 8 + 5 = 13, so the total is $13, matching the rule.
Teacher notes and timings
- Rough timing: Lesson 1 evaluating a rule (section 1), Lesson 2 finding the rule from a table (section 2), Lesson 3 plotting and reading the graph (section 3), Lesson 4 real-world rate problems and the exit ticket (section 4 and assessment).
- AC9M7A04 and AC9M7A05 previously had no dedicated Year 7 lesson; the nearest existing unit, Grade 8 linear functions above, is a genuinely different (and more advanced) lesson that only linked to this worksheet as 'the closest direct practice available', not core content.
- Keep the vocabulary consistent with Grade 8 linear functions ahead of time (gradient, y-intercept) so students already have the words when slope is formalised next year.
- Present mode and print both work: use the Print button for a clean teacher copy or a student handout, and project the page to teach straight from the graph.