At a glance. One lesson of about 60 minutes. The first lesson of Unit 3 assumes coordinates in four quadrants and the idea of a constant rate from Year 8. Students decide whether a table is linear from the change in y for equal steps in x, including falling and flat relationships and tables whose x steps are unequal.
Calculator: no calculator. Nothing here needs one. The changes are small whole numbers and the point is the pattern between them.
What students will learn
Know
- A relationship is linear when equal steps in x give the same change in y.
- The constant change is the rate of change, and it can be positive, negative or zero.
- The points of a linear relationship lie on one straight line.
Understand
- Going in one direction is not the same as constant change: y = x² rises every step and is not linear.
- Two points are enough to fix a linear relationship, because every step in between shares the change equally.
- A table, a graph and a rule are three views of the same relationship.
Do
- Find the change in y for each step of x and decide whether a relationship is linear.
- Fill in a missing value or extend a linear table using the constant change.
- Handle unequal steps in x by dividing each change in y by its change in x.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the test for a linear relationship and a linear table (questions 1 to 7).
- Criterion B. Investigating patterns: the figure. Ask students to predict the changes row for each relationship before choosing it, then describe the pattern in the changes for y = x² (+1, +3, +5, +7) in their own words.
- Criterion C. Communicating: justifying a verdict with the changes written out, as in the worked example and question 9.
- Criterion D. Applying mathematics in real-life contexts: clean-up totals, taxi fares, a rain barrel and a falling pond level (the hook, the taxi, the rain barrel and question 4).
The one thing to do with the figure
Make students predict the change row before they move the slider. “Which of these will have the same number in every box of the change row?” Keep a tally of predictions, then settle each one with the figure.
Stop on y = x² and the doubling pattern: both rise at every step, so they are the cases that test “going up means linear”. Ask where the points leave the dashed guide line and why.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook: Team A and Team B. Collect predictions and reasons; do not settle them yet. |
| 10 min | The figure, one relationship at a time, with predictions of the change row first. |
| 10 min | Key idea; the taxi fare worked example; students finish the rain barrel in pairs. |
| 5 min | Question 1, the check. If many choose A or B, go back to y = x² in the figure. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | −5; Quadrant II; 9 cups. |
|---|---|
| Worked example | Linear: the fare rises 400 naira per km. |
| Faded example | Linear: −15 litres per hour, empty after 6 hours. |
| 1. Is this table linear? | C. A and B use direction or an average instead of checking that every change is the same; D thinks equal x steps are a problem. |
| 2. Constant change from a table | 5. The table is linear with a constant change of 5. |
| 3. Which table is linear? | C. A has changes 3, 5, 7; B has 2, 4, 8; D has 2, 3, 4. Only C has a constant change (−2). |
| 4. Negative change | −15. 105 − 120 = −15, 90 − 105 = −15, 75 − 90 = −15. |
| 5. Missing value in a linear table | 18. The change is 7 each step: 4, 11, 18, 25, 32. |
| 6. Value further along a linear table | 23. 3, 7, 11, 15, 19, 23. |
| 7. Unequal steps in x | 3. Each row pair gives 3 per unit of x. |
| 8. Working backwards in a linear table | −2. The change is 4 per step; two steps back from 6 is −2. |
| 9. Almost constant is not constant | B. A accepts a pattern with an exception; C uses direction only; D invents a rule about x = 0. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| 7 or 4 in question 2 | Gave a y value or the number of steps instead of a change. | Ask: subtract which two numbers? Write the first subtraction on the board and let them continue. |
| A or B in question 1; A, B or D in question 3 | Used direction, or a growth pattern, as the test. | Write the change row under the table and ask whether every box holds the same number. |
| 15 or 45 in question 4 | Dropped the sign of a fall, or used the whole fall instead of one step. | Ask whether the depth went up or down, and over how many hours the 45 happened. |
| 7 or 14 in question 5 | Answered with the change, or the gap across two steps. | Ask: which x is the blank at? How many changes take you there from the last known row? |
| 20 or 19 in question 6 | Forgot the starting value, or counted four steps for five. | Draw the steps between x = 0 and x = 5 as arrows and count them. |
| 6, 9 or 18 in question 7 | Did not divide the change by the size of the x step. | Add a column for the x change and ask for the change in y per 1 in x on each row. |
| 2 or 10 in question 8 | Went back one step, or forward. | Mark x = 0, 2 and 5 on a number line with the y values and move one step at a time. |
| A, C or D in question 9 | Accepted a mostly regular pattern, used direction, or thought x must start at 0. | Ask whether 13 to 17 is the same change as 10 to 13, and what that does to the points. |
Questions worth asking
- Can a linear relationship go down? Yes: the change is negative. Listen for “the line slopes down” or “minus 15 each hour”.
- Can the change be zero? Yes: y = 5 is linear, a flat line. Listen for “the same every time”.
- How many rows do you need to be sure a table is linear? To be sure from data you need to check all the changes; two rows alone only fix a line, they do not test one.
- Is a table with x = 1, 2, 4, 8 and a constant change in y linear? No, unless the change is 0: the same change over x steps of 1, 2 and 4 gives rates c, c/2 and c/4, which differ. Divide each change by its x step before comparing.
Differentiation
A way in for anyone stuck
- Give students a table with a blank change row to fill in before any question, and always ask them to finish it.
- Use only x steps of 1 at first: questions 1 to 6 can all be done by subtracting neighbours.
- Colour the change cells that match; the eye finds the break in a pattern quickly.
Stretch
- Question 7, then: how could you decide whether three points lie on one line using only changes? (Compare the change per unit of x across the two gaps.)
- Invent a table with x steps of 2, 3 and 5 that is linear with a rate of change of −4 per unit of x.
Students learning in English as an additional language
“Constant” means “stays the same” and “linear” comes from the word line. Say the sentence “the change is the same every time” each time a table is called linear.
| Word | Plain meaning | Example |
|---|---|---|
| relationship | a rule that links one quantity to another | the fare depends on the distance |
| linear | having a constant change, so the graph is a straight line | y = 2x + 1 is linear |
| rate of change | how much y changes for each step of 1 in x | 400 naira for each km |
| constant | staying the same every time | a constant change of 5 |
| table of values | pairs of x and y written in rows | x = 0, 1, 2 and y = 3, 5, 7 |
| change in y | the difference between two neighbouring y values | 12 − 7 = 5 |
Sentence starters for explaining aloud or in writing:
- The change in y is … every time, so the relationship is …
- It is not linear, because the changes are …
- For each step of 1 in x, y goes … by …
Two things not to say
“Linear means it goes up.” Many non-linear relationships rise, and some linear ones fall.
“Plot it and see if it looks straight.” A graph can look nearly straight; the change row is exact.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Changesx = 0, 1, 2, 3 and y = 2, 8, 14, 20. Find the change in y each step.
6. - VerdictIs y = 1, 2, 4, 7, 11 for x = 0 to 4 linear?
No: the changes are 1, 2, 3, 4.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA school's recycling bin weighs 12 kg, 17 kg, 22 kg and 27 kg after 0, 1, 2 and 3 weeks. What is the weekly change, and how much will it hold after 6 weeks?
5 kg per week; 12 + 6 × 5 = 42 kg. - Earlier work (Year 8)Which quadrant is (−5, −2) in, and which is (2, −7) in?
Quadrant III; Quadrant IV.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA linear table has y = 11 at x = 1 and y = 2 at x = 4. What is y at x = 0, and at x = 6?
Change −9 over 3 steps is −3 per step. At x = 0 it is 14; at x = 6 it is −4. - Unequal stepsx = 0, 2, 5 and y = 1, 7, 16. Is it linear? If so, what is the change per unit?
Yes: 6 over 2 is 3 and 9 over 3 is 3. The rate is 3.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to the slider, then the arrow keys) and by touch. Under it, “Show me the steps” gives the same six cases in words.
No device? Write each table on the board with a blank change row and fill it in together.