At a glance. One lesson of about 60 minutes. A contradictory pair of reports opens the hook. The figure fixes one line and lets students change the other until they have seen all three cases, and the panel's gradients let them state the rule themselves.
Calculator: no calculator. Nothing here needs one. The ratios between coefficients are small whole numbers or one half.
What students will learn
Know
- A system of two linear equations has one solution, no solution or infinitely many.
- Different gradients give one solution; same gradient with different intercepts give none; the same line gives infinitely many.
- In standard form, compare the coefficients and the total as a ratio.
Understand
- Parallel lines have no point in common, and the same line has every point in common.
- The number of solutions can be predicted before any method is used.
- The same gradient alone is not enough to say there is no solution.
Do
- Predict the number of solutions from the gradients and intercepts.
- Rewrite an equation to compare it with another.
- Find a coefficient or total that gives a stated number of solutions.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the three cases and how each is recognized (questions 1 to 8).
- Criterion B. Investigating patterns: find, with the figure, every second equation of the form ax + 3y = c that gives each case, and say what a, b, c must satisfy.
- Criterion C. Communicating: justify a count using the words gradient, intercept, parallel and the same line.
- Criterion D. Applying mathematics in real-life contexts: the contradictory reports in the hook and what no solution means for a business.
The one thing to do with the figure
Ask for a prediction before each change. “What happens to the line if I double every term?” Then: “What if I double only the left side?” Students should be able to say “same line” and “parallel” before the figure shows it.
Let them find a second equation of their own for each case, then swap with a partner who has to name the case without looking at the graph.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook: two reports; students predict and argue. |
| 10 min | The figure: find all three cases; fill in the three-case card. |
| 10 min | Worked examples in y = mx + c and standard form; the faded example in pairs. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 8, on paper or on screen. |
The answers
| Warm-up | y = −(2/3)x + 2; yes; yes. |
|---|---|
| Worked example 1 | No solution. |
| Worked example 2 | Infinitely many solutions. |
| Worked example 3 | One solution, (3, 11). |
| Faded example | No solution. |
| 1. Parallel lines in standard form | B, no solution. A assumes every system has a point; C confuses parallel with the same line; D is not possible for straight lines. |
| 2. Count solutions from gradients | 1. The gradients differ; the lines meet at (0, 5). |
| 3. Parallel pair | B. A swaps the numbers; C shares the intercept; D compares a sloping line with a flat one. |
| 4. The gradient that makes lines parallel | 5. The gradient of the second line is 5. |
| 5. Standard form: same line | C. 1.5 × (2x + 4y = 10) is 3x + 6y = 15: the same line. |
| 6. The total that makes the same line | 30. 3 × 10. |
| 7. A coefficient that makes lines parallel | 4. 4x + 6y is double 2x + 3y; 12 ≠ 9. |
| 8. What no solution means | A. B gives one solution; C gives infinitely many; D describes a vertical line, which still meets most lines. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Assumed every pair of equations meets, or confused parallel with the same line. | Rewrite both in y = mx + c and compare gradients, then intercepts. |
| 0 or 2 in question 2 | Thought equal intercepts mean no crossing, or counted lines as solutions. | Plot the two lines: both pass through (0, 5), so they cross. |
| A, C or D in question 3 | Matched the numbers rather than their positions, or compared a sloping line with a flat one. | Underline the number in front of x in each equation. |
| 3, −2 or −5 in question 4 | Used an intercept or the opposite of the gradient. | Say “gradient” and point to the number multiplying x. |
| A, B or D in question 5 | Compared coefficients but not the total. | Divide the second equation by 3: x + 2y = 5, which is half the first. |
| 10, 3 or 14 in question 6 | Copied the old total, gave the multiplier, or added instead of scaling. | Write the multiplier beside the arrow, then apply it to both parts. |
| 2, 6 or 8 in question 7 | Copied a coefficient or added two, instead of scaling. | Find the factor from the y coefficients: 3 to 6 is × 2. |
| B, C or D in question 8 | Mixed up the three cases. | Sketch the three pictures: crossing, parallel, on top of each other. |
Questions worth asking
- Can you tell the number of solutions without solving? Yes, from gradients and intercepts. Listen for the three cases.
- Why can two straight lines not meet exactly twice? Two points fix a line, so two shared points means the same line.
- What does “no solution” mean for the co-op? The two reports contradict each other: the prices cannot satisfy both.
- What does “infinitely many” mean? The two facts say the same thing: you do not have enough information.
Differentiation
A way in for anyone stuck
- Give a three-case card: one, none, infinitely many, each with a sketch and an example.
- Write every equation in y = mx + c form before comparing.
- Let students test a candidate answer by drawing the two lines on squared paper.
Stretch
- Question 7, then: find b so that 2x + 3y = 6 and 4x + by = 12 has exactly one solution.
- Why is there exactly one value of c that makes ax + by = c the same line as 2x + 3y = 6, given a and b?
Students learning in English as an additional language
“Parallel” means “going the same way and never meeting”. “The same line” means two equations that describe one line. “Infinitely many” means “too many to count and never ending”.
| Word | Plain meaning | Example |
|---|---|---|
| gradient | how steep a line is | in y = 4x − 1 the gradient is 4 |
| intercept | where a line crosses the y-axis | in y = 4x − 1 the intercept is −1 |
| parallel | lines with the same gradient that never meet | y = 4x + 5 and y = 4x − 1 |
| infinitely many | more than any number you could count | every point on a line |
| coefficient | the number multiplying a letter | in 6x the coefficient of x is 6 |
| consistent | the equations can be true at the same time | x + y = 6 and x − y = 2 |
Sentence starters for explaining aloud or in writing:
- The gradients are … and …, so …
- The first equation multiplied by … is …, which matches / does not match the second.
- There are no solutions because the lines are …
Two things not to say
“Same gradient means no solution.” It also means the same line when the intercepts match.
“Two equations always give one answer.” That is the belief the lesson takes apart.
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.
- CountHow many solutions: y = 3x + 1 and y = 3x + 4?
None: parallel. - CountHow many solutions: y = x + 2 and y = 2x + 2?
One, at (0, 2).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA shop says: 2 notebooks and 3 pens cost 18 pesos. Another shop says: 4 notebooks and 6 pens cost 36 pesos. Do the two reports give the prices?
No: they say the same thing, so there are infinitely many possible prices. - Earlier workSolve 3(x + 2) = x + 14.
x = 4.
3Challenge
Reasoning, generalizing and working backwards.
- Reasoningx + y = 5 and 2x + 2y = k. For what k is there no solution, and for what k infinitely many?
None when k is not 10; infinitely many when k = 10. - Working backwardsWrite two equations with exactly one solution at (2, 3).
Any correct pair, for example x + y = 5 and x − y = −1.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
All three sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under the figure, “Show me the steps” gives the same states in words.
No device? Plot both lines from a table of two points each on squared paper.