At a glance. One lesson of about 60 minutes. A comparison of two prices introduces the idea before any method. A table finds the shared pair, then the figure shows it as a crossing and shows that equal rates give no crossing.
Calculator: no calculator. Nothing here needs one. Every table value is a small whole number, and the crossing is found by comparing, not by computing.
What students will learn
Know
- A system of linear equations is two or more linear equations in the same unknowns.
- The solution of a system is a pair (x, y) that makes every equation true.
- On a graph, the solution is where the lines meet.
Understand
- Every point on a line is a solution of that line's equation, so a point on both lines solves both.
- A table, a graph and a point of intersection are three views of one fact.
- A graph can show when there is no crossing at all.
Do
- Find the solution of a system from a table of values.
- Read the solution from the graph of two lines.
- Interpret the crossing point of two lines in a situation.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: a solution as a pair, found by table or by graph (questions 1 to 8).
- Criterion B. Investigating patterns: use the figure to find what must be true of the rates for the lines to be parallel.
- Criterion C. Communicating: write the solution as a pair and check it in both equations.
- Criterion D. Applying mathematics in real-life contexts: the charging stalls in the hook, the plans in question 3 and the school stall in question 6.
The one thing to do with the figure
Run the table before the figure. Students find 5 hours and 30 taka with a table, then see the same point as a crossing. Ask them to set line A to y = 4x + 10 and line B to y = 6x on the figure and say what they see.
Then ask for a pair of lines that never meet. Let them find equal rates by themselves. The next lesson names the three cases.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook; students predict, then build the table in pairs. |
| 10 min | The figure: reproduce the table result, then explore parallel lines. |
| 5 min | Key idea; the warning about giving one number. |
| 5 min | Faded example: the table for y = x + 12 and y = 4x. |
| 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 | 14; the gradient; x = 5. |
|---|---|
| Charging stalls | 5 hours, 30 taka. |
| Faded example | (4, 16). |
| 1. Meaning of the solution | A. B splits the pair between the equations; C swaps x and y; D confuses a crossing with the same line. |
| 2. Read y at the crossing | 11. 2 × 4 + 3 = 11 and 4 + 7 = 11. |
| 3. Meaning of a crossing in context | A. B has the order of the two plans reversed; C swaps x and y; D treats the gap as fixed. |
| 4. Find the crossing from a table | 2. Both rows show 12 at x = 2. |
| 5. Close a gap between two lines | 4. Gap 6, gain 3 a step, so 2 steps after x = 2. |
| 6. Break-even as a crossing | 4. 40 ÷ 10 = 4 items. |
| 7. Find the crossing from two points on each line | 2.4. 2x + 2 = −3x + 14; 5x = 12; y = 6.8. |
| 8. A solution satisfies both equations | B, (3, 7). A satisfies only the first equation; C swaps the pair; D satisfies only the second. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Treated the solution as split between the equations, swapped x and y, or mixed it up with the same line. | Substitute the point into each equation and see both are true. |
| 4, 8 or 7 in question 2 | Gave x, left out the constant, or used the constant instead of the value. | Write “at x = 4, y = …” and substitute. |
| B, C or D in question 3 | Misread which plan is cheaper before the crossing, or swapped x and y. | Work out each plan at 0 months and at 10 months before deciding. |
| 12, 6 or 3 in question 4 | Gave a y value or a value from the table instead of x. | Underline x in the question and read the row above the matching numbers. |
| 2, 6 or 9 in question 5 | Gave the number of steps, the gap, or added the rates. | Draw the two lines on paper and see the gap close. |
| 40, 1.6 or 2.67 in question 6 | Divided by the wrong amount. | Ask what one item adds to the gap, then how many items cover the start-up. |
| −12, 12 or 6.8 in question 7 | Gradient sign error, stopped before dividing, or gave y. | Write y = mx + c for both lines before equating. |
| A, C or D in question 8 | Tested one equation only, or swapped the pair. | Test each option in both equations, and cross out as soon as one fails. |
Questions worth asking
- How do you know the crossing is a solution of both equations? Listen for “every point on a line is a solution of that line”.
- Can two lines cross twice? No: two points fix a straight line, so two crossings means the same line.
- Why is a table not always enough? The crossing might not be on a whole number, or might be off the table.
- What does 5 hours and 30 taka mean for the customer? For 5 hours both stalls cost the same; longer, A is cheaper.
Differentiation
A way in for anyone stuck
- Give a pre-drawn table with the first two rows filled in.
- Use whole-number rates and starts, with a crossing on the first screen.
- Let students check an answer by substituting before asking for help.
Stretch
- Question 7, then: find the y-coordinate and check it in both equations.
- Design two plans that cross at (6, 40): one with a fee and one without.
Students learning in English as an additional language
“System” here means “a set of equations that go together”. “Solution” is the answer that works in all of them. “Cross” and “intersect” mean the same on a graph.
| Word | Plain meaning | Example |
|---|---|---|
| system of equations | two or more equations with the same unknowns | y = 2x + 1 and y = 3x − 2 |
| solution | values that make every equation true | (3, 7) solves both equations above |
| point of intersection | the point where two lines meet | the lines cross at (3, 7) |
| parallel | lines that never meet | y = 3x + 6 and y = 3x + 20 |
| rate | how much y changes for each step in x | 6 taka an hour |
| coordinates | the pair (x, y) that names a point | (5, 30) is 5 across and 30 up |
Sentence starters for explaining aloud or in writing:
- At x = … the first line gives … and the second gives …
- The lines cross at (…, …), which means …
- I know it is right because I put both values into both equations.
Two things not to say
“Just find where they cross.” Without the pair and the check, students report one number.
“Look at the graph and read it.” A graph gives an estimate unless the crossing is on a grid corner.
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.
- Check a pairIs (2, 7) a solution of y = 3x + 1 and y = x + 5?
Yes: 3 × 2 + 1 = 7 and 2 + 5 = 7. - Read a tableA gives 4, 7, 10, 13 and B gives 16, 13, 10, 7 for x = 0 to 3. Where do they meet?
x = 2, y = 10.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA taxi charges 30 pesos plus 5 a kilometre. Another charges 8 a kilometre with no starting fee. After how many kilometres do they cost the same?
10 km, at 80 pesos. - Earlier workFind the y-intercept and gradient of y = 3x − 4.
−4 and 3.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningTwo lines both pass through (0, 5) and have different gradients. Where do they cross?
At (0, 5), the shared point. - Working backwardsWrite two equations that cross at (2, 9).
Any correct pair, for example y = 3x + 3 and y = 4x + 1.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
All four 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 on squared paper with a table of three values each.