What c does to a line
In y = mx + c, the number on its own moves the line up or down without changing its steepness, so lines with the same m are parallel.
Two phone plans with the same price per gigabyte but different monthly fees never meet: their graphs are parallel. The fee is c.
In the last lesson stayed at 1 and you changed , the steepness. Now stays at 2 and you change , the number on its own.
- Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
- Try: Try it on the board and compare the result with your prediction.
- Investigate: Work out why it happens, one step at a time.
- Modify: Change the problem on purpose and see what happens.
- Practise: A few on your own, to check you have it.
- Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
A bigger c
Work out when for each rule. Then think about what happens at .
Change c
Change and compare the solid line with the dashed line, . Watch the step, and the two rows of the table.
Set to 4. Was your prediction right?
Use the board. This ticks when you get there.
Reading c
Here is how to read from a rule.
- Where does this line cross the -axis, and how does it compare with ?
- The -axis is where . Put 0 in place of .
- is 0 there, so is just : the line crosses at 4.
- Compare the two rules at .
- is 3 more, and it is 3 more at every , because only changed.
- The line moved up 3 without turning: lines with the same are parallel.
Read Ana's working one line at a time. Which line does not follow?
Which number has to match?
Where does the line cross the -axis? Give the value of .
Ana decides whether two lines are parallel. Click the line where her reasoning goes wrong.
Change c on purpose
Each task puts a point on the board. Set so the line passes through .
Need a hint?
T is on the y-axis. Where must the line cross it?
Now take the line through . What is ?
Need a hint?
Work out 2x at T: 2 × 2 = 4.
Another hint
T is at y = 4, so how much is left for c?
is not on the -axis. Work out at first.
Need a hint?
At T, x = 1, so 2x is 2.
Another hint
y at T is 6. What do you add to 2 to get 6?
Two sliders now. Make a line parallel to the dashed one, through .
Need a hint?
Parallel means the same steepness. What is m for the dashed line?
Another hint
Then set c so the line crosses the y-axis at T.
Set so the line passes through .
Use the board. This ticks when you get there.
Set so the line passes through .
Use the board. This ticks when you get there.
Set so the line passes through .
Use the board. This ticks when you get there.
Make a line parallel to the dashed line that passes through .
Use the board. This ticks when you get there.
Read the line
Where does cross the -axis? Give the value of .
The line passes through . What is ?
Review: how far does go up for each step right? (A negative answer means it goes down.)
A parallel line
Choose where goes, then find the that takes the line through it. Check: is your line parallel to the dashed one?
Stretch: write the rule of the line parallel to that passes through .
Need a hint?
Drag P first. Then work out 2x at P.
Another hint
c is the y of P minus 2x.
Drag anywhere, then set so a line parallel to passes through it.
Use the board. This ticks when you get there.
For teachers
Students vary c in y = 2x + c on a graph board, find that c moves the line up or down and is where it crosses the y-axis, and that lines with the same m are parallel (AC9M8A04, node 2). About 45 minutes.
Open the teacher guide: answers, solutions, slides and a worksheet.
Students will be able to
- I can describe what happens to the line y = mx + c when c changes.
- I can read where a line crosses the y-axis from its rule.
- I can tell from their rules whether two lines are parallel.
Before class
Do the Run and Modify boards yourself first. This lesson follows "What m does to a line": students should already read m as "up for each step right".
Here m stays at 2 and only c changes, so every line is parallel to the dashed y = 2x + 1. The last Modify task is the first time both sliders appear: it is the stretch point of the lesson.
The lesson says "where the line crosses the y-axis". The primer has the formal word, y-intercept, if you want to introduce it at the end.
Running the lesson
- Predict: take a show of hands. The common wrong answer is 'it gets steeper': students expect any bigger number to steepen the line. Do not correct it yet.
- Run: ask students to compare the step on the solid line with the dashed line as they move c. It stays 'up 2'. Then read across a column of the table: each y is the same amount more.
- Investigate: model the worked example live. Put x = 0 into y = 2x + 4 and say 'on the y-axis, x is 0, so 2x is 0 and y is just c'. Then compare both rules at x = 1.
- Modify: before 'Not on the axis', ask: why can't we just read c off the graph here? (T is not on the y-axis.)
- Practise: the parallel question plants the same numbers in different places. Ask a student who chose it to work out the steepness of each line.
- Make: collect two students' answers and check them as a class: work out 2x at P and subtract from y.
When students get stuck
- A student says a bigger c makes the line steeper. Ask: How far up is the step on each line? Fix: Point at the two steps: both are 'up 2'. Only the starting height on the y-axis changed.
- A student reads m as where the line crosses the y-axis. Ask: What is y when x is 0? Put 0 in the rule. Fix: Work it through: 3 × 0 − 5 = −5. The x term disappears at x = 0, so only c is left.
- A student says lines with the same numbers are parallel (Ana's mistake). Ask: Which number sets the steepness? Fix: Compare m only: same m means same steepness, so parallel. Different m means the lines cross somewhere.
- A student cannot find c when T is not on the y-axis. Ask: What is 2x at T? How much more is y? Fix: Write y = 2x + c with T's numbers in: 6 = 2 × 1 + c, so c = 4. (This is the first equation-solving step of the next lesson.)
Stretch and support
Stretch: Write the rule of the line parallel to y = 2x + 1 through (5, 0). Then: can two different lines with the same m ever cross? Why not?
Support: Keep the Run board open for the Investigate questions: set c to the number in the rule and read where the line crosses the y-axis. Use squared paper to draw the two parallel lines.
Where it leads
Next: solving equations with graphs. 2x + 3 = 11 becomes the point where y = 2x + 3 meets y = 11, which joins this unit to the Equations unit. Then modelling: two plans with different m and c, and where their lines cross.
Where it leads in class: a fixed starting amount plus a rate (a call-out fee and an hourly charge, a deposit and weekly savings). c is the starting amount; m is the rate.
What each step is for
- a bigger c: Committing to what c will do before moving it shows whether students have separated c from m.
- change c: Moving c while m stays fixed slides the line up and down: the step stays 'up 2', and the line never turns.
- reading c: c is where the line crosses the y-axis (the value of y when x is 0). Lines with the same m are parallel; only c tells them apart.
- change c on purpose: One change at a time: a negative c, c = 0 (through the origin), a point that is not on the y-axis, then both numbers at once for a parallel line.
- read the line: Reading c, recognising parallel lines and finding c from a point are one skill; the review question keeps m from the last lesson fresh.
- a parallel line: Choosing a point and then finding the c that reaches it, with m fixed, reverses the lesson: from the picture to the rule.
- Describe what happens to the line y = mx + c when c changes.
- Read where a line crosses the y-axis from its rule.
- Tell from their rules whether two lines are parallel.