🛰️

8th Grade algebra · Ages 13–14

Lines and Slope

Every straight line has a rule. Grade 8 learns to read it, draw it, and find where two lines cross.

y = mx + bthe unknown

The one idea

A straight line is a rule: start at b, and for every step right, go up m.

y = mx + b looks abstract but it is the taxi fare from Grade 7: b is the fee you pay once, m is what each mile adds. When your child is stuck on a graph, ask “where does it start?” and “how much does it go up each step?”.

📚 The 4 lessons ✏️ Practice

This page is the right one if they can already…

Solves two-step equations, including negatives
Plots coordinates in all four quadrants
Writes an equation from a story with a fixed fee and a rate

Not yet? Start at 7th Grade: Two Steps Back — algebra only works one rung at a time.

The four 8th Grade algebra lessons

In order — each one leans on the last. Every example is broken into single moves and ends by checking itself.

From table to line

A rule gives a table of pairs. Plot the pairs and, for this kind of rule, they always sit on a straight line.

y = 2x + 1
(0, 1) (1, 3) (2, 5)

Worked example

Draw y = 2x + 1.

1
Pick some x values and work out y.
x = 0 → 1, x = 1 → 3, x = 2 → 5
2
Plot the points.
(0, 1), (1, 3), (2, 5)
3
Join them with a ruler and extend both ways.
a straight line

Answer: A line through (0, 1) going up 2 for every 1 across.

Check: Pick a point on your line, say (3, 7): 2(3) + 1 = 7 ✓

⚠️ The usual mistake

Joins the dots with a wobbly curve, or stops at the last point.

They think only the points in the table belong to the graph.

Fix: Try x = 1.5: y = 4. That point is on the line too. Every point on it follows the rule.

💬 Say this

“Does every point on your line follow the rule? Check one you didn’t plot.”

🏠 Try it at home

Plot your savings: start with $10, add $5 a week. Where is the line in week 6?

Slope is steepness

Slope = how far up ÷ how far across. Big slope means steep; negative slope means downhill.

slope 2
(1, 3) (3, 7)

Worked example

Find the slope of the line through (1, 3) and (3, 7).

1
How far up? Subtract the y’s.
7 − 3 = 4
2
How far across? Subtract the x’s in the same order.
3 − 1 = 2
3
Up divided by across.
4 ÷ 2 = 2

Answer: Slope = 2

Check: From (1, 3), go across 1 and up 2 → (2, 5), again → (3, 7) ✓

⚠️ The usual mistake

Gets ½ by doing across ÷ up.

Two numbers, two ways to divide — and no reason to pick one.

Fix: “Rise over run”: you rise out of bed before you run. Up goes on top.

💬 Say this

“How far up? How far across? Up goes on top.”

🏠 Try it at home

Measure a ramp or a slide: height up ÷ length across. Which is steepest in the playground?

Start and step: y = mx + b

b is where the line starts (where it crosses the y-axis). m is the step: how much y changes for each 1 across.

y = x + 2
y = −x + 4

Worked example

A phone plan costs $10 a month plus $5 per gigabyte. Write the rule and find the cost for 4 GB.

1
The once-only cost is b.
b = 10
2
The per-gigabyte cost is m.
m = 5
3
Write the rule, then put in x = 4.
y = 5x + 10 → 5(4) + 10 = 30

Answer: y = 5x + 10, and 4 GB costs $30.

Check: The graph starts at $10 and climbs $5 for every gigabyte ✓

⚠️ The usual mistake

Mixes up m and b, writing y = 10x + 5.

Both are “the numbers in the rule”.

Fix: Ask “what happens when x = 0?” — that’s the start, and it’s b.

💬 Say this

“Where does it start? How much does each step add?”

🏠 Try it at home

Find three costs at home with a fixed part and a per-unit part. Write each as y = mx + b.

Where two lines meet

Two lines cross at the one point that follows both rules. Set the rules equal to find it.

y = x + 1
y = −x + 5
(2, 3)

Worked example

Where do y = x + 1 and y = −x + 5 meet?

1
At the crossing, both y’s are the same.
x + 1 = −x + 5
2
Add x to both sides, then take 1 off both sides.
2x = 4 → x = 2
3
Put x = 2 into either rule.
y = 2 + 1 = 3

Answer: They meet at (2, 3).

Check: Second rule: −2 + 5 = 3 ✓ — the point works in both.

⚠️ The usual mistake

Finds x = 2 and stops.

The equation feels finished once x is found.

Fix: A point needs two numbers. Find y too, then check it in both rules.

💬 Say this

“The crossing point follows both rules. What’s true about the y’s there?”

🏠 Try it at home

Gym A: $20 joining + $10 a month. Gym B: $40 joining + $5 a month. When do they cost the same?

✏️ Practice that never runs out

Every question is new. A wrong answer gets a hint; a second one shows the worked solution, so no one stays stuck.

Use the rule

Put x into y = mx + b and work out y.

Find y when x = 3.

y = −x + 8

Find the slope

Rise over run: how far up ÷ how far across.

Find the slope of the line through these points.

(0, −3) and (1, −7)

The words, in plain English

Slope (m): Rise ÷ run — how steep a line is.; Slope 2: up 2 for every 1 across.
y-intercept (b): Where the line crosses the y-axis.; y = 2x + 1 crosses at 1.
Linear: Makes a straight line.; y = 5x + 10
Function: A rule giving one output for each input.; y = 2x + 1
System of equations: Two rules that must both be true.; y = x + 1 and y = −x + 5

🌍 Algebra in an ordinary day

Comparing two phone plans or gyms — where do the lines cross?
Ramps and roofs have slopes
Speed: miles per hour is a slope on a distance-time graph

🏁 By the end of 8th Grade

Graphs a line from its rule
Finds slope from two points or a graph
Writes y = mx + b from a real situation
Solves a system by finding where two lines meet

Where this fits

One idea, twelve rungs. The hiding spot just gets cleverer.

← One rung down

7th Grade: Two Steps Back

One rung up →

9th Grade: Algebra 1

All 12 grades More maths, grade by grade →

Frequently Asked Questions

What algebra should a 8th Grade child learn?

A straight line is a rule: start at b, and for every step right, go up m. By the end of the year they should be able to: graphs a line from its rule; finds slope from two points or a graph; writes y = mx + b from a real situation; solves a system by finding where two lines meet.

Isn’t 8th Grade too early for algebra?

No — lines and slope is standard 8th Grade maths in most curricula. This page breaks it into single moves so it never feels like a jump.

What is the most common 8th Grade algebra mistake?

Gets ½ by doing across ÷ up. Two numbers, two ways to divide — and no reason to pick one. The fix: “Rise over run”: you rise out of bed before you run. Up goes on top.

What if my child is stuck?

Go down one rung. Every lesson here builds on the 7th Grade page, and the “ready for this page if” list at the top tells you exactly what they need first. In practice, a wrong answer gives a hint, and a second wrong answer shows the full worked solution.

Is it free?

Yes. All 4 lessons, the pictures, the worked examples, the balance scale and the practice questions are free with no signup. The practice never runs out — every question is generated fresh and checked.

Lines and Slope

This page is the right one if they can already…

The four 8th Grade algebra lessons

✏️ Practice that never runs out

Use the rule

Find the slope

The words, in plain English

🌍 Algebra in an ordinary day

🏁 By the end of 8th Grade

Where this fits

Frequently Asked Questions

What algebra should a 8th Grade child learn?

Isn’t 8th Grade too early for algebra?

What is the most common 8th Grade algebra mistake?

What if my child is stuck?

Is it free?

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