Straight lines, curves and sequences
A graph turns an equation into a picture. Reading gradient, intercepts and shape lets you answer questions that would be laborious algebraically.
Lesson overview
What you'll learn in this lesson
Manipulate algebraic expressions, solve equations and interpret graphs of functions.
Key learning points
- • Straight-line graphs
- • Recognising curves
- • Sequences
This lesson at a glance
- 30 minutes
- 15 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 15
Watch first
Straight lines, curves and sequences
An English teaching video on this topic from Cognito. Watch it, then scroll down for the lesson.
In a nutshell
Manipulate algebraic expressions, solve equations and interpret graphs of functions.
Learning cycle
Part 2 of 15
Learning cycle 1 of 2
Part 1 · Straight-line graphs
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 3 of 15
Learn
Straight-line graphs
In y = mx + c, m is the gradient and c the y-intercept. Gradient is the change in y divided by the change in x between two points. Parallel lines share a gradient; perpendicular gradients multiply to −1, so a line perpendicular to y = 2x + 1 has gradient −½.
Explore the idea
Part 4 of 15
Learn
Recognising curves
A quadratic gives a parabola, positive coefficient opening upwards. A cubic has a characteristic double bend, a reciprocal graph two separate branches with asymptotes, and an exponential rises ever more steeply while never reaching zero. Sketch questions reward the correct shape and intercepts, not perfect plotting.
Quick check
Part 5 of 15
Quick check
The gradient of a line perpendicular to y = 4x − 1 is
Quick check
Part 6 of 15
Quick check
The nth term of 5, 8, 11, 14 is
Reset break
Part 7 of 15
Pause
That's sitting 1 of 2 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Learning cycle
Part 8 of 15
Learning cycle 2 of 2
Part 2 · Sequences
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 9 of 15
Learn
Sequences
For a linear sequence the nth term is (common difference)n plus an adjustment: 5, 8, 11, 14 has difference 3, and 3n gives 3, 6, 9, so the nth term is 3n + 2. For quadratic sequences, a constant second difference of 2a identifies the n² coefficient. Always check your rule on the first and a later term.
Quick check
Part 10 of 15
Quick check
A graph with two separate branches and asymptotes is
Quick check
Part 11 of 15
Quick check
A quadratic sequence is identified by a constant
Explore the idea
Part 12 of 15
Worked example
Worked answer: find the equation of the line through (1, 5) and (4, 14) (3 marks)
Gradient m = (14 − 5) ÷ (4 − 1) = 9 ÷ 3 = 3 (1). Substituting the point (1, 5) into y = 3x + c gives 5 = 3 + c, so c = 2 (1). The equation is y = 3x + 2 (1). Verifying with the second point: 3(4) + 2 = 14, correct. Subtracting the coordinates in inconsistent order is the usual error.
Challenge round
Part 13 of 15
Game · Sort it
Which of these are true?
Drag each card into the right column. Tap a card first if dragging is fiddly.
True
Not true
Mastery quiz
Part 14 of 15
Marked quiz
End of lesson quiz: Straight lines, curves and sequences
4 questions, marked with the reasoning shown. No timer.
1. The gradient of a line perpendicular to y = 4x − 1 is
2. The nth term of 5, 8, 11, 14 is
3. A graph with two separate branches and asymptotes is
4. A quadratic sequence is identified by a constant
Lesson round-up
Part 15 of 15
Lesson round-up
Ready when you are
Quiz score
Not sat
Games
Not played
Points this lesson
0
Best run
0 in a row
Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?
Ask LunaDo the work
Your written work for this lesson
This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.
The cost of a school trip
A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.
If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.
The school decides to charge £20 and to use anything left over for a second trip later in the year.
Written for Good Learning Co.
Reading questions
- Work out the total cost of the trip if 50 pupils go.
- Show why the price per pupil falls when more pupils go, using numbers.
- At £20 each, how many pupils must go before the trip makes money? Show your working.
Working out
Working outDo the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.
10–15 questions
Worked example, explained
WritingCopy one question out and write underneath, in words, why each step works.
1 question, 4–5 sentences
Problem in words
Working outWrite your own real-life problem that uses today's method, then solve it.
1 problem + solution
Saved as you type.
Part 1 of 15 · Watch & discover