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      Features of linear relationships

      I can recognise that linear relationships have particular algebraic and graphical features as a result of the constant rate of change.

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      Lesson 5 of 15
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      • Year 8

      Features of linear relationships

      I can recognise that linear relationships have particular algebraic and graphical features as a result of the constant rate of change.

      Lesson details

      Key learning points

      1. A linear graph can be described by its features.
      2. All the coordinates on the line fit the relationship.
      3. The relationship between the coordinates can be described algebraically.
      4. These linear relationships have particular features.
      5. You can move between the algebraic statement and the graphical representation and back using coordinates.

      Keywords

      • Linear - The relationship between two variables is linear when they change together at a constant rate and form a straight line when plotted.

      Common misconception

      Only equations in the form $$y=mx+c$$ are linear. All equations are linear.

      There are many forms that linear equations can take, however they always share a common feature. The variables have exponents of $$1$$.

      We know $$y=4x$$ to be linear; explore $$4y=x$$. Finish and plot the coordinates: $$(x,0), (x,1), (x,2)$$ and so on. $$y=x+4$$ is linear, what about $$x+y=4$$? Finish and plot $$(0,y), (1,y), (2,y)$$ and so on. Draw attention to the $$1$$ exponents.
      Teacher tip

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

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      6 Questions

      Q1.
      The sequence $$7,13,19,25,31, ...$$ is arithmetic (linear) because:

      is it increasing.
      the terms are all positive.
      Correct answer: it has a common, constant difference between terms.
      the terms are all odd numbers.

      Q2.
      Which of these sequences is arithmetic (linear)?

      Correct answer: $$5,10,15,20,25, ...$$
      $$6,10,15,21,28, ...$$
      Correct answer: $$6,1,-4,-9,-14, ...$$
      Correct answer: $$2.1,2.4,2.7,3,3.3, ...$$
      $$1,2,4,8,16, ...$$

      Q3.
      Which of the below values will be a $$y$$ coordinate in this table of values for the relationship $$y=4x+5$$?

      An image in a quiz
      $$-5$$
      Correct answer: $$-3$$
      $$-1$$
      $$4$$
      Correct answer: $$5$$

      Q4.
      Here are the second and fifth terms of an arithmetic sequence. Which are the missing terms?

      An image in a quiz
      $$4$$
      Correct answer: $$6$$
      $$12$$
      Correct answer: $$14$$
      Correct answer: $$18$$

      Q5.
      What is the $$10^{\text{th}}$$ term of the sequence $$2n^3$$?

      $$60$$
      $$200$$
      Correct answer: $$2000$$
      $$8000$$

      Q6.
      What will the $$y$$ coordinate be when $$x=-2$$ for the relationship $$y=10-3x$$?

      $$-16$$
      $$-4$$
      $$4$$
      Correct answer: $$16$$

      Assessment exit quiz

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      6 Questions

      Q1.
      The relationship between two variables is linear when they change together .

      with a constant multiplier
      Correct answer: at a constant rate
      with the same difference

      Q2.
      Does this table of values represent a linear relationship?

      An image in a quiz
      No. The difference between $$-2$$ and $$3$$ in $$y$$ is different.
      No. $$x$$ changes by $$+1$$ whereas $$y$$ changes by $$+5$$.
      Correct answer: Yes. The $$y$$ values change constantly by $$+5$$ as $$x$$ changes by $$+1$$.
      We can't know until we plot it and draw a line.

      Q3.
      What is the constant rate of change in this relationship?

      An image in a quiz
      $$1$$
      $$2$$
      $$4$$
      $$+1$$ in $$x$$ and $$+2$$ in $$y$$
      Correct answer: $$+1$$ in $$x$$ and $$+4$$ in $$y$$

      Q4.
      Does this table of values represent a linear relationship?

      An image in a quiz
      No. The $$x$$ values don't change by $$1$$.
      No. The $$y$$ values are decreasing, not increasing.
      Correct answer: Yes. The $$y$$ values change by $$-3$$ for every change of $$+2$$ in $$x$$.
      We can't know until we plot it and draw a line.

      Q5.
      Which of these rules will form a linear relationship?

      Correct answer: $$y=6x-7$$
      Correct answer: $$y=7-6x$$
      $$y^3=7-x$$
      $$y=6x^2-7$$
      Correct answer: $$6y=7-x$$

      Q6.
      Match the coordinate to the linear relationship it fits.

      Correct Answer:$$(3,12)$$,$$y=5x-3$$
      Correct Answer:$$(3,-12)$$,$$y=3-5x$$
      Correct Answer:$$(6,-3)$$,$$y = {1\over 3}x-5$$
      Correct Answer:$$(6,10)$$,$$y = {5x\over 3}$$
      Correct Answer:$$(8,1)$$,$$5y=x-3$$

      Lesson appears in

      UnitMaths / Graphical representations of linear equations

      Maths

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