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      Problem solving with surds

      Lesson details

      Learning outcome

      I can use my knowledge of surds to solve problems.

      Key learning points

      1. Surds are a useful format to leave an answer as.
      2. Surds can help with the accuracy of future calculations.
      3. Surds can be seen in many areas of maths.

      Keywords

      • Surd - A surd is an irrational number expressed as the root of a rational number.

      Common misconception

      Pupils may be uncertain how to solve problems with surds.

      Many problems can be solved through reasoning what you know, such as fully simplifying a surd.

      Teacher tip

      This lesson brings together previous learning from other units and asks pupils to use their new knowledge of surds within their previous learning. Pythagoras' theorem is a useful context as it prepares pupils for 3D problems.

      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

      6 Questions

      Q1.
      Order the fractions from smallest to largest.

      1 - $$2 \over 7$$
      2 - $$1 \over 3$$
      3 - $$4 \over 9$$
      4 - $$3 \over 5$$
      5 - $$5 \over 8$$

      Q2.
      What is the value of $$a + b - c$$ when $$a$$ = 7, $$b$$ = 21 and $$c$$ = 2.4

      Correct Answer: 25.6

      Q3.
      In order to rationalise $$(2 - \sqrt {3})$$ you would multiply by:

      Correct answer: $$(2 + \sqrt {3})$$
      $$(-2 + \sqrt {3})$$
      $$(3 - \sqrt {2})$$
      $$(3 + \sqrt {2})$$

      Q4.
      True or false? Pythagoras' theorem can only be used to find the length of the hypotenuse.

      True
      Correct answer: False

      Q5.
      When is a surd in it's simplest form?

      Correct answer: when the radicand is an integer and has no perfect square factors greater than 1
      when the radicand is an integer
      when the radicand is a perfect square

      Q6.
      Solve $$x\sqrt {180} - 30 = x\sqrt {80}$$ and write your answer in the form $$a\sqrt {b}$$. State the value of $$b$$.

      Correct Answer: 5

      6 Questions

      Q1.
      True or false? Surds are a way of giving an answer exactly.

      Correct answer: True
      False

      Q2.
      Arrange the surds from smallest to largest.

      1 - $$1 \over \sqrt {5}$$
      2 - $$2 \over \sqrt {7}$$
      3 - $$\sqrt {3}$$
      4 - $$\sqrt {7}$$

      Q3.
      The perimeter of a rectangle is $$12 + 8\sqrt {2}$$. One of the lengths is $$6 + \sqrt {2}$$. Find the area.

      $$10\sqrt {2}$$
      $$20 + 60\sqrt {2}$$
      $$3\sqrt {2}$$
      Correct answer: $$6 + 18\sqrt {2}$$

      Q4.
      The length of the line between (3,8) and (7,13) can be written in the form $$\sqrt {a}$$. What is the value of $$a$$?

      Correct Answer: 41

      Q5.
      Which of these shapes has a greater area?

      An image in a quiz
      Shape A
      Correct answer: Shape B
      They both have the same area

      Q6.
      Which of these shapes has a greater perimeter?

      An image in a quiz
      Shape A
      Shape B
      Correct answer: They both have an equal perimeter

      To help you plan your 10 maths lesson on: Problem solving with surds, download all teaching resources for free and adapt to suit your pupils' needs...