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      The distributive law with two or more binomials

      Lesson details

      Learning outcome

      I can apply the distributive law to two or more binomials.

      Key learning points

      1. The distributive law can be used with expressions containing surds.
      2. This can be extended to finding the product of two or more binomials.
      3. Any binomial could contain surd terms.

      Keywords

      • Distributive law - The distributive law says that multiplying a sum is the same as multiplying each addend and summing the result.

      Common misconception

      Pupils may suggest using FOIL (or similar) to expand both brackets and make errors whilst doing so.

      The grid method is derived from the area model and therefore the area model supports expanding both one binomial and two binomials.

      Teacher tip

      Pupils may benefit from using one representation throughout their learning on multiplication. This will help them to see that this is all connected and it is not disjointed topics.

      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.
      Write $$7(2\sqrt {5} + 3\sqrt {3})$$ in the form $$x\sqrt {5} + y\sqrt {3}$$. What is the value of $$x$$?

      Correct Answer: 14

      Q2.
      Write $$7(2\sqrt {5} + 3\sqrt {3})$$ in the form $$x\sqrt {5} + y\sqrt {3}$$. What is the value of $$y$$?

      Correct Answer: 21

      Q3.
      Write $$5\sqrt {3}(2\sqrt {8} + 3\sqrt {2})-\sqrt {3}(\sqrt {8} + 2\sqrt {2})$$ in the form $$a\sqrt {b}$$. What is the value of $$b$$?

      Correct Answer: 6

      Q4.
      Write $$5\sqrt {3}(2\sqrt {8} + 3\sqrt {2})-\sqrt {3}(\sqrt {8} + 2\sqrt {2})$$ in the form $$a\sqrt {b}$$. What is the value of $$a$$?

      Correct Answer: 31

      Q5.
      Write $$\sqrt {5}(3\sqrt {5} + 2)-(3\sqrt {5} + 2)$$ in the form $$a-\sqrt {b}$$. What is the value of $$b$$?

      Correct Answer: 5

      Q6.
      Write $$\sqrt {5}(3\sqrt {5} + 2)-(3\sqrt {5} + 2)$$ in the form $$a-\sqrt {b}$$. What is the value of $$a$$?

      Correct Answer: 13

      6 Questions

      Q1.
      Expand $$(2\sqrt {3} + \sqrt {3})(\sqrt {3} - 1)$$

      $$3 + 3\sqrt {3}$$
      $$9 + 3\sqrt {3}$$
      $$3 - 3\sqrt {3}$$
      Correct answer: $$9 - 3\sqrt {3}$$

      Q2.
      Expand $$(\sqrt {2} + 1)(\sqrt {2} - 1)$$

      Correct answer: $$1$$
      $$1 + 2\sqrt {2}$$
      $$3$$
      $$3 + 2\sqrt {2}$$

      Q3.
      Expand $$(\sqrt {5} + \sqrt {2})(\sqrt {5} - \sqrt {2})$$

      $$3 + 2\sqrt {10}$$
      Correct answer: $$3$$
      $$7 + 2\sqrt {10}$$
      $$7$$

      Q4.
      Expand $$(3\sqrt {7} - 2\sqrt {3})(\sqrt {7} + \sqrt {3})$$

      $$15 + 5\sqrt {21}$$
      Correct answer: $$15 + \sqrt {21}$$
      $$27 + \sqrt {21}$$
      $$27 + 5\sqrt {21}$$

      Q5.
      Expand $$(\sqrt {10} + 2\sqrt {6})(\sqrt {10} - 2\sqrt {6})$$

      Correct answer: $$-14$$
      $$-14 - 4\sqrt{60}$$
      $$-14 - 4\sqrt{15}$$
      $$-14 + 4\sqrt{15}$$

      Q6.
      Expand $$(4\sqrt {2} - \sqrt {5})(4\sqrt {2} + \sqrt {5})$$. What is the result?

      Correct Answer: 27

      To help you plan your 10 maths lesson on: The distributive law with two or more binomials, download all teaching resources for free and adapt to suit your pupils' needs...