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      The product of two binomials

      Lesson details

      Learning outcome

      I can use the distributive law to find the product of two binomials.

      Key learning points

      1. The distributive law can be used to find the product of two binomials.
      2. An area model can be used to explore the underlying structure.
      3. Both of the terms in one bracket must be multiplied by both terms in the second.

      Keywords

      • Binomial - A binomial is an algebraic expression representing the sum or difference of exactly two unlike terms

      Common misconception

      Missing out partial products

      Relating back to numerical examples and showing that 12 × 34 is not just 10 × 30 + 2 × 4. Using algebra tiles and area models can help to support student's understanding.

      Teacher tip

      It will be really valuable for pupils to see the connection between using an area model to multiply two two-digit numbers and multiplying a binomial. Show the two side by side as much as possible.

      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.
      $$2x$$ and $$5x$$ can be called __________ terms.

      equivalent
      identical
      inverse
      Correct answer: like
      reciprocal

      Q2.
      What expression could these algebra tiles represent?

      An image in a quiz
      Correct answer: $$x^2 + 6x + 9 $$
      $$6x^2 + 9 $$
      $$7x + 9$$
      $$4x^2 + 9x$$
      $$x^2 + 15x$$

      Q3.
      Which is the correct area model to show the expansion of $$x(x + 3)$$?

      An image in a quiz
      An image in a quiz
      An image in a quiz
      Correct Answer: An image in a quiz
      An image in a quiz

      Q4.
      Expand $$x(x + 8)$$.

      $$2x + 8$$
      $$10x$$
      $$9x^2$$
      $$x^2 + 8$$
      Correct answer: $$x^2 + 8x$$

      Q5.
      Expand $$x(x - 3)$$.

      $$x^2 + 3x$$
      Correct answer: $$x^2 - 3x$$
      $$3 - x^2$$
      $$2x^2 - 3x$$
      $$x^2 - 3$$

      Q6.
      Which of these calculations are the same as $$11 \times 14$$ ?

      $$(10 \times 10) + (1\times 4)$$
      $$(11\times 7) + (11\times 2)$$
      Correct answer: $$(1 \times 14) + (10 \times 14)$$
      Correct answer: $$(4\times 11) + (10\times 11)$$
      $$(4\times 11) + (1\times 14)$$

      6 Questions

      Q1.
      Which of these expressions are binomials?

      Correct answer: $$x + 4$$
      $$ x^2 + 2x + 2$$
      $$2x + 3x$$
      Correct answer: $$x - 6$$
      $$xy$$

      Q2.
      Here is an area model for $$(x + 2)(x + 4).$$ Which of these expressions is the correct expanded form?

      An image in a quiz
      $$x^2 + 2x + 8$$
      $$x^2 + 4x + 4$$
      Correct answer: $$x^2 + 6x + 8$$
      $$x^2 + 8x + 6$$
      $$3x^2 + 4x + 8 $$

      Q3.
      Here is an area model for $$(x - 3)(x - 1).$$ Which of these is the correct expanded form?

      An image in a quiz
      $$x^2 -2x + 3$$
      Correct answer: $$x^2 -4x + 3$$
      $$x^2 -4x - 3$$
      $$x^2 + 4x + 3$$
      $$x^2 + 4x - 3$$

      Q4.
      Which product of two binomials could be represented by this area model?

      An image in a quiz
      $$(x - 15)( x - 8)$$
      $$(x - 15)( x + 8)$$
      $$(x - 10)( x + 12)$$
      $$(x + 12)( x - 10)$$
      Correct answer: $$(x + 15)( x - 8)$$

      Q5.
      Which is the correct expanded form of $$(x - 3)^2$$ ?

      $$x^2 - 9$$
      $$x^2 + 9$$
      $$x ^2 - 6x - 9$$
      Correct answer: $$x^2 - 6x + 9$$
      $$x ^2 + 6x - 9$$

      Q6.
      Match each product of two binomials to its correct expansion.

      Correct Answer:$$(x + 7)(x + 3)$$,$$x^2 + 10x + 21$$

      $$x^2 + 10x + 21$$

      Correct Answer:$$(x + 7)(x - 3)$$,$$x^2 + 4x - 21$$

      $$x^2 + 4x - 21$$

      Correct Answer:$$(x - 7)(x + 3)$$,$$x^2 - 4x - 21$$

      $$x^2 - 4x - 21$$

      Correct Answer:$$(x - 7)(x - 3)$$,$$x^2 - 10x + 21$$

      $$x^2 - 10x + 21$$

      Correct Answer:$$(x + 7)^2$$,$$x^2 + 14x + 49$$

      $$x^2 + 14x + 49$$

      Correct Answer:$$(x - 7)^2$$,$$x^2 - 14x + 49$$

      $$x^2 - 14x + 49$$


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