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Year 10 Surds warmup

See also: math-lessons-resources, teaching-mathematics, surds

Prepare a warmup activity for a Year 10 surge class. Focus on simplifying surds. Part of a sequence of lessons revising for an imminent final exam.

Related resources

Aim#

Be able to answer questions that ask

Simplify the following expressions. Write your answers in the simplest exact surd form.

For example

  1. $\(3\sqrt(27) =\) -- take out any obvious square factors

    • \(= 3 \times ( \sqrt(9) \times \sqrt(3) )\)
    • \(= 3 \times 3 \times \sqrt(3)\)
    • \(= 9\sqrt(3)\)

  2. \(7\sqrt(2) + 3\sqrt(8) =\) -- take out any obvious square factors, addition of like squares

    • \(7\sqrt(2) + 3(\sqrt(4 \times 2)) =\)
    • \(7\sqrt(2) + 3(\sqrt(4) \times \sqrt(2)) =\)
    • \(7\sqrt(2) + 3 \times 2 \times \sqrt(2) =\)
    • \(7\sqrt(2) + 6\sqrt(2) =\)
    • \(13\sqrt(2)\)

  3. \(-6\sqrt(3) + 5\sqrt(12) =\)

  4. \(\dfrac{\sqrt(35)}{\sqrt(7)} =\)

Design considerations#

Motivation#

  • Approximation is bad
  • Some connection with trignometic ratios
  • rationalising the denominator arises elsewhere: algebra, calculus, and later mathematics

Necessary knowledge#

  • Declarative knowledge

  • What are surds

    • What is a like surd and unlike surd
    • What is a conjugate surd

  • What does simplify mean?

  • What does simplest exact surd form mean?

  • Distributive law and special products

    • Distributive law will be helpful

      • \(a(b + c) = ab + ac\)
      • \(a(b - c) = ab - ac\)

      • \((a + b)(c + d) = ac + ad + bc + bd\)

        • The difference of squares identity is very useful for simplifying surds

      • \(a^2 - b^2 = (a + b)(a - b)\)

  • Basic rules

    • For positive numbers, squaring and taking a square root are inverse processes

      • \((\sqrt(a))^2 = a\)
      • \(\sqrt(a^2) = a\)

    • Multiplication

      • \(\sqrt(a) \times \sqrt(b) = \sqrt(ab)\)
      • \(a\sqrt(b) \times c\sqrt(d) = ac\sqrt(bd)\)

    • Division

      • \(\sqrt(a) \div \sqrt(b) = \sqrt(a/b)\)
      • \(a\sqrt(b) \div c\sqrt(d) = \dfrac{a}{c}\sqrt(b/d)\)

    • Addition and subtraction of like surds

      • \(a\sqrt(b) \pm c\sqrt(b) = (a \pm c)\sqrt(b)\)

  • Procedural knowledge

    • Which one or more of the

Implementation#

Mathematics is patterns and relationships#

  • Some idea to reinforce this point - perhaps on the slide with the warm up activity

    • The Golden Ratio is one of the earliest and most famous surds discovered by the Greeks (and perhaps others?)

  • Related to surds, algebra, or logs/exponentials in someway

  • Maybe a step too far

Questions

  1. \(\sqrt(32)\) - as a simple application of taking out any square factors (probably not)
  2. \(5\sqrt(12)\) - take out square factors + multiplication
  3. \(8\sqrt(5) + 3\sqrt(20)\) - take out square factors + addition of like surds
  4. \(-3\sqrt(3) + 2\sqrt(27)\) - take out square factors + addition of like surds
  5. \(\dfrac{\sqrt(63)}{\sqrt(7)}\)
  6. \(\dfrac{\sqrt(5) - 3}{\sqrt(3) + 1}\)