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Grade 10 · Core Mathematics · Lesson 5 of 14

Nearby Bases for Mental Squaring

A Grade 10 Core Mathematics CBE lesson from the sub-strand Quadratic Expressions and Equations. In it, learners choose a nearby base for mental calculation.

Strand Numbers and AlgebraSub-strand Quadratic Expressions and EquationsLesson code 10CM-13-L540 minutes · 2 slides

KICD learning outcome (c)

Apply quadratic identities in numerical cases.

By the end of this lesson, learners should be able to

  1. Choose a nearby base for mental calculation.
  2. Track the cross-product and adjustment terms.
  3. Check a square using a second method.

Key vocabulary

Nearby base
A convenient number close to the number being squared.
Three hundred is a nearby base for 307.
Cross-product
The middle term 2ab in a square identity.
For 300−4, the cross-product is 2×300×(−4)=−2400.

Lesson preview

Slide 1 · Choose a nearby baseOpening · 18 min
A number line shows the adjustment from a base of 300 to 307, alongside the three-term identity for squaring 307.
Figure 1: A number line shows the adjustment from a base of 300 to 307, alongside the three-term identity for squaring 307.

A round base turns a difficult square into three manageable terms. The adjustment must retain its sign and be stated before calculation. The positive adjustment controls the cross-product and final square. Use Figure 1 to see how the base and adjustment are placed into the identity.

  • Choose 300 as the convenient base.
  • Write 307 as 300+7.
  • Identify the positive adjustment.
  • Predict the size of the answer.
Worked example. Write 307=300+7. Calculate 300²=90000 and 2×300×7=4200. Add 7²=49 to obtain 94249. Answer: 307²=94249..

Check question: When calculating 3072 using a quadratic identity, why is 300 chosen as the base?

  1. Its square and its product with 7 are simple to calculate mentally.
  2. It is the closest square number to 307.
  3. It removes the need to calculate the middle cross-product term.
  4. It ensures the final answer is simply 3002+72.
Show answer

Its square and its product with 7 are simple to calculate mentally. Using a round number like 300 means the terms 3002 (90,000) and 2×300×7 (4,200) are very easy to compute. This reduces mental arithmetic while preserving the exact answer when adding the 72 (49).

Slide 2 · Compare a negative adjustmentCheck · 22 min
A comparison of two squared numbers demonstrates how positive and negative adjustments change the sign of the middle cross-product term.
Figure 2: A comparison of two squared numbers demonstrates how positive and negative adjustments change the sign of the middle cross-product term.

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

  • Both slides with diagrams and class activities
  • Teacher notes, SNE adaptations and assessment checks
  • Homework and the other 13 lessons in this sub-strand
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All 14 lessons in Quadratic Expressions and Equations

  1. 1Building Quadratic Expressions from DimensionsForm quadratic expressions from different situations
  2. 2Four Products on a NoticeboardForm quadratic expressions from different situations
  3. 3Deriving a Square Identity by AreaDerive the quadratic identities from the concept of area
  4. 4Restoring a Difference-Square AreaDerive the quadratic identities from the concept of area
  5. 5Nearby Bases for Mental SquaringApply quadratic identities in numerical cases
  6. 6Factor Pairs and GroupingFactorise quadratic expressions from different situations
  7. 7Forming an Area EquationForm quadratic equations in different situations
  8. 8A Consecutive-Integer EquationForm quadratic equations in different situations
  9. 9Solving by FactorisationSolve quadratic equations by factorisation
  10. 10Exact Roots with Signed FactorsSolve quadratic equations by factorisation
  11. 11A Triangular Sign ModelApply quadratic equations to real life situations
  12. 12Planning a Rectangular TankApply quadratic equations to real life situations
  13. 13Exploring Quadratic Models for a Community MuralExplore the use of quadratic equations in real life situations
  14. 14Exploring a Community Greenhouse FrameExplore the use of quadratic equations in real life situations