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Kenya's Competency Based Education, Grades 7 to 12

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A real Grade 7 Mathematics lesson, the lesson plan a teacher gets with it, and four short quizzes from across the grades.

Learner view Grade 7 › Mathematics › 1.0 Numbers › Factors

GCD by the factor method

A dairy has a 36-litre drum and a 60-litre drum of milk. What is the largest jerrycan that empties both with nothing left? That can is the GCD. Agriculture uses the same idea when one scoop must measure two different bags of feed.

Definition

The Greatest Common Divisor (GCD) of two or more whole numbers is the largest whole number that divides each of them exactly. It is also called the HCF (highest common factor). By the factor method: prime-factorise, then multiply the shared primes using each smaller power.

36 60 = = 2 2 3 3 2 2 3 5 ××× ××× not shared shared factors 2 × 2 × 3 GCD = 12 in both numbers - keep in one number only - drop
Shared primes, smaller powers, multiply.
GCD by prime factors
  1. Write each number in power form.
  2. Keep only the primes that appear in every list.
  3. For each of those primes, take the smaller power.
  4. Multiply. That product is the GCD.
  5. Check: the GCD divides each number, and no larger common factor does.
Worked example · 36 L and 60 L of milk

36 = 22 × 32   and   60 = 22 × 3 × 5.
Shared: 22 and 3 (not 32, not 5).
GCD = 22 × 3 = 12.
A 12 L jerrycan: 36 ÷ 12 = 3 and 60 ÷ 12 = 5. A 15 L can fails on 36; an 18 L can fails on 60.

GCD words in a story: largest, equal pieces, no remainder, a measuring can that fits both. If the story says next time together or smallest pack, that is the LCM - the next lesson.

Quick check

What is the GCD of 24 and 40? Which jerrycan empties 24 L and 40 L of milk with none left?

Show the answer

24 = 23 × 3 and 40 = 23 × 5. Only 23 is shared, so the GCD is 8 - an 8 L jerrycan.

The full lesson also has a hands-on card activity, practice tasks with feedback and a marked quiz.

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Teacher view Lesson plan · Grade 7 Mathematics · 40 minutes

GCD by the factor method

Key inquiry question

How do we apply the GCD and the LCM in day-to-day activities?

Objectives

By the end of the lesson the learner should be able to:

  • find the GCD of two or three numbers by the factor method;
  • apply the GCD to sharing, cutting cloth and measuring containers;
  • reject a common factor that is not the greatest.

Introduction

Two drums, 36 L and 60 L. Ask learners to guess a can: 6 L works but is not the largest. 12 L is. 18 L fails.

Lesson development

  1. 10 minDefinition and diagram. Green chips for the shared primes.
  2. 8 minMilk worked example. Take the smaller powers.
  3. 8 minCloth and class share. Kitenge strips of 48 cm and 36 cm; 48 books and 36 pencils.
  4. 10 minCard GCD, including three numbers. Agriculture scoop for 18 kg and 30 kg bags.

Conclusion

GCD(24, 40) = 8. The 8 L can is the dairy answer.

Assessment

Practice tasks 4 and 5 are the real-life items. Watch for 6 offered as the GCD of 36 and 60 - it divides both but is not the greatest.

Common errors

  • Multiplying all the primes, which is closer to the LCM.
  • Taking the higher power (24 from 48 and 36 gives 48, which does not divide 36).
  • Stopping at any common factor (2 or 6) instead of the greatest.
  • Using the GCD for a meeting-time story.

Support and extension

Support: list the factors of 12, 18 and 24 first, circle the common ones, then pick the greatest as a check.

Extension: GCD(24, 36, 60) = 12 - three numbers still use the smallest shared powers.

Teacher notes

Link to Agriculture: the largest container that measures two substances. Listing factors is a check for small numbers; insist on the factor method as the Grade 7 algorithm.

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