Review: division, factors, multiples and rational numbers
Discover Mathematics P6: Term 1, Units 1–7, Review
GoalRevise long division, prime factors, the GCF and LCM, fractions with unlike denominators, and rational numbers.
New words
GCF · the greatest common factor of two numbersLCM · the least common multiple of two numbersrational number · a number that can be written as a fraction a/babsolute value · the distance of a number from 0, written | |
Explanation
In Units 1 and 2 we learned five big ideas. Long division: estimate first, then divide, multiply, subtract and bring down; a remainder is smaller than the divisor and we decide what it means in the story. Prime factors: write a number as a product of primes with a factor tree. GCF and LCM: from the primes of two numbers, the shared primes give the GCF and all the primes, shared ones once, give the LCM. The GCF also lets us write a sum as a(b + c), and the LCM is the common denominator for adding fractions. Rational numbers: a number line goes below 0, opposites are the same distance from 0, the number further right is greater, and the absolute value is the distance from 0.
★Remember
Estimate, divide, check. Shared primes → GCF; all primes (shared once) → LCM. Right on the number line = greater. |x| = distance from 0.
Worked examples
A charity has 2,500 LE. A food parcel costs 36 LE. Estimate: 2,500 ÷ 36 is about 70. The division gives 69 with remainder 16, because 36 × 69 = 2,484 and 2,500 − 2,484 = 16. The charity can make 69 parcels and 16 LE is left over (it cannot pay for another parcel).
The primes of 36 and 60 are 36 = 2 × 2 × 3 × 3 and 60 = 2 × 2 × 3 × 5. The shared primes are 2, 2, 3, so the GCF is 12. The LCM uses all the primes with the shared ones once: 2 × 2 × 3 × 3 × 5 = 180. Check: 12 × 180 = 2,160 = 36 × 60. Then 24 + 36 = 12 (2 + 3) uses the GCF 12 of 24 and 36.
Add 5/6 + 3/8 with the LCM of 6 and 8, which is 24. 5/6 = 20/24 and 3/8 = 9/24, so the sum is 29/24 = 1 5/24. Next, put −2.5, 3/4, −1/2, 0 and 1.2 in order. On the number line the order from the least is −2.5, −1/2, 0, 3/4, 1.2. The absolute value |−2.5| is 2.5, the distance from 0.
Class activity
Number hunt. Choose two numbers between 10 and 50. Write the prime factors of each, find their GCF and LCM, and check that GCF × LCM equals the product of the two numbers. Then say a negative temperature that you might see in a cold place and its opposite.
Practice
1
Divide 4,128 ÷ 24 and check.
172, because 24 × 172 = 4,128.
2
Find the GCF and the LCM of 84 and 90 from their prime factors.
84 = 2 × 2 × 3 × 7 and 90 = 2 × 3 × 3 × 5. The shared primes are 2 and 3, so the GCF is 6. The LCM is 6 × 2 × 7 × 3 × 5 = 1,260.
3
Write 24 + 36 as GCF × (sum).
The GCF is 12, so 24 + 36 = 12 (2 + 3) = 12 × 5 = 60.
4
Find 7/12 − 3/8.
The LCM of 12 and 8 is 24. 14/24 − 9/24 = 5/24.
5
Put 0.5, −0.75, −3/2, 1 in order from the least to the greatest, and say which has the greatest absolute value.
−3/2 is −1.5, so the order is −3/2, −0.75, 0.5, 1. The greatest absolute value is |−3/2| = 1.5.
