Week 2 · Writing sums with the GCF; the LCM and fractions with unlike denominators · 3 lessons
1
Writing a sum as the GCF times a smaller sum
Discover Mathematics P6: Term 1, Unit 1, Lessons 3–4
GoalWrite a sum of two numbers as the GCF times a smaller sum, and read such an expression back into a story.
New words
distributive property · a (b + c) = a × b + a × cGCF · the greatest number that divides both numbersequal groups · groups with the same number of items
Explanation
Sometimes we pack two kinds of things into identical boxes with nothing left over. The most boxes we can make is the GCF of the two numbers. For 8 pencils and 12 pens the GCF is 4, so there are 4 boxes, each with 2 pencils and 3 pens. That gives 8 + 12 = 4 × (2 + 3). The GCF goes outside the brackets, and inside the brackets are the numbers of items in ONE box (each number divided by the GCF). This works because of the distributive property: 4 × 2 + 4 × 3 = 4 × (2 + 3). It also works backwards: 6 × (5 + 4) means 6 groups, each with 5 and 4 items, so 30 + 24 = 54.
ƒFormula
8 + 12 = 4 × (2 + 3)
The GCF goes outside the brackets
Worked examples
Salma packs 8 pencils and 12 pens into identical gift bags with nothing left over. The GCF of 8 and 12 is 4, so she makes 4 bags. Each bag gets 8 ÷ 4 = 2 pencils and 12 ÷ 4 = 3 pens. So 8 + 12 = 4 × (2 + 3) = 4 × 5 = 20 things.
Sum
GCF
Inside the brackets
Total
18 + 27
9
2 + 3
45
A shop has 18 white and 27 brown eggs and wants identical baskets. The GCF of 18 and 27 is 9, so there are 9 baskets with 2 white and 3 brown eggs each. 18 + 27 = 9 × (2 + 3) = 9 × 5 = 45 eggs.
Part
Rolls
Cakes
All
One tray
5
4
9
6 trays
30
24
54
Read an expression back into a story. 6 × (5 + 4) can be 6 trays, each with 5 rolls and 4 cakes. Rolls: 6 × 5 = 30. Cakes: 6 × 4 = 24. Total: 30 + 24 = 54, which is also 6 × 9.
Class activity
Packing game. Take 12 buttons and 18 beads. Find the GCF, then pack identical little bags with nothing left over. Write the sum with the GCF outside the brackets and check by counting everything.
Practice
1
Write 16 + 24 with the GCF outside the brackets.
GCF = 8, so 16 + 24 = 8 × (2 + 3) = 40.
2
Write 30 + 45 with the GCF outside the brackets.
GCF = 15, so 30 + 45 = 15 × (2 + 3) = 75.
3
Tell a story for 7 × (3 + 6) and find its value.
For example, 7 boxes each with 3 red and 6 blue pens: 21 + 42 = 63, which is 7 × 9.
4
Khaled writes 12 + 20 = 2 × (6 + 10). It is true, but what is missing?
He did not use the greatest common factor. The GCF is 4, so 12 + 20 = 4 × (3 + 5) = 32.
5
A craft class has 36 red beads and 48 blue beads. What is the greatest number of identical necklaces, and what is in each?
GCF = 12, so 12 necklaces with 3 red and 4 blue beads each: 36 + 48 = 12 × (3 + 4) = 84.
