Mathematics 8 · Handsworth 2026–27 · Unit 1

Operations with Fractions

Ms. Ras Mulinta
Handsworth Secondary
Guided notes

Fractions are how we describe parts of a whole — slices of pizza, parts of an hour, a share of a recipe. In this unit you build real fluency: adding, subtracting, multiplying, and dividing fractions, and handling them inside order of operations. Take your time. Each idea has a worked example, then a few problems to try. Answers are at the end of each section so you can check yourself right away.

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The toolkit words we will use

A fraction a/b has a numerator (top, the parts you have) and a denominator (bottom, parts in one whole). A proper fraction is less than one (3⁄4); an improper fraction is one or more (5⁄4); a mixed number mixes a whole and a fraction (11⁄4). To simplify, divide top and bottom by the same number until they share no common factor.
Switching forms
Mixed → improper: multiply the whole by the denominator, add the numerator. 21⁄3 = (2×3 + 1)/3 = 7⁄3.
Improper → mixed: divide. 7⁄3 = 7 ÷ 3 = 2 remainder 1 = 21⁄3.

1 · Adding & subtracting fractions common denominators

You can only add or subtract fractions when the pieces are the same size — same denominator. If the denominators differ, rewrite each fraction over a common denominator (a number both denominators divide into), then add or subtract the numerators. Keep the denominator. Simplify at the end.
Worked example — unlike denominators
Find 2⁄3 + 1⁄4.
Common denominator of 3 and 4 is 12.
2⁄3 = 8⁄12  (top and bottom ×4)    1⁄4 = 3⁄12  (top and bottom ×3)
8⁄12 + 3⁄12 = 11⁄12
= 11⁄12
Mixed numbers: add the whole parts and the fraction parts separately. If you are subtracting and the top fraction is smaller, borrow one whole (rewrite it as a fraction) first.
Worked example — subtracting mixed numbers with borrowing
Find 31⁄4 − 15⁄6.
Common denominator of 4 and 6 is 12:   33⁄12 − 110⁄12.
3⁄12 is smaller than 10⁄12, so borrow 1 from the 3:   33⁄12 = 2 + 12⁄12 + 3⁄12 = 215⁄12.
215⁄12 − 110⁄12 = 15⁄12
= 15⁄12

Check yourself

  1. 1⁄5 + 2⁄5
  2. 3⁄4 − 1⁄6
  3. 21⁄3 + 11⁄2
Answers
1. 3⁄5    2. 9⁄12 − 2⁄12 = 7⁄12    3. 35⁄6 (since 2⁄6 + 3⁄6 = 5⁄6)

2 · Multiplying fractions straight across

To multiply, go straight across: numerator times numerator, denominator times denominator. No common denominator needed. Simplify at the end — or cancel a common factor first to keep numbers small. For a mixed number, change it to an improper fraction before multiplying.
Worked example
Find 2⁄3 × 3⁄5.
Straight across: (2×3)/(3×5) = 6⁄15.
Simplify by dividing top and bottom by 3: 6⁄15 = 2⁄5.
= 2⁄5
Worked example — a mixed number
Find 11⁄2 × 2⁄3.
Change 11⁄2 to 3⁄2:   3⁄2 × 2⁄3 = 6⁄6 = 1.
= 1

Check yourself

  1. 1⁄2 × 4⁄5
  2. 3⁄4 × 2⁄9
  3. 21⁄2 × 3⁄5
Answers
4. 4⁄10 = 2⁄5    5. 6⁄36 = 1⁄6    6. 5⁄2 × 3⁄5 = 15⁄10 = 3⁄2 = 11⁄2

3 · Dividing fractions multiply by the reciprocal

To divide by a fraction, flip the second fraction (its reciprocal) and multiply. The reciprocal of 2⁄5 is 5⁄2. A short way to remember it: keep, change, flip — keep the first, change ÷ to ×, flip the second.
Worked example
Find 3⁄4 ÷ 2⁄5.
Keep, change, flip: 3⁄4 × 5⁄2.
Straight across: (3×5)/(4×2) = 15⁄8.
15⁄8 = 17⁄8.
= 17⁄8

Check yourself

  1. 1⁄2 ÷ 1⁄4
  2. 2⁄3 ÷ 4⁄5
  3. 3 ÷ 1⁄2
Answers
7. 1⁄2 × 4⁄1 = 4⁄2 = 2    8. 2⁄3 × 5⁄4 = 10⁄12 = 5⁄6    9. 3 × 2⁄1 = 6

4 · Order of operations with fractions BEDMAS still rules

Fractions follow the same order as everything else: brackets, exponents, divide and multiply (left to right), then add and subtract (left to right). Do not just work left to right — multiplication and division come first.
Worked example
Find 1⁄2 + 1⁄3 × 3⁄4.
Multiplication first: 1⁄3 × 3⁄4 = 3⁄12 = 1⁄4.
Now add: 1⁄2 + 1⁄4 = 2⁄4 + 1⁄4 = 3⁄4.
= 3⁄4
Worked example — with brackets
Find (1⁄2 + 1⁄4) ÷ 3⁄8.
Brackets first: 1⁄2 + 1⁄4 = 2⁄4 + 1⁄4 = 3⁄4.
Then divide: 3⁄4 ÷ 3⁄8 = 3⁄4 × 8⁄3 = 24⁄12 = 2.
= 2

Check yourself

  1. 1⁄4 + 1⁄2 × 1⁄2
  2. (2⁄3 − 1⁄6) × 2
Answers
10. 1⁄2×1⁄2 = 1⁄4, then 1⁄4 + 1⁄4 = 1⁄2    11. (4⁄6 − 1⁄6) = 3⁄6 = 1⁄2, then ×2 = 1

If you want to go further

The reciprocal trick for division works because dividing by a number is the same as multiplying by how many times it fits into one. Try explaining 1⁄2 ÷ 1⁄4 = 2 in words: "how many quarters fit in one half?" Two. The math matches the picture.

Mathematics 8 · Unit 1 — Operations with Fractions · Ms. Ras Mulinta · Handsworth Secondary 2026–27 · BC Math 8 content: operations with fractions (addition, subtraction, multiplication, division, and order of operations). Pairs with the Unit 1 practice set and answer key.