062i - Adding and Subtracting Fractions

Lesson 62 — Adding and Subtracting Fractions

Strand: Number | Descriptor: AC9M7N06 | Duration: 45 minutes

Learning Intentions

• To add and subtract fractions with related and unrelated denominators.
• To understand why a common denominator is necessary.

Success Criteria

I can:

1. Explain why fractions need a common denominator before adding.
2. Find the lowest common denominator of two fractions.
3. Add and subtract proper fractions and mixed numbers.
4. Give answers in simplest form.

Warmup

(6 minutes — why we can’t just add tops and bottoms, pairs)

A student claims .

1. Roughly how big is ? And ? So roughly how big should the answer be?
2. How big is ?
3. What does this tell you about the student’s method?
4. Try the same wrong method on . What do you get, and why is that absurd?

Answers: 1. About , so the answer should be about — more than ; 2. — smaller than ; 3. Adding tops and bottoms gives an impossible answer; 4. — adding a half to a half cannot leave you with a half.

The point: estimation (Lesson 60) instantly exposes the error. Keep estimating throughout today.

Activities

Activity 1 — Explicit Instruction: Common Denominators (14 min)

Why a common denominator? You can only add like things. Halves and thirds are different units — like adding metres to minutes. Rewriting both as sixths makes the units match.

I do — related denominators. . One denominator divides the other, so convert only one:

I do — unrelated denominators. . Use the LCM of the denominators (Lesson 7):

Check against the warmup estimate: ✓

I do — subtraction. . LCM of and is :

Choosing the denominator — insist on the LCM, not just any common multiple. works over , but the LCM keeps numbers small:

Note the answer exceeds — sensible, since and together clearly pass a whole.

We do: ; ; .

Activity 2 — Mixed Numbers (12 min)

I do — two methods for , side by side.

Method A — wholes and parts separately:

Method B — convert to improper fractions first:

Which to use? Method A is quicker for addition; Method B is safer for subtraction with borrowing, e.g. :

You do:

(Answers: ; ; ; .)

Activity 3 — Inquiry: the Fraction Wall Puzzle (8 min)

Pairs.

1. Find two different fractions that add to exactly .
2. Find three different fractions (all different denominators) that add to exactly .
3. Find two fractions with different denominators whose difference is .
4. Challenge: find three different unit fractions (numerator ) that add to exactly .

Socratic scaffolding for Q4:

Prompt	Purpose
Understand: what is a unit fraction?	Numerator :
Start big. Could be one of them?	Probably — the three must average , so one should be larger.
With taken, what must the other two total?	.
Two different unit fractions summing to ?	Try : the remainder is — a unit fraction!
State the answer.	.
Verify over 6.	✓
Looking back	Is it the only one? (With all three different, yes — this is the unique solution.)

Answers: 1. e.g. ; 2. e.g. or ; 3. e.g. or ; 4. .

Checks for Understanding

(5 minutes — exit ticket)

1. Calculate: (a) (b) (c) .
2. Calculate .
3. Calculate .
4. Reasoning. Explain why , not .
5. Estimate first, then calculate: .

Answers: 1. (a) (b) (c) ; 2. ; 3. ; 4. Adding two quarters gives two of the same unit — quarters. The denominator names the unit and does not change; only the count (numerator) adds; 5. Both are near , so estimate about — slightly under. Exact: .

Common Misconceptions

Misconception	How to pre-empt it
Adding numerators and denominators: .	The warmup demolishes this by estimation before the method is taught.
Adding denominators even with a common denominator: .	The denominator is the unit name. Two quarters plus one quarter is three quarters, as surely as 2 apples + 1 apple = 3 apples.
Using any common multiple instead of the LCM, then failing to simplify.	LCM keeps numbers small; always simplify the final answer regardless.
Converting only one fraction.	Both must be rewritten over the common denominator. Show both conversions explicitly.
Subtracting the smaller part from the larger regardless of order in mixed numbers, e.g. treated as .	Use improper fractions for any subtraction where the first part is smaller.
Leaving answers like unconverted or unsimplified.	Final answers as simplified mixed numbers, every time.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). What is ?

Answer

Each term halves the remaining gap to — the sum approaches but never reaches .

E2 (AMC Junior style). A tank is full. After of the tank’s capacity is drained, what fraction remains?

Answer

E3 (Challenge). Find the missing fraction: .

Answer

E4 (Challenge). Ana ate of a pizza, Ben ate , and Cara ate . What fraction is left?

Answer

E5 (Challenge). Show that for , then use the pattern to compute instantly.

Answer

✓; ✓; ✓. The long sum telescopes: everything cancels except the ends, leaving .

Homework

1. Calculate, simplifying where possible: (a) (b) (c) (d) (e) .
2. Calculate: (a) (b) (c) (d) .
3. Calculate: (a) (b) (c) (d) .
4. Find the missing fraction: (a) (b) .
5. A jug is full. After of the jug’s capacity is poured out, what fraction remains?
6. Leo ran km on Monday and km on Tuesday. How far in total?
7. A plank is m long. A piece m is cut off. How much remains?
8. Reasoning. Explain, using estimation, why must be less than but more than .
9. Challenge. Find three different fractions with different denominators that add to exactly .

Answers: Q1 — (a) (b) (c) (d) (e) . Q2 — (a) (b) (c) (d) . Q3 — (a) (b) (c) (d) . Q4 — (a) (b) . Q5 — . Q6 — km. Q7 — m. Q8 — each is just under , so the sum is just under ; and each exceeds , so the sum exceeds . (Exact: .) Q9 — e.g. .