Adding and subtracting fractions

Parts can be counted together only when they are the same size. Cut both fractions into the same parts first: 1/3 + 1/4 = 4/12 + 3/12 = 7/12. The lowest common multiple of the bottoms is the smallest choice.

A recipe needs 1/3 of a cup of milk and 1/4 of a cup of cream. Will it all fit in a half-cup measure? You cannot tell until the parts are the same size.

You can add 2 apples and 3 apples, but 2 apples and 3 pears do not make 5 of anything. Fractions are the same: $\tfrac{2}{7} + \tfrac{3}{7} = \tfrac{5}{7}$ (sevenths and sevenths), but thirds and quarters are different sizes.

To add or subtract fractions with different bottoms, cut both into the same parts first, using equivalent fractions. The new bottom is a common denominator: a common multiple of the two bottoms, and the lowest common multiple is the smallest.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

A third and a quarter

Is the answer more or less than $\tfrac{1}{3}$ on its own?

What is

\tfrac{1}{3} + \tfrac{1}{4}

\tfrac{2}{7}

\tfrac{1}{7}

\tfrac{7}{12}

\tfrac{2}{12}

TryTry it on the board and compare the result with your prediction.

Cut them the same

The board says when a bar does not split evenly into $k$ parts.

Try it yourselfYour edits are saved in this browser.

Change $k$ , the number of parts in each whole, until both fractions split evenly. Then read the sum. Was your prediction right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Common denominators

Here is the method, then a mixed numeral and a decimal.

Worked exampleBefore each step, say what you think comes next.

$\tfrac{1}{3} + \tfrac{1}{4} = \tfrac{4}{12} + \tfrac{3}{12}$ The LCM of 3 and 4 is 12: write both in twelfths.
$= \tfrac{7}{12}$ Same parts: add the tops, keep the bottom.
$2\tfrac{1}{2} - \tfrac{3}{4} = \tfrac{10}{4} - \tfrac{3}{4} = \tfrac{7}{4} = 1\tfrac{3}{4}$ A mixed numeral: write it as quarters first.
$3.45 + 2.70 = 6.15$ Decimals: line up the points (the same parts: tenths, hundredths).

Now try one: $\tfrac{2}{5} + \tfrac{1}{3}$ .

Read Ella's working. Could the answer be less than $\tfrac{1}{2}$ ?

Work out $\tfrac{2}{5} + \tfrac{1}{3}$ . Give your answer in simplest form.

Ella adds $\tfrac{1}{2} + \tfrac{1}{3}$ . Click the line where her method goes wrong.

Work out $3.45 + 2.7$ .

ModifyChange the problem on purpose and see what happens.

One change at a time

Each task changes one thing.

Need a hint?

24 works. Is there a smaller number that 4 and 6 both divide?

Try it yourselfYour edits are saved in this browser.

Find $\tfrac{3}{4} - \tfrac{1}{6}$ with the lowest common denominator.

Use the board. This ticks when you get there.

Work out $\tfrac{5}{6} - \tfrac{1}{4}$ in simplest form.

Work out $1\tfrac{1}{2} + \tfrac{2}{3}$ .

Edge case: what is the lowest common denominator for

\tfrac{1}{4} + \tfrac{3}{8}

?412 = 4 + 88, since 8 is already a multiple of 432 = 4 × 8

PractiseA few on your own, to check you have it.

Add and subtract

Work out $\tfrac{3}{4} + \tfrac{1}{6}$ in simplest form.

Work out $3\tfrac{1}{3} - 1\tfrac{1}{2}$ .

A plank is 2.4 m long. 0.65 m is cut off. How long is the rest, in metres?

Review: find the LCM of 6 and 10.

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

A sum of one

Then find a pair where neither bottom is a multiple of the other.

Try it yourselfYour edits are saved in this browser.

Need a hint?

Try 1/2 and something else, or 1/3 and something else.

Another hint

Choose k so both bars split, then count the parts.

Make two fractions with different bottoms that add to exactly 1.

Use the board. This ticks when you get there.

For teachers

Students cut two fraction bars into the same number of parts to add and subtract fractions with different denominators, use the LCM as the lowest common denominator, and extend the idea to mixed numerals and decimals (AC9M7N06, node 1). About 45 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

I can add and subtract fractions with different denominators using a common denominator.
I can add and subtract mixed numerals and decimals.
I can explain why the tops and the bottoms are not simply added.

Before class

Do the Run board yourself first. Students need equivalent fractions (Equivalent fractions) and the LCM (HCF and LCM).

The one idea: parts can be counted together only when they are the same size, so cut both fractions into the same parts first. The common mistake is adding the tops and the bottoms (Ella's mistake); a quick check is that the sum must be bigger than each fraction.

Decimals are the same idea: line up the places so tenths are added to tenths.

Running the lesson

Predict: students who chose 2/7 added across. Ask: is 2/7 bigger than 1/3?
Run: ask which values of k work, and what they have in common.
Investigate: for the decimals task, write 2.7 as 2.70 and ask why that is allowed.
Modify: for the edge case, ask why only one fraction needs to change.
Make: collect pairs; then challenge for one where neither bottom divides the other.

The hard part (on purpose)

Finding a common denominator without guessing. Ask students to list a few multiples of each bottom number and look for the first match, instead of telling them the LCM.

When students get stuck

A student adds the tops and the bottoms (Ella's mistake). Ask: Is your answer bigger than each fraction you added? Fix: Use the bars: change both into the same parts, then count.
A student changes the bottom but not the top (1/3 = 1/12). Ask: What did you multiply the bottom by? Did the top get the same? Fix: Equivalent fractions: the same multiplication on the top and the bottom.
A student subtracting mixed numerals takes the wholes and the parts separately and gets stuck at 1/3 − 1/2. Ask: What if you write both as improper fractions first? Fix: Convert to improper fractions with a common denominator, then subtract.

Stretch and support

Stretch: Find three different fractions with different bottoms that add to 1.

Support: Use the board for every sum; fraction walls and a list of multiples help find the common denominator.

Where it leads

Next: Multiplying and dividing fractions (node 2), where the rule is different: no common denominator needed.

Where it leads: algebraic fractions (Year 10) and probability with fractions.

What each step is for

a third and a quarter: Committing to a sum for 1/3 + 1/4 shows whether they add the tops and the bottoms (2/7), which the bars then contradict.
cut them the same: Changing the number of parts until both bars split evenly shows that a common multiple of 3 and 4 makes the parts countable.
common denominators: A common denominator makes the parts the same size; Ella adds tops and bottoms as if the parts already matched. Mixed numerals and decimals follow the same rule: line up like with like.
one change at a time: One change at a time: subtracting on the board with the lowest common denominator, subtracting without the board, a mixed numeral answer, then the case where one denominator is a multiple of the other.
add and subtract: Fractions, a mixed numeral and decimals in context, mixed; adding across is the planted slip; the review keeps the LCM fresh.
a sum of one: Choosing two different fractions that add to exactly 1, and showing it with a common denominator, works the method backwards.

Now you can

Add and subtract fractions with different denominators using a common denominator.
Add and subtract mixed numerals and decimals.
Explain why the tops and the bottoms are not simply added.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson13. Multiplying and dividing fractionsa/b × c/d is c/d of a/b: multiply the tops and multiply the bottoms. Multiplying by a fraction less than 1 makes a number smaller. Dividing asks how many fit: 3 ÷ 1/4 = 12.

Source: content/maths/number/adding-fractions/lesson.md