Fractions as Sums of Unit Fractions

Lesson 1 of 3: Building Fractions From Unit Fractions

In this lesson:

• Every fraction is built from smaller, single pieces
• Adding and subtracting fractions and mixed numbers

Your Learning Objectives for Today

By the end of this lesson, you can:

1. Show a fraction as a sum of unit fractions
2. Decompose a fraction more than one way
3. Add and subtract like-denominator fractions
4. Add and subtract like-denominator mixed numbers
5. Solve word problems with a model and equation

Three Bites of Pizza, One at a Time

Quick recall: in , what does the 8 tell you? What does the 3 tell you?

• A pizza is cut into 8 equal slices
• You eat one slice, then another, then a third
• Can you write that as an addition equation?

Building the Equation One Bite at a Time

• Bite 1:
• Bite 2: more — now
• Bite 3: more — now

Meet the Unit Fraction Idea

• A unit fraction has a 1 on top — one single piece of the whole
• means 3 copies of the unit fraction
• Every fraction is a sum of unit fractions

The Same Idea on a Number Line

• Divide 0 to 1 into sixths
• Make five jumps of each

More Than One Way to Break It Apart

• (three single pieces)
• (one piece, then a two-piece group)
• Can you find yet another grouping?

A Bigger Fraction Past One Whole

• = five copies of
• , so

Decomposing a Mixed Number Fully

• means 2 wholes and 1 more eighth
• Add the pieces:

Your Turn: Decompose Seven Sixths Three Ways

Find at least three different ways to write as a sum of same-denominator fractions.

Hint: one of your three ways can use a group bigger than a unit fraction.

From Taking Apart to Putting Together

You've been decomposing fractions — breaking one apart.

• Next: joining two fractions together
• Same pieces, opposite direction

Joining Parts of the Same Fence

• Maya paints of a fence in the morning
• She paints more in the afternoon
• Total:

Predict: Is This Addition Correct?

A classmate writes:

• (a) Correct
• (b) Incorrect

Decide before the next slide — and think about what the classmate might have done.

Reveal: The Denominator Doesn't Change

• The classmate added 6 + 6 = 12 in the denominator — that's the error
• The denominator names the piece size; it doesn't get added
• 2 sixths + 3 sixths = 5 sixths, just like 2 cats + 3 cats = 5 cats

Subtraction Means Separating the Parts

• A full water bottle is
• You drink
• What's left:

On a number line, this looks the same: start at , jump back , land on .

A Sum That Passes One Whole

• of a gallon in one jug, in another
• Total:
• — same amount, two names

Quick Check: Find the Missing Piece

What has to be added to two fifths to reach four fifths?

How Do We Add These?

Aiden ran miles on Saturday and miles on Sunday.

• These aren't single fractions anymore — each has a whole-number part too
• How should we add them?

Approach A: Wholes and Fractions Separately

• Add the whole numbers: 2 + 1 = 3
• Add the fractions:
• Combine:

Predict: Is This Answer Finished?

• (a) Yes, that's the final answer
• (b) No, something still needs to happen

Decide before the next slide.

Reveal: Regroup the Extra Whole

• is more than one whole:
• So the answer regroups to
• Just like carrying: 13 ones becomes 1 ten and 3 ones

Approach B: Convert and Add

• and
• Add:
• Convert back: — matches Approach A

Predict: Can You Subtract This Directly?

• (a) Yes — just subtract
• (b) No — something is going to block that step

Try the fraction subtraction yourself before the next slide.

Reveal: Decompose a Whole First

• doesn't work — 1 eighth isn't enough
• Break a whole into eighths:
• Now subtract:

Word Problems Bring It Together

Every word problem needs:

• A visual model — shows what's happening
• An equation — records the computation
• A check: do the fractions share the same whole?

A Recipe Needs More Sugar

A recipe needs cup of sugar. Dina already added cup.

• Equation:
• Solve: cup
• The story sounds like adding, but finding the missing amount takes subtracting

Cutting a Board Down to Size

A board is feet long. Marcus cuts off feet.

• is smaller than — decompose first:
• Subtract: feet remaining

Reading a Book Over Three Days

Ava reads of her book Monday, more Tuesday, and more Wednesday.

• Equation:
• Ava has read of her book

Predict: Is That Really of a Pizza?

Sam eats of a large pizza and of a small pizza.

• (a) Yes — of a pizza
• (b) No — something about this addition doesn't apply

Decide before the next slide.

Reveal: The Wholes Don't Match

• A fourth of a large pizza is not the same amount as a fourth of a small pizza
• only works when both fourths come from the same whole
• Always ask first: do these fractions refer to the same whole?

Your Turn: Solve It Yourself

A relay team must swim of a mile. The first swimmer covers of a mile.

How far must the second swimmer go?

No hints this time — decide the operation, draw a model, write the equation, and solve.

Four Things That Trip People Up

• The denominator never gets added — count pieces, not numbers
• After adding, check: numerator bigger than denominator? Regroup a whole
• Can't subtract the fraction part? Break a whole into pieces first
• Always check: do the fractions share the same whole?

What You Now Know About Fractions

✓ A fraction is a stack of same-size unit-fraction pieces
✓ Adding and subtracting is just joining and separating those pieces
✓ Mixed numbers regroup and decompose the same way whole numbers do

Next Up: Adding the Same Fraction Repeatedly

You've been adding fractions one piece at a time. Next lesson asks what happens when you add the same fraction to itself over and over — and gives that repeated addition a name: multiplication.