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Understand a Fraction as a Sum of Unit Fractions

Students understand that any fraction a/b (where a > 1) can be expressed as a sum of unit fractions (1/b). They use fraction models and number lines to build this understanding and write fractions as repeated addition of unit fractions.

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Understand a Fraction as a Sum of Unit Fractions

Grade 4 · Understanding Fraction Addition and Subtraction · question 1 of 3

0 of 20

20 rungs. A star at the top.

1/2 + 1/2 + 1/2 = 2

How it is taught

0:00 of 3:09

Any fraction is a stack of same-size unit pieces.

The bottom number names the piece; the top counts them.

Read it: A fraction is its unit pieces

A bar is cut into 10 pieces, and 1 piece is 1/10. Add 1/10 and 1/10 and the run is 2 pieces long: 1/10 + 1/10 = 2/10.

  1. Name 1 piece by the bottom number: 1/10.
  2. Write the same piece once for each time it is added: 1/10 + 1/10.
  3. Count the pieces in the run and keep the bottom number: 2 pieces of 1/10 make 2/10.

Each 1/10 is a piece of the same bar, so adding them makes a longer run, not smaller pieces.

The top number counts the pieces in the run and the bottom number names their size: 2/10 is 2 pieces of 1/10.

Adding the same piece 2 times is 2 copies of it, so 1/10 + 1/10 is also 2 × 1/10.

A fraction is its unit pieces added up: the top number says how many, and the bottom number says what size.

Adding the bottom numbers too, and calling 1/10 + 1/10 the same as 2/20.

2/20 is 2 pieces of a bar cut into 20, half the size of the run here. The pieces are still tenths, so the sum is 2/10.

1/5 + 1/5 + 1/5 = ___/5 — 3. 3 copies of 1/5 make 3 pieces of 1/5: 3/5.

3/4 = 1/4 + 1/4 + 1/4

Today we'll see how any fraction is built from smaller, equal pieces called unit fractions.

Here's three-fourths. The bottom number, four, tells us the whole is cut into four equal parts.

Look at our bar — four equal pieces, nothing shaded yet.

Shade one piece. That's one-fourth — our unit fraction.

Shade another. One-fourth plus one-fourth makes two-fourths.

Shade one more piece. Three-fourths!

What does 34 really mean?A fraction is a sum of unit fractions.344 = the whole is split into 4 equal parts14141414 + 14 = 2414 + 14 + 14 = 3434 = 14 + 14 + 14three unit fractions added together01142434+14+14+1434ANSWER34= 14 + 14 + 14
Grade 4 · Building fractions from unit pieces

So three-fourths equals one-fourth plus one-fourth plus one-fourth.

Let's see the same idea on a number line from zero to one.

From zero, hop one-fourth to the right.

Hop another one-fourth — we're at two-fourths.

Hop one more time — we land right on three-fourths.

Three-fourths is three unit fractions added together. Any fraction works the same way!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal
1 star — one skill mastered, every rung of its ladder climbed
Climbing now
Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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