FRACTION WORKSHOP

Same whole, different slices

Slice a whole, join slices together, and see how many of one fit inside another.

Fraction

3 of 4 equal slices

Decimal

0.75

position on the number line

Percent

75%

of one whole

Simplest form

already simplest

Mode

Denominator (slices)4

Numerator (served)

3/4

View

Equivalent family

3/4

6/8

9/12

The shaded length is the same in every row: same amount, new name.

Setting up the scene…

Concepts

What is a fraction?

We cut a whole into equal parts. The denominator (bottom) says how many equal parts; the numerator (top) says how many of them we take.

\frac{3}{4}

: three of four equal slices. The tart, the bar and the number line show the same amount three ways.

\frac{a}{b} = a \times \frac{1}{b}

Why must the parts be equal?

Cut a pizza into one huge piece and three tiny ones and none of them is

\frac{1}{4}

. In a fraction, how many parts matters, and so does that the parts are the same size. That's why every cut in the workshop makes equal slices. Careful: 3 cuts split a bar into 4 pieces — count pieces, not cuts.

Fractions on the number line — and past 1

Split 0 to 1 into

b

equal steps and each step is

\frac{1}{b}

;

\frac{a}{b}

a

steps from 0. When the numerator passes the denominator we go beyond 1:

\frac{5}{4}

is one whole and one quarter,

1\tfrac{1}{4}

. A fraction is one number —

\frac{5}{4}

lives between 1 and 2, not near 4 or 5.

\frac{5}{4} = \frac{4}{4} + \frac{1}{4} = 1\tfrac{1}{4}

Equivalent fractions: same amount, new name

Press Cut every slice in two: the number of slices and the number served both double, yet the served part of the tart doesn't change at all.

\frac{1}{2} = \frac{2}{4} = \frac{4}{8}

. Multiplying (or dividing) top and bottom by the same number keeps the amount; adding the same number does not.

\frac{a}{b} = \frac{a \times k}{b \times k}

Simplest form

Merge joins neighboring slices in groups. In

\frac{6}{9}

, both 6 and 9 divide by 3: merge slices in threes and you get

\frac{2}{3}

. When top and bottom share no factor except 1, the fraction is in simplest form.

\frac{6}{9} = \frac{6 \div 3}{9 \div 3} = \frac{2}{3}

Three ways to compare fractions

Same denominator: same-size slices, so the bigger numerator wins. Same numerator: same number of slices, so the smaller denominator has bigger slices,

\frac{3}{5} > \frac{3}{8}

. Otherwise use a benchmark (distance to

\frac{1}{2}

or 1) or rewrite both with a common denominator.

Common denominator

We can't directly compare or add slices of different sizes. Re-slice both using the least common multiple of the denominators and every slice becomes the same size. For

\frac{2}{3}

and

\frac{3}{5}

the LCD is 15:

\frac{10}{15}

\frac{9}{15}

— now just count slices.

\frac{2}{3} = \frac{10}{15}, \quad \frac{3}{5} = \frac{9}{15}

Adding and subtracting: same-size slices first

1. Rewrite with a common denominator (same-size slices). 2. Join or remove slices — the denominator stays, because the slice size doesn't change. 3. Simplify; if the numerator passes the denominator, the pieces spill into a second tart and you get a mixed number.

\frac{1}{2}+\frac{1}{3}

is never

\frac{2}{5}

\frac{2}{5}

is smaller than

\frac{1}{2}

alone!

\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}

Multiplying = part of a part

\frac{a}{b} \times \frac{c}{d}

means "

\frac{a}{b}

\frac{c}{d}

". On the square tray, shade

\frac{a}{b}

of the width blue and

\frac{c}{d}

of the height yellow: the glowing overlap is the product.

a \times c

cells glow out of

b \times d

\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

Why can a product be smaller?

With whole numbers, multiplying makes bigger:

3 \times 4 = 12

. But multiplying by a fraction less than 1 takes only part of something.

\frac{2}{3} \times \frac{3}{4} = \frac{1}{2}

, smaller than both factors. Multiplying by 1 changes nothing; multiplying by more than 1 makes bigger.

Dividing: how many fit?

\frac{a}{b} \div \frac{c}{d}

asks: "how many

\frac{c}{d}

fit into

\frac{a}{b}

?" Lay copies of the divisor over the dividend one by one. If the last copy doesn't fit completely, the part that fits counts too:

\frac{3}{4} \div \frac{1}{2} = 1\tfrac{1}{2}

. Dividing by a fraction less than 1 makes the result bigger:

2 \div \frac{1}{4} = 8

Why flip and multiply?

One whole holds

\frac{d}{c}

copies of

\frac{c}{d}

(one whole holds 4 quarters).

\frac{a}{b}

\frac{a}{b}

of a whole, so it holds

\frac{a}{b} \times \frac{d}{c}

copies. Keep the dividend; flip only the divisor.

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Skills and curriculum

Unit fractions and parts of a whole

Grades 3–4

Partition a whole into equal parts; read 1/b as one part and a/b as a parts, and place fractions on a number line.

Not started

MEB Grade 3 · Numbers and Quantities (2) · MAT.3.1.9, MAT.3.1.10, MAT.3.1.11MEB Grade 4 · Fractions · M.4.1.6.1, M.4.1.6.2CCSS 3.NF.A.1CCSS 3.NF.A.2

Equivalent fractions

Grades 4–5

Find fractions naming the same amount by scaling numerator and denominator by the same factor.

Not started

MEB Grade 5 · Numbers and Quantities (2) · MAT.5.1.3CCSS 3.NF.A.3bCCSS 4.NF.A.1

Comparing fractions

Grades 4–5

Order fractions using common denominators, common numerators, or benchmarks such as 1/2.

Not started

MEB Grade 4 · Fractions · M.4.1.6.2, M.4.1.6.4MEB Grade 5 · Numbers and Quantities (2) · MAT.5.1.4CCSS 3.NF.A.3dCCSS 4.NF.A.2

Adding and subtracting fractions

Grades 4–6

Add and subtract fractions by rewriting them with a common denominator, then simplify.

Not started

MEB Grade 4 · Operations with Fractions · M.4.1.7.1, M.4.1.7.2MEB Grade 6 · Numbers and Quantities (2) · MAT.6.1.7CCSS 4.NF.B.3CCSS 5.NF.A.1

Multiplying fractions (area model)

Grades 5–6

Model a/b × c/d as part of a rectangle and explain why a product can be smaller than its factors.

Not started

MEB Grade 6 · Numbers and Quantities (2) · MAT.6.1.7CCSS 5.NF.B.4CCSS 5.NF.B.5

Dividing fractions

Grades 6–6

Model a/b ÷ c/d as "how many of one fraction fit in the other?"

Not started

MEB Grade 6 · Numbers and Quantities (2) · MAT.6.1.7CCSS 6.NS.A.1

Mission 1/19Build0/19 ✓

One slice of three

Cut the tart into 3 equal slices and serve one of them:

\frac{1}{3}

Try it in the simulation — the mission completes on its own when you reach the goal.

Same whole, different slices

Concepts

Skills and curriculum

One slice of three

All missions

How it works: Same whole, different slices

Keyboard and mouse