B. Number · Number Sense: Numbers to 10 000, Fractions and Decimal Tenths

Comparing and Ordering Fair Shares

B1.5 A1About 40 minutes

Learning goals

I can find each person's share when things are shared fairly by 2, 3, 4, 5, 6, 8 or 10 people.
I can use drawings to compare shares from two different sharing situations.
I can explain why more sharers means smaller pieces.
I can order fractions using drawings and smart strategies.

Words to know

fair share
Everyone gets the same amount.

sharer
A person (or animal!) who gets a share.

portion
The amount one sharer gets.

compare
Decide which is more, which is less, or if they are equal.

Remember from Grade 3

In Grade 3 you shared things fairly. You learned that when 3 pancakes are shared by 4 people, you can cut each pancake into 4 equal pieces. Each person gets one piece of each pancake: 3/4 of a pancake.

The number of things becomes the numerator. The number of sharers becomes the denominator. This year, you'll compare shares from two different sharing situations.

Let's start with a story

At a winter party, 1 sub sandwich is shared by 4 friends at Table A. At Table B, another sub, the same size, is shared by 6 friends.

Who gets more, a friend at Table A or at Table B?

Table A: each friend gets 1/4 of a sub.
Table B: each friend gets 1/6 of a sub.

More people sharing the same amount means smaller pieces. So 1/4 > 1/6. Table A friends get more.

Two subs of the same size. Top: cut into 4 equal parts, 1/4 shaded. Bottom: cut into 6 equal parts, 1/6 shaded. 1/4 is longer.

Same-size wholes

We can only compare fractions fairly when the wholes are the same size. Half of a giant pizza is more than half of a tiny one!

Worked example

Example 1: Same number of things, different sharers

3 pancakes are shared by 5 kids. Another 3 pancakes (same size) are shared by 8 kids. Which kids get more?

3 pancakes ÷ 5 kids: each kid gets 3/5 of a pancake.
3 pancakes ÷ 8 kids: each kid gets 3/8 of a pancake.
Both have 3 pieces, but fifths are bigger than eighths.
So 3/5 is more than 3/8.

3/5 > 3/8. The kids in the group of 5 get more.

3/5 of a pancake compared with 3/8 of the same size pancake. 3/5 is longer.

Worked example

Example 2: Different things and different sharers

3 pizzas are shared by 4 people. 2 pizzas (same size) are shared by 3 people. Who gets more?

3 pizzas ÷ 4 people = 3/4 of a pizza each.
2 pizzas ÷ 3 people = 2/3 of a pizza each.
Draw both on same-size bars (below).
Another way: 3/4 is missing 1/4 to make a whole. 2/3 is missing 1/3. A fourth is smaller than a third, so 3/4 is missing less. It is closer to 1 whole.

3/4 > 2/3. The people sharing 3 pizzas among 4 get more.

3/4 and 2/3 on same-size bars. 3/4 reaches a bit farther.

Worked example

Example 3: Shares bigger than one

5 granola bars are shared by 4 kids. Another 5 bars are shared by 6 kids. Compare the shares.

5 bars ÷ 4 kids: each kid gets one whole bar, and the extra bar is cut into 4: 1 1/4 bars (or 5/4).
5 bars ÷ 6 kids: there aren't enough bars for each kid to get a whole one. Cut each bar into 6: each kid gets 5/6 of a bar.
1 1/4 is more than 1 whole. 5/6 is less than 1 whole.

1 1/4 > 5/6. The 4 kids each get more.

Smart strategies to compare fractions

Same denominator: compare the numerators. 3/8 < 5/8 (more of the same-size pieces).
Same numerator: the smaller denominator is bigger. 2/3 > 2/5 (same number of pieces, but thirds are bigger).
Compare to 1/2: 3/8 is less than half (half of 8 is 4). 4/6 is more than half (half of 6 is 3). So 3/8 < 4/6.
Compare to 1 whole: see how much each one is missing, like in Example 2.
Draw it on same-size bars. This always works!

Worked example

Example 4: Order three shares

Order from least to greatest: 2 apples shared by 10 kids, 2 apples shared by 3 kids, 5 apples shared by 8 kids.

Shares: 2/10, 2/3 and 5/8.
2/10 is much less than half (half of 10 is 5). It is the least.
2/3 is missing 1/3 to make a whole. 5/8 is missing 3/8. 3/8 is a little more than 1/3 (look at the bars below), so 5/8 is missing more and is the smaller one. You can check by drawing same-size bars.

2/10, 5/8, 2/3

2/10, 5/8 and 2/3 on same-size bars. 2/10 is shortest and 2/3 is longest.

Just for fun

What if 1 pizza were shared by 100 friends? Each friend would get 1/100 of a pizza: about one bite and a dab of sauce. More sharers, smaller slices!

Hands-on activity

Paper pancake sharing

You need: paper, a drinking glass or small lid to trace circles, scissors (ask a grown-up to help), a pencil.

Trace and cut out 6 paper circles the same size. These are pancakes.
Situation A: share 2 pancakes among 3 stuffed animals or family members. Fold and cut to make fair shares. What does each sharer get? (2/3)
Situation B: share 3 pancakes among 4 sharers. What does each get? (3/4)
Put one share from A on top of one share from B. Which is more?
Write a sentence with > or <, like 3/4 > 2/3.

Math talk: "How could you tell which is more without cutting?" If you're frustrated when cutting doesn't come out perfectly, take a breath. Close is okay; the idea is what counts!

Your turn

1 pie shared by 2 people, or 1 pie (same size) shared by 8 people. Which share is bigger?

4 muffins shared by 5 kids. What is each kid's share?

Compare with > or <: 3/10 □ 3/4

Compare with > or <: 5/8 □ 3/8

2 sandwiches shared by 3 kids, or 2 sandwiches shared by 6 kids. Which kids get more?

Is 4/6 more or less than 1/2? How do you know?

Order from least to greatest: 1/5, 1/2, 1/10

More practice: warm-up

Which is bigger: 1/3 or 1/4?

Which is bigger: 2/5 or 4/5?

3 oranges shared by 4 kids: each kid gets...?

1 loaf of bread shared by 10 people: each person gets...?

Is 1/8 more or less than 1/2?

More practice: on your own

5 pizzas shared by 6 people vs 5 pizzas shared by 8 people. Who gets more each?

1 cake shared by 3, or 2 cakes shared by 5. Compare the shares.

7 granola bars shared by 4 kids. How much does each kid get? Is it more or less than 1 bar?

Order from greatest to least: 2/3, 2/10, 2/5

Compare 3/8 and 3/5 using "compare to 1/2".

Challenge

4 pizzas are shared by 5 kids at one table and 5 pizzas by 6 kids at another. Which kids get more? Explain.

Choose a number of sharers (2, 3, 4, 5, 6, 8 or 10) so that 3 pies give each person less than half a pie.

Mila says 2/3 of a small pizza and 2/3 of a large pizza are the same amount. What would you tell her?

Check your understanding

Quiz

1 pizza is shared by 3 people. Another same-size pizza is shared by 5. Which share is bigger?

2 pies are shared by 5 people. How much does each person get?

Which is true?

Which share is more than half?

Order from least to greatest: 1/2, 1/8, 1/4

5 bars shared by 4 kids. Each kid gets...

For parents and teachers

Big idea: B1.5 asks Grade 4 students to use drawings and models to represent, compare and order the portions that result from two different fair-share situations with 2, 3, 4, 5, 6, 8 or 10 sharers. Key ideas: more sharers of the same amount means smaller portions; comparisons only make sense when the wholes are the same size.

Common difficulties:

Comparing only the denominators and thinking 1/8 > 1/3 because 8 > 3.
Drawing wholes of different sizes when comparing. Draw bars under each other, same length.
Comparing 2/3 and 3/4 is hard. The "how much is missing from a whole" strategy helps: 3/4 is missing 1/4, 2/3 is missing 1/3.

How to help: Make sharing real: cut toast, muffins or playdough for different numbers of people and ask "Would you rather be at this table or that one?" Strand A: encourage your child to try more than one strategy and to stay calm when cuts aren't perfect.