Grade by Grade
B. Number · Number Sense: Numbers to 200 and Fractions

Comparing Fair Shares

B1.6About 30 minutes
Learning goals
  • I can explain why more sharers means a smaller share.
  • I can compare two fair-share answers using drawings.
  • I can compare mixed numbers by looking at the wholes first.
  • I can compare fractions with the same size parts, like one third and two thirds.

Words to know

compare
To find out which is more, which is less, or if they are the same.
share
The amount each person gets.
larger / smaller share
A bigger amount / a littler amount for each person.

Which table would you sit at?

At a birthday party there are two tables. Each table has 3 pizzas to share.

  • At the blue table, 2 children share 3 pizzas.
  • At the red table, 6 children share 3 pizzas.

If you are very hungry, which table should you sit at? Let's learn to compare fair shares.

Remember from Grade 1

In Grade 1 you saw that when more people share the same whole, each piece is smaller. One half is bigger than one fourth.

shared by 2shared by 3shared by 6
The same cookie shared by 2, 3 and 6 people. One half is greater than one third. One third is greater than one sixth.

More sharers, smaller shares

When the same amount is shared:

  • fewer sharers means a bigger share
  • more sharers means a smaller share

So one half is bigger than one third, one third is bigger than one fourth, and one fourth is bigger than one sixth.

1 whole1 wholehalveshalfhalfthirdsthirdthirdthirdfourthsfourthfourthfourthfourthsixthssixthsixthsixthsixthsixthsixth
A fraction wall. Every row is the same whole, and one part of each row is coloured. One half is the biggest part and one sixth is the smallest.
Worked example

Example 1: The pizza tables

Blue table: 2 children share 3 pizzas. Red table: 6 children share 3 pizzas. Who gets more pizza, a child at the blue table or the red table?

  1. Blue table: 3 pizzas between 2. Each child gets 1, and the last pizza is cut in halves. Each gets 1 and a half pizzas.
  2. Red table: 3 pizzas among 6. Cut each pizza in half: 6 halves. Each child gets one half of a pizza.
  3. 1 and a half is more than one half. Also, 2 sharers is fewer than 6 sharers.

A child at the blue table gets more pizza.

Blue table: 2 children1212Each child gets1 and a half pizzasRed table: 6 children123456Each child getsone half of a pizzaThe numbers show which child gets each piece.
Same 3 pizzas at each table. With 2 children, each child gets 1 and a half pizzas. With 6 children, each child gets only one half.

Comparing mixed numbers

To compare shares like 2 and a half and 1 and two thirds, look at the whole number part first.

  • 2 wholes is more than 1 whole, so 2 and a half is greater.

If the whole numbers are the same, compare the fraction parts. 3 and a half is greater than 3 and one fourth, because one half is greater than one fourth.

Worked example

Example 2: Muffins

Group A: 2 friends share 5 muffins. Group B: 3 friends share 5 muffins. Compare the shares.

  1. Group A: 2 each (4 muffins), 1 left, cut in halves. Each gets 2 and a half.
  2. Group B: 1 each (3 muffins), 2 left. Cut each leftover muffin into thirds: 6 thirds, so each friend gets two of them. Each gets 1 and two thirds.
  3. Compare the wholes: 2 is greater than 1.

Group A gets more: 2 and a half is greater than 1 and two thirds. Same muffins, fewer sharers, bigger share.

Worked example

Example 3: Same sharers, more things

2 kids share 7 crackers. Another 2 kids share 9 crackers. Which kids get more each?

  1. 7 between 2: 3 and a half each.
  2. 9 between 2: 4 and a half each.
  3. 4 is more than 3.

The kids sharing 9 crackers get more. With the same number of sharers, more things means a bigger share.

Worked example

Example 4: Same-size parts

Which is more: one third of a pan of lasagna or two thirds of the same pan?

  1. Both are thirds, so the parts are the same size.
  2. Two thirds is 2 of those parts. One third is just 1 part.

Two thirds is more.

Compare the same-size wholes

Fractions can only be compared fairly if the wholes are the same size. One half of a tiny cookie is less than one fourth of a giant pizza! In this lesson, the wholes being compared are always the same size.

Just for fun

What if 100 people shared one cookie? Each share would be so tiny you'd need a magnifying glass to find it! More sharers, smaller shares.

Hands-on activity

Make a fraction wall

You need: 5 paper strips the same length (cut from one sheet of paper), crayons, and scissors (with a grown-up).

  1. Leave one strip whole. Label it 1 whole.
  2. Fold one strip in half. Label each part one half.
  3. Fold one strip into thirds (like a letter). Label the parts.
  4. Fold one strip in half, then in half again: fourths.
  5. Fold one strip into thirds, then in half: sixths.
  6. Line them up under each other. Which part is biggest? Which is smallest?
  7. Use your wall to answer: Is two fourths more or less than one third?

Math talk: "Why does the sixths strip have the smallest parts?"

Your turn

1

Which is bigger, one half or one fourth of the same sandwich?

2

Which is smaller, one third or one sixth of the same pizza?

3

Which is more, 3 and a half or 2 and a half?

4

Which is more, two fourths or three fourths of the same cake?

5

4 friends share 1 pie. 6 friends share another pie that is the same size. Who gets a bigger piece?

6

Put in order from smallest to biggest: one half, one sixth, one third

More practice: warm-up

1

Which is bigger, one half or one third?

2

Which is more, 2 and a half or 1 and a half?

3

Which is more, one sixth or five sixths of the same cake?

4

True or false: More sharers means a bigger share.

5

Which is less, one fourth or one half of the same orange?

More practice: on your own

1

Which is more: 2 and one third, or 2 and a half?

2

Team A: 3 kids share 6 orange slices. Team B: 2 kids share 6 orange slices. Which team's kids get more each?

3

Ella gets one sixth of a cake. Ben gets one third of the same cake. Ella says she got more because 6 is bigger than 3. Is she right?

4

2 kids share 5 pancakes. 4 kids share 5 pancakes. Find each share. Which is bigger?

5

Put in order from smallest to biggest: 1 and a half, one half, 2 and one fourth

Challenge

1

Would you rather have one half of a sub sandwich or two sixths of the same sub? Explain.

2

6 friends share some granola bars, and each gets one half of a bar. How many granola bars were there?

3

Blue group: 3 kids share 4 sandwiches. Red group: 4 kids share 5 sandwiches. Which group's kids get more each? Explain.

Check your understanding

Quiz

1.

The same pizza is shared fairly. Which share is the biggest?

2.

Which is more?

3.

Which is more?

4.

5 cookies are shared. Which group gets the biggest share each?

5.

Which is more of the same cake: two sixths or five sixths?

6.

Which list goes from smallest to biggest?

For parents and teachers

Big idea: When the same amount is shared, more sharers means smaller shares. With the same number of sharers, more items means bigger shares. To compare mixed numbers, compare the whole numbers first, then the fraction parts.

Common difficulties:

  • Thinking a bigger bottom number means a bigger piece (one sixth > one third). Use a fraction wall or real food to show that sixths are smaller.
  • Ignoring the whole number part when comparing mixed numbers.
  • Comparing parts of different-sized wholes. Keep the wholes the same size.

How to help: At snack time ask, "Would you rather share this with 2 people or 4 people? Why?" Make the paper fraction wall together and keep it on the fridge.