B. Number · Number Sense: Numbers to 1000 and FractionsComparing Fair Shares
B1.6 A1About 30 minutes
Learning goals- I can explain why sharing the same amount among more people gives smaller shares.
- I can compare fractions like one fifth and one eighth, or two tenths and seven tenths.
- I can compare mixed numbers, like 3 and one fourth and 2 and four fifths.
- I can use a drawing to show which share is bigger.
Words to know
compare
To decide which share is more, which is less, or if they are the same.
same-size wholes
Wholes that are exactly the same size. We need these to compare fractions fairly.
mixed number
A whole number and a fraction together, like 2 and one half.
Which table?
At a birthday party, there are two tables.
- Table A: 5 kids share 2 pizzas.
- Table B: 8 kids share 2 pizzas.
All the pizzas are the same size. If you love pizza, which table should you sit at?
At Table A, each kid gets two fifths of a pizza. At Table B, each kid gets two eighths. Both have 2 pieces, but the pieces are different sizes!
Fifths are bigger than eighths, so two fifths is more than two eighths. Sit at Table A!
Rule 1: Same amount of food, more people → smaller shares
The same 2 pizzas shared by 8 kids gives each kid less than when they are shared by 5 kids.
Rule 2: Same number of people, more food → bigger shares
5 kids sharing 3 pizzas get three fifths each. That is more than two fifths each when they share 2 pizzas.
Rule 3: The more equal parts in a whole, the smaller each part
One half > one fourth > one eighth. One fifth > one tenth.
Same-size wholes
Only compare fractions of same-size wholes. One half of a small pizza might be less than one fourth of a giant pizza!
Worked exampleExample 1: Same food, different number of people
9 cookies are shared by 2 kids, or by 4 kids. Find each share and compare.
- By 2 kids: 4 each uses 8 cookies. Cut the last one in half: 4 and one half each.
- By 4 kids: 2 each uses 8 cookies. Cut the last one into fourths: 2 and one fourth each.
- Compare the whole numbers first: 4 is more than 2.
Sharing by 2 kids gives the bigger share: 4 and one half is more than 2 and one fourth.
Worked exampleExample 2: Same number of parts
Which is more: three tenths or seven tenths of the same pan of brownies?
- Both are tenths, so all the pieces are the same size.
- 7 pieces is more than 3 pieces of the same size.
Worked exampleExample 3: Mixed numbers
Which is more: 3 and one fourth or 2 and four fifths?
- Compare the whole numbers first.
- 3 wholes is more than 2 wholes.
- Two and four fifths is not even 3 wholes, because four fifths is less than one whole.
3 and one fourth is more.
Worked exampleExample 4: Same whole number
Which is more: 2 and one half or 2 and one fifth?
- The whole numbers are the same: 2 and 2.
- Compare the fractions: one half is bigger than one fifth (a whole cut into 2 parts has bigger parts than a whole cut into 5).
- So 2 and one half is more.
Be a good friend
Sometimes friends get different amounts because there are different numbers of people at each table. That isn't anyone's fault! If you notice someone has less, you can kindly offer to share, or help find a fair way.
Just for funWould you rather have one half of a pizza or one hundred hundredths of a pizza?
Take the hundredths! One hundred hundredths is the whole pizza. Just be ready for some very tiny slices.
Hands-on activityStrip races
You need: your fraction strip kit from the last lesson (or new paper strips the same length) and a pencil.
- Put one fifth above one eighth. Which is longer?
- Compare two fifths and two eighths. Which is longer?
- Compare three tenths and seven tenths.
- Compare one half and one tenth.
- For each pair, write a sentence: "___ is more than ___."
- Line up one half, one fourth, one eighth in order from biggest to smallest. What pattern do you see?
Math talk: "Why does a bigger number of parts mean a smaller piece?"
Your turn
1Which is bigger: one third or one sixth of the same sandwich?
One third. Thirds are bigger pieces than sixths.
2Which is bigger: one fifth or one tenth?
3Which is more: three eighths or five eighths?
Five eighths. Same-size pieces, and 5 is more than 3.
41 pizza is shared by 4 people, or by 2 people. Which way gives each person more?
By 2 people. Each gets one half, which is more than one fourth.
510 kids share 4 pies, or 10 kids share 6 pies. Which way gives each kid more?
6 pies. Each kid gets six tenths, which is more than four tenths.
6Which is more: 2 and one half or 3 and one tenth?
3 and one tenth. 3 wholes is more than 2 wholes.
77 bars are shared by 2 kids, or by 3 kids. Which way gives each kid more?
By 2 kids. By 2: 3 and one half each. By 3: 2 and one third each. 3 and a half is more.
More practice: warm-up
1The same amount of food is shared by more people. Do the shares get bigger or smaller?
2Which is bigger: one half or one eighth?
3Which is more: two fifths or four fifths?
4Which is more: 3 and one half or 1 and one half?
5Which is smaller: one sixth or one fourth?
More practice: on your own
1Ana says one eighth is bigger than one fourth because 8 is bigger than 4. Is she right?
No. Eighths cut the whole into more pieces, so each piece is smaller. One fourth is bigger.
22 pizzas are shared by 3 kids, or by 6 kids. Find each share. Which is more?
By 3 kids: two thirds. By 6 kids: two sixths. Thirds are bigger than sixths, so two thirds is more.
38 kids share 5 sandwiches, or 8 kids share 3 sandwiches. Which gives more each?
5 sandwiches: five eighths is more than three eighths.
415 crackers are shared by 4 kids, or by 5 kids. Find each share and compare.
By 4: 3 each uses 12, then 3 left cut into fourths = 12 fourths, so 3 and three fourths each. By 5: 3 each exactly. 3 and three fourths is more.
5Ben gets one half of a small pizza. Kai gets one fourth of a large pizza. Can you tell who has more just from the fractions? Why?
No. The pizzas are not the same size, so we can't compare the fractions fairly. We would need to look at the actual pieces.
Challenge
1Order from smallest to largest: one tenth, one half, one fifth, one eighth (same-size wholes).
one tenth, one eighth, one fifth, one half. The more parts in the whole, the smaller each part.
212 muffins are shared by 5 people, or by 8 people. Find each share. Which is more?
By 5: 2 each uses 10, then 2 left. Cut each into fifths: 10 fifths, so 2 more fifths each: 2 and two fifths. By 8: 1 each uses 8, then 4 left. Cut each into eighths: 32 eighths, so 4 more eighths each: 1 and four eighths. Sharing by 5 gives more (2 and a bit is more than 1 and a bit).
34 people share some pizzas. What is the fewest number of pizzas so that each person gets more than 2 whole pizzas?
9 pizzas. 8 pizzas give exactly 2 each. With 9 pizzas, each person gets 2 and one fourth, which is more than 2.
Quiz
1. Which is the biggest piece of the same whole?
2. The same food is shared by more people. Each share is:
4. 3 pizzas are shared by 4 people, or by 6 people. Which gives each person more?
6. Why can't we compare one half of a small pizza with one fourth of a large pizza just by the fractions?
Big idea: Children compare shares by reasoning about the number of items and the number of sharers, and they prove it with drawings. The curriculum (B1.6) asks Grade 3 students to use drawings to compare the results of fair-share problems.
Common difficulties:
- The bigger-number trap: thinking one eighth > one fourth. Keep coming back to the picture: more pieces means smaller pieces.
- Comparing fractions of different-size wholes.
- Mixed numbers: compare whole numbers first, then the fractions.
How to help: At meals, ask: "Would you rather have one fourth or one eighth of this pizza?" Use the fraction strip kit to check. Talk about fairness and kindness when sharing (Strand A: building healthy relationships).