Common Mistakes with Halves and Thirds
Learn
Everyone makes mistakes when learning something new! In this lesson, we will look at common mistakes students make with fractions. Learning about these mistakes will help you avoid them.
Why Learn About Mistakes?
When you know what mistakes to watch for, you can check your work and catch errors before they cause problems. Smart learners study mistakes!
Common Mistakes to Avoid
Mistake #1: Forgetting That Parts Must Be Equal
Wrong:
"This shape is split into halves."
The parts are NOT the same size!
Right:
"This shape is split into halves."
Both parts ARE the same size!
Mistake #2: Thinking More Parts Means Bigger Pieces
Wrong:
"1/3 is bigger than 1/2 because 3 is bigger than 2."
Right:
"1/2 is bigger than 1/3 because cutting into fewer pieces makes each piece larger."
1/2
1/3
Mistake #3: Counting Parts Wrong
Wrong:
"I ate 1 piece out of 3, so I ate 1/2."
Right:
"I ate 1 piece out of 3, so I ate 1/3."
Mistake #4: Confusing "Half" and "Halves"
Wrong:
"I cut the apple into 1 half."
Right:
"I cut the apple into 2 halves." (Then each piece is 1 half.)
Mistake #5: Not Checking if Shapes Start the Same Size
Wrong:
"Half of a small cookie equals half of a big pizza."
Right:
"Half of a small cookie is much less than half of a big pizza because the whole things are different sizes."
Practice
Find the mistake in each problem. Then explain what is wrong.
Problem 1
Mia says: "I cut my paper into 2 pieces, so I have halves." Is Mia always correct?
Show Answer
NOT always correct! Mia only has halves if the 2 pieces are EQUAL (the same size). If one piece is bigger than the other, they are not halves.
Problem 2
Jake says: "One third is bigger than one half because 3 is more than 2." What is Jake's mistake?
Show Answer
Jake has it backwards! When you divide something into MORE parts, each part is SMALLER. One half (1/2) is bigger than one third (1/3).
Problem 3
Sara divided a rectangle into 3 parts. One part is very small, and two parts are big. Can she say she has thirds?
Show Answer
No! For thirds, all 3 parts must be the SAME size (equal). Sara just has 3 unequal parts.
Problem 4
Tom ate 2/3 of a cookie. He says he ate most of the cookie. Is he right?
Show Answer
Yes! 2/3 is more than half. If you eat 2 out of 3 equal pieces, you ate more than half the cookie.
Problem 5
Emma says: "I want the bigger half!" What is funny about what Emma said?
Show Answer
There is no "bigger half"! If the pieces are truly halves, they are EQUAL - both the same size. If one is bigger, they are not real halves.
Problem 6
A circle is drawn. Leo draws one line through the middle. Does he have halves?
Show Answer
It depends! If the line goes through the CENTER of the circle, yes, he has halves. If the line is off-center, the pieces are not equal, so not halves.
Problem 7
Kim has half of a small pizza. Ben has half of a large pizza. Kim says they have the same amount. Is Kim correct?
Show Answer
No! Ben has more pizza because his whole pizza was larger. Half of a big pizza is more than half of a small pizza.
Problem 8
A student writes: "1 third + 1 third = 2 sixths." Find the mistake.
Show Answer
The mistake is in the answer. 1/3 + 1/3 = 2/3 (two thirds), not two sixths. The bottom number stays the same when adding fractions with the same bottom number.
Problem 9
Amy colored 1 out of 2 equal parts of a shape. She says she colored 1/3. What is wrong?
Show Answer
Amy colored 1/2, not 1/3. There are 2 equal parts, so the bottom number should be 2. She colored 1 of the 2 parts, so the fraction is 1/2.
Problem 10
Carlos says: "If I fold a paper twice, I get halves." Is this correct?
Show Answer
Not exactly! One fold can make halves (2 equal parts). If Carlos folds twice, he might get 4 equal parts (fourths or quarters), not halves. It depends on how he folds.
Check Your Understanding
Answer these questions to show you understand common mistakes.
1. What is the most important rule for halves and thirds?
Show Answer
The parts must be EQUAL (the same size).
2. Which is bigger: 1/2 or 1/3? Why?
Show Answer
1/2 is bigger because when you cut something into fewer pieces (2 instead of 3), each piece is larger.
3. If you and a friend share something equally, what fraction do you each get?
Show Answer
You each get 1/2 (one half).
Next Steps
- Review any mistakes you have made in the past
- Always check: Are the parts equal?
- Take the Unit Quiz to test everything you learned!