1. Sam and Joe are arguing as they split up a chocolate bar
Sam and Joe are arguing as they split up a chocolate bar. Sam says 1/5 is more than 1/3. Joe says no, one third is bigger than one fifth. Who is right?
Use a proof to settle the argument.

2. The J in Joe should be up higher. Joe is not Joes because not Because
Sam and Joe are arguing as they split up a chocolate bar. Sam says 1/5 is more than 1/3. Joe says no, one third is bigger than one fifth. Who is right?
Use a proof to settle the argument.
The diagrams would fit on top of each other. It is the same size whole. The thirds are larger than the fifths. On third is more candy than one fifth.
So Less is more means there are less pieces but each pieces is larger when there are thirds. There are more pieces but each piece is smaller when there are fifths.
The J in Joe should be up higher. Joe is not Joes because not Because
Bigger not Biger pieces not pices
Let’s conclude with the statement: If they are talking about the same size chocolate bar, one third is more than one fifth therefore Joe is correct.

3. Draw a proof to settle the argument.
Sam and Joe are arguing as they split up a chocolate bar. Sam says 1/5 is more than 1/3. Joe says no, one third is bigger than one fifth. Who is right?
Draw a proof to settle the argument.
What is wrong with the diagram?
Even if you do not understand the problem, the diagram is incorrect. Why?
What information needs to be in the diagram? What fractions? How do you draw fractions? Does this picture make sense? Does it help explain the situation.. What do we know

4. Draw a proof to settle the argument.
Sam and Joe are arguing as they split up a chocolate bar. Sam says 1/5 is more than 1/3. Joe says no, one third is bigger than one fifth. Who is right?
Draw a proof to settle the argument.
When you divide a circle into fifths you need 5 equal pieces. Those pieces are not equal.
Same with the thirds, the pieces are not equal.
What information needs to be in the diagram? What fractions? How do you draw fractions? Does this picture make sense? Does it help explain the situation.. What do we know

5. Draw a proof to settle the argument.
Sam and Joe are arguing as they split up a chocolate bar. Sam says 1/5 is more than 1/3. Joe says no, one third is bigger than one fifth. Who is right?
Draw a proof to settle the argument.
Now you can see the answer is wrong. One fifth is not a bigger piece therefore one fifth is not a bigger number.
What information needs to be in the diagram? What fractions? How do you draw fractions? Does this picture make sense? Does it help explain the situation.. What do we know

6. Fractions are numbers. One third has a numerator of one and a denominator of three. Together the numerator and denominator make the number one third You can place one third on a number line.

7. One fifth has a numerator of one and a denominator of five
One fifth has a numerator of one and a denominator of five. Together the numerator and denominator make the number one fifth
You can place one fifth on a number line. Five fifths fit inside the one.

9. Explain your diagram to me? What are the circles?

10. If the circles are the chocolate bars, it would make more sense to me to draw rectangular chocolate bars. If the circles represent the chocolate bar they are demonstrating which piece is more chocolate bar: one third or one fifth?

11. One third represents a piece of the chocolate bar, not one out of 3.
One fifth represents a piece of the chocolate bar, not 1 out of 5.

12. A piece that is one third of the bar is bigger than a piece that is one fifth of the bar.

13. Do these diagrams make sense?
Do they fit the problem?
Did the student explain his or her thinking?

14. Do these diagrams make sense?
Do they fit the problem?
Did the student explain his or her thinking?

15. I see fifths, but did the students know?
I suspect he or she was struggling with how to show the “pieces” of a chocolate bar as fractions. I see 10 pieces but one fifth is circled. The circle is divided into 5 parts.

16. I see thirds but did the students know?
Always label your diagram so there is no need for the teacher to guess what you are trying to show.

17. I suspect he or she is trying to compare thirds and fifths on this one circle because I see the names Sam and Joe written in the pieces, but then erased.

18. I did not understand what the numbers up here were for.
Why did he or she decide to talk about 1, 2, 3, 4, 5?
And notice he or she underlined 3 and 5.

19. Sam is right in fractions but not numbers.
Jos right in numbers but not fractions.
Both of them are right.

20. Is there a serious issue with not understanding that fractions are a part of a whole. When we talk 1 out of 3 and 1 out of 5 do we leave them completely confused?
1 out of 3 encourages whole number thinking.
One third should encourage seeing a “part” or set?

21. The diagrams are wrong. They do not show parts of a whole
The diagrams are wrong. They do not show parts of a whole. They have been drawn as single squares connected to each other. They have been labelled as individual counts.
The imagery is all focused on whole number counting.
Each collection is circled but not considered to be parts of a whole.
More and less are used to refer to how many things they see not comparing the size of the pieces.
There is no feeling of a whole divided 2 different ways.
I am curious about the emphasis on “way” bigger and “way” smaller.

22. Sam is right becaus if you split a pizza in 3 halfs the pizza would have less slices so they would be bigger. So is biggers than 1 5

23. I really like the use of the arrow and the identification of Example
I really like the use of the arrow and the identification of Example. Math explanations are often created by providing and explaining an example.
In this case the students decided to talk about pizza and slices as parts of a whole instead of staying with the chocolate bar and parts of a whole. I find this curious but indicates a confidence with understanding how fractions relate.
How problematic is it that the pizza fraction became “slices”. The parts of a whole vision of one third of a bar became parts of a set but we do not refer to the size of the whole set.

24. Fractions as Parts of a Set

25. This is a diagram of one

26. then each section or part is ?.
1/3

27. I divided the 1 into 3 equal parts.
1/3

28. I ÷ 3
1/3

29. If I fold the paper to show the three parts
I demonstrate that I ÷ 3 = 1 3
1 3
1/3

30. Each section or part is one third. 1 3
1/3

31. I unfold the paper to show that the whole “one” can be renamed as: 3 3
1 3
1 3
1 3
1/3

32. Now imagine this is a diagram of 9 horses….

33. Then how many horses are in of the 9 horses?
1 3
Then how many horses are in of the 9 horses?
3/3

34. Can you equally divide up the 9 horses?
3/3

35. 1 3
1 3
1 3 3 3 3
3/3

36. Now this is a diagram of one

37. I divided the one into 4 equal parts…
1/4

38. 1 4
1 4
1 4
1 4
1/4

39. 1 4
1 ÷ 4 =
4/4

40. Unfold the whole to see
1 4
4 4
1 4
1 4
1 4
1/4

41. Now imagine this is a diagram of flock of 12 ducks…..

42. How many ducks is in of the flock of 12?
1 4
How many ducks is in of the flock of 12?
3/3

43. How many ducks is of the flock of 12?
1 4
How many ducks is of the flock of 12? 3 3 3
3/3 3

44. I went to the pound. There were 20 animals.
One fourth (1/4) of them were dogs.
How many dogs is that?
How many were not dogs?

45. 3/3

46. 3/3

47. There were 36 mini tarts in the box.
My brother ate 1/3 and I ate ¼
How many tarts are left?

48. There were 72 cookies in the box.
My brother ate 1/3 and I ate ¼
How did the answers change?