You Make the Call
Since all of you are planning to be teachers, it’s never too early to start learning to grade. Here are responses from the homework assignment. Decide which ones are right and which are wrong. Some criteria to keep in mind:
• Does the answer use fraction thinking and not rely solely upon whole number reasoning?
• Does the answer use the appropriate image for iterating or partitioning?
• Does the answer sufficiently justify that the process gives the desired fraction?
• Is the answer complete, or are there parts missing?
Problem: Say what 9/4 means from an iterating perspective.
1. Take 4 pieces to make a whole—get 9 identical pieces to make 9/4.
2. 9/4 means 9 one-fourths, where 1/4 is the amount that if I had 4 copies of that amount and put them together, I would get a whole.
3. 9/4 would be if I had 9 pieces that were each equal to 1/4.
4. Take a smaller piece and see if you can piece it together 9 times and be the same length as 9/4 keep trying until you get the right size piece. 9/4 is 9 of these little 1/4 pieces put together.
5. If you take a whole and by trial and error, find a piece that will go into the whole 4 times. Then duplicate one of these pieces 9 times.
Problem: Say what 9/4 means from a partitioning perspective.
6. 9/4 is the amount you get when cut a whole into 4 equal parts and then take 9 pieces of the same size.
7. You get the fraction 9/4 when you take the whole and divide it up into nine equal pieces. You then take the whole and add it together twice along with one of the nine equal pieces.
8. Cut 9/4 into 9 individual pieces which would be equal to 1/4.
9. 9 1/4 pieces, where a whole cut into 4 pieces to create the 1/4 pieces.
10. If you take 9/4 and break it into nine equal parts, and take one unit and break it into four equal parts, one part from the nine and one from the four will equal each other; they will both equal 1/4 of a whole unit.
Problem: Say what 3/4 of 12 things is from an iterating perspective.
11. If you take 3 things, 4 times, you get the whole set of 12, so 3 things is one fourth. Since you need three fourths, you take one fourth 3 times, and then you have three fourths of the whole (9 things).
12. By taking 3 things and copying it 4 times I get 12 (whole). So I know that if I take 3 things and copy it only 3 times I will get 3/4.
13. With iterating we see that taking a block and stacking it 12 times will give me 12/12. Taking 3 of 4 blocks gives me 3/4. Since there are 12 blocks total I would have to do it 3 times to give me 9 blocks of total. This would be 3/4.
14. Take equal groups of the objects and fit them together to see how many need to be in each group to equal the entire set. The result is four groups with three in each set. Now take three of those groups.
Problem: Say what 3/4 of 12 things is from a partitioning perspective.
15. I would divide the 12 into groups of four, which would equal 3 groups. Then I would take one group of 4 and set aside 3 pieces from the group of four. I would then take 3 pieces from the other groups which would equal 9 pieces from the 12 things, which is 3/4 of the set.
16. 12 objects separated into 4 equally sized groups is three objects in each group. Three of those four groups is 3/4.
17. Divide the 12 things into four equal piles. There will be three things in each pile or each 1/4. Adding 3 of the piles together is adding 3 one-fourths and makes 9 total things, which are 3/4 of 12.
18. This is 3/4 because it’s what I get from putting three of the 1/4 sized groups together.