#### A Math Forum Project

 ESCOT Problem of the Week: Archive of Problems, Submissions, & Commentary

Student Version

### Fractris - posted April 2, 2001

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Fractris

In the Fractris game, you will be trying to finish as many rows as possible using different combinations of fractions. The computer will start you off with a random fraction and then it will be up to you to add your fractions to fill up the row exactly to score points. You will gain more points if you can avoid using the same fraction twice.

To start playing, click on the Run button. You will see a block fall immediately on the left of the grid. To add to that block, select a fraction on the right side by clicking on it. You should then see a block fall into place. Keep adding blocks until you have filled the row. If you fill your row up without using any fraction more than once, it will disappear so you can start a new row. Otherwise, your row will remain and you will start on the next row up. You'll have 250 seconds to finish as many rows as possible!

To play the game again, press the Reset button, then press the Run button.

### Questions

1. If the computer sends down a 1/3 block, how can you finish the row with the fewest number of blocks and without using the same size block twice?

2. If the computer sent down 1/5, would you be able to fill the row? If so, how could you do it with the fewest blocks? If not, explain why not and tell how close you could get to completing the row.

3. What do all the fractions in the Fractris game (1/2, 1/3, 1/4, 1/6, 1/12, 5/12) have in common?

Bonus: What are all the different combinations of the fractions 1/2, 1/3, 1/4, 1/6. 1/12, and 5/12 that will sum to 1 without using any fraction twice? Explain how you know that you have found all the ways.

Teacher Support Page

First, one extra equation, 1/12 + 1/6 + 1/4 + 1/2 = 1, was correct in the bonus. I gave credit even if this was not included since I started grading before I came across it. However, if it was included but was not one of the original four in the answer key, then I denied credit. I figure since we didn't come up with it originally, I can't penalize the ones I graded later.

The biggest problem was with question 3 and finding the term of least common denominator, or at least explaining it correctly. Many students stated that they were all factors of 12 or were divisible by the primes of 2 and 3. Fractions aren't really divisible by whole numbers, and the prime factors of 2 and 3 only apply if the denominator is changed to 12. Many students simply stated that they all had a common denominator and were divisible by the primes, without indicating...

A) the LCD found;
B) that the primes of 2 and 3 only apply after the fractions have been converted.

Some students didn't connect the LCD principle at all or used 60 instead of 12.

Questions 1 and 2 generally went well, although some students were misled by question 2 and simply assumed that it must be possible without including why they 'knew' so. The bonus was fairly unsuccessful, but mostly because of question 3. If the earlier connection had been made, the bonus would have been easier.

I tried to explain my difficulty with the wording used in the explanations for question 3, but did not deny credit if this was the only thing wrong with the solution. One of the schools, Caroline Davis, must have done the problem as a class since all of the submissions contained the 2,3 prime factor explanation. I understand their idea but it was not explained or presented well. The students did not seem to understand that the numerator is divisible by a whole number, but the denominator can be made into a "common denominator" in order to add or subtract various fractions. It's a subtle point, but the group effort on the problem did not foster the correct thought.

### Highlighted solutions:

 From: Katie L., age 13 School: Taipei American School, Taipei, Taiwan

```QUESTIONS

1. If the computer sends down a 1/3 block, how can you finish the row
with the fewest number of blocks and without using the same size
block twice?

The 1/2 and the 1/6 block could finish the row.

2. If the computer sent down 1/5, would you be able to fill the row?
If so, how could you do it with the fewest blocks? If not, explain
why not and tell how close you could get to completing the row.

No, if the computer sent down a 1/5 block then you wouldn't be able
to fill the row up. If you changed all of the denominators to 60,
then no combination of fractions would be able to fill up the row.
Also, it's lowest common multiple isn't 12, which proves that it is
impossible to be able to fill up the row. If it is a multiple of
twelve, that means that the numerator will be a fraction of twelve.
If they are all fractions of twelve, then some of them will
definitely fit together to equal 1. The closest that you can come to
filling up the row is putting down a 1/2 and a 1/4 block. Added
together, it all becomes 57/60, which is the closest you can get to
filling up a row if the computer sent down a 1/5 block.

3. What do all the fractions in the Fractris game (1/2, 1/3, 1/4,
1/6, 1/12, 5/12) have in common?

All of the fractions have a lowest common multiple of 12, and they
all somehow add up to one (like in the game, different combinations
equalled to 1).

Bonus:
What are all the different combinations of the fractions 1/2, 1/3,
1/4, 1/6. 1/12, and 5/12 that will sum to 1 without using any
fraction twice? Explain how you know that you have found all the ways.
```

 From: Mary S., age 13 School: Taipei American School, Taipei, Taiwan

```QUESTIONS

1. If the computer sends down a 1/3 block, how can you finish the row
with the fewest number of blocks and without using the same size
block
twice?

If the computer sent down a 1/3 block, the fastest way to fill up the
row would be to put down a 1/2 block, then a 1/6 block.

2. If the computer sent down 1/5, would you be able to fill the row?
If
so, how could you do it with the fewest blocks? If not, explain why
not
and tell how close you could get to completing the row.

If the computer sent down a 1/5 block, then it would be impossible to
fill up the row. When you change all of the fractions to have the
number 60 as their denominator, then no matter what combination of
fractions you put down, you'll won't be able to fill up the row.
However, you can come very close to filling the row by putting down a
1/4 and a 1/2. This way, the added sum comes to 57/60, which is the
closest you can come to filling the row.

3. What do all the fractions in the Fractris game (1/2, 1/3, 1/4,
1/6,
1/12, 5/12) have in common?

All of them are factors of 12.

Bonus:
What are all the different combinations of the fractions 1/2, 1/3,
1/4,
1/6. 1/12, and 5/12 that will sum to 1 without using any fraction
twice? Explain how you know that you have found all the ways.
```

 From: Julius L., age 13 School: Taipei American School, Taipei, Taiwan

```QUESTIONS

1. If the computer sends down a 1/3 block, how can you finish the row
with the fewest number of blocks and without using the same size
block
twice?

You can use a 1/2 block and a 1/6 block to fill it up.

2. If the computer sent down 1/5, would you be able to fill the row?
If
so, how could you do it with the fewest blocks? If not, explain why
not
and tell how close you could get to completing the row.

No, you can not. The denominator in 1/5 is not a multiple or factor
of the denominator in the other fractions. You can get as close as
57/60 of the row completed.

3. What do all the fractions in the Fractris game (1/2, 1/3, 1/4,
1/6,
1/12, 5/12) have in common?

They all have denomonators that are factors of 12.

Bonus:
What are all the different combinations of the fractions 1/2, 1/3,
1/4,
1/6. 1/12, and 5/12 that will sum to 1 without using any fraction
twice? Explain how you know that you have found all the ways.

1/2 + 1/3 + 1/6
1/2 + 1/4 + 1/6 + 1/12
1/2 + 5/12 + 1/12
1/3 + 5/12 + 1/6 + 1/12
1/3 + 1/4 + 5/12

I found out all the points by multiplying all the numbers by 12 and
then adding them up to see which combination add up to 12.
```

 From: Albert W., age 13 School: Taipei American School, Taipei, Taiwan

```QUESTIONS

1. If the computer sends down a 1/3 block, how can you finish the row
with the fewest number of blocks and without using the same size
block
twice?

2. If the computer sent down 1/5, would you be able to fill the row?
If
so, how could you do it with the fewest blocks? If not, explain why
not
and tell how close you could get to completing the row.

3. What do all the fractions in the Fractris game (1/2, 1/3, 1/4,
1/6,
1/12, 5/12) have in common?

Bonus:
What are all the different combinations of the fractions 1/2, 1/3,
1/4,
1/6. 1/12, and 5/12 that will sum to 1 without using any fraction
twice? Explain how you know that you have found all the ways.

1) Because a row equals 1 all together, you can find the least number
of blocks needed by subtracting 1/3, the size of the block dropped,
from 1. Our answer is 2/3. The fraction 1/2 also equals 3/6 and the
fraction 2/3 also equals 4/6. Therefore we are left with 1/6 left. We
can see that there is a 1/6 block so therefore, the least number of
blocks is 2. The 1/2 block and the 1/6 block.

2) If the computer sends down a 1/5 sized block, you will be not able
to fill up the row. To find this out, you need to find the LCD (least
common denominator) of all the fractions including the 1/5 block. We
find that this number is 60. Therefore, 1/5 also equals 12/60.
Converting the other blocks...

1/2 = 30/60
1/3 = 20/60
1/4 = 15/60
1/12 = 5/60
5/12 = 25/60

To fill up the whole row, we need to get 60/60. Since we already have
12/60, we still have 48/60 left. We find this not possible with the
fractions we have above because they are all in multiples of five.

3) All blocks in the Fractis game have a denominator with a multiple
of 12.
```

### 26 students received credit this week.

Carlos A., age 13 - Caroline Davis Intermediate School, San Jose, CA
Ojita B., age 13 - Caroline Davis Intermediate School, San Jose, CA
Raul B., age 14 - Caroline Davis Intermediate School, San Jose, CA
Vivian C., age 13 - Taipei American School, Taipei, Taiwan
Linda D., age 12 - Caroline Davis Intermediate School, San Jose, CA
Nick G., age 13 - Caroline Davis Intermediate School, San Jose, CA
Jessica H., age 12 - Caroline Davis Intermediate School, San Jose, CA
Nina H., age 14 - Taipei American School, Taipei, Taiwan
Asim K., age 13 - Caroline Davis Intermediate School, San Jose, CA
Nasser K., age 13 - Caroline Davis Intermediate School, San Jose, CA
Thibault K., age 13 - Taipei American School, Taipei, Taiwan
Elizabeth L., age 13 - Caroline Davis Intermediate School, San Jose, CA
Julius L., age 13 - Taipei American School, Taipei, Taiwan
Katie L., age 13 - Taipei American School, Taipei, Taiwan
Nicholas M., age 14 - Caroline Davis Intermediate School, San Jose, CA
Sergio M., age 14 - Caroline Davis Intermediate School, San Jose, CA
Shone M., age 13 - Caroline Davis Intermediate School, San Jose, CA
Chau N., age 12 - Caroline Davis Intermediate School, San Jose, CA
Grace P., age 12 - Caroline Davis Intermediate School, San Jose, CA
Gilbert S., age 13 - Caroline Davis Intermediate School, San Jose, CA
Mary S., age 13 - Taipei American School, Taipei, Taiwan
Sasha S., age 13 - Welsh Valley Middle School, Narberth, PA
Phillip V., age 14 - Caroline Davis Intermediate School, San Jose, CA
Albert W., age 13 - Taipei American School, Taipei, Taiwan
Lee W., age 12 - Caroline Davis Intermediate School, San Jose, CA
Grace Hsiang Wen Y., age 14 - Taipei American School, Taipei, Taiwan