Lamon's Student Work

Using Gen's Work (Fifth Grade):

Connection in Student Work to Ratio & Proportion

Showing Gen how ratio and proportion correlates to the way she solved this problem would be quite simple. For each portion of Gen's work- 3 p for 9 kids, 30 p for 90 kids, 6 p for 18 kids, I would tell her to simplify each separate row in terms of pizza to kids and write each as a ratio to the right of the row. Once Gen did this, she would see that each row, when simplified, has the common ratio of 1:3. I would then explain to Gen that this shows us that we are dealing with a proportional ratio problem. I would then ask Gen, since we know that our ratio X:108 in simplest terms will be proportional to 1:3, how can we solve to find X? Gen would then realize that by dividing the 108 by 3, we would arrive at an answer for X that would make the X:108 proportional to 1:3. Once doing the division, Gen could also compare the answer 108/3 to the answer she got through adding 30+6 and see that both yield the same answer.

Reframing in terms of Ratio & Proportion:

Gen uses the tactic of multiplication and addition to solve the problem, which is an acceptable method to use for this problem and a method which results in the correct answer. Another method Gen could use to solve this problem however could be utilizing the idea of proportional ratios. Since Gen is given that 3 pizza's serve 9 people, Gen could write down the ratio 3:9, which then could be simplified 1:3. Since we know that this is the ratio of pizza to people, Gen could then state that for every one pizza, three people are served.

If Gen knows that one pizza serves three people, she could solve to find how many pizzas serve 108 people through the 1:3 ratio.

Since X is our number of people, Gen could set up the ratio X:108.

We know that 1 pizza serves 3 people and that the size of pizza's we are serving are all going to be the same. Therefore, we can say that X:108 will be proportional to 1:3.

Gen can solve to find the total number of pizzas needed for 108 people by dividing 108 by 3. This division results in the answer 36.

This 36 tells us that we would need 36 pizzas to serve 108 people. The final ratio of the problem comes out to be 36:108, a ratio Gen could simplify to show its proportional value to 1:3.

Connection in Student Work to Fractions

To connect fractions to how Gen solved this problem I would have her write out the fraction for her first row of pizzas to people- 3/9. I would then tell her to simplify this, getting 1/3.

I would then tell Gen to do the same for the following two rows, resulting in her answers again being 1/3.

Once getting to the final portion, I would have Gen write X (the unknown number of pizzas)/108. Since all the previous fractions have come out to 1/3, we can assume that the X/108 when simplified should also come out to 1/3.

To solve for this, Gen could either use her knowledge to try to figure out what number divides into 108 three times or she could set up the division problem 1/3/ X/108 to find the value for X.

Reframing in Terms of Using Fractions

Another method that could be used to solve this problem would be to break the problem down into fractional parts.

Gen could break the problem down into pizzas divided by people to find the fractional value of the problem (3 pizzas/9 people).

Gen could then simplify the fraction 3/9 to its simplest terms, 1/3.

Now that Gen has the fraction of 1 pizza over 3 people, she can divide by the fraction of X pizzas over 108 people to solve for the number of pizzas needed to serve 108 people.

Gen would need to use the reciprocal of X/108 to solve the division problem- so would multiply 1/3 x 108/X to yield the answer of 108/3x. The 3 would then be divided out to arrive to the answer- X=36.

Gen then could represent this answer in fractional form to show its proportional value to 1/3--> 36 pizzas/108 people.

For 108 people, 36 pizza's would need to be served.

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