Discuss each of the three solutions in Examples we saw earlier in terms of whether or not they make sense of the original problem . Why might a teacher want to delay presenting the solution in Example we saw earlier?

a. Construct diagrams of alternative ways the twenty four members can be seated and show how they might share equitably the eighteen pizzas. b. Here is a seating arrangement where pizzas and members were moved from one table to three different tables in such a way that everyone got a fair share of pizza: In how many different ways could eighteen pizzas and twenty four members have been arranged at each table so everyone got a fair share of pizza? How much pizza does each person get? Is there more than one way to distribute pizzas among members? c. Return to part (a). Discuss your work in terms of equivalent ratios.

In a one-hour TV slot, one week's program started with a three-minute review of last week as a lead-in. Commercials took thirty percent of the total time for the slot. What percent of the total show time was left for new developments?

We often misinterpret common phrases using percents. For example, twenty five percentage larger than thousand two hundred dollars is correctly interpreted to mean one thousand and five hundred dollars, and $25 \%$ as large as $1200$ is correctly interpreted to mean $\$ 300$. However, $300 \%$ larger than 1200 and $300 \%$ as large as 1200 are often interpreted to mean the same thing, although $300 \%$ larger than $\$ 1200$ is $\$ 4800$ and $300 \%$ as large as $\$ 1200$ is 3600 . Once again, drawings we saw earlier can help make the needed distinction. You make drawings of $300 \%$ larger than $\$ 1200$ and $300 \%$ as large as $\$ 1200$.

Student one answer: "Okay,fifty percent of one hundred and sixty is eighty . Half of eighty is fourty so that is twenty five percent. So if you add eighty and fourty , you get one hundred and twenty . But that's too much because that's $75 \%$. So you need to minus $10 \%$ (of 160 ) and that's 16 . So, 120 take away 16 is 104 ." Student two answer: "The answer is 104 . First $\mathrm{I}$ did $50 \%$ which was 80 . Then $\mathrm{I}$ did $10 \%$ of 160 which is 16 . Then I did $5 \%$ which was 8 . I added them $(16+8)$ to get 24 , and added that to 80 to get 104 ." Use either student's procedure described above to find $45 \%$ of 180 .

The orange drink in Pitcher A is made by mixing one can of orange concentrate with three cans of water. The mixture in Pitcher B is made by mixing two cans of orange concentrate with six cans of water. Which will taste more "orangey": the mixture in Pitcher A or the mixture in Pitcher B or are they the same? a. Give an argument to support your answer to the above question. b. How would you, as a teacher, respond to a student who says that the mixture in Pitcher $\mathrm{A}$ is more orangey because less water went into making it? c. In a fifth-grade class some students reasoned like the student in part (b). Other students in the same class argued that the mixture in Pitcher B is more orangey because it has more orange concentrate. How would you settle the argument? What is wrong with the reasoning of the students in each group? d. There was another student who argued as follows: Pour one can of orange concentrate and one can of water into Pitcher A. Take two cans of orange concentrate and two cans of water and pour them into Pitcher B. The two mixtures are equally orangey because they are made with equal parts orange and water, that is, $50-50$. Now there are only two cans of water left to go into Pitcher $\mathrm{A}$ and four cans of water left to go into Pitcher B. Because the mixtures are the same strength, when you add four cans of water to the mixture in Pitcher $\mathrm{B}$, it will be more watery than the mixture in Pitcher $\mathrm{A}$, which gets only two more cans of water. Therefore, the mixture in Pitcher A is more orangey. How would you deal with this student's thinking? e. How can "oranginess" be quantified in this situation to facilitate the comparison?

The boss says, "You remember when business was bad last year, I had to cut everyone's pay by ten percent ? Well, business is better, so I can raise your pay by ten percent now. That will put you back to where you were before the cut." Is the boss correct?

Consider the following problem : A new housing subdivision offers rectangular lots of three different sizes: a. $75$ feet by $114$ feet b. $455$ feet by $508$ feet c. $185$ feet by $245$ feet If you were able to view these lots from above, which would appear most nearly square? Which would appear least square? Explain your answers.

Be sure to think about what attribute or characteristic of the lots we are interested in and in what ways this attribute can be quantified.

You are running a contest to see if people can tell which of three different-sized batches of coffee (from the same type of coffee bean) is strongest. As the manager of the contest, how would you measure the strength of each batch?

This problem can easily be solved mentally. Compare your mental strategies that depend on good number sense. Then for each problem, set up a proportion for which one ratio has hundred as the value of its second quantity. Compare this method.thirty is what percent of fourty five ?

Consider the following problem : A new housing subdivision offers rectangular lots of three different sizes: a. $75$ feet by $114$ feet b. $455$ feet by $508$ feet c. $185$ feet by $245$ feet If you were able to view these lots from above, which would appear most nearly square? Which would appear least square? Explain your answers.

Be sure to think about what attribute or characteristic of the lots we are interested in and in what ways this attribute can be quantified.

The orange drink in Pitcher A is made by mixing one can of orange concentrate with three cans of water. The mixture in Pitcher B is made by mixing two cans of orange concentrate with six cans of water. Which will taste more "orangey": the mixture in Pitcher A or the mixture in Pitcher B or are they the same? a. Give an argument to support your answer to the above question. b. How would you, as a teacher, respond to a student who says that the mixture in Pitcher $\mathrm{A}$ is more orangey because less water went into making it? c. In a fifth-grade class some students reasoned like the student in part (b). Other students in the same class argued that the mixture in Pitcher B is more orangey because it has more orange concentrate. How would you settle the argument? What is wrong with the reasoning of the students in each group? d. There was another student who argued as follows: Pour one can of orange concentrate and one can of water into Pitcher A. Take two cans of orange concentrate and two cans of water and pour them into Pitcher B. The two mixtures are equally orangey because they are made with equal parts orange and water, that is, $50-50$. Now there are only two cans of water left to go into Pitcher $\mathrm{A}$ and four cans of water left to go into Pitcher B. Because the mixtures are the same strength, when you add four cans of water to the mixture in Pitcher $\mathrm{B}$, it will be more watery than the mixture in Pitcher $\mathrm{A}$, which gets only two more cans of water. Therefore, the mixture in Pitcher A is more orangey. How would you deal with this student's thinking? e. How can "oranginess" be quantified in this situation to facilitate the comparison?

A student says, "You claim that a rectangle one hundred and five feet by hundred feet is more nearly square than one that is ten feet by five feet. But they're both off by the same amount, five feet. Why do you say the larger one is more nearly square?" What do you say?

What is the advantage of using percents in situations such as these? One discussion class has twenty eight females and seventeen males. A large lecture class has hundred and six females and sixty two males. In which class is the female population more dominant?