Division ProblemsPROBLEM: Is 8 ÷ 2 defined or undefined? Use the missing factor model to explain youranswer.ANSWER: If 8 ÷ 2 is defined then there is a unique whole number c such that2 × c = 8 (1)Since the only solution to equation (1) is c = 4, it follows that 8 ÷ 2 = 4.===========================================PROBLEM: Is 8 ÷ 2 defined or undefined? Use the repeated subtraction model to explainyour answer.ANSWER: Let us examine 8 ÷ 2 by using the repeated subtraction model. Suppose thatyou have a pile of 8 coins. We need to remove groups of two coins from the pile of 8 coinsuntil all the coins are gone. If 8 ÷ 2 is defined, then it is defined to be the total numberof groups we remove. Removing one group of two coins will leave us with a pile of 6 coins.Removing another group of two coins will leave us with a pile of 4 coins. Removing anothergroup of two coins will leave us with a pile of 2 coins. Removing yet another group of twocoins will leave us wil a pile of 0 coins. Since we had to remove 4 groups of two coins inorder to remove all the coins from the original pile of 8 coins, it follows that 8 ÷ 2 = 4.===========================================PROBLEM: Is 8÷ 2 defined or undefined? Use the partition model to explain your answer.ANSWER: Let’s examine 8 ÷ 2 according to the partition model. Suppose that we have8 sandwiches and 2 lunch boxes. We need to place each sandwich in a lunch box, and weneed to make sure that each lunch box contains the same number of sandwiches. If we cando this in a unique way, then 8 ÷ 2 will be defined to be the number of sandwiches in eachlunch box The only way to evenly distribute all of the sandwiches into the lunch boxes is toplace 4 sandwiches in each lunch box. Therefore 8 ÷ 2 = 4.===========================================PROBLEM: Is 56 ÷ 0 defined or undefined? Use the missing factor model to explain youranswer.ANSWER: If 56 ÷ 0 is defined, then there is a unique whole number c such that0 × c = 56. (2)Clearly equation (2) does not have any solutions. Therefore 56 ÷ 0 must be undefined.===========================================1PROBLEM: Is 56 ÷ 0 defined or undefined? Use the repeated subtraction model to explainyour answer.ANSWER: Let’s examine 56 ÷ 0 according to the repeated subtraction model. Supposewe start with a group of 56 coins. We need to remove groups of zero coins from the pile of56 coins until all the coins are gone. But removing groups of zero coins from the pile willalways leave 56 coins in the pile. That is to say, we will never be able to remove all the coinsfrom the pile in this way. Therefore 56 ÷ 0 is undefined.===========================================PROBLEM: Is 56÷0 defined or undefined? Use the partition model to explain your answer.ANSWER: Let’s examine 56 ÷ 0 according to the partition model. Suppose we have 56sandwiches and 0 lunch boxes. To evaluate 56 ÷ 0 we would need to put each sandwich ina lunch box and make sure that each lunch box has the same number of sandwiches. Thenwe would need to count the number of sandwiches in each box. Since there are 0 bags, wecannot put each sandwich in a lunch bag. Therefore 56 ÷ 0 is undefined.===========================================Note: 0÷ 0 is a difficult concept, and rightly so since it it undefined. I think that the missingfactor model provides the simplest explanation as to why 0 ÷ 0 is undefined. I have used allthree models here, but I think the last two explanations are pretty convoluted.PROBLEM: Is 0 ÷ 0 defined or undefined? Use the missing factor model to explain youranswer.ANSWER: If 0 ÷ 0 is defined, then there is a unique whole number c such that0 × c = 0. (3)There are many solutions to equation 3. For example c = 0 and c = 5 are both solutionsto equation 3. Therefore equation 3 does not have a unique solution and, hence, 0 ÷ 0 isundefined.===========================================PROBLEM: Is 0 ÷ 0 defined or undefined? Use the repeated subtraction model to explainyour answer.ANSWER: Let us examine 0 ÷ 0 using the repeated subtraction model. Suppose that youhave a pile of 0 coins. We need to ask ourselves how many groups of 0 coins must be removedfrom the pile so that no coins are left in the pile. Since the initial pile does not contain anycoins, I could remove 0 groups of 0 coins and leave a pile with no coins. So at first glance itmight appear that 0 ÷ 0 is equal to zero, but let’s not be hasty. I could also, for example,remove 3 groups of zero coins and leave a pile with no coins. From this example, it appearsthat 0 ÷ 0 could also be equal to 3. Therefore, there is not a unique solution for 0 ÷ 0, and,hence, 0 ÷ 0 is undefined.2===========================================PROBLEM: Is 0÷ 0 defined or undefined? Use the partition model to explain your answer.ANSWER: Let us examine 0÷ 0 using the partition model. Suppose you have 0 sandwichesand 0 lunch boxes. We need to place each sandwich in a lunch box so that each lunch boxcontains the same number of sandwiches. Then we need to count the number of sandwichesin each lunch box. The total number of sandwiches in each lunch box would be our answer.However, we cannot count the number of sandwiches in each lunch box because we do nothave any lunch boxes. Therefore 0 ÷ 0 is