Multiplication and Division Categories
Grouping and Partitioning Problems
This category shares some similarities with the join category for addition and subtraction. Again, there can be three different unknowns depending how the problem is set up. When the compare category was analyzed three problems could be created. Two were subtraction and one was addition. Similarly with this category, three problems can be created, two for division and one for multiplication.
In interpreting these problems, it is important to take into account that there are two interpretations of division: measurement and partitive. Measurement division is when the number of the groups is unknown, and partitive is when the size of the group is unknown. It is important for students to learn that, although these problems may seem quite different, they are both solved by the same operation, division.
Let's say Zachary makes bags of lollipops for his friends. He puts 4 lollipops in each bag and makes 5 bags. How many lollipops did he use altogether?
This problem asks for the total, which indicates that it is a multiplication problem. So the equation would be:
4 x 5 = []
In the next problem, the number of bags is unknown. This is measurement division.
Zachary has 20 lollipops. He makes some bags of lollipops for his friends. He puts four lollipops in each bag. How many bags can he make?
Equations representing this problem are
20 / 4 = [] or [] x 4 = 20.
As the second equation suggests, this problem could also be written as a missing factor problem using the multiplication.
A third type of problem calls for partitive division because the number in each group is unknown.
Zachary has 20 lollipops. He makes 5 bags. How many lollipops will he put in each bag?
Equations representing this problem are
20 / 5 = [] or [] x 5 = 20.
Rate Problems
There is also a group of problems known as rate problems. These are different from the previous group because they do not deal with objects that can be counted. They are a little more abstract, especially for children. The rate problems can also be divided into the same types: multiplication, measurement division, and partitive division.
A monkey eats 3 pounds of bananas a day. How many pounds of bananas will the monkey eat in 5 days?
This is a multiplication problem. We can represent it by
3 x 5 = [].
The next two problems will involve division.
A monkey eats 3 pounds of bananas a day. At this rate, how many days will 15 pounds of bananas feed the monkey?
This is a measurement division problem because the measurement or the number of days is missing. It can be schematized by the equations
15 / 3 = [] or 3x [] = 15.
A monkey eats 15 pounds of bananas in 5 days. If the monkey eats the same amount each day, how many pounds does the monkey eat in one day?
This problem is partitive division because the unknown is the amount or the part that the monkey eats in each day. The equation below will solve this problem.
15 /5 = [] or 5x [] = 15
Price Problems
These are similar to the previous group, but the rate involves a price. Oranges sell for 6 cents each. How much would 7 oranges cost? This is a multiplication problem and the equation to solve it is:
6 x 7 = []
In the measurement division problem, the number of oranges is unknown.
Oranges sell for 6 cents each. How many can you buy for 42 cents?
Equations that capture this equations are
42/ 6 = [] or 6 x [] = 42.
The last one in this trio is the partitive division example.
You have 42 cents and you buy 7 oranges. How much does each orange cost?
Equations for this one are
42 /7 = [] or 7 x [] = 42
Multiplicative Comparisons
In these problems one quantity is described in relation to the other. For example:
The bowling ball weighs 5 times as much as the basketball. If the basketball weighs 4 pounds, how much does the bowling ball weigh?
Multiplication is used to solve this type of problem. It is represented by the equation
4x 5 = [].
The equation looks like the other multiplication problems, but it is different from all of the other problems presented so far. The key concept is that one quantity is presented in relation to the other. One quantity is described in terms of how many times larger it is compared to the other.
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