I'm just finishing up a unit on multiplication and division of whole numbers with my fifth grade class. We have focused a lot of time on developing different strategies for problem solving within these operations. With it being my first year in fifth grade, I had no idea how difficult it would be to help them be more flexible. I have a group of very procedural-minded mathematicians and getting them to look at things differently can be difficult. Sometimes it feel almost impossible! One thing that I think made a HUGE difference in encouraging them to be more flexible is changing up the problem types.
Multiplication ProblemsMultiplication (x times y) doesn't always indicate x groups of y. 3 groups of 12 cupcakes or 15 groups of 21 beads is great, but you don't always have discreet objects to put into groups in real life multiplication. While the solutions aren't that different with some of these, it's important for students to work with different conceptions of multiplication. Here are some of the types of multiplication problems I used besides just grouping problems.
Measurement Problems- Because are no objects that can be counted, these are a little more abstract. They can be solved in similar ways even though there technically aren't any countable objects.
- How many miles does a car travel in 6 hours at an average speed of 72 miles per hour?
- The zoo has 2 bears. The black bear weighs 243 pounds. The polar bear weighs 3 times as much as the black bear. How much does the polar bear weigh?
Area Problems- These problems use two factors that are interchangeable. They have no distinct, independent roles and they don't allow student to solve by grouping. They also don't have any discreet objects.
- I planted a garden that is 14 meters by 9 meters. How big is the area of my garden?
Array Problems- These problems are different from area problems because arrays are made up of discrete objects. My textbook uses pictures of arrays, but my student rarely use them themselves. However, I think using arrays are the best way to demonstrate the commutative property of multiplication. So to get students to use arrays, you need a problem that lends itself to thinking of objects in rows.
- Last night my daughter's choir concert was really crowded. Every seat was taken. I wondered how many people were there so I did some counting. There were 28 rows and each row had 54 chairs in it. How many people were at the concert?
- The special at the ice cream shop includes 2 scoops of ice cream and any 2 toppings in either a cup or a waffle cone. Mom says we can go to the ice cream shop for my birthday party but everyone has to order the special. I didn't want to tell my friends that they only had one option, but Dad said there are tons of options when you order the special. How many options will my friends have?
When students look at a division problem such as 1425 divided by 12, you can't tell what type of division problem it is. I think that's fine when I just want them to get a lot of practice with division. But if I want to force them to think about the problem differently, it has to be put into context. Are they solving by making 12 groups or making groups of 12?
Partitive Division- These problems require students to solve for how many items are in each group.
- The district athletic department is preparing for their annual celebration dinner. 1,425 athletes have signed up to come. It's being held on the football field and by putting food on both sides of the field, they will have just enough room for 12 rows of tables. How many athletes will they need to put at each table?
Measurement Division- This type of problem requires students to make groups of 12.
- The PTA made 1,425 cupcakes for the school bake sell. If they package them in boxes of 12, how many boxes will they have?
What about the remainder? What you do with the remainder depends on the context of the problem. Sometimes you do nothing with the remainder.
- The school is planning a huge celebration. The cafeteria bought 14 dozen eggs to make cakes. If each cake takes 5 eggs, how many cakes can they make?
Other times, you need to add another trip up to take care of the remainder.
- 168 people have been invited to a dinner on the 19th floor of a building. The elevator can only hold 9 people at a time. If everyone takes the elevator, how many trips will be needed to get everyone up to the dinner?
And in some instances you will need to deal with a fractional part.
- During art class, the students will be making sculptures of birds. Mrs. Smith brought in 30 pounds of clay for the class. There are 25 students in the class. How much clay will each student get?