Multiplication and Division
Multiplication: Grouping Problems
Megan has 5 bags of cookies. There are 3 cookies in each bag. How many cookies does Megan have all together?
Division: Measurement Division and Partitive Division
Partitive Division – Partitioning the total number of objects into a given number of groups (the number of objects in each group is unknown).
Megan has 15 cookies. She put the cookies into 5 bags with the same number of cookies in each bag. How many cookies are in each bag?
Measurement Division – “Measuring” a total amount using a “standard” (the number of objects in each group is the standard; the number of groups is unknown).
Megan has 15 cookies. She puts 3 cookies in each bag. How many bags can she fill?
Other Multiplication and Division Problems
Rate problems
Your family uses 45 gallons of water every day. How much water do you use in one month (30 days)?
John rides his bike at 10 miles per hour for 3.5 hours. How far has he gone?
John rides his bike from Lansing to Dansville, a distance of 24 miles. It takes him 2 hours. How fast is he riding?
A baby sitter earns 4 dollars per hour for baby-sitting. How many hours will he have to baby-sit to earn 24 dollars?
Price problems
You buy 5 pizzas for a party. Each pizza costs $8.75. How much do you pay altogether?
You buy 5 pizzas for a party. Your total bill is $45. How much did each pizza cost?
Multiplicative Comparison problems
The first grade class has a hamster and a gerbil. The hamster weighs 3 times as much as the gerbil. The gerbil weighs 9 ounces. How much does the hamster weigh?
“The differences among all of these problems can be relatively minor, and it is not necessary to be concerned about whether a problem falls in one particular class or another. The critical differences are between Multiplication, Measurement Division, and Partitive Division problems. It is the differences among these three basic types that are reflectyed in children’s thinking.
We have introduced Rate, Price, and Multiplicative Comparison problems because it is important to provide opportunities for children to solve a variety of problems involving different kinds of quantities. Whereas the basic multiplication and division problems involve only whole numbers, Rate, Price, and Multiplicative Comparison problems can be extended to include multiplication and division of fractions. Consider the problems about the baby elephant and its weight gain:
A baby elephant gains 4 pound each day. How many pounds will the baby elephant gain in 8 days?
A baby elephant gains 4 pound each day. How many days will it take the baby elephant to gain 32 pounds?
A baby elephant gained 32 pounds in 8 days. If she gained the same amount of weight each day, how much did she gain in one day?
It could have gained ¾ pound each day or 3½ pounds, and the time period could have been ½ day or 7.25 days. Thus, including these problems with whole numbers in the primary grades lays the groundwork for developing an understanding of multiplication and division of fractions in later grades.” Children’s Mathematics: Cognitively Guided Instruction, pp. 48-49.
Symmetric problems
Area and Array problems
You are setting up chairs for a concert in a gym. You put the chairs in 15 rows, with 26 chairs in each row. How many chairs did you set up altogether?
Combination problems
Ginny and Patty are at the ice cream store. They have a choice of 12 different ice cream flavors and 3 different cones (sugar cone, waffle cone, and safety cone). How many different combinations of single scoop ice cream cones could they make, using all 12 flavors and all three kinds of cones?
Problem set 4: Building place value understanding
Mel has 2 boxes of crayons. There are 10 crayons in each box. He also has 4 extra crayons. How many crayons does he have in all? (write the number)
Jim picked 54 flowers. He put them into bunches with 10 flowers in each bunch. How many bunches of flowers did Jim make? Were there any flowers left over? (how does your answer relate to the number 54?)
Problem sets for Multiplication and Division