Discovery lesson: cooperative learning
Objective: students are able to explain the associative property for multiplication
Materials: 24 wooden blocks for each group, paper and pencil
Use a round-robin format for this activity. Each team will form a rectangular solid with their 24 blocks. (There are several ways this can be done.)
Team members use round-robin to take turns telling how they see the blocks in the solid. One might say, “I see two blocks high, three blocks wide, and four blocks deep.”
Another member writes that combination on paper: 2 by
3 by 4. Another might say, “I see two blocks high, four blocks wide, and three blocks deep.” This is recorded:
2 by 3 by 4. Members name and record all combinations they see.
When teams are finished, have each one show its solid and the combinations they saw in it. Here are three ways
24 solids can be grouped and sentences for a 2 by 3 by 4 solid.
3 4 2 24
4 2 3 24
2 4 3 24
3 2 4 24
4 3 2 24 2 3 4 24
Other solids that may be made and discussed as 2 by 6 by 2, 1 by 3 by 8, 1 by 2 by 12, 1 by 1 by 24. (If only one or two solids are made by the children, see if they can make the unmade ones before writing and discussing their sentences.)
Have teams work with other numbers of cubes, such as
36 or 48. Have groups that used the same number of cubes compare their solids and sentences with each other.
Discovery lesson: cooperative learning
Objective: students will apply both commutative and associative properties to solve a problem involving multiplication
Materials: “menu board” for an ice-cream parlor, paper and pencils
Use a work-share format for this activity. Each team member is to work alone at first, then members share their work. The problem is to come up with as many different ways to make ice-cream-sundae-and-cookie combinations as possible.
Present this problem: An ice cream parlor has this menu for sundae-and-cookie combinations.
Ice Cream
Vanilla
Chocolate
Strawberry
Rocky Road
Mocha Fudge
Sundaes and Cookies
Sauces
Chocolate
Marshmallow
Strawberry
Toppings
Walnuts
Coconut
Cherry
Cookies
Vanilla
Oatmeal
Your problem is to determine how many different kinds of sundae-and-cookie servings can be made using one of each of the sundae ingredients and a cookie for each serving. Each team member is to work for five minutes to consider ways to solve the problem, then all of you get together to share your work. Remember, each person works alone, then you share your work. When you share, work out a scheme to organize your work so you keep track of what you do. You have ten minutes for sharing.
When fifteen minutes have elapsed, have teams share their work with the class. Teams may use the list-all-thepossibilities problem-solving strategy and have large lists that contain their combinations. Some may have used the break-it-into-a-simpler-problem strategy. Instead of listing all possibilities, they simplified the problem and found that it was a Cartesian product situation that could be solved by multiplication. Other teams may have had other or no strategies.
During sharing, identify different strategies and discuss how each one works. Point out that some strategies are more efficient than others. In this instance, the best strategy is multiplication: 5 3 3 2 90.
Order the factors different ways to help students see that they can use pairs of factors in any combinations they choose as they do the multiplication.