Flexibility with Multiplication and Division Situations

1. Flexibility with Multiplication and Division Situations SARIC RSS Mini-Conference 2014 Laura Ruth Langham Hunter AMSTI-USA Math Specialist

2. Learning Outcomes • Identify the meaning of quantities in word problems and how students can interact with these different meanings • Identify different types of multiplication and division situations and their purpose • Apply new learning to high level tasks to make sense of quantity and remainders

3. CCRS Math Practice Standards • Make sense of problems and persevere in solving them. • Reason abstractly and quantitatively. • Construct viable arguments and critique the reasoning of others. • Model with mathematics. • Use appropriate tools strategically. • Attend to precision. • Look for and make use of structure. • Look for and express regularity in repeated reasoning.

4. Modeling Math • A recipe calls for 2 cups of flour for a cake. How many cups of flour will we need if we are going to make 5 cakes? • What pictorial figures might your students draw to represent this situation? • How are our representations similar and different? • What do the differences mean? How might this shape instruction?

5. Students have difficulty understanding the meaning of the answer to multiplication and division problems (including the meaning of the remainder). (Schwartz, 1988 Cited in Lamon; Silver, VT visit 2006)

6. Meaning of Quantity • Multiplication and division problems often contain quantities with different meanings. Solutions often result in quantities with new meaning. (Schwartz, 1988 Cited in Lamon; Silber, CT visit 2006) • Joe rides his bike 13 miles per hour. How long will it take him to ride 52 miles? Show your work.

7. Meaning of Quantity • Anticipate student responses (correct and incorrect) for your problem. Chart ideas. • Rotate to the next poster and identify any strategies or misconceptions the first group hasn’t identified. • Take a gallery walk and discuss with your group which problems seem more difficult for students and why.

8. Multiplication and Division Situations • Create a word problem that demonstrates a specific multiplication or division situation. Do not solve it. • Trade word problems with someone at your table. Solve your new word problem on copy paper.

9. Multiplication and Division Situations 5th Grade: Unit fraction language can be used in compare situations 3rd Grade 4th Grade

10. Equal Groups • In Equal Groups, the roles of the factors differ. One is the number of objects in a group, and the other is a multiplier that indicates the number of groups. • Thus there are two kinds of division situations depending on which factor is the unknown.

11. Arrays and Area • In the array situations, the roles of the factors do not differ. One tells the number of rows. The other tells the number of columns. Array situations can support the generalization of the commutative property. • Area problems are where regions are partitioned into square units. • The area model is used for single-digit multiplication/division (3rd), multi-digit multiplication/division (4th), and decimal multiplication/division (5th and 6th).

12. Compare • In multiplicative comparison, the underlying question is what factor would multiply one quantity in order to result in the other. • The use of times as much to describe the comparison. The terms three times more than and three times less than are now appearing frequently in newspapers. It is recommended to discuss these complexities with students but to test on more defined mathematical terms.

13. Multiplication and Division Situations • In solving, do you see any similarities between strategies? What manipulatives and strategies would you expect students to use for the different problem types? • Which types do you often use in your classroom? Which types might you incorporate into your classroom more?

14. Why did you choose that operation? • Key words or phrases • “how many in each group? so I must divide” • The size of the numbers • “one’s big and one’s small, so I must divide” • The numbers are compatible • “seven divides 21, so I must divide” Reasons such as these are never valid. Knowing why an operation is an appropriate choice for a solution strategy is an important part of establishing a robust understanding of mathematics.

15. Division Situations Suppose that Tina has 43 oranges that she is packing into identical empty boxes. Tina experiments and discovers that 8 oranges completely fill a box. After completely filling as many boxes as possible, how many “extra” oranges will Tina have? How many more oranges and boxes will Tina need to pack every orange and completely fill every box with no “extra” oranges? If Tina packs all the oranges in boxes, what is the smallest number of boxes that she will need? What is the largest number of boxes that she will need?

16. Division Situations In many situations, problem solvers need to understand how the divisor, the quotient, and the remainder can be used in the context. Even when they have calculated the quotient and remainder, they need to interpret each of them in the context of the situation and understand how to use each to find an answer. (“Developing Essential Understanding of Multiplication and Division” NCTM)

17. What about the remainder? • CDs are on sale for $4.00. Peter has $30.00 in his wallet. How many CDs can he buy? • Dad divided 30 large cookies onto 4 plates. How many cookies were on each plate? • Mr. Smith gave his four children $30.00 to share equally. How much money did each child get? • There are 30 children in a 4th grade class. Each table seats 4 children. How many tables will be needed?

18. What about the remainder? Create a word problems involving 14 ÷ 4 for each of the following situations. • The result was 3 groups of 4 with 2 left over. • The result was 4 groups of 3 with 2 left over. • The result was 4 ½ groups. • The result was 4.50 groups.

19. Assessment: High Level Task Every year a carnival comes to Hallie's town. The price of tickets to ride the rides has gone up every year.

20. Reflection • How did you have to make sense of the quantities you used to solve the problem? • Did you choose a certain strategy? Was it different than your tablemates? • Did you form a generalization while solving the last question? • Did you use the CCRS practice standards?

21. Learning Outcomes • Identify the meaning of quantities in word problems and how students can interact with these different meanings • Identify different types of multiplication and division situations and their purpose • Apply new learning to high level tasks to make sense of quantity and remainders

22. Thank You for Attending! Contact Information Laura Ruth Langham Hunter LLangham@southalabama.edu