See my Do Now in my Strategy folder that explains my beginning of class routines.
Often, I create do nows that have problems that connect to the task that students will be working on that day. Today I want students to make reasonable estimates for multiplying mixed numbers.
Some students may round the mixed numbers to the nearest whole number and then multiply. Others will simply use the given whole number and ignore the fraction. Other students may come for a range, using a low and high estimate. The important thing is that students are creating estimates and explaining the thinking behind them. I call on students to share their estimates and their thinking.
I have students move to work with their partners. We work through this problem together. I quickly ask students to raise their hand if they like to run. I have partners do a quick Think Pair Share what their favorite way to exercise.
I have a volunteer read the problem. I have students work independently to create an estimate. Students may follow the same thinking they used in their do now, or they may try a different strategy that they heard from another student. I call on a couple students to share out their estimates. I ask students why they are multiplying. I emphasize that making an estimate is a great way to check the reasonableness of your answer once you’ve found it.
I ask, “How could we create a rectangle to show 4 3/5 x 2 2/3?” Some students may connect with the rectangle models they have made in the previous lessons (Representing Fraction Multiplication and Representing Fraction Multiplication Day 2). I don’t want students to test out their own idea, so I model how we can create a rectangle with the dimensions of 4 3/5 and 2 2/3. I explain that the rectangles don’t have to be drawn to scale, but they should look relatively accurate. For example, the part of the rectangle that is 7 meters long should be longer than the part of the diagram that is 2 meters. See the resource “Unit 4.9 Juliana’s Problem Example”.
I ask a student to explain our shortcut for multiplying fractions. We go through and find the product of each individual rectangle. Once that is through, I ask students what we need to do next? I want students to recognize that we need to add all of the products together. I ask, “What is an efficient way to do this?” One strategy would be to turn 8/3 and 6/5 into a mixed numbers. Then, we can add the whole numbers. Then we need to deal with the left over fractions. I want students to realize that we can create equivalent fractions that are fifteenths and then add them together. I ask students if we can simplify our answer. We revisit our estimates. How did we do? Does our answer make sense?
At my school, students learn how to use the box method to multiply whole numbers in 5th grade. This method breaks down each factor by place value. You use a box to show each product and then add the products together.
For instance, 43 x 27 is broken up into (40 + 3) x (20 + 7). I wrote this problem on the board and asked students how they would use the box method to find the product. I then connected this to multiplying mixed numbers. We split each mixed number into a whole number and a fraction and then create the box to model our multiplication. Students were quickly able to see the connection between the methods.
I have students work on the practice problems with their partners. Students are engaging in MP4: Model with mathematics, MP5: Use appropriate tools strategically, MP6: Attend to precision, and MP7: Look for and make use of structure. I Post A Key so students can check their work as they go. I walk around and monitor student progress and behavior. I want to see how well students make estimates and create rectangle diagrams.
If students struggle, I may ask some of the following questions:
For problem 1, some students may see that they can easily convert the fractions to decimals and then multiply them together to find the area. This is great! I make sure that students still draw and label the rectangle. They will not be able to use this method with the practice problems that involve thirds.
If students successfully complete their work, they can choose whether to play “Smaller Answer Wins” or “Score the Difference”. Students will need rules, game sheets, and dice.
For Closure I ask students to share out how they figured out the answer to Anita’s bran question. Then I ask students whether Homelie’s idea in problem 3 worked. I want students to realize that her answer would be too small. I want students to be able to look at their rectangle diagram for problem 2 in order to see what multiplication problems were missing. If students don’t mention it, I make the connection to the distributive property.
I ask students if there is a shortcut that would work. Some students may share that you can change the mixed numbers into improper fractions and then multiply them like they would regular fractions. The challenge is then turning the improper fractions back into mixed numbers. Students are engaging in MP8: Look for and express regularity in repeated reasoning.
In the do now, many students rounded the factors to find an estimate. Other students multiplied the whole numbers and then multiplied the fractions and added the two products together. I made sure to bring up Homelie’s shortcut idea during the closure. I asked students if it would work and to show why or why not. Some students used the box method to show that Homelie was missing two products in the problem. Other students were still thinking that Homelie’s idea could work. I had the students figure out the answer using Homelie’s idea and using the box method. Students were able to see that Homelie’s shortcut ended up being smaller than the box method. She did not take into account ¾ x 3 and 2 x 1/3. By asking this question, I was able to see what students were thinking and how much they understood about multiplying mixed numbers.