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 compare and contrast multiplying whole numbers and multiplying fractions. I remind students to use the problem ½ x 1/3 as an example. This activity takes longer than a typical do now because I want students to spend time talking and thinking about these statements.
Students participate in a Think Pair Share. We go through each statement. For each statement I call on a student to share their ideas with the class. Then I ask students to respond to that student’s thinking. Students are engaging with MP3: Construct viable arguments and critique the reasoning of others.
I want students to see how statements 3, 4, and 5 work for fractions. For statement 1, I give students the example of 6 x ½ to show that it can be shown as repeated addition. For statement 2, it is not common for people to say “one-half groups of one-third”. Instead we can say, “one-half of a group of one-third”. I give 4 x ½ as an additional example. It makes sense to say “4 groups of ½” and “1/2 of a group of 4”. For statement 6, some students may want to add to it so it reads, “…the product is larger than the factors, unless one of the factors is zero, one, or a fraction”. I present 4 x 3/2 with a diagram. Three halves is a fraction, but it is greater than one resulting in a product that is larger than 4. I want students to see that the statement needs to read, “…unless one of the factors is zero, one, or a fraction smaller than one”.
I have students move to their groups. I have a volunteer read the information on the page. I call on a student to summarize the task. I give them the Group Work Rubric. As students work I walk around and monitor student progress and behavior. Students are engaging in MP1: Make sense of problems and persevere in solving them, MP2: Reason abstractly and quantitatively, and MP5: Use appropriate tools strategically.
I encourage students to draw their ideas on their own paper to help them communicate with their group members. Some students may already have an algorithm that they want to use, but I remind them that it is also important that they can use a variety of diagrams to model the problem. If you cannot use a diagram and explain it to someone else, it is likely that you don’t really understand what is going on in the problem.
If students struggle, I have them start with the rectangle model. I ask them what fraction is already shaded. I ask them to explain the next step? Do you think the answer will be larger or smaller than 4/7? Why? I want students to recognize that since they are multiplying two fractions that are less than one, their answer is going to be smaller than both factors.
If some groups complete the problem quickly, I ask them to choose a different diagram and show how they could use it to confirm their answer.
With five minutes left, we come back together. I have a volunteer model how one diagram could be used to solve 4/7 times 2/3. We do this with each diagram. I ask students to revisit their estimates in problem one and compare them with their answer. Does your answer make sense? How do you know? Again, I want students to realize that because they are multiplying two fractions that are smaller than one, their answer is smaller than each of the factors.
I have students work on questions 1 and 2 independently. If they have questions, they can check in with their group members. I walk around and monitor student progress and behavior. I want to see what kind of diagrams students are using to represent each problem.
If students struggle, I may ask some of the following questions:
If students successfully complete their work, they can work on the “Diagram Challenge”.
For Closure I write 2/3 x 4/5 on the board. I ask students, “What is the product of this problem?” I tell students to find some space in their packet to jot down ideas. Students participate in a Think Write Pair Share. I call on students to share their answers.
Then I ask, “How can you figure out the size of a part of a part without having to draw a diagram?” Some students may see that if they used a square diagram that they broke up the whole into 15 pieces. They split the rectangle into 5 pieces one way and 3 pieces the other and 3 x 5 = 15. Other students may observe that four of the fifths are lightly shaded, but then each of these is broken into thirds. This results in 8 of the smaller parts being shaded, or 4 x 2 = 8. The product is 8/15. Students are engaging in MP8: Look for and express regularity in repeated reasoning.
I review the connections and conclusions that students have made. Then I ask them if they think it will work for all fraction multiplication. I have them look back at the problems in the packet to see if their rules work. I remind students that these rules will serve as a short cut for multiplying fractions, but it is still important that they are able to make estimates and diagrams for fraction multiplication problems.
I collected the tickets to go to see what students understood about fraction multiplication and what gaps in understanding they had. I corrected the tickets to go and grouped them in the following way:
Novice: Unit 4.8 Novice.jpg
These students were able to accurately use the shortcut for fraction multiplication. They struggled to create rectangular models for these problems. This tells me that although they know the procedure, they do not really understand what is going on when you multiply fractions. This particular student looks like he created the models after he figured out the answer. Instead of shading the factors (one vertical and one horizontal) he creates 30ths and shades three rows. For problem 2, he was able to shade 2/3 and create columns of sevenths, but he did not shade 2/7. He needs more practice and instruction on how to create rectangular models and how they connect to multiplication problems.
Approaching Mastery: Unit 4.8 Approaching Mastery.jpg
These students were able to use the shortcut to multiply the fractions. They also created models to show the fraction multiplication. These students made at least one mistake with their model. This student in particular created a model of ½ x 2/5, instead of ½ x 3/5. Her answer is 3/10, which shows me that she can use the shortcut, but that she didn’t check to make sure that her model showed the same answer. These students were also able to identify that problem 4 represents ¾ x 1/3 = 3/12 or ¼.
Proficient: Unit 4.8 Proficient.jpg
These students were able to create accurate rectangular models of the fraction multiplication problems. They were also able to use the shortcut to verify their answer. For problem 4, these students were able to recognize that the rectangle showed 1/3 x ¾ = 3/12 or ¼.
For this situation, I did not include an “advanced” category because the content did not require students to explain or analyze their work. Nearly all of the students were able to accurately use the shortcut. Most students were able to create accurate models using rectangles. I plan on returning these ticket to gos to students to that they can compare and analyze their work. I plan on showing students rectangular models and asking them to determine the factors and the product. I also want students to be able to come up with a word problem that matches the situation. For example, if my model shows ½ x ¾, a story could be that I took home ½ of the math quizzes to grade over the weekend. Over the weekend I only graded ¾ of the quizzes I had. What fraction of all of the quizzes did I grade? This will help the Novice and Approaching Mastery students to develop understanding about what is going on and how to model it.