In preparation for the lesson on multiplying mixed numbers I want students to review changing a mixed number to an improper fraction and multiplying fractions. The Do Now is a review of these concepts.
1. Write 2 3/5 as an improper fraction.
2. Write 5 1/8 as an improper fraction.
3. Find 1/4 of 2/5
4. Find 6/7 x 3/4 x 2/3
Students may struggle with problem number four, multiplying 3 fractions. I will suggest that they think about what they would do if they didn't have fractions and just had three numbers to multiply.
After about 5 - 7 minutes, I will randomly call students to the board to show their work and answers to the Do Now problems. These students will explain their work for the class. It is important for me and their classmates to hear their thought process. If a students' work is incorrect, I will ask a classmate to share their work.
I will use area models and questioning to help students develop an algorithm for multiplying mixed numbers.
Mr. Smith buys a 2½ pound wheel of cheese. His family eats 1/3 of the wheel. How much cheese have they eaten?
What operation is needed to solve this problem? How do you know? Students should respond that since they've eaten a fraction of the wheel, it requires multiplication. 2½ x 1/3
Can we multiply the mixed numbers straight across? Many students may think that this is an appropriate strategy. I will use an Multiplying Mixed Numbers Area Model.png to illustrate why we would get a different answer. Students have worked on area models for multiplying fractions, however area models for multiplying mixed numbers may be challenging for them; for this reason it is helpful to model it with them. How can we represent 2½? We will shade in 2 squares and ½ of another square. How can we show 1/3 of this? We will shade a third of each one of the squares. What is our answer, based on the area model? Students should remember that the answer is the overlapping portion. Since we have 3 separate squares, what should we do with each overlapping section? It is important for students to realize that we need to add each section to find our answer. 1/3 + 1/3 + 1/6 = 5/6 Is this the same answer we would have if we multiplied straight across?
Is 5/2 x 1/3 the same as the original problem? What did I do to the mixed number? Students should realize that it's the same problem, but the mixed number has been changed to an improper fractions. When solving, students should realize that it's the same answer as the area model.
Developing an Algorithm
When we need to multiply mixed numbers, should we multiply straight across? What should we do first? Based on the previous example, students have discovered that to multiply mixed numbers they need to change them to improper fractions first. Then, they can multiply straight across.
Students will be given a few problems to practice multiplying mixed numbers. Because the area models may be difficult for some students, I will suggest that they use the algorithm we developed and then if they choose to, they can draw an area model.
As students work, I will circulate throughout the class, monitoring students' progress.
After about 5 minutes, students will share their work and answers with their group. If there are any discrepancies, we will review the problems as a class.
To promote, students' understanding of the relationship between multiplying fractions and multiplying mixed numbers, I will pose a few questions to the class.
What is the same and what is different about multiplying fractions and multiplying mixed numbers?
How do you know your solution is reasonable?
I will select a few low level and high level math students to share their answers in order to assess understanding.