For several days now, students have been learning and reviewing the basics of fractions. Before we move on to algorithms for adding and subtracting mixed numbers, I want to review the topics of improper fractions and mixed numbers. Since the most common mistake with these types of problems is often computational, I decided to challenge them with larger numbers.
Change the following to improper fractions.
1) 327 17/25
2) 1941 4/31
Change the following to mixed numbers.
Students may need more than the usual 5 - 10 minutes to work out these problems.
In previous lessons (Adding and Subtracting Fractions Activity), students have been working on developing strategies for adding and subtracting fractions. The focus of this lesson is for students to develop algorithms based on their strategies and observations.
I will show students example 1 on the board and give them a scenario of explaining how to solve it to a fictitious student, John, who hasn't worked with fractions. To explain their steps, students should be using appropriate vocabulary such as, unlike denominators, LCM, equivalent fractions, ...
Example 1: 1/5 + 1/2
Can John add the numerator and denominator to get 2/7? Why not? When given fractions with unlike denominators, what should he do first? Through questioning, students will develop the first step.
Step 1 - Find the least common multiple of the denominator.
Many students may notice that if you multiply the denominators for this example you will find the LCM. If you multiply the denominators, will you always find the LCM? Let students discuss this question until they agree that multiplying will not always give you the LCM.
After finding the LCM is John ready to add? What should he do next? Students may struggle with the vocabulary term equivalent fractions and instead use words like convert or same. Let students discuss the correct vocabulary that leads into step 2.
Step 2 - Find the equivalent fractions
Is John ready to add? Should he add both the numerators and the denominators? Why not?
Step 3 - Add the numerators and keep the denominators
Are we done? Is step 3 always the last step? What will we have to do sometimes?
Step 4 - Simplify, if possible.
Following example 1, I will present students with example 2 and ask if they can use the same algorithm for subtracting mixed numbers. I will have them work on example 2 with their groups to see if the algorithm works.
Example 2 - 16 3/4 - 2 4/9
Did the algorithm work for example 2?
Following example 2, I will present the class with a third example and ask the same question. Will the algorithm work for example 3? I will let students work through the example and see if they notice that the fractions can't be subtracted. As students are working, I will walk around. I will stop them when the majority of the students have reached step 2. Some students may notice that there is a "problem", while others may try to subtract. We will discuss their observations and strategies for this type of problem. After discussion, I will summarize their ideas into 2 methods.
Example 3 - 12 1/2 - 5 3/4
Method 1 - Borrowing
This method shows students how they can borrow 1 from the whole number and add it back to the fraction. Students may have difficulty with understanding the concept of borrowing, so I will make the comparison of borrowing a library book. "When you borrow a book from the library, you have to return it in the same condition. Think of it as borrowing 1 book from their collection and returning in excellent condition, such as 4/4 or 8/8. You don't want to return part of a book. Returning 3/4 of a book is not acceptable. You need to return 4/4 of a book."
Method 2 - Change to Improper Fractions
I will use example 3 again to show students another method. I will refer to a previous lesson where we reviewed changing mixed numbers to improper fractions. Instead of borrowing, students may want to change the mixed numbers to improper fractions and then subtract. As a class, we will work through the steps.
If a student subtracts mixed numbers by borrowing and another student changes them to improper fractions, will they have the same answer? When do you have to decide whether to borrow or change the mixed numbers to improper fractions?
Since subtracting mixed numbers is a difficult concept for students, I want to assess their understanding of the lesson. I will give each student an index card for them to solve the below problem on. They will have 5 minutes to work on the problem. Before leaving the class, students will hand me their exit ticket.
Find the difference. Simplify your answer completely.
22 1/8 - 5 3/7 =
These exit tickets will provide me with information in planning the next lesson. For example, I may need to clarify a common mistake being made or re-teach a step.