## Lesson: Adding and Subtracting Fractions and Mixed Numbers Developing the Concept

Students have been introduced to adding and subtracting fractions and mixed numbers. They also have a visual image of what is happening in those operations. This lesson will develop those concepts and relate them to adding and subtracting whole numbers.

Materials: paper and pencil

Preparation: none

Prerequisite Skills and Background: Students should be able to find equivalent fractions by finding the least common denominator and simplify fractions by using the greatest common factor.

• Say: Today we are going to extend our work on adding and subtracting fractions and mixed numbers. Let's look at adding two mixed numbers. On the board, write 3 + 2.
• Ask: What do we need to do in order to add these two numbers?
Students may say that they could add the 3 and 2 and get 5. Then they need to find a common denominator to add the two fractional parts.
• Ask: What do you need to do to find a common denominator? We can multiply 4 and 5 and get 20 as our common denominator.
• Say: Good. Twenty is a common denominator—in fact, it is the least common denominator for these two fractions. What do we do next?
Students should say they need to change and into equivalent fractions with denominators of 20. This is done by multiplying the numerator and denominator of by 5 and the numerator and denominator of by 4. Be sure to stress that you are really multiplying each by 1 in the form of and .
• Say: We have found equivalent fractions with a common denominator, so we can add. 3 + 2 = 3 + 2 = 5.
• Say: Now try the following problems at your desk. I'll come around and help those who need it.
Write the following problems on the board: + , 3 + 4, and 2 + 1.
Ask for volunteers to come to the board to show their work and explain what they have done.
• Ask: Who can summarize for us what we need to do in order to add two fractions or mixed numbers?

List steps such as these, based on what students answer.

1. If the fractional parts do not have the same denominator, change them to equivalent fractions with a common denominator, preferably the LCD.
2. Add the fractional parts and simplify, if possible.
4. Finally, add the sum of the whole numbers to the sum of the fractions.
• Say: Now that we know how to add two fractions or mixed numbers, let's look at subtracting two fractions or mixed numbers.
On the board, write 4 − 2.
• Ask: What is the first thing we might do to solve this problem?
Students should say that they need to change and to equivalent fractions with a common denominator.
• Ask: How do we find a common denominator for these two fractions?
Students should suggest multiplying the two denominators to get a common denominator of 40.
• Say: Yes, we could do that, but how could we find the least common denominator?
Suggest that students find the LCM for 10 and 4 by finding the prime factorization for the two numbers.
• Say: We know that 4 = 2 x 2 and 10 = 2 x 5, so the LCM would be 2 x 2 x 5, or 20. On the board, write 4 = 2 x 2 and 10 = 2 x 5. Mention that the LCM for 4 and 10 is 20.
• Say: Therefore, = x = and = x = . Since > , we need to regroup 4 as 3 = 3. Thus, 3 − 2 = 1.
• Say: Now try solving the following problems at your desk, and we'll discuss them when you are done.
Write the following problems on the board: 8 − 4, , and 4 − 2.
Walk around the room to help the students who need it. After all students have had time to do the problems, have volunteers come to the board and explain how to do the problems.
• Say: Who can summarize for us how to subtract two fractions or mixed numbers?

List steps such as these, based on the students' suggestions.

1. If the fractions don't have the same denominator, change them to equivalent fractions with a common denominator. Use the LCD.
2. Look to see if you need to regroup in order to subtract the fractional parts. If you do, regroup before subtracting.
3. Subtract the fractional parts, and then subtract the whole-number parts.
4. The answer is the sum of the two.

Wrap-Up and Assessment Hints
Having students estimate the sum or difference of two fractions before adding or subtracting will help them recognize unreasonable answers. For example, if they add and , they should recognize that both fractions are less than , so the sum should be less than one. Also, since both fractions are just a little less than , the sum should be close to 1. Students should recognize that is not a reasonable answer. Ask students questions such as, “Why do we have to have common denominators before we add or subtract two fractions?” and “Why do we only add the numerators and not the denominators when adding two fractions with like denominators?” By answering questions like these, students will improve their understanding of the concepts.

To challenge some of your best students, ask them open-ended questions such as, “If the sum of two fractions is , what could those two fractions be?” ( and , and , and so on) or “If the difference of two fractions is , what could those two fractions be?” ( and , and , and so on)