As an introduction to fractions, the do now is an exercise of identifying fractions that can be reduced.
Students will be given the following directions:
Circle the fractions where the numerator and denominator are relatively prime.
I will review the answers with students. What observations can you make about the circled fractions?
Students may reply with:
This will lead into a lesson of when and how fractions can be reduced.
In this lesson, I will show students two methods of simplifying fractions. The first method is for students who have had difficulty finding the GCF in previous lessons, but are comfortable with the divisibility rules. The second method is for students who are not only comfortable with finding the GCF, but also they may be able to do so with mental math.
We will discuss both methods and work through an example for each method.
Method 1 - Find a Common Factor
Ex. 1 - Simplify 40/48
Step 1 - Find a common factor for the numerator and denominator.
I will remind students that to find a common factor, the divisibility rules can be used. What is a common factor of 40 and 48? Most students will say 2, because they are both even numbers. On the board, I will use 2 to show students the repetitive steps of using the lowest common factor.
Step 2 - Divide both the numerator and denominator by the common factor.
Students should have an answer of 20/24. Can the fraction be reduced again? How do you know?
Step 3 - Repeat the process until there aren't any more common factors.
The final answer will be 5/6. How do you know when you've completely simplified a fraction? What observations can you make?
Students may have several answers:
Although the above observations are all correct, I want students to think about which one is true for all simplified fractions. If necessary, I will give some more examples of simplified fractions. Students should come to the conclusion that if the numerator and denominator have a gcf of 1, relatively prime, then the fraction is simplified completely.
Method 2 - Find the GCF
Ex. 2 - Simplify 36/84
Step 1 - Find the GCF of the numerator and denominator.
Students have 2 methods for find the GCF (see Finding the GCF lesson). Some students may try to find the GCF using mental math and think the answer is 4. Although I want to develop students' mental math skills, I will challenge them to prove their answer is correct by finding the GCF using one of the methods we worked on in class. Students should find that the GCF is 12.
Step 2 - Divide the numerator and denominator by the GCF.
Students will have an answer of 3/7. Can this fraction be simplified any further? Why not? Again, students should come to the conclusion that the numerator and the denominator are relatively prime.
Students are seated in groups of four. I will pair students together based on a previous GCF assessment; a high level math student will be paired with a lower level student. I've found that this promotes a collaboration where the low level student can learn from the high level student and the high level student has the opportunity to reinforce and deepen their understanding of the concept.
I will post 5 problems on the board for all students to work on independently. They should choose the method that they are more comfortable with. When they have completed the problems, they will exchange their notebook with their partner. Their partner must verify the answer and simplify the fractions using a different method.
After 10 minutes, we will review the answers. Did anyone have a different answer when you checked your partner's work? Did you find the reason for the difference?
I will hand out index cards for each student to answer the following prompt:
List 3 things that a fellow student might misunderstand about the topic.
Before students leave class, I will collect the cards. These cards will be used as an assessment for students understanding of the topic. Students often write answers, such as