Prime Factorization and GCF

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Objective

SWBAT calculate the GCF of 2 integers, including negative numbers, by using prime factorization.

Big Idea

Students work with bar models and prime factorization trees to solve word problems

Do Now

10 minutes

Students enter silently according to the Daily Entrance Routine. Their Do Now assignments are already on their desk and they begin working silently. Directions on the board ask students to raise their hand once they’ve finished. I ask them to retrieve a Senteo clicker from the back and enter their answers. For #3, I have a SMARTBoard page ready with specific directions to write the answer as an improper fraction.

Students have 5 minutes to complete these three problems independently and enter their answers. After generating results we review the lowest scored questions given the amount of time left for this section. 

Guided Practice

15 minutes

I distribute Guided Practice papers and have students read the Orchestra word problem independently (Source for the 1st two paragraphs). Then I ask for a reader to read it out loud. Then I ask a selection of 3 – 4 students to tell me what the question wants us to find, each in their own words. I usually select students at varying levels of academic achievement, allowing them to build off each other’s responses (i.e “do you agree with him?” “do you have anything to add?”).

Then students will need to read the following paragraph independently. I will also ask that they read what is in the box labeled “Help with Solving”. Students will be given 2 – 3 minutes to read and 1 – 2 minutes to summarize what they’ve read to their neighbors. Then I will ask a series of comprehension questions:

  • What’s the difference between a common factor and the greatest common factor?
  • One way to find the GCF is to make a list, or “go shopping for factors”, but why might this method not be useful? For what kinds of situations?
  • If we use the numbers in the Orchestra problem to calculate a GCF, what will this number represent in that context?

The following video shows how we use bar models and prime factorization trees to solve the word problem.

 

The most difficult part about finding the GCF through prime factorization is taking the product of the common factors. Some students often count the factors more than once. For example, some students see nine 2s and use all of them to calculate the GCF.

Other students simply forget the operation is multiplication and use addition instead. For these students it is important to continuously check their work when showing prime factorization trees. Students work to find the GCF independently. If finished early, they will need to list the first 13 prime numbers and list them in the table on the left corner of the paper. During the last 5 minutes of this section students work at the board to summarize and explain the answer to this question.

Partner & Independent Practice

20 minutes

Class work is distributed. Students must work with their neighbor for the first 10 minutes of this section. They are encouraged to use bar models and prime factorization trees to solve. Most students will need more space than is provided on the sheet.

During this time I am walking around asking students question or giving them feedback about their work:

  • What does each bar in your model represent? Label them.
  • What does the GCF represent? What is being divided?

Students work independently during the last 10 minutes of this section. I am also asking students who are finished or nearly finished to display their work on the board during this time. It will allow other students to check their own work or ask questions about what is going up on the board. 

Closing

10 minutes

During the last 10 minutes I have 1 – 2 students explain their answers at the board. Students who display their work on the board may have solved a different way or a student who showed their work in a neat and clear manner. Students elect the work they want explained. The goal is to have students talk about the concept and process of finding GCF, specifically the application to real world problems. As they talk about the concepts, they build understanding and I gain knowledge of any holes in understanding I still need to work on.

 

Homework will be distributed at the end of class.