Factors: Prime Factorization
Lesson 2 of 11
Objective: SWBAT find the factors for the numbers, 21 - 40.
Today's Number Talk
For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation. For this Number Talk, I am encouraging students to represent their thinking using an array model.
Task 1: 18/9
For the first task, 18/9, some students simply drew a 2 x 9 rectangle and placed 18 on the inside. Other students divided the 18 into two 9s: 18:9. During this lesson, students experimented with different equations (some correct, some incorrect) to represent their thinking. Regardless, experimentation is an important part of constructing and testing arguments (Math Practice 3).
Task 2: 900/9
During the next task, some students chose to divide the 900 up into (500 + 400)/(5 + 4) = 100: 900:9.
Task 3: 918/9
For 918/9, I was excited to see many students using the first two tasks to solve this task: 918:9. I also modeled this strategy on the board to encourage all students to begin using decomposing in order to solve a division problem.
Task 3: 1,836/9
For the final task, most students decomposed the 1,836 into 900 + 900 + 18 + 18 to find the quotient: 1836:9.
To begin today's lesson, I introduced the goal: I can find the factors for 21-40. I explained: Yesterday, you identified the factors for all of the numbers up to 20. Today, we will move on to the numbers 21-40. A sense of excitement filled the air. My students love when tasks slowly become more challenging!
Prime Verses Composite Numbers
Before jumping into our lesson, I wanted to teach students about prime and composite numbers. Instead of telling them the meanings, I wanted them to discover the meanings for themselves! I passed out Prime vs Not Prime Numbers to each group of of 2-3 students. I have students strategically placed in desk groups for easy pairing in all subject areas!
Next, I asked students divide their white boards in half. On either side of their white boards, students wrote the following questions: What is a prime number? What is not a prime number? Students began examining the prime numbers. Some students began adding the numbers together. Others, Odd vs Even, pointed out, "At first I thought all prime numbers were even, but then I saw the two."
After a few minutes, I decided to slowly provide students with more examples and non examples of prime numbers to help expedite the investigation: Five is a prime number. A minute later I said: But 8 is not a prime number. Students added each of these clues to their sheets. With each clue, students gasped, "Oh! Now I know!" I would go running over to their desks and would often say, "Keep thinking! You're almost there." I also asked students to get out their colored tiles (below) and to begin modeling the prime numbers with arrays: Conferencing with Student.
This was when students figured it out! Here, Student Explaning Prime Numbers, a student explains his discovery! I was proud of my students for practicing Math Practice 1 as they were continually "analyzing givens, constraints, relationships, and goals" as well as continually asking, “Does this make sense?”
We discussed the student's definitions of prime and not prime numbers (Student Explanation Example A. and Student Explanation Example B. I then revealed the following vocabulary posters: Prime Numbers and Composite Numbers.. I loved watching students make connections between their investigation, reasoning, and the formal definitions.
We quickly reviewed the vocabulary posters from yesterday's lesson: multiplication, factors., Factor Pairs., and multiples. I asked students to turn and talk: Explain the difference between a factor and a multiple. I knew that encouraging student conversations would increase the comprehension rate of the newly learned information. I also knew that this would support Math Practice 3: Construct viable arguments and critique the reasoning of others.
Cupcake Baker & Building Context
To help students understand a real-world application (Math Practice 4: Model with Mathematics) in which factors really do matter, I acted out the role of a cupcake baker. I put on a Baker's Cap and set out Mini Cupcakes. I then explained: Remember how we talked about the importance of understanding factors if you are a cupcake baker?! We discussed how understanding factors helps a baker decide how to arrange cupcakes in a cupcake tray (or rectangular box). Yesterday, we also talked about the importance of placing cupcakes in perfect rectangular arrangements (or arrays). I loved how one student pointed out that the cupcakes could tip over and roll around otherwise!
Using the Prime Factorization Equation to Find Factors
I asked students to come up to the front carpet with their white boards. I began by explaining: Today, we are going to use prime factorization equations to find the factors of numbers. Here's the prime factorization equation for 12: 2 x 2 x 3 = 12.
I then modeled how to multiply the prime factors within the prime factorization equation to find the factor pairs for 12: Finding Factor Pairs for 12. I explained: Look at how we can multiply two prime factors, such as 2 x 2 to get 4. What times 4 is 12? (Three!) What is left in our prime factorization equation when we cover up the 2 x 2? (Three!) We could also multiply the 2 x 3 to get 6. What times 6 is 12? (Two!) What is left in our prime factorization equation when we cover up the 2 x 3? (Two!)
I then followed the same steps to model Finding Factor Pairs for 36. Students were amazed!
Using a Factor Tree to Identify the Prime Factorization Equation
I explained and modeled on the Prime Factorization Poster: When you use the Prime Factorization method, you always follow the following steps. Students completed the same steps on their white boards.
1. Write the target number, 24, at the top your board.
2. Ask: What two factors, when multiplied together, equal 24? Because this method is called PRIME factorization, the goal is to find all the prime factors for 24. I always ask: Does 2 go into 24 evenly? This is because 2 is a prime factor and it's easy to multiply and divide with! So... does 2 go into 24 evenly? Students responded, "Yes! 2 x 12!" I modeled how to write 2 x 12 on the Prime Factorization Poster.
3. Whenever you are using the prime factorization method, I'd like you to circle all of the prime numbers. Does anyone see a prime number in our factor tree so far? (Two!) What about the 12? Is 12 prime or composite? (Composite!) How do you know? (Because it has more than two factors)
4. Whenever you have a composite number left, ask, "What goes into this number evenly?" So... what goes into 12 evenly? (Two... 2 x 6!) What should we circle? (The 2!) Why? (The 2 is prime!)
5. What goes into 6 evenly? (2!.... 2 x 3) What should we circle? (The 2 and the 3!) Why? (Both 2 and 3 are prime!)
6. Now we are ready to write the prime factorization equation! I modeled how to write 2 x 2 x 2 x 3 = 24. I also tested the equation with students while pointing at each factor: Let's see if it works! What's 2 x 2? (4) What's 4 x 2? (8) What's 8 x 3? Students excitedly said, "24!"
Connecting the Prime Factorization Method with the U-Turn Method
At this point, I asked students to return to their desks. I asked team leaders to pass out our colored papers in page protectors. Students labeled one page "Prime Factorization" and the other "U-Turn." Next, I modeled how to use the prime factorization method and U-Turn Method together to find the factors for 24. As I modeled this on the board: Modeling Prime Factorization & U-Turn, students completed the same steps at their desks: Student Example of Prime Factorization & U-Turn for 24.
U-Turn Method for 24
1. Draw a t-chart.
2. Write the target number, 24, on top of the line.
3. Rule number one is... students finished my sentence, "Always start with 1!"
4. Ask yourself: How many times does 1 go into 24? Or, in other words: What times 1 is 24?Students said, "24!" Okay! Perfect! Write 24 in the right column, across from the one.
5. We continued on, writing 2 (left column). I covered up the 2 in the prime factorization equation for 24 (2 x 2 x 2 x 3 = 24). I asked: If I cover up the 2, what is left? (2 x 2 x 3) What is 2 x 2? (4) What is 4 x 3? (12) I then modeled how to go back to the u-turn chart to write 12 in the right column across from the factor 2.
6. We then wrote 3 in the left column of the u-turn chart. I asked: Do you see a 3 in the prime factorization equation for 24? (Yes!) Let's cover up the 3 and see what's left! What is 2 x 2? (4) What is 4 x 2? (8). Going back to the u-turn chart, I asked: So what times 3 is 24? (8) We wrote 8 in the right column across from the factor 3.
7. I wrote 4 in the left column of the u-turn chart and asked students: Does anyone see how we can multiply two factors together in the prime factorization to get to 4? Students said, "Yeah! We can multiply the 2 x 2. What would be left? (2 x 3... which equals 6!) We then wrote 6 in the right column of the u-turn chart across from the factor 4.
8. I wrote 5 in the left column of the u-turn chart and asked students: How about 5? Does anyone see how we can multiply two factors together in the prime factorization to get to 5? One student said, "Yes! 2 and 3!" Another students said, "But that's multiplication!" I then asked the question again, stressing the word "multiply": Does anyone see how we can multiply two factors together in the prime factorization to get to 5? (No!)
I passed out Factor Chart B so that students could begin documenting the factors. While projecting the factor chart, I asked: What were the factors for 24 again? I modeled how to in the chart.
For guided practice today, I wanted to complete the first row of the Factor Chart B altogether. We first started with finding the factors for 21 using the prime factorization and u-turn methods. I modeled each problem on the board while students modeled the prime factorization and u-turn strategies at their desks: Student Example of 21 and Student Example 22.
During this time, we also completed the top row of Factor Chart B together: Completed Chart.
During this time, I released more and more responsibility to students by providing less and less direction with each task. By the time we finished the top row, students were ready to continue on their own!
For student practice time, students worked in groups of 2-3. Assigning partners is always easy as I already have students strategically sitting in groups of students based on abilities, skills, and behavior.
Students continued using the prime factorization and u-turn methods on their colored pages while finding the factors for all the numbers in the second row (21-30) of Factor Chart B.
During this time, I conferenced with all students and monitored student understanding by making observations and asking questions:
- Can you explain how you can use the prime factorization equation?
- Is ___ a prime or composite number?
- How do you know?
- Does that always work?
- How do you know when you should make a u-turn?
Here is an example of a student conference: Conferencing About 32.
As students finished, they compared their answers with others at the back table.
To bring closure to this lesson, I celebrated students who were on task, working hard, persevering, and finding creative ways to solve problems.
I also projected the first two rows of this following document (Factors for all Numbers Under 100.pdf) for students to check their work.
Here's an example of a student's Completed Factor Chart.