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: 8/4
For the first task, 8/4, most students drew a rectangle with an 8 in the middle, placed the number 4 on one side, and placed a question mark on the other side of the array. One student explained: 4 times 2 is 8 so 8 divided by 4 equals 2.
For this task, students were a bit baffled because it equaled the same as the last task. Students immediately began trying to make sense of the situation! One student explained, "It's like you take a 2 x 4, which equals 8 and you double it, but you keep the 2 on one side. That way you have a 2 x 8, which equals 16. I loved watching students grapple with finding an equation to match the array representation.
Task 3: 32/8
For the next task, 32/8, students explained, "Just double the 2 x 8 array to get to 32. Then the answer doubles too: (2 x 8) + (2 x 8) = (4 x 8). So the answer is 4."
Task 4: 800/8
During this task, students explained, "You can make a rectangle with 800 on the inside. Then put 8 on one side. The other side is unknown. It equals 100 because 100 x 8 = 800."
Task 5: 832/8
For the final task, students used previous tasks to determine the quotient. Many students drew two arrays. One array was an 8 x 100 = 800. The other array was an 8 x 4 = 32. Students then explained that 832/8 is the same as (800/8) + (32/8) = 100 + 4.
To begin today's lesson, I introduced the goal: I can find the factors for 41-70. I explained: Over the past two lessons, you identified the factors for all of the numbers up to 40. Today, we will move on to more challenging numbers, 41-70.
Before jumping into our lesson, I wanted to introduce students to some conjectures that I have heard students come up with over the past couple of days using a Conjectures Poster. I also included a couple conjectures based on common misconceptions:
1. Larger numbers have more factors.
2. All odd numbers are prime.
3. All even numbers are composite.
4. One is a factor of all numbers.
5. If a number is a multiple of 2, then 2 will be one of its factors.
I explained: Today, I want you to keep these conjectures in mind. Some of them are true while others are false. If you can provide evidence to either support or disprove one of the conjectures today, I would like to know! Immediately, hands shot up. Many students were eager to prove and disprove the conjectures. I asked students to wait until later on in the lesson. I wanted students to really think about the conjectures and possible evidence!! This activity will most certainly 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 using Cupcakes and Baker's Cap Putting on the baker's cap, I explained: Today, I wanted you to truly understand that factors are not just numbers that we work with in math. Factors are important in every day life as well. Can anyone explain why factors would be so important to a baker? Students volunteered one at a time. They came up to the front table and manipulated the cupcakes to explain their thinking. I also put the baker's cap on the student to help them get into character! One student explained that factors help a baker determine the size of box needed. Another student explained the importance of using rectangular arrays so that cupcakes don't tip over in the box. I asked another student to come up and make as many arrays as possible with 11 cupcakes. At the same time, I asked other students to take out 11 colored tiles out of their bags to complete the same task. A wonderful conversation followed: How many arrangements are there for 11 cupcakes? (1 array) Why would bakers make 12 cupcakes instead? (They would have more packaging options). What are the arrangements for 12 cupcakes? (12 x 1, 3 x 4, 2 x 6)
Reviewing Factors & Multiples
Although I had already taught students the difference between multiples and factors using vocabulary posters, I wanted to review these math concepts using a quick higher-level thinking task that required students to compare and apply. Instead of simply asking students to provide the meaning of a factor and multiple, I asked students to draw a line down the center of their white boards and to label one side "Factors of 6" and the other side "Multiples of 6." I encouraged students to use their colored tiles. During this time, I conferenced with students to check for understanding and to reteach these concepts. Then, altogether, we discussed the factors of 6 (1,2,3,6). I wrote these in a box on the board and pointed out that there was plenty of room left in the box. Then, we identified the multiples of 6 (6, 12, 18, 24...). Again, I wrote these in a box on the board as students suggested them. After a while, we began to run out of room. I said: Oh my! I'm running out of room! Why is that? One student said, "Because there are millions of multiples!" Of course, this was the perfect time to sing our Factors & Multiples Song!
Reviewing Prime & Composite Numbers
Next, I asked students to divide their white boards in half. On one side, I asked students to write "Prime Numbers" and on the other side, I asked students to write "Composite Numbers." I explained: To review prime and composite numbers today, I would like you to work with your group members to identify all the numbers between 1 and 20 as prime or composite. Many students immediately began working. Others referred to our math vocabulary posters, Prime Numbers and Composite Numbers for a little help. I added the words "greater than one" on our prime poster to make the definition more precise. After a few minutes, we came back together as a class and categorized each number, starting with 2: Is 2 prime or composite? (prime) How do you know? (It only has two factors... one and itself.) What about 3? Is 3 prime or composite? We continued until all whole numbers 1-20 were categorized correctly. This was the perfect time to share the following video. I chose to show this video after this little review activity so that students were given the opportunity to grapple with these concepts.
After the video, I revealed the I'm Prime Chant poster so that we could refer to this poster to identify prime numbers throughout today's lesson.
Reviewing the Prime Factorization Method
Prior to the lesson, I placed colored paper inside sheet protectors and passed out two to each student. At this point, I asked students to write "Prime Factorization" at the top of one paper and "U-Turn" at the top of the other paper. I then used the Process Grid. on the board to review how to find factors using these methods while students completed the tasks on their colored papers: Student's Prime Factorization & U-Turn.
1. Write the target number, 18, at the top your board.
2. I asked: What two factors, when multiplied together, equal 18? Because this method is called PRIME factorization, the goal is to find all the prime factors for 18. I always ask: Does 2 go into 18 evenly? Students responded, "Yes! 2 x 9!" I modeled how to write 2 x 9.
3. Whenever you are using the prime factorization, we always want to circle the prime numbers. Does anyone see a prime number that we can circle? Students responded, "2!"
4. We continued this process until we were able to identify the prime factors for 18 using an equation: 2 x 3 x 3 = 18.
Reviewing the U-Turn Method
We then moved on to the U-Turn for 18.
1. Draw a t-chart.
2. Write the target number, 18, on top of the line.
3. Rule number one is... Students finished my sentence, "Always start with one!"
4. Ask yourself: How many times does 1 go into 18? Or, in other words: What times 1 is 18?Students said, "18!" Okay! Write 18 in the right column, across from the one.
5. We continued on, writing 2 (left column) ... and then 9 (right column) and then 3 (left column) .... and then 6 (right column). I counted down the left side: 1... 2... 3.... What number should we try next? After ruling out the numbers 4 and 5 as factors, we then took a u-turn!
Connecting the Methods
I included an Array column to the process grid to help students connect the prime factorization method and the u-turn method to cupcakes and arrangements. Without asking students to model this at their desks, I simply showed them the factors in array form: Arrays for 18. Again, I wanted today's focus to be on the prime factorization and u-turn methods.
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 method on their colored pages while completing Factor Chart C.
During this time, I conferenced with all students and monitored student understanding by making observations and asking questions:
While students were finding factors, I also encouraged them to challenge the previously presented conjectures on the poster. Several students eagerly volunteered to provide either evidence or counter-evidence that proved or disproved each conjecture. For example, one student explained, "Larger numbers sometimes have more factors, but not always. For example, 9 has two factor pairs: 1 x 9 = 9 and 3 x 3 = 9. Eleven has one factor pair: 11 x 1. Nine has more factors than 11 so the thought was wrong."
Here's what the poster looked like afterwards: Corrected Conjectures Poster. You'll notice that we didn't get to the last conjecture.
To bring closure to this lesson, I celebrated students who were on task, working hard, persevering, and finding creative ways to solve problems.
We also took the time to discuss the Corrected Conjectures Poster. Students seemed to love the opportunity to explain their thinking to the class! I loved listening to them construct viable arguments and critique the reasoning of others (Math Practice 3).