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: 10/5
For the first task, most students drew a rectangle with a 10 in the middle, placed the number 5 on one side, and placed a question mark on the other side of the array. Some students even used variables: 10:5.. Students explained: 10 divided by 5 equals 2 because 2 x 5 = 10.
For this task, many students doubled the 2 x 5 array to get (2 x 5) + (2 x 5). These students explained: 2 + 2 = 4 so 20/5 equals 4: 20:5.
Task 3: 40/5
For the next task, 40:5, students explained, "Just double the 4 x 5 array because 20 + 20 = 40." More and more students began to use correct division equations: (20/5) + (20/5) = 40/5 =8.
Task 4: 400/5
During the next task, some students drew a 5 x 40 array which equals 200. Then, they doubled this array to get (200/5) + (200/5) = 80: 400:5..
Task 5: 440/5
For the next task, most students made a 5 x 80 array, which they already knew equals 400. Then, they added on another 5 x 8 array, which they already knew equals 40. Then, they showed how 80 + 8 = 88: 440:5.
Task 6: 880/5
For the final task, students simply doubled the 5 x 88 array: 880:5. It was great seeing students looking for and making use of structure (Math Practice 7).
To begin today's lesson, I introduced the goal: I can find the factors for 71-100. I explained: Over the past three lessons, you identified the factors for numbers up to 70. Today, we will move on to more challenging numbers, 71-100.
Reviewing Multiples & Factors
To review multiples and factors, I simply referred to Factors & Multiples Song and said: When we talk about factors, we are referring to the numbers you multiply in a multiplication problem. With the equation, 2 x 3 = 6, the 2 and the 3 are both factors. Remember, (singing) factors are just a few-ew-ew. However, when we look at the multiples of 6, we skip count by 6: 6... 12... (with gesturing, students joined in)... 18... 24... 30... 36... Remember, Singing many millions of multiples! Factor are just a few-ew-ew. Many millions of multiples! Factors are just a few!
Reviewing Prime & Composite Numbers
Next, I referred to the posters, Prime Numbers and Composite Numbers as well as the I'm Prime Chant.. Turn and talk: What is a prime number? Soon, students were singing the chant and I couldn't help but join in, "I'm prime! P-R-I-M-E... The only the factors factors are 1 and me." Surprisingly, after yesterday's lesson, some students had looked up the song on you tube and were able to continue singing, "Factors that divide evenly!"
Reviewing the Prime Factorization Method
At this point, I explained: Yesterday, most students didn't have time to investigate the last row on the factor chart (Factor Chart C). Today, we will complete this row together before you move on to completing the last factor chart that covers the numbers 71-100 (Factor Chart D).
I paired students up and asked one partner to write "Prime Factorization" at the top of his/her whiteboard and the other partner to write "U-Turn" at the top of his/her white board. I then explained: Today, I want to you to take turns representing your thinking with each of these methods. After we find the factors for one number, I'd like you to switch so each partner has the opportunity to practice each method.
I began with the first task, 61. Students referred to the I'm Prime Chant. poster and quickly said, "61 is prime! That means only 61 and 1 are the factors!"
We then moved on to 62. I modeled the Prime Factorization for 62:
1. Write the target number, 62, at the top your board.
2. I asked: What two factors, when multiplied together, equal 62? Because this method is called PRIME factorization, the goal is to find all the prime factors for 62. I always ask: Does 2 go into 62 evenly? After some time, students responded, "Yes! 2 x 31!" I modeled how to write 2 x 31.
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!" With time, a student then pointed out, "And 31 is prime too!"
4. We then wrote out the prime factorization equation for 62: 2 x 31 = 62.
Reviewing U-Turn Method
We then moved on to the U-Turn for 62.
1. Draw a t-chart and write the target number, 62, on top of the line.
2. Rule number one is... Students finished my sentence, "Always start with one!"
3. Ask yourself: How many times does 1 go into 62? Or, in other words: What times 1 is 62?Students said, "62!" Okay! Write 62 in the right column, across from the one.
4. We continued on, writing 2 (left column) ... and then 31 (right column).
Instead of asking students to try completing Factor Chart D during this time, I challenged students to find which number on the page has the most factors. After giving the students the opportunity to make predictions, I explained: Today, I want you to continue working with your partner and taking turns using the prime factorization method and u-turn method. Who would like to start with finding the factors for 71? Who would like to begin with 100? Today, you and your partner can take turns choosing which numbers you would like to investigate! Without any prompting, students immediately began testing each number and finding the total factors!
During this time, I conferenced with students (Finding the Factors for 100). I would often provide a little advice or ask guiding questions to help support and stretch student thinking:
Sometimes I ask,
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 took the time to ask students: Which number between 70 and 100 has the most factors? Students were very excited to share the results to their investigation! Although they didn't have time to investigate all the numbers, students were able to rule out prime numbers and seemed to focus most on the even numbers.
After allowing each group the opportunity to guess, I showed students the Factors for all Numbers Under 100. The numbers, 72, 84, 90, and 96 all had 6 factor pairs (12 factors each). I heard several students say, "I knew it!" "We were right!"