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: 12/3
For the first task, I modeled how to use an array to represent division. Students created their own models on their white boards: 12:3. I always try to begin with simpler tasks for several reasons:
1. I want to build confidence in math. The more confident students are, the more likely they will persevere when the tasks become more complex.
2. By using simpler tasks to begin with, students focus on making sense of the array model and how to represent division instead of the problem itself.
3. Finally, I want students to see how simpler problems, such as 12/3, can help solve more complex problems, such as 42/3.
Task 2: 30/3
During the next task, most students modeled 3 x 10 = 30 using an array. Several students tried showing their thinking in another way, such as (3 x 5) + (3 x 5) = 30: 30:3.
Task 3: 42/3
During the next task, we discussed 42/3. I modeled how to use the first two tasks to solve this task on the board: Modeled 42:3 on the Board. Then, students solved the same tasks on their own boards: 42:3.
Task 4: 84/3
We then moved on to 84/3. Here, a student shows how 84/3 = 2 (42/3): 84:3. This is exactly what I was hoping to see students doing!
Task 5: 384/3
To begin today's lesson, I introduced the goal: I can find the factors for numbers under 20. I explained: Before we can find the factors, we need to learn more about what a factor actually is!
Factors Versus Multiples
I passed out Factors vs Multiples to each group of 2-3 students. I have students strategically placed in desk groups for easy pairing in all subject areas! I wanted students to investigate the difference between factors and multiples. Instead of telling students the difference between these mathematical terms, I wanted them to discover the meanings for themselves!
Next, I asked students divide their white boards in half. On either side of their boards, students wrote the following questions: What is a factor? What is a muliple?. Students excitedly began examining the factors and multiples for twelve. I loved listening to the conversations between students: Great Conversations!. I took every opportunity I had to Encourage Precision (Math Practice 6).
Then, we discussed the student's definitions of factors and multiples. At times, students would say, "We find factors by adding or multiplying." I would encourage more exact definitions by asking: Do you add or multiply the 3 and the 4 to get to 12? Students edited their own definitions as our conversation continued.
Now that students were given the time to discover the difference between factors and multiples on their own, I knew this was a great time to introduce and discuss the following vocabulary posters: Multiplication, Factors, Factor Pairs, and Multiples. After each new vocabulary word, I asked students to turn and talk: Explain the difference between a product and a sum or... Explain the difference between a factor and a factor pair or... Does 12 have more factors or more multiples? 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.
Multiples & Factors Song
This was a great moment in the lesson to introduce the Factors & Multiples Song (sang to I've Been Working on the Railroad). All of the writing on the right hand side of the poster was added as we sang each verse of the song. The goal was to add meaning and build connections. For my visual learners, I drew a tree. I explained how I've always imagined the factors as roots and the millions of multiples as the countless leaves on each branch. I also color-coded the song: factors (brown roots) and multiples (green leaves).
I then made another analogy and drew one more visual: Factors are a few! They fit on a horseshoe! Multiples are mounting! It's like skip counting. The students loved the analogies and they helped turn two mathematical terms into friendly, understandable concepts.
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 made a Baker's Cap using sentence strips and a white garbage bag. I set out Mini Cupcakes and showed students the Cupcake Trays I had made out of cardboard and aluminum foil. I then explained: As a baker, factors really come in handy when determining how to package my cupcakes. For example, if I have 7 cupcakes, the only way that I can make a rectangular arrangement is by putting all 7 cupcakes in one row (I demonstrated this as I spoke.). I know this because the only factors for 7 are 7 and 1. Turn and talk: Why would you want to package cupcakes in rectangular arrangements (arrays)? After a minute, I then explained and demonstrated: Every square inch of packaging costs me money, so do you think it's important to make sure the cupcakes take up all the space within a container? Are there any other reasons why rectangular arrangements are important when packaging cupcakes? A student also pointed out that the cupcakes could tip over and roll around!
Arranging 12 Cupcakes
I showed students twelve cupcakes and explained: I need help finding all the rectangular arrangements for these 12 cupcakes. Then I can make some decisions as baker on how I'd like to package and sell these cupcakes! I asked team leaders to pass out colored tiles to each student. I also passed out Factor Chart A so that students could begin documenting the factors for each number, 1-20. Here's what a student's desk looked like at this point: Student's Desk. You'll notice that I had students write "U-Turn" at the top of their whiteboards. I decided to wait until after this activity to introduce the "U-Turn" method. Instead, students made a list of "Factor Pairs for 12."
I also modeled student thinking on the board using magnetic unifix cubes:
After giving students time to find rectangular arrangements for twelve, I asked students to share their ideas. One student said, "We can make a 2 x 6." I modeled Modeled 2 x 6 on the board and asked a student to fill the 2x6 Tray. After the tray was filled, I walked around the room, showing all the students the tray. Students were eager to see! We then moved on to discuss and model the other arrangements for 12. It didn't take long for students to Model all three Arrangements and demonstrate their understanding of Factor Pairs for 12. Each time we discussed a new array, students excitedly volunteered to either model their thinking on the board (Student Modeling 12 x 1.) or to arrange cupcakes in the trays: 3 x 4 Tray and 12 x 1 Tray.
Teaching the U-Turn Method
Now that students truly understood how to find all the factor pairs for 12 using colored tiles, I wanted to introduce another tool to help them find factors, the U-Turn method. I asked students to come up to the front carpet with their white boards. I showed students about a 30-second portion of the following video clip so that they would all know what a U-Turn is in real life:
I didn't want to waste a moment so I immediately began teaching after showing this video clip!
1. Draw a t-chart. I used a "t-chart" reference as I have seen students get this method mixed up with the in and out box method. I wanted to make as clear of a distinction as possible.
2. Write the target number on top of the line.
3. Rule number one is... always write down 1 first. (I'll ask students to explain the importance of this rule later on when they've developed a deeper understanding of factors.)
4. Ask yourself: How many times does 1 go into 12? Or, in other words: What times 1 is 12? Students said, "12!" Okay! Perfect! Write 12 in the right column, across from the one.
We continued on, writing 2 (left column) ... and then 6 (right column) and then 3 (left column) .... and then 4 (right column). I counted down the left side: 1... 2... 3.... What number should we try next? Students responded, "Four! But we already have 4!" I said, Exactly! And whenever you get to a number that you've already written down... that's when you get to take a... U-TURN! I taped a picture of a U-Turn sign on the poster.
To practice the U-Turn method a few more times, we discussed and modeled the U-Turn for 24 (composite number) and the U-Turn for 7 (prime number). With each U-Turn method, I always asked: And what is rule number one? "Always write down 1 first!"
Altogether, we discussed and modeled our thinking using colored tiles and the U-turn method while finding the factors for all the numbers in the first row (1-10) of the Factor Chart. I Modeled U-Turns on the Front Board while Students Modeled U-Turns on White Boards. Step-by-step, I Recorded Factor Pairs on the Board.while students completed their own charts.
While completing the first row, two students amazingly came up with conjectures without any prompting! I had originally planned on including a list of conjectures in today's lesson, but making this list had slipped my mind! For the first Conjecture 1, the student discovered a counter example and decided that he wanted to make a few changes. He also decided that he need to continue testing the "truth" of his conjecture! Another student was also inspired to create a Conjecture 2. In this video, he also changes his conjecture based upon the evidence that he collected! These are prime examples of Math Practice 3: Construct Viable Arguments!
For student practice time, students worked in groups of 2-3.
Students continued using the U-turn method on their white boards while finding the factors for all the numbers in the second row (11-20) of the Factor Chart. I loved watching students immediately begin filling in 1 x 11.... 1 x 12... 1 x 13 as I knew they were using repeated reasoning (Math Practice 8).
During this time, I conferenced with all students and monitored student understanding by making observations and asking questions:
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.