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: 48/6
For the first task, students decomposed 48 in a variety of ways. For example, some students decomposed 48 into four twelves: 48:6 = 4 x 12:6. Other students decomposed the 48 into two twenty-fours: 48:6 = 24 x 2:6
Task 2: 12/6
For the next task, some students made a connection between task 1 and task 2: Connecting Tasks. Here, a student showed 12:6 = 6 x 2:6. Sometimes I wonder if there is much for my students to gain from such simple problems as 12/6. Then, students like this one, 12:6 = 3x4:6., prove me wrong! I loved how she came up with 4 x 1/2 = 2! We definitely had to share this with the rest of the class!
Task 3: 60/6
Task 4: 600/6
During this task, we didn't take the time to model as most students automatically knew that 600/6 = 100.
Task 5: 648/6
Throughout every number talk, I continually model student thinking on the board to inspire other students. This also requires students to use math words to explain their thinking instead of relying on a model to represent the math. As students solved each task, I wrote the answers on the board to encourage students to use prior tasks to solve the more complex tasks: Listed Tasks.
For this lesson, students will be building Multiple Towers to compare the multiples of single-digit numbers. Eventually, students will line up all of their towers on the board to observe patterns (Students Placed Multiple Towers on the Board). Even though the standards don't specifically state the importance of finding common multiples, it is important for students to analyze multiples on a deeper level than just identification. Also, this lesson will engage students in Math Practice 7: Look for and make use of structure. In addition, if students understand that two and four have common multiples, this builds the foundation for understanding the distributive property: 4 x 5 = 2 (2 x 5).
To begin today's lesson, I introduced the goal: I can find common multiples for single-digit numbers. I asked: Does anyone remember what common means? Reflecting upon a previous lesson on common factors, students responded, "Shared!" I explained: Today, we will be finding common multiples (under 100) between two numbers, but first, I'd like for you to investigate common multiples with unifix cubes!
Unifix Cubes Activity
Today, we are going to make Multiple Towers! I'd like for each group of two-three students to create a multiple tower for a single-digit number. For example, if you were making a Multiple Tower for 2, you might take two orange cubes and place them together with two green. Then you might add on two blue. Your goal today is to get as close to 50 cubes without going over. Anytime I get out the unifix cubes, students light up! I had one student come running up to me and say, "I love this! Can we do this more often?"
I assigned each group a number between 2 and 10. Even though the focus of our lesson today was on single-digit numbers, I wanted to include 10 as it is a benchmark number that helps students make sense other numbers. Here, students are building a Multiple Tower for 2.
Looking for Patterns
Next, Students Placed Multiple Towers on the Board and we all gathered on the carpet to begin Looking for Patterns. I truly wanted student to engage in Math Practice 7: Look for and Make use of Structure. During this time, I also wanted to encourage students to construct viable arguments (Math Practice 3).
What happened next was just wonderful! One by one students went to the board and explained their observations. At first, students explained patterns with colors, "It goes red, orange, blue, red orange, blue." The color of the cubes actually wasn't what I was looking for students to point out. However, this is a typical "beginning" observation. Often students begin by pointing out the most noticeable (and often not math-related) patterns first. Then, they build up to observing on a deeper, mathematical level.
Then, students began to see mathematical patterns: Most vs. Least Multiples! Each time a student shared, another student became inspired! Even students that struggle with math were going up to the board!
Another student then pointed out how It's Like a Line Graph. Then she explains a conjecture, "The smaller the number, the more multiples there are under a given number."
Here, the students begin Discussing Divisibility. I loved how students pointed out that 54 is the closet multiple of nine to 50, but 45 is the closest multiple of nine under 50.
Next a student pointed out that the Factors of 10 are Factors of 50. This was such great observation as students were beginning to look at common factors and multiples!
This student was then inspired to begin Comparing Multiples of 5 & 10. She points out that there are 12 groups of 4 in 48 and 6 groups of 8 in 48. Then, she explains that the same pattern works with the multiples of 5 and 10.
Finally, to connect this activity with today's goal, I decided to Rearrange the Multiple Towers. This student explained the connections between the multiple towers perfectly. I knew that this was a great time for students to begin finding common multiples using their Venn Diagrams.
Then, I reviewed: Remember, a Venn Diagram is used to compare two topics. We have already used this Venn Diagram to compare the factors of two numbers. Today, we are going to compare the multiples of two numbers. Can anyone tell me where the common multiples will go in the diagram? Students responded, "In the middle, where the circles overlap!"
Counting on the Number Line
I created a Venn Diagram on the Board. At first I placed 2 on one side and asked students to count by twos up to 100 on their number lines: Counting by 2s. Next, I placed 6 on the other side of the Venn Diagram on the board. At this point, students counted by 6s up to 100: Counting by 6s. Using this number line model, I wanted students to see how multiples of 2 overlap multiples of 6.
At this point, students completed the Venn Diagram for 2 & 6.. As a class, we completed the Venn Diagram on the board. At first students grappled with the idea that there aren't any multiples of 6 that aren't also multiples of 2. After conversing as a class, this idea made more sense to students.
Comparing the Multiples of 3 and 7
We then completed the same process in order to compare the multiples of 3 and 7. Here, Counting by 3s & 7s., students observed how the multiples of 3 and 7 overlap on the number line. Then, with their partners, students constructed a Venn Diagram for 3s & 7s..
Originally, I had planned for students to also compare the multiples of 4 and 8 as well as 5 and 9. However, the rich conversations we had as a class were more important that completing more tasks!
To bring closure to this lesson, I celebrated students who were on task and meeting expectations. Students then cleaned up and returned their math materials.