Making Conjectures about the Multiples of 0-4
Lesson 7 of 11
Objective: SWBAT determine if a number is a multiple of a given number.
Today's Number Talk
For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation.pdf. For this Number Talk, I am encouraging students to represent their thinking using an array model.
Task 1: 18/6
For the first task, even though students knew 18/3 = 6, many students found creative ways of representing their thinking using an array. Some students decomposed the 18 into two 9s: 18:6 = 2(9:3). Others decomposed the 18 into three 6s: 18:6 = 3(6:6).
Task 2: 36/6
Task 3: 72/6
During this task, some students duplicated the array for 36/6. Some students also divided 18 by 6 four times: 72:6 = 4(18:6).
Task 4: 672/6
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.
To begin today's lesson, I introduced the goal: I can determine if a number is a multiple of a given number. I explained: Over the past few lessons, you have learned how to identify factors. Today we will begin working with multiples!
Reviewing Multiples & Factors
To review Multiples, I pointed out that the word, multiple, is the root word for multiplication. I then wrote 2 x 3 = 6 on the board and said: Two times three equals six, right? What do we call the 2 and the 3. Students hesitated for a moment and then said, "Factors!" (This goes to show that constantly reviewing vocabulary is important.) I then asked: What is the 6 called? Again, with some time, students said, "Product!" I continued: Yes, you're right. The answer to a multiplication problem is called a product. However, we can also call the 6 a multiple. It is a multiple of 2 and a multiple of 3: Product & Multiple..
We then sang and discussed our Factors & Multiples Song to help solidify the difference between factors and multiples. I then reiterated: Multiples... millions! Factors... few!
Next, I passed out the Multiples Chart to each student. I reminded students that "multiples are mounting... just like skip counting." The students and I found the first 10 multiples of 1 and 2 together. Then students were ready to complete the chart on their own. I figured there was no better way to understand multiples than skip-counting! I Conferenced with Students during this time, but was only able to connect with a few as students finished this task very quickly (10 minutes).
In order to provide students with the opportunity to color hundreds charts and analyze patterns with multiples, I created a Google Presentation using Google Drive Documents called Multiples Presentation prior to the lesson. Here are specific directions explaining How to Create a Google Presentation for Student Practice. Next, I shared this presentation with students using their student Google email accounts. Students then copied the shared presentation and saved it in their math folders under the Google Drive. We've done this many times so it only takes a couple minutes!
Once all students were successful at copying the presentation and making it their own, we reviewed today's goal on Slide 1, Goal: I can determine if a number is a multiple of a given number. We highlighted and discussed important words in the goal: "determine" and "multiple."
On the next slide, we reviewed the meaning of a conjecture: Slide 2, What is a Conjecture? We discussed that a conjecture is a statement that we believe to be true based upon current evidence and how conjectures can be proven true or disproven using new evidence. I included conjectures in this lesson to provide students with the opportunity to practice Math Practice 3: Construct viable arguments and critique the reasoning of others.
Guided Practice: Multiples of 0
We moved on to the next slide, Slide 3, Multiples of 0 . I asked students to highlight all of the multiples of 0. One student immediately said, "You can't do it!" A discussion erupted between students without any prompting on my part. One student said, "Zero times any number is 0!" Another student said, "Yeah, there aren't any multiples of 0." And another student said, "Mrs. Nelson, we can't highlight any numbers!"
After agreeing that we shouldn't highlight any numbers, we moved on to the next page, Slide 4, Conjecture for Multiples of 0. I asked, What is a conjecture that we can make about multiples of zero? One student said, "No number is a multiple of zero." I then explained: Today, when you make a conjecture, I also want you to include evidence. So who can provide evidence for this conjecture? Another student said, "For example, 0 x 1 = 0."
Guided Practice: Multiples of 1
We then went to Slide 5, Multiples of 1. Students said, "We need to highlight all of them!" I asked: Are you sure? Every single number in the hundreds grid is a multiple of 1? Whenever I ask this question, inevitably at least one student says, "No... wait..." as they take one more look. As a class, students decided that all multiples of 1 should be highlighted.
Next, we moved on to Slide 6, Conjectures for Multiples of 1. One student said, "All numbers are multiples of 1." Another student said, "All numbers but zero." I asked students to turn and talk about the multiples of 1. After some time, a student offered the following conjecture and example: Every number is a multiple of 1, except for 0. For example 1 x 4 = 4. Anything times 0 equals 0.
Ready for Partner Practice
At this point, students were ready to continue on with partners.
To mix it up a bit today, I pulled glitter sticks (sticks with glitter and student names) to assign partners. At times, I would hold on to a student's stick until I pulled a partner that he/she would be successful with!
Students continued on to the next slide, Slide 7, where they found the multiples of 2 with their partners and then wrote a conjecture about the multiples of 2 on Slide 8. I asked students to work with their partners to complete slides 7-12 (multiples of 2-4).
During this time, I conferenced with each partnership to check on student understanding. I would also ask clarifying questions to push student thinking and to encourage precision (Math Practice 6).
Multiples of 2
For multiples of 2, students came up with a variety of conjectures. Here, a student explains how to quickly highlight all the multiples of 2. He explains that any number that has 2, 4, 6, 8 or 0 in the ones place is a multiple of 2: Even Numbers are Multipels of 2.
Multiples of 3
Next, students highlighted and analyzed the multiples of 3. Here, a student explains how Any Multiple of 6 is a Multiple of 3.
Multiples of 4
Here's an example of student work: Student Work Example.
Discussing & Testing the Conjectures for Multiples of 2
To bring closure to this lesson, we discussed the conjectures students had come up with for the multiples of 2. After each student shared his/her conjecture, I asked students to look at their own hundreds charts to determine the correctness of the shared conjecture. We also discussed which conjectures would be the most helpful with identifying multiples of 2.
I asked students to use the conjectures that we had just discussed to identify the multiples of 2 in the first section: Student Finding Multiples of 2.
Discussing & Testing the Conjectures for Multiples of 3
Then, we discussed student conjectures for the multiples of 3. Again, after discussing conjectures, students used these conjectures to identify the multiples of 3 in the second task: Student Finding Multiples of 3.
Discussing & Testing the Conjectures for Multiples of 4
Finally, we discussed the student conjectures for the multiples of 4 and students applied these conjectures to the practice page in order to find the multiples of 4 in the third task: Student Finding Multiples of 4.
I was so proud of my students for investigating multiples and for applying their new understandings of multiples in this way!