In this lesson I want my students to gain familiarity with factors and multiples, and analyze and generate patterns. Since, Common Core requires students to cluster some domains together; I want to start by making a list of multiples of whole numbers.
I invite students to the carpet to discuss and explain how they should be thinking. I write the number 3 on the board. Can someone tell me the multiples of 3? 1 and 3 are both multiples of 3. Ok! Which of the numbers in your list are multiples of 6? 1 and 3 are both multiples of six. What pattern do you see in where the multiples of 6 appear in the list? Some students say 3 is a multiple of both 3 and 6. Exactly! To move them a little deeper into understanding, I say which numbers in the list so far are multiples of 7? Students pause, and are unable to answer. Can any of you predict when multiples of 7 will appear in the list of multiples of 3? Explain your reasoning. This task will help students notice that unlike 6, there is no factor of 3 in 7 and so not every multiple of 7 has a factor of 3: in order to be a multiple of both 3 and 7, a number must be a multiple of 21. (MP7, 8) look for and make use of structure and express regularity in repeated reasoning.
I give students a sheet of scratch paper so they can analyze a bit on their own. I ask them to write the first ten multiples of 3.
Then I ask them to write the first ten multiples of 6 on their paper
I ask them combine the multiples of both 3 and 6
I ask them to highlight the multiples of 6
I ask them to observe the multiples of both 3 and 6 to see if they could notice any pattern. Some students notice that every other number in the sequence is a multiple of 6. I wanted them to understand why, so I create a picture that would allow them to visually see that 2 groups of three make 1 group of six. I repeat this activity using other number pairs. I want to make sure students are understanding the task before moving them deeper into the lesson.
In this lesson we will be focusing on the following Mathematical Practices:
MP.2. Reason abstractly and quantitatively.
MP.7. Look for and make use of structure.
Material:Factor Illustration of 3 and 6.docx
In this portion of the lesson I want to work with my students a little bit more. To make sure that students stay focused on the purpose of this lesson, I use the same numbers used in the warm-up part of this lesson. I think it is wise that we spend some more time discovering how and why factor and multiples are related. To do this, I refer back to the illustration of 3 and 6. I remind students that it takes 2 groups of three make 1 group of six. I pass out counters, so students can take about five minutes or so to build groups of threes. As students are working, I ask how many groups of threes do it takes to make 1 group of six. Students seem to understand a bit better when they are actually creating the groups of threes on their own.
I say, I notice you guys are using an even number of threes, and grouping them with another group of threes to make sixes. I wonder what would happen if you build five groups of threes. I give the students a moment or two to build. OK! How many sixes can you make with 5 groups of threes? As students are trying to figure this out, some notice that they have a group of threes left over. I point back to the illustration and say, an odd number of threes leaves a purple group which does not match up with a white group (or vice versa).
Now! Remember we discussed how seven is related to 3, but not 6 earlier! The only number in the list that is a multiple of 7 is 21 which is 7 X 3. If we were to write the list of multiples of 7 I wonder how many would be multiples of 3. I ask students to help me write the multiples of 7
Then I ask them to extend the list of multiples of 3 we did earlier, so that we can compare and determine the pattern.
I give students a moment or two to analyze the pattern to see if they can see that the first four multiples of 7 appear in the list of multiples of 3 are 21, 42, 63, and 84. As students are analyzing the pattern, I encourage them to turn and talk to their neighbor about what they notice. I use students responses to determine if they are ready to move deeper into the lesson.
In this portion of the lesson, I want students to have a little time using the given concepts on their own. I ask students to move with their assigned partner. I remind students of the purpose of this lesson. For instance: make a list of multiples, which of the numbers in your list are multiples of the number?, What pattern do you see in where the multiples of the number appear in the list?, Which numbers in the list are multiples of the other numbers? Can you predict when multiples of the number will appear in the list of multiples of 3? Explain your reasoning. Some students ease through the question and explanation, but some students seem to struggle a bit. I encourage them to use illustrations to help them determine whether or not certain numbers are factors of the given numbers.
I set the timer for 10 minutes so that students can have time to investigate properties of the given numbers with their partner. After that time is up, I ask students to pair up with another paired group to discuss similarities and differences in their work. I circle the room to see what students are thinking. I see that some students are looking for patterns with the illustrations they created, and other students are noticing the repeated pattern in discovering multiples of whole numbers. I may chime in to see if I can extend their reasoning. I do this by repeating some of the questions used earlier in the lesson. Hopefully after scaffolding them through with questions, students be able to automatically think their way through.
I use students responses to determine if students need additional support.