I want my students to really focus on the difference between and factor and an multiple. Throughout the lesson I will gradually release different concepts to see where the students go with it.
I begin this lesson by explaining that multiples are the product of two factors. For example 12 is a multiple of 3 because 3 times 4 is 12. 12 is also a multiple of 4 because 3 times 4 is 12. 6 is a multiple of 2 because 2 times 3 is 6. 6 is also a multiple of 3 because 3 times 2 is 6. It is easy to find the multiples of numbers when using a multiplication chart. I display a large chart on the board for the students to see. I explain that is I want to know the multiples of 3, I look at the 3 and continue skip counting of going down the line. I model 3,6,9,12,15,18,21., etc. I ask students to give me the multiples of 5. 5,10,15,20,25,30,35,40.
In other words, a multiple is the product of any given number and any other whole number. I go on to explain that we are going to find the lowest common multiple in a given set of numbers. I model Let's look at 6 and 12. I am going to write the multiples for these numbers on the board
6: 6 12 18 24 30
12: 12 24 36
I explain that the least common multiple for those two numbers are 24.
To help struggling students determine the difference between a multiple and a factor, I use a multiplication chart. We use the chart to help us fill out the T-chart of what is the factor and what is the multiple. Multiplication chart.ppt T Chart explanation.ppt
In this lesson we will focus on the following Mathematical Practices:
MP.2. Reason abstractly and quantitatively.
MP.7. Look for and make use of structure.
I break students into pairs and give them t-charts to list their multiples under each side. I give them numbers such as 2 and 12... 5 and 6..... 9 and 12, and so on. I explain to students that I want them to place one number on the left side of the t-chart and the other number on the right side. They will continue to find the multiples of those numbers until they find a common multiple. I ask students to explain again what a multiple is. A number that can be made by multiplying a number by another number. What is a common multiple? Numbers that are multiples of both pairs. It is the lowest number that both numbers have in common. The students worked well with finding the common multiples. I ask students how they determine which number is a factor/multiple.
Students noted that writing the multiplication expression for numbers with several factors and for numbers with a few factors help them make conjectures about the numbers. Basically they say the repeated pattern in some of the numbers they explored.
I pose word problems for students such as Mark goes swimming every 3 days. Tammy goes swimming every 4 days. What days will they both be at the pool?
We write the multiple for 2 and the multiples for 3
What are the multiples that these two have in common? 6, 12, and 18
Ok, so they will meet at the pool on days 6, 12, and 18. What is the soonest day that they will meet? Day 6.
I change up the numbers to see if students grasp the concept. We continue this until I am sure that this lesson is mastered. After that students completed a multiple assessment on their own. As they were working, I circle the room to see what they are thinking. For instance, I ask students to think of another way to determine the multiple ; how do you know? can you illustrate it?