What students already know: Students of this problem were able to solve the questions- How many 6 packs are in 156 sodas and how many of each of the 6 flavors were able to fit into the machine- by using visuals, patterns, and the beginnings of multiplication and division.
1. Mimicking the "Reasoning about Multiplication and Division" video, I would advance students knowledge on multiplication and division by writing an exhaustive list of the products of 6, beginning with 0 and ending at 156. I would have the class participate with me, starting at 0 and adding 6, then starting at 6, adding another 6 and so on until reaching 156.
0 6 12 18 24 etc..
Objective: By consecutively adding 6, the students, without yet realizing it, are listing the multiples of six. Finding multiples through addition allows students to utilize math that they are already confident in, resulting in them clearly understanding where the numbers listed come from.
2. Once reaching 156, I would ask the students what we could call the exhaustive list of numbers.
After ideas were discussed, I would then have the students use the "jump" method found in both "Reasoning about Multiplication and Division" and "Discover Number Patterns with Skip Counting" to find how many jumps of 6 it took to get to 18. When starting from 0, it takes 3 jumps to get to 18. I then would progress to ask the students what 3x6 was?
Objective: Utilize the method of skip counting and then connecting the number of skips to a multiplication problem to get the students brains stirring and thinking about how multiplication connects to the amount of skips taken to get to the desired number.
3. Allow students to see that the 3 skips it takes to get to 18 can be found by finding 6x3.
From there I would go to explain the term multiple.
Multiple: Product of a given number and another whole number
Our example: 6- so any product resulting from 6 and another whole number will be a multiple of 6.
Ask students again: Given this definiton of multiples, what can we call the list of numbers that we found?
Objective: Explain what a multiple is and connect it to the previous work students had done.
4. Explain to the students that the list we wrote out to 156 were all multiples of 6.
6 is the multiple of 6x1
12 is the multiple of 6x2
18 is the multiple of 6x3
and so on...
Ask students- "156 is the multiple of 6 and what number?"
Answer: 26
Students can find this multiplying 6x26 or counting how many "jumps" of 6 it took to get to 156 from our exhaustive list.
Objective: This explanantion allows students to have a better grasp on the concept of multiples and what multiples of a number are.
7. Ask the students the question- How many 6 packs are in 156 sodas?
The students had already found the answer to be 26. Now with their knowledge of multiples, what could we say about the correlation between 6, 26, and 156.
Answer: A multiple of 6 is 156. 156 is the multiple of 6 and 26. Therefore, we know that if there are 156 sodas arranged into 6 packs then there must be 26 packs because when listing multiples of 6, 156 is only achieved by multiplying 6 and 26.
8. Following this little lesson on multiples I would interact with the children and begin asking them questions surrounding the concept of multiples.
Ex:
Can you give me three numbers that are multiples of 2?
3, 6, 9, 12 are all multiples of what number?
Is the number 3 a multiple of 2?
1. Given that 156 is a large number, I would branch off from the soda machine question for a little bit to initially begin the introduction of factors.
Write the number 4 on the board.
Ask- "What numbers can I multiply to get to 4?
On the white board, write out the students answers in factor tree form.
Objective: Expose students to a factor tree.
2. Explain to the students what a factor is.
Factor- Two numbers that multiply together to produce the desired/stated number.
Explain that 1, 2 and 4 are factors of 4 because:
Next, I would work through a larger factor tree with the students, lets say 20. Ask the students what we can multiply to get to 20.
1x20, 2x10, 5x4
Then ask the students, given the factor tree (not shown here), what the factors of 20 are?
Answer: 1, 2, 4, 5, 10, 20
Objective: Start with simple numbers to introduce the students to the concept of factors.
3. Going back to one of our soda machine questions- How many 6 packs are in 156 sodas?
We can find the answer to this by making a factor tree for 156.
Write out 156 and go through every multiple with the students to find the factors of 156. Due to 156 being a large number, this will take some time without a calulator and participation with the students.
Once writing out the factor tree for 156, how can we get the answer asked above from this factor tree?
Answer: We can get the answer by looking at what multiplies by 6 to get us to 156.
Ask the students: So, what number when multiplied by 6 will give us the 156 sodas?
Answer: 26
Objective: Work with the students to arrive at the factors for 156. By working with the students on the board, they will be able to visually see and understand how we are getting the factors. By implementing the question, they can see how the factor tree can be used in problems in the classroom.
4. Tie together the concept of factors with the lesson just given on multiples.
Ask the students: "If factors are numbers that are multiplied together to get to 156, is 156 a multiple of all of its factors?
Answer: Yes
Explain: Factors are two numbers that multiply together and produce 156.
Therefore, 156 is a multiple of all of its factors because (show students this work visually):