For our June meeting, we were lucky to have Usha Kotelawala, Director of Math Education for CUNY’s LINCT to Success, as a guest presenter. Usha started the meeting by talking a little about her thought process in choosing today’s problem. In discussing CAMI with Usha, Eric had raised the issue of how to order problems through a semester, so that the mathematics is sequenced and scaffolded for students and students learn through problem-solving. In response to this question, Usha brought us a problem she recommends as the first in a sequence on algebraic reasoning. The problem, an exploration of consecutive numbers, is from Fostering Algebraic Thinking, by Mark Driscoll (excerpt attached above).
Usha (in italics): So, the word, “consecutive.” What does it mean?
One after the other. 1, 2, 3.
Can you give me another three consecutive numbers?
2, 4, 6.
Ah, these are consecutive even numbers, but for our purposes here, these are not consecutive numbers.
10, 11, 12.
Can someone give us five consecutive numbers?
14, 15, 16, 17, 18.
Usha handed out the first page of the worksheet on consecutive numbers (attached above) and asked us to work individually for a while. The worksheet contains some samples to get you started. One and 2 are two consecutive numbers that sum to 3. Three consecutive numbers that add to 6 are 1, 2 and 3.
A few questions/observations came up:
- Are negative numbers allowed? I’m not going to answer that right now…
- Do they have to be whole numbers?
- Most of the spaces stay blank.
Usha stopped us after about 5 minutes. Are there any patterns that you’re noticing? Don’t share them yet, but note what they are.
Usha handed out the 2nd page and then the 3rd page of the worksheet as we continued to work independently.
Usha then gave us a few minutes to see if we could express the patterns we saw as a rule.
Let’s take a commercial break.
I want to interrupt with a story from the history of mathematics. Do you remember one of those students who always had the answer and kept raising their hands? This is a story about a young boy named Carl Gauss who lived in Germany. His teacher got exasperated with him and wanted to give him something to keep him busy, so he asked him to add up the numbers 1 through 100. Gauss lined up the numbers and saw pairs of numbers that added to 101 (1 + 100, 2 + 99, etc.) and there were 50 pairs.
End commercial break.
Usha: Does Gauss’ idea help you add the numbers 1 through 6 more quickly?
Some no’s & some yes’s.
Usha: Just a side note: I am protective of quiet students. I think it’s important to include quiet, reflective time for everyone to try their own ideas. We also know that exciting ideas can come out of collaboration. How do we balance this as teachers?
I think what I’ll do is ask, how many people are ready to work with a partner? Who wants to keep working on their own? If you’re ready to work with a partner, move over to this side of the room. Everyone else can stay where they are and keep working on your own.
As people were looking for patterns on their own or in groups, Usha kept putting out questions, like:
- Look at the sums of three consecutive numbers. How can you predict if three consecutive numbers would add up to a number?
- 63. Are three consecutive numbers possible? No’s & yes’s. Think about it for a minute and we’ll come back together.
Usha asked the rest of the participants to find a partner and start working together.
Question from the group: In a series of consecutive numbers, is the middle number always the average?
Whole Group Discussion
What about the number 63? How many consecutive sums can you work out for 63?
- Two consecutive numbers: 31, 32
- Three consecutive numbers: 20, 21, 22
Betty said to see if 63 could be written as the sum of three consecutive numbers she drew three lines:
___ + ___ + ___ = 63
She ignored the three and divided the 60 into three 20s. then she saw that she could write the sum of 63 as 20+21+22.
What about 84? What would Betty’s method look like if we tried it with 84?
Usha: Let’s look at 63 together. Can it be made of 4, 5, 6 or 7 consecutive numbers?
Stephanie’s explanation for how she found 7 consecutive numbers adding up to 63:
First method: I’m looking for the base number. The total is 63. 63 divided by 7 equals 9. Line up the numbers. Take 2 away from 9. Add 2 to 9. Take 1 away from 9. Add 1 to 9. Line up the numbers. 6, 7, 8, 9, 10, 11, 12.
n + (n +1) + (n +2) + (n + 3) + (n + 4) + (n + 5) + (n + 6)= 63
Add all the n’s to get 7n.
Add all the numbers to get 10.
7n + 21 = 63
7n = 42
n = 6 (So, 6 is the first number in the series: 6, 7, 8, 9, 10, 11, 12)
Stephanie’s second method added some clarity as to where the triangular numbers came from in Mark’s observations.
Mark used a series of equations to determine if 63 could be written as the sum of different numbers of consecutive numbers.
n + n +1 = 63
2n = 62
n = 31, n + 1 = 32
He followed a similar procedure for 3, 4 and 5 consecutive numbers. Then he noticed a pattern that allowed him to create the following expressions:
2 consecutive numbers –> 2n + 1
3 consecutive numbers –> 3n + 3
4 consecutive numbers –> 4n + 6
5 consecutive numbers –> 5n + 10
6 consecutive numbers –> 6n + 15
7 consecutive numbers –> 7n + 21
8 consecutive numbers –> 8n + 28
9 consecutive numbers –> 9n + 36
Mark pointed out that the constant in each expression is always a triangular number, one behind the number of consecutive numbers. For example, in the expression 4n + 6, 6 is the 3rd triangular number (1, 3, 6, 10…). In the expression 6n + 15, 15 is the 5th triangular number (1, 3, 6, 10, 15, 21…).
To figure out (1) if a number could be could be written as the sum of consecutive numbers and (2) what those numbers are, Mark used the following method:
Let’s say you wanted to know if there was a way to write 63 as the sum of 4 consecutive numbers. Create the expression as described above – so for 4 consecutive numbers, 4n + 6. Set that equal to 63 and solve.
4n + 6 = 63
4n = 57
Since 57 is not evenly divisible by 4, 63 can not be written as the sum of 4 consecutive numbers.
Let’s try 6.
6n + 15 = 63
6n = 48
48 can be divided evenly by 6. It gives us 8. 63 can be written as the sum of the 6 consecutive numbers beginning with 8 (8+9+10+11+12+13 = 63)
Solange noticed a pattern when it came to which numbers could be produced by the sums of consecutive numbers. For example, consider the numbers that can be written as the sum of two consecutive numbers: 1, 3, 5, 7… every second number. Now consider the numbers that can be written as the sum of three numbers: 3, 6, 9… every third number. Now, look at the numbers that can be written as the sum of five consecutive numbers: 5, 10, 15… every fifth number. So for all odd numbers, Solange realized she could determine if a number can be written as a sum of n consecutive numbers by testing to see if the desired number can be evenly divided by n. So for example, 63 can be written as the sum of seven consecutive numbers because 63 can be divided evenly by 7.
Solange noticed a slightly different pattern when it came to even numbers of consecutive numbers. It partly followed the same pattern as the odds, which is to say that the numbers that can be written as the sums of four consecutive numbers are 2, 6, 10, 14 (i.e. every 4th number). Similarly, the numbers that can be written as the sums of six consecutive numbers are 3, 9, 15… (i.e. every 6th number). The difference from the odd is the starting point. The first number that can be written as the sum of three consecutive numbers is 3. The first number that can be written as the sum of five consecutive numbers is 5. Not so with the evens. The first number that can be written as the sum of two consecutive numbers is 1. The first number that can be written as the sum of four consecutive numbers is 2. The first number that can be written as the sum of six consecutive numbers is 3. Solange saw that, for evens, she needed to adjust her method to take into account the “starting number”. She set up the following expressions to test if you want to know if a particular number can be written as the sum of an even number of consecutive numbers:
- To see if a number can be written as the sum of two consecutive numbers, use x+1 over 2.
- To see if a number can be written as the sum of four consecutive numbers, use x+2 over 4.
- To see if a number can be written as the sum of six consecutive numbers, use x+3 over 6.
When x = the number you are testing
Throughout the meeting, Usha added extension questions:
- Can you find 27 consecutive numbers that add to 63?
- Can you think of 4 consecutive numbers that add to 2? (This forced us to use negative numbers.)
- Can you think of a general formula that would work for any sum and any number of consecutive numbers?
Final Comments on Teaching This Problem
In closing the meeting, Usha asked if this problem would work in our classrooms:
- In order to get started, students don’t need to know any algebra. They just have to add numbers and look for patterns.
- This problem is accessible to students at any level and students can really engage with the math of the problem at their own pace.
- To simplify the task, Usha suggested that the worksheet could be modified by removing some of the columns on the right if we think it would be too intimidating for some of our classes.
In Attendance: Lucinda, Eric, Solange, Mark, Linda, Stephanie, Stephen, Betty,
Programs Represented: BMCC, CUNY LINCT to Success, CUNY Adult Literacy PD Team, LaGuardia Community College, ParentJobNet, YALP