# Write an essay about the sums of consecutive integers.

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Question description

Here are examples of sums of consecutive integers.

• 3 + 4 = 7
• 4 + 5 + 6 + 7 = 22
• 2 + 3 + 4 = 9

In the first, the number 7 is written as the sum of two consecutive integers. In the second, 22 is written as the sum of four consecutive integers. In the last, 9 is written as the sum of 3 consecutive integers.

Initial exploration:

1. Try to write each number from 1 to 35 as the sum of consecutive integers. Can you find more than one way for some of the numbers?
2. Explore what numbers can and cannot be made by sums of consecutive integers!
Record 2-3 discoveries that you can share with the class.

For your write-up of this Anchor Task, record and explain TWO discoveries you’ve made about the sums of consecutive integers. You should use some examples to illustrate each discovery and then make a convincing argument about why you think the discovery is true. You may draw upon the ideas of your classmates.

For this anchor task, engage in one or more of the following additional challenges.

1. Record and justify a third discovery about the sums of consecutive integers.
2. Work backwards! Without doing any calculations, predict whether the following numbers can be made with 2 consecutive integers, 3 consecutive integers, 4 consecutive integers, etc. Explain your predictions.
• 45
• 57
• 62
• 75
• 80
3. Write a shortcut for writing the following numbers as the sum of two or more consecutive numbers. Describe the shortcuts you created and tell how you used them to write each of the numbers below as sums of consecutive integers.
• 45
• 57
• 62
• 75
• 80

Turn in a hard copy of (1) your solution (two discoveries about consecutive sums from your individual thinking and your time sharing in small groups and whole-class) and (2) your reflection at the start of class on Tuesday, 5/2.

You will write a 1-1.5 page (typed, 1.5-spaced) reflection on your mathematical learning both from your work on the problem and from our class discussion of it. You may insert photos of solutions and models into your reflection. To fully meet expectations your reflection must include the following:

• Clear thinking about the mathematics of this task. What mathematical ideas do you understand better or differently after having solved and discussed this problem? How did your understanding change as you solved and discussed the problem? (Note that this reflection isn’t a restatement of how you solved the problem, but a reflection on what you thought about over the course of the problem. That is, you’re not writing “What I did to solve this problem was….” Instead, you’re writing something like: “This problem was about … And after working on it, I am thinking a bit differently about…” And if you understood the problem solidly from the beginning, this might be an opportunity to reflect upon the mathematical ideas that you heard your peers considering as they solved the problem or to consider how you might adjust or extend the problem to start to consider other mathematical ideas.)
• Specific reference to the strategy of at least one of your peers and how it connects to your own thinking. (This might be a place to include photos or drawings of strategies or models.) Which solutions, models, or explanations shared in the discussion struck you as particularly useful or interesting? Why? You will explain the details of at least one of your peers’ thinking and speak to the specific ways it was similar to or different from your own thinking. (That is, instead of saying something like, “I thought that Maya’s strategy was really cool,” and leave it at that, you might say, “I thought that Maya’s strategy was really efficient because what she did was... That was different from how I thought about the problem because… But her strategy works because…”).
• Engagement with some of the following questions:

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Daily assignment—Consecutive sums Consecutive numbers are the numbers that follow each other in order, without gaps, and from small number to big number. For example, 1, 2, 3, and 4 are consecutive numbers. The consecutive sum is adding all the consecutive numbers together. The first discovery of consecutive sum is that the every odd number start from 3 is the consecutive sum of two consecutive numbers. The reason of starting from 3 is 1+2=3. I found that the odd number is the consecutive number because 3=1+2, 5=2+3, and 7=3+4. This can be proved below. a+a=2a a+(a+1)=2a+1 a and a+1 are consecutive numbers. 2a is an even number. Then 2a+1 is an odd number. The second discovery is that the sum of three consecutive numbers are 3’s multiple, starting from 6 because 1+2+3=6. We can keep counting to prove it. 2+3+4=9, 3+4+5=12, and 4+5+6=15. We can find that the sum is always equal to the 3 times the second number from the three consecutive numbers. (a-1)+a+(a+1)=3a a+a+a=3a (a-1)+(a+1)+a=a+a+a=3a

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