For example, can 26 be written as the sum of 4 consecutive numbers?

To use Solange’s method, we would add 2 to 26 and then divide by 4, which gives us 7. Because it goes in evenly, it passes the test and so we know 26 can in fact be written as the sum of 4 consecutive numbers. But what is that 7? Going further, the consecutive numbers that add up to 26 are 5, 6, 7, and 8. Solange’s method gave us the third number in the sequence. We did the same thing with 46 and again, the result was the 3rd consecutive number in the sequence – 10, 11, 12, 13.

Our first conjecture was that Solange’s method would give us the number in the sequence that was one less than the number of consecutive numbers we were looking for. While testing if numbers could be written as the sum of 4 consecutive numbers, the number that came out of Solange’s expression was the 3rd. So we thought maybe if we were looking for numbers that can be written as the sum of six consecutive numbers, maybe her method would give us the 5th number. So we tried it out with 27. 27 plus 3 divided by 6 is 5, which is the fourth number in the sequence (2+3+4+5+6+7 = 27). We tried again with 39, and got 7, which is the fourth consecutive number in the sequence (4+5+6+7+8+9=39.)

So what is the pattern? Will Solange’s expression always give us one of the consecutive numbers? And which number in the sequence will it give us?

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]]>CCR.Math.Practice.MP7 Look for and make use of structure.

In terms of this practice, I would emphasize the area model and the partial product method. I’m not really sure how to bring the others in, though I think the Egyptian and Russian Peasant methods are fascinating and certainly relate to binary numbers, distribution, in/out tables…

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“Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.”

]]>CAMI folks, I highly recommend the webinar Amy and Connie Rivera did last year on area models for LINCS:

Uncovering Coherence in Area Models: https://www.youtube.com/watch?v=bYvmMcBSDlk&feature=youtu.be

]]>I do think Amy’s comment below is important for us to think about in the classroom. Which of these models are most useful for our students? I agree that arrays and the area model is the place to start, especially since it can help us lead into a discussion of the distributive property, eventually multiplying polynomials.

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