Week 2 Math Task

Complete the Negative Space Task on p 223. Answer questions 1 – 3 on that page. Share your questions, thoughts, sketches, process, mistakes, or solutions in the discussion board.

It is challenging to post images to this discussion board.  One way that you can do this is to save your image as a file, post it to the documents tab in this group, and then refer to it in your discussion post.  Thanks for working with this challenge!

Comments

Ah, the crab claw visual pattern, my old friend. 

I've actually seen this pattern before, at a meeting of our math teachers' circle (NYC Community of Adult Math Instructors - CAMI) in Feb. 2015. We were exploring the problem in the context of problem-posing, and since that came up in some of last week's conversation I wanted to share a link to our notes from that meeting - Problem-Posing with Visual Patterns. The focus of that meeting was on what is could look like to let students generate and pursue their own questions when observing change in a visual pattern. We looked at the same sequence (also figures 2, 3 and 4). We all wrote our own questions, and one group (Solange and Eric) did focus their attention on the negative space and the earlier figures. That meeting ended up becoming part of a lesson in the CUNY HSE Math Curriculum Framework that Patricia mentioned. (The lesson is in Unit 8, on page 183. We used different a visual pattern though). If you're really interested, several of us actually wrote an article for the COABE journal on both CAMI in general and that one meeting in particular.

Anyway, it's been a while since I've looked at the pattern with my problem-solving brain, and I tried to look at it with fresh eyes. I could not figure out how to post a document, but I think I found a way to share my work. I am on Twitter (@mtrushkowsky) and I sent out a tweet with my work. I think that you can follow the link below to view it, even if you don't have a Twitter account.

By the way, is there anyone else in the group who is on Twitter? I've noticed a few of us have been posting using the hashtag #LINCSbookgroup. Even if you are not on Twitter, you can view any tweets that have that hashtag by clicking here

yours in productive struggle, 

Mark

 

 

 

 

 

 

I so loved this problem as I love working with expanding patterns. My first attempt was to do an in-out table just to check to see if the pattern was growing linearly (which I doubted given the shape). I was right it wasn't growing at a constant rate but it was increasing by odd numbers which told me this was going to involve a 2nd degree expression.  Then I went back to the pictures to see any patterns. I wrote what I saw as a numerical expression for each figure to see what repeated, etc.

 Figure 2 = 3 + 2^2 + 4

 Figure 3 = 4 + 3^2 + 5

Figure 4 = 5 + 4^2 + 6  

This pattern at least allowed me to find the 100th term I noticed the first numerical term was one more than the figure #, the middle term was the figure # squared and the last term was 2 more than the figure number. For 100 it would then be 101 + (100)^2 + 102 or 10,203                           

I could have written my algebraic expression but wanted to look at it a different way. I broke the shape into 3 rectangles, the top, the large rectangle to the left and the bottom right 2 which stayed constant. 

For figure 2:   3 + (3x2) + 2    

Figure 3:  4 + (4 x 3) + 2

Figure 4: 5 + (5x4) + 2

Then I noticed a common factor so I rewrote each expression

Figure 2: 3(1+2) + 2  or 3(3) + 2 = 3^2 + 2

Figure 3: 4(1+3) + 2 or 4(4) + 2 = 4^2 + 2

Figure 4: 5(1+4) + 2 or 5(5) + 2 = 5^2 + 2

For the nth term then (n+1)^2 + 2 or expanded n^2 + 3n + 2

Working back with the pictures to negative one was challenging but fun again noticing what was happening in the picture each time and also using the algebraic expression to check to see if I was right.

 

 

 

 

 

 

 

I am learning as I go here.  Members do not have access to post to the documents tab; I am not sure of any other way for members to post images that you create to this discussion.  Images are an important part of sharing our work in Math Tasks!  If you have an image that you'd like to post, please send it to me indicating that in the subject line and I will post it as a file to the documents tab.  Thanks for your patience! :)

I had the same experience with How close to 100. A group of teachers actually played the game in partners last week. It was interesting, but we all felt there was something missing from the rules. It would help to see someone playing it. Or we could figure out rules that would make it more interesting: Limit the number of turns each partner gets, or 3 strikes and you're out....

This is my first time reading Mathematical Mindsets from front to back, chapter by chapter. I've already read a lot of the book piecemeal and when I got to chapter five this past weekend to read it for our book discussion group, I saw that I had already worked through the Negative Space Task and come up with a function. I remember what jumped out at me when I first looked at this pattern was the perfect square in the middle of each figure that correlated with the figure number. Given that, I came up with x^2 + 2x + 3, which worked for the figures pictured. I'd also thought about what Figure # -1 would look like, trying to work backwards, and I came up with a picture of one box sitting on top of another. And I reckoned that Figure 100 would look like 100^2 with 101 squares on the top of the perfect square and 102 squares on the bottom of the perfect square. A total of 10,203 squares. Assuming the function I came up with is accurate or at least one way the pattern could be represented as a function.

What strikes me as I look at it now, though, is that if I fill in the empty spaces of each figure, they also make a perfect square. So it would go something like:

Figure #2 = 4^2 - 5

Figure #3 = 5^2 - 7

Figure #4 = 6^2 - 9

So, I'm thinking that this might work:    (x +2)^2 - 2x - 1 

I spent some time trying to figure out fig. -1 using the negative space way of seeing. I came up with the same expression as Patricia: (n + 2)^2 -2x - 1. 

(n + 2)^2 --> growing square in the center

-2x --> growing negative rectangle on the right

-1 --> constant negative unit in the top left

I also came up with 2 boxes for fig. -1, but they were next to each other, not on top of each other. The orientation of the boxes probably depends on your way of seeing and how you deconstruct the figure.

When I first looked at the figures I noticed that figure 2 was a 4X4 square with pieces missing, figure 3 was a 5X5 with pieces missing, and figure 4 was a 6x6 square with pieces missing, so the "starting figure" in each case was a square with sides equal to 2 more than the figure number and area of, where f=figure number, f2  , then each square had one little square missing in the upper left corner, so there was f2 -1.  All that was left was to see that each piece had a rectangle with a base of 2 missing.  Then looking at the height, it was noticed that the height was the same as the figure number.  The area of the missing rectangle was 2f.  Putting all that together, the relationship was obvious that the 100th figure would be a 102 X 102 square with one little square in the left corner missing and a rectangle on the right side missing that measured 2 X 100. So that's what it would look like and it's area would be (f+2)2  -1 - (2f).  Of course that can be simplified, but not simplifying gives a better picture of what I saw. It follows that it would have area of 10,203 square units.

Plugging in f = -1, we get (-1 + 2)2 -1 - 2(-1) so the figure would have an area of 2, but what would it look like? Hmmm...start with a (-1+2) square, then take away a square...nothing left.  Now since I know that -2(-1) is +2, I can add 2 squares.  The end figure would simply be a 2 x 1 rectangle.  Drawing the diagram makes a little more sense, but this was just too much fun!