This is a guest blog post by Danny Phelan, who is a student at Wake Technical Community College in Raleigh, NC, studying for an Associates Degree in Arts. Danny’s eventual goal is becoming a novelist and screenwriter, as well as editor for fantasy, science fiction, and horror novels.
When I was in high school, I noticed a curious pattern: one plus three equals four, plus five equals nine, plus seven equals sixteen, and so on. Each whole square was a certain number greater than the next, and the difference between the squares would increase by two every time. When I mentioned this to a math teacher, she either did not see it that way or misunderstood my phrasing, for she dismissed it as simple coincidence, not a natural law. After class, however, I was not convinced. Stubbornness is both a vice and a virtue of mine, and I continued to puzzle over how a series of odd numbers, increasing by two each time, connected to the succession of perfect squares.
Eventually, I drew the pattern out on scrap paper and saw something interesting: each square on a grid increases by the addition of new units to a side; being two-dimensional, they increase by two new units each time–one for each dimension.
We came up with an equation that represented the pattern. Since we want to look at the difference between squares (how much you add to get the next whole square), we added 1 to r, which stands for the square root. There’s the 2 pattern!
(r+1)2 = r2+2r+1
But I was still not fully satisfied. Something felt missing. I reasoned that if one could calculate differences between squares, surely a similar pattern might hold true in three dimensions–that is, whole cubes. The formula would be more difficult to visualize, but fortunately, my friend Ray owned a bowl of small cubes which we used to crack this puzzle. Three new squares would need to be added with each successive layer, and by holding cubes together, we discovered that the edges between sides would also need to be accounted for.
We determined that the formula is:
(r+1)3 = r3+3r2+3r+1
Applied fascination, when shared with other curious thinkers, spurs the creation of wonders.
My family and I have been enjoying natural math games and classes with Maria for a while now. We love doing math in an unstructured way and prefer to focus on concepts. But sometimes, you just need to practice the math facts.
So I was excited when we were asked to review the game Math-a-Round Sigma. Math-a-Round is a simple addition and subtraction game. Players need to get all the numbers to sum to 10. The fun part is that you may have to add or subtract a lot of numbers to get there. You can see my daughter’s rows are pretty long.
My six-year-old and my three-year-old played it first. They loved how hands on the game was. It was fun to watch them doing so much math in their heads and on their hands, but they were anxious to figure out when they had reached 10. It didn’t take long for the six year old to come up with some strategies for deciding whether to subtract or add the numbers.
Here is her “review”:
It was powerful to get insights into my children’s number sense and math abilities. The six year old never subtracted until the numbers went over ten. It was easier for her to decide which number to subtract at that point. It was intriguing to listen to her thought process. My 11 year old refused to go over the number ten, until we pointed out that 18-8 is 10! Even the three-year old shocked me by knowing all the numbers by sight! Here’s his “review” of the game!
While we were playing the game the older kids kept giving advice and doing the math in their heads. They were chomping at the bit to get a turn playing the game. First thing the next morning, they had it out and were playing it again!
As a mom, I love games that all my kids can play; even the three-year-old was learning from the game. When the 11 year old plays against the six year old, we make his rules harder so they can play together. We even tried putting a number slider upside down and treating it like a negative number.
We love the physical nature of the game. It’s a fun, fast-moving way to practice mental math. My kids don’t really think about it as a math game, they just had fun playing!
I also love how easily it stores because of the awesome design. There are two tracks (blue and red). The red is the bottom of the storage container and the blue is the container lid. All the pieces store inside.
Fast paced, strategic thinking, lots of math learning, easy to store, great design, adaptable for all ages.
Easy to lose those little parts, so keep it away from kids under 3.
This is the story of the fifth and final meeting of the Sunday Natural Math Circle doing activities inspired by calculus.
The first activity was making a flip book out of cards. To figure out the size of designs, we used LEGO blocks. How many blocks fit on the card in a row? Will it still fit if you re-position it? For a 3×5 inch card, a line of four blocks fit every which way.
Eli, Mark, and Chris are trying and testing things for the project. Our instructions tend to be open, such as “try using a clip”. This way participants get in the habit of problem-solving.
Yash made a measuring square to approximate positions of shapes on paper.
Maria demonstrating how to fold small squares of paper (measured out to be no more than 4 LEGO blocks across). We folded them into 4, 6, and 8 layers through the middle, then cut across. What shapes do we get?
Eli’s checking out his shape.
Chris and Mark, Krishnan and Yash, and Maria and Eli are making 6-fold shapes, which are challenging. While people have no problems estimating folds into 2 parts (and then 4, 8, 16…) – folding into 3 parts is hard.
It’s also challenging, with any fold, to cut the paper in such a way that both sides are about the same. It’s tempting to cut at the right angle to the nearest side instead. If at first you don’t succeed, cut and cut again!
4-fold and 6-fold shapes glued onto cards. Is the 4-fold shape a square or a diamond?
Mark is drawing figures on his shapes.
Smile! Mark is ready to animate the flip book.
The participants are peaceful, working at their own pace drawing, cutting, measuring, and animating ready flip-books.
After finishing the flip book, the group moved to the next table for the math snack time – always a happy time. Children wanted to straighten a banana like last week, which Yash wanted to see since he wasn’t there.
Mark and Eli demonstrated the technique from the last week: rotate every other slice 180 degrees.
Next, we are going to play with an apple. Can we straighten this circular slice?
Children are reusing the banana technique, rotating every other section of apple the other way.
Instead of a circle, they got an approximate rectangle.
Maria asked grown-ups if they can see Pi in this, and they wanted to borrow her “math goggles” for the purpose. So the adults asked children to play elsewhere, since they were getting tired, and held their own math circle.
If you straighten out the peel from a banana slice, you see the length of the circumference.
How long is it? Pi is the ratio of the circumference (banana peel) to the diameter (the slice of banana cut across).
That outrageously simple way to see Pi had participants thinking seriously on what their school math classes really meant. After all, that happened way back in the previous millenium!
The peel is three slices across, and a bit more. The circumference is three diameters and a bit. Pi=3.14… And not any kind of Pi, but banana Pi!
What about apple Pi? When we alternate the pieces, the rectangle is half a circumference long and one radius wide. The ratio of its length to its width is Pi again! This time, apple Pi. Next time, it would be interesting to slice different round fruits, to straighten the circles into rectangles, and to observe how all the rectangles look like scaled-up versions of the same shape, since their length:width ratio is always Pi.
Returning to the table, we are about to make round things out of straight and rectangular paper.
We are folding the paper as if we’re making fans in an alternating zig-zag pattern, just like we alternated fruit slices: it all comes together.
Can you curve the straight paper?
These folded papers will become wings of creatures. Mostly dragons. Everyone drew their creatures, then cut them out:
The wings go into a slit cut in the middle of the creature’s body.
Picasso would be proud.
Group pictures – make silly faces for the press!
Photos by Erin Song, captions by Erin Song and Maria Droujkova, Math Spark by Kalid Azad, Shelley Nash, and Maria Droujkova, edited by Ray Droujkov.