Nanako came a second summer from Japan, June 16-21, 2008 to work with Don!
The following is Nanako's work with Don, as they did it, then scanned and remembered by Don. Please have patience, there are many pictures to download!
Monday June 16, 2008
After greeting Nanako and her Mum outside, they all went down to the mathroom. Don had copies of Nanako's math curriculum downloaded from her school website, Nanako's work from last summer and her notebooks from her 4th grade work at her school. Don decided to see what Nanako could do with fractions, decimals and percents; he made the chart below in her notebook that she had used last summer. He wrote in 1/2 in the first row, .47 in the second row and 33 1/3 % in the third row. Nanako's job was to fill in the equivalent names in that row. She did fine on the first 2 rows, 1/2 = 0.5 = 50% and .47 = 47/100 = 47%. Then Nanako had a difficult time with 33 1/3 %.
They talked about her answer of 33.333 as a decimal and 3333/100 as a fraction and ended up with .333... = .33 1/3 as a decimal and 33.333.../100 = 33 1/3 / 100 then tried to simplify this. In the next to the last line above, she made equivalent fractions for 33 1/3 / 100 (thirty three and one third hundredths) as 50ths, 200ths, then Don suggested multiplying top and bottom by 3 and she ended up with 100/300 = 50/150 = 25/75 = 1/3. Don showed Nanako she could use 100/300 and divide top and bottom by 100 to get 1/3 directly. In the process she took half of 33 1/3 to get 16 1/2 + 1/2 of 1/3 = 16 2/3 using the picture above. So Nanako did 1/2*1/3 = 1/6 and 1/2 + 1/6 = 2/3, which was good stuff!
Don asked Nanako if she bought a shirt for 6000 Yen and received a 33 1/3% discount, how much would she pay for it? The discount would be 1/3*6000= 2000, then she would pay 6000-2000 = 4000 Yen (100Yen=~$1). Another way to look at that was, if the discount was 1/3, then she would pay 2/3 of the regular price or 2/3*6000 = 4000 Yen.
Then Don tried to get Nanako to
do some percents in her head. Like take 10% of 200. 10%=1/10, so divide 200 by
10 = 20. Then 5% of 200 is 1/2 of 10% of 200 = 10 and 15% of 200 is (10%+5%) of
200 = 20 +10 = 30. They talked about 10% of 
= 20 which is 1/10 *
= 20 , multiplying both sides by 10, you get
= 200. At one point, after Don wrote that 1/10 = 10%, Nanako wrote that 1/5 = 5%
and 1/15=15% and 1/20 = 20%. But with a couple of questions from Don, she
figured these out to get the correct answers.
To keep Mum from getting nervous about Nanako's mistakes, Don asked her to solve the equation 2x + 5 = x - 7, that is, find a number that would make this sentence true. Mum proceeded to do the normal thing, subtracting 5 from both sides (except she thought of "moving things to the other side")- see her arrows. She got to x = x/2 - 6. Then Don realized that Mum had the correct answer x = - 12.
At this point Don decided to show Mum how to iterate x/2 - 6. Don asked her to try a number in for x; she chose - 4 above. She did the arithmetic, - 4/2 - 6 = - 8. She then put - 8 in for x; she did the arithmetic, - 8/2 - 6 = - 10. See her answers above to the right, forming a sequence - 4, - 8, - 10, - 11, - 11.5, - 11.75, - 11.875, ... which approaches - 12 as the limit! And Mum saw a pattern in the differences, (that Don never saw before-great!) : - 4, - 2, - 1, - 1/2, - 1/4, - 1/8, which is another sequence, this time approaching 0 as a limit!
Then Mum tried 5 to put in for x and Don suggested she use a calculator to do the arithmetic. This sequences of x's also approaches - 12 as the limit!
So, in solving a simple linear equation as Mum did, one can iterate a function X/2 - 6, and get an infinite sequence which approaches a limit (-12 in this case), which is the answer to the problem!!! Very exciting!
Mum also worked on quadratic equations like x2 - 5x + 6 = 0 answers {2,3} and proceeded to find the 2 secrets in solving these equations. She even found the answers to x2 - 25x + 24 = 0 and x2 - 6 1/2x + 9 = 0 and made up three quadratic equations for Don to figure out. [The math genes certainly run in the family!!]
Gauss' method for adding the counting numbers from 1 to 100, and how Nanako expands this problem.
The story goes like this: When Gauss was in first grade, his teacher, in trying to give him a hard problem to keep him busy, asked him to add the numbers from 1 to 100.
When Don asked Nanako to make up a sum using her rule
S= n(F+L)/2
she made up 2+4+6 (even numbers!), S=3(2+6)/2 = 12 which was correct. At this point Mum and Don made up some numers to add and used negative numbers (below). And her rule worked. It didn't work if the differences were not the same!
Nanako played the Peg game (see chapter 6)
The object of this puzzle is to interchange the blue and the red pegs. The rules are 1) you can move to a hole that's next to a peg; 2) you can jump, but only one peg and it must be of the other color, and 3) you can't move backwards. You must start with the empty space in the middle and end that way. You can use golf tees, as I do, or you can use two different kinds of coins or bottle caps or pieces of colored paper as the pieces.
It takes time for people to figure out the pattern of how the pieces move, without having to start over. She was able to do this fairly quickly. Then Don made a table below, where x was the number of pairs of pegs, and y was the number of moves to interchange the pegs. She started with one pair, and it took 3 moves, starting with 2 pairs, took 8 moves
Nanako saw the pattern 1x3=3,
2x4=8, 3x5=15.. could be written as 1(1+2)=3 and 2(2+2)=8 and 3(3+2)=15..
Nanako found the rule x*(x+2)=y
with some help from Don. In the
process Don made sure she could multiply negative and positive numbers (see
above at the bottom of the page in the center). She then graphed this function
to get a parabola.
They talked about the graph having symmetry and the axis of symmetry was x = -1.
Don had Nanako get the area of triangles with the same base and height. She found the area in each case was 3.
This work is from chapter 6 of Don's worksheet book.
Don talked to Nanako about comparing things 1.) by subtraction, like if Don is 79 years old and Nanako is 9 years old, how much older is Don than Nanako? Don is 79-9 = 70 years older than Nanako. and 2.) by ratio or division, like how many times as old is Don than Nanako? Don is 79/9 = about 9 times as old as Nanako.
They used the 10 different length Cuisenaire rods above, from the 1x1x1 white rod (1 cubic cm.) to the 1x1x10 orange rod (10 cubic cm.). Don had Nanako find the surface area of each rod (in square cm.), the volume of each rod (in cubic cm.), then find the SA to Vol ratio (SA/Vol) for each rod.
Don asked Nanako to find a pattern to find the surface area from the length of the rod. She saw that the first SA was 6 and 2*3=6, then 2*5=10 and 2*7=14..but neither Don nor Nanako saw how the length was involved at that time. She also saw that the SA went up 4 each time and that the SA was 4 times the Length (the lateral area for each rod), + 2, for the 2 ends of every rod, so the SA was 4x1 + 2= 6, 4x2 + 2 = 10, and 4x3 + 2 = 14 and she generalized to SA = 4xL + 2. The Volume was simple, it was always equal to the same number of cubic cm. as the # of Length units.
In the process of getting the SA/Vol ratios Don and Nanako talked about the division, of like 14/3, which she wrote as 14R2. Don talked with her about sharing 14 cookies between 3 people- each person would get 3, with 2 cookies left over. With a picture above on the right, of the 2 left over cookie she realized each person would get 4 and 2/3 or 4 2/3 cookies.
[Don made sure Nanako understood that if 14/3 = 4 2/3, then what was 3x 4 2/3? on her paper above near the bottom, you can see how Don did this by adding 4 2/3 three times. and they got into 3x2/3= 2, and 3x5/3 = 5 and n* Y/n = Y she wrote. Also Nanako (in the lower left corner above) did 3x5/3 = 5 an entirely different and wonderful way. For each of the 5/3 she put 1 (= 3/3), then thought (5-3)/3 = 2/3, so she thought of 5/3 as 1+2/3, added the 3 1's and 3 of the 2/3 to get 3 + 6/3= 3+2=5, the answer to 3x 5/3.
So for the length L of the rod, the SA/Vol ratio is (4xL + 2)/L and dividing by L, this could be written as SA/Vol = 4 + 2/L. So they had an infinite sequence of ratios (L could get bigger and bigger). Don asked Nanako, what happens to 2/L as L gets bigger and bigger. The discussion went like this:
Why was Don doing this with Nanako? Besides the fact that 1.) they got an infinite sequence which approaches a limit, 2.) she found the surface area of a rectangular solid, 3.) she had to divide to get a mixed number, 4.) she multiplied a whole number by a mixed number, 5.) she looked for and found patterns,
They talked about the fact that a small animal (the white rod) has the largest SA/Vol ratio. Now what does the SA have to do with anything.., actually the SA is a measure of the skin area. When the small animal is out in the sun, it loses moisture too fast and can die. The picture below is from an old Scientific American article. The word desiccation means to dry up.
This shows that rodents, to stay alive, become nocturnal animals!
This idea also explains why we use grated cheese on spaghetti instead of a big glob of cheese. The grated cheese, with a much larger surface area, melts easier, and can get all around the spaghetti- mmmm good!
Thursday, June 19, 2008
Don helped Nanako get the Fibonacci numbers from a pine cone from his front yard and a sunflower stalk. See chapter 7.
They got 3 and 8 from the sunflower stalk and 5 and 8 from the pine cone; the Fibonacci numbers are 3,5,8 and from these Nanako got 1, 1, 2, 3, 5, 8, up to 144 below.
She then got the ratio of consecutive Fibonacci numbers as mixed numbers and decimals.
She was able to get 1 8/13 as a decimal, then Don suggested she use a calculator to do the rest of the ratios. Nanako saw that she was getting an alternating, infinite sequence to 4 decimal places on a TI 84 plus! She also became aware that the 10th Fibonacci ratio 89/55 above showed up in the 11th, but as 1 55/89 (1 + the reciprocal of 89/55).
Then she graphed this infinite, alternating sequence. This was hard because the numbers were very close to each other, so they used mm. graph paper to help a little.
Don showed Nanako and Mum how Mathematica can get 100 digits for The Golden Mean quickly:
N[(Sqrt[5]+1)/2,100] =
1.618033988749894848204586834365638117720309179805762862135\
448622705260462818902449707207204189391137...
Friday, June 20, 2008
Don asked Nanako to graph the equation 2x + 3 = y. She found pairs of numbers for x and y that will make the sentence true. She made a table to show these pairs of numbers, then plotted these on the graph with dots ..
She saw a pattern in the y-numbers in the table- they go up 2. She saw that the points on the graph go 1 to the right and 2 up and Don asked her if there was a 2 in the equation, and sure enough it was there 2x + 3 = y, YES!!! They looked at the 3 in the equation and she saw the 3 on the graph when x=0, where the line crosses the y-axis. Then she graphed the equation 5x+3 = y with ''s. Don asked how the graph of this equation would be different; she predicted correctly that the pattern would go 1 right and 5 up.
Don then did the reverse, he gave Nanako 3 graphs and had her write the equation for each graph below. They talked about the slope of the line 3, in the first case below, 0 if the line is horizontal, and negative, like -3/2 for graph #3, is the line goes up to the left. And they also talked about the y-intercept, where the line crosses the y-axis, when x=0 and is the adding number.
They then worked on geometric transformations with matrices (see Don's book Changing Shapes With Matrices, which Don gave to Nanako):
Don started Nanako with the "grocery-store" arithmetic to multiply the matrices. Nanako went to the grocery store on Monday to buy 3 boxes of Blueberries, 2 boxes of Strawberries and 4 packages of Grapes. This is shown in the row matrix on the left. The Price matrix shows the price of the Blueberries is 10 cents per box (all prices are in cents to make the multiplication simple), the price of the Strawberries 50 cents per box, and the price of the Grapes 60 cents per package.
The question is how much will Nanako spend on Monday? And how does she find this? She said 3x10 + 2x50 + 4x60 = 370 cents. In other words, one goes to the right in the row matrix and down in the column matrix, multiplying the number of each item by the price of the item, then adding the products.
Then Don added Tuesday to the row matrix, and Nanako found the total cost for Monday and Tuesday.
Once Nanako could multiply the matrices, Don made the black "dog" on the graph paper below, showing its 9 points. He then had Nanako choose 4 numbers from -1, 0 and 1 only, for her transformation matrix (2 rows x 2 columns); she chose {0, 1 and 1, -1}. The her job now was to take each point on the dog, like point #1 at (1,1), make this a matrix [ 1 1 ] and multiply this matrix by her transformation matrix (remember going right on the 1st matrix and down the first column in the 2nd matrix; 1x0 + 1x1 = 1 ; then going right in the first matrix and down the 2nd column in the 2nd matrix 1x1 + 1x -1 = 0 to get [ 1 0 ], the new point #1 at (1,0). So each point on the old "dog" goes to a new point to get the transformed "dog". How was the old dog changed?
Try some transformation matrices yourself using the java applet created by ies in Japan for Don's website, as done in his book Changing Shapes With Matrices.
Saturday, June 21, 2008
Nanako and Don painted in watercolor (Nanako's ideas, after eating pancakes for breakfast!)
This was a sensational week working with Nanako and her Mum!!! Thank you Dad for allowing Don to do this.
Don made and sent the following card to Nanako for her 10th birthday in July, 2008 as an attached file:
Don received this response from Nanako :