Natalie, a 7 year old from California, worked with Don for 5 days + she continues to work with Don in his Math By Mail program!

[See what Natalie did at age 8, when she returned to work with Don, August, 2001].

Day 1: Sunday June 18, 2000 for 1 1/2 hours, an individual session.

Throughout this session, Natalie repeatedly remarked that she wanted to do hard math and she was bored in school. She will be going to a new school in the Fall in which parents have a big say in what the curriculum should be.

Natalie worked on the infinite series 1/2 + 1/4 + 1/8 ... coloring in an 8x8 square on 1/2" graph paper (see ch. 1 in the sample problems). When she colored in 1/2 of 1/2 of 1/2 = 1/8 and was able to add 1/2 + 1/4 + 1/8 = 7/8, Don asked her to write the next four partial sums.  So she had 1/2, 3/4, 7/8, 15/16, 31/32, 63/64, 127/128... She said the top numbers are all odd, the bottom are all even.  They talked about these numbers were increasing. Then Don asked if the sum would ever get to 2. Natalie said no. Would the sum ever get to 1? It's getting closer to 1, and Natalie said "each number is 1 fraction less that 1"! This was very perceptive and showed her ability to see patterns. Don wrote this as 127/128 = 1 -  1/128. Don asked Natalie what would happen if we did 1/3 + 1/9 + 1/27 +... She said  it will go to 3 of the fractions less than 3! 

At this point Don went off to exponents because the denominators of the fractions were powers of 2 and Natalie hadn't done this. Don started on the left column first:

Then using examples with numbers she was able to generalize:

A^R *A^T =  A^R+T  ,   A^R / A = A^R-T  ,   and  (A^R) ^T = A^R*T     

_________________________________________

Don gave Natalie a 1 baht coin from Thailand. He asked her "If 40 baht = $1 (close to the daily exchange rate), how much would each baht be worth in cents?"  Natalie thought about this and said 5x20 = 100. Then after a little discussion about ?x40=100 and 40 is double 20..and the number had to be less than 5 and 1/2 of 5 and she came up with 2 1/2' and Don wrote 2 1/2' = $0.02 1/2  .

________________________________________

Don got Natalie solving some equations:

2x + 3 = 17   (what number multiplied by 2 then add 3 = 17?). She said 7. Then Don gave her 2x + 3 = 18 and she said 7 1/2 quickly. Don tried to get her to believe 2 x 7 1/2 = 14 1/2, but she wouldn't accept that; she said one has to take 2- 7's + 2 -halves, and 2x7=14 and 2x 1/2 = 1, so 2x 7 1/2 = 15. Don gave her with the equation 3x + 5 = 19. She said 4 is too small and 5 is too big. Because she got interested in the peg game (see ch. 6 of the sample problems) Natalie took this equation back to the hotel to think about it.

_______________________________________

Natalie worked hard trying to do the peg game with 4 pegs on each side. At some point Don suggested that when you have a hard problem, it is sometimes better to make up an easier problem like this, solve the easier problem, then go to a harder one. So he suggested she use one peg on each side, which she did quickly. Then she got to do 2 pegs on each, but struggled with 3 on each side.

Don had, among a myriad of students' works on the ceiling and walls of the math room,  some squares students had divided into 2 congruent pieces. Natalie's Mom borrowed scissors and took some squares for Natalie to cut.

Time ran out as Natalie picked up the tower puzzle (see ch. 6), so she borrowed that for the evening; they would be returning  for 2 group sessions in the morning and one in the evening on Monday. Her Mom had difficulty getting Natalie out the door! ;-)

_________________________________________

Day 2: Monday June 19, 2000 in 3 group sessions (about 2 1/4 hours):

Don showed Natalie what squaring a number meant (3 squared = 3²  = 3x3 = 9 ..) and she knew that 3 - 4 = ^-1. The using the rule for substituting- in any one open sentence or equation, whatever number you put in for one x, you have to put the same number in for all the x's. Don started her on x² - 5x + 6 =0 with the directions, try a small number, positive, no fractions, put it in for x and try to make the sentence true. (See the sample problems from chapter 8 for the sequence of quadratic equations Don uses). Natalie proceeded to find the secrets and solved about 3 quadratic equations fairly quickly. That evening she made up a quadratic equation for Kelsey, a 9th grader!

Don started Natalie off on graphing linear equations, like 2x+ 3 = y. (See the sample problems from chapter 6 for the graphing). Natalie saw some patterns: the y-numbers go by 2 as the x-number goes up by 1; the points go up 2 as you go to the right 1. Then Don asked if there was a 2 in the equation..sure 2x + 3 = y. Then he asked her what the pattern would be if the equation was 5x + 3 = y. Natalie, with no hesitation said the points would go up 5! She proceeded to find pairs of numbers to make this true and graphed them; and she was right. Her Mom mentioned the adding number.. where was that on the graph?

At this point Don started using the '5 magic number cards' with Kelsey and before we knew it all 4 students wanted Don to guess their number using the number cards. They all used 3x5 cards to copy the 5 number cards.

As Natalie's homework, Don gave her 3 linear graphs for each of which Natalie was supposed to write an equation.

One thing Don noticed about Natalie, as she worked on her assignment, she always was looking at and listening to what Don and other students were working on. She realized she could answer many of Don's questions he asked other students as well! Of course Don asked her not to say anything, and she and Don smiled as they shared this silent realization.

_______________________________________

Day 3: Tuesday June 20, 2000 in 2 afternoon group sessions (1 1/2 hours):

Natalie had worked with her Mom on the graphing before she came this afternoon. Natalie had a question about the one graph Don gave her which had an equation of 1/3 x + 4 = y. It went right 3 and up 1. Don drew the straight line and showed her that if she went over 1 she would go up an amount, then over 1 to the right again, and go up the same amount, then go 1 more to the right and go up the same amount again. Going up this last time she moved  up 1 as she moved 3 to the right. She realized she'd have to go over 1 to the right and go up 1/3, so the slope was 1/3. From working with her Mom, she knew the adding number 4, was where the line crossed the y-axis. When she wrote the equation, Don had her check a point to see if it's coordinates made the sentence true, and they did.

Natalie worked on the tower puzzle and the peg game. Then Don, using a protractor and ruler helped Natalie construct a 70 degree spiral. This turns out to be close to the Nautilus shell which is 79.5 degrees.

Day 4: Wednesday June 21, 2000 in 2 morning group sessions and 2 evening group sessions (3 hours):

While Natalie was working with Don, her mother was checking out Don's library to see what she would like to get for Natalie. She even copied a 3 out-of-print books at a local copy shop. She would also come over to help Natalie sometimes while Don worked with other students. He didn't mind that at all. Natalie at one point told Don that she was not interested in the 'pokemon' phenomenon, because it took away time she could spend on math!

Don wanted Natalie to have the experience of graphing parabolas. He started her on x² = y. He drew the axes, but did not number them. He made a table starting with x = ^-3, ^-2, ^-1, 0 1, 2, and 3. So Don asked her what  ^-3 x ^-3 was and Natalie said 9. So Don did a pattern in numbers with her:

2 x ^-3 = ^-6  she knew this

1 x ^-3 = ^-3

0 x ^-3 = 0  no problem. Now Don asked her "as the first number goes down 1 what happens to the answer?" She said the answer goes down 3. But When Don drew a number line, she realized that going from ^-6 to ^-3, goes up 3, and from ^-3 to 0, this goes up 3. So when they continued the pattern, it was no problem that ^-1 x ^-3 = 3, and ^-2 x ^-3 = 6 and finally for the problem we were looking for ^-3 x ^-3 = 9 ! Then she was able to fill in the y values correctly, and proceeded to graph the pairs of numbers. Don left her to go to another student and when he returned, found Natalie had not put the numbers on the axes correctly. The numbers on the x-axis were in the spaces and she started the y-axis at (^-2, 0). He corrected this and she plotted the points correctly after that. [Don realizes that students have to make mistakes before they learn, and there is no computer program that allows a student to make these mistakes!]

Don then asked her if she saw any patterns: she saw the y-numbers went 9, 4, 1, 0 , 1, 4, 9. Don showed Natalie that the points From (0,0) to the right go right 1, up 1, right 1, up 3, right 1, up 5, what happens next? right 1 up 7, then right 1 up 9..what kind of numbers are these? 1, 3, 5, 7, 9.. odd numbers.. why? Don showed on the graph that 1+3= 4= 2²   and 1+3+5= 9= 3² and she wrote 1+3+5+7= 16= 4²  and Don asked what the sum of the first 10 odd numbers is, and she said 10² = 100. Don brought out that the parabola was symmetrical with the y-axis; that is if a mirror was along the y-axis the right side of the graph would be the mirror image of the left side.

Don asked Natalie to move the parabola x² = y up 2 units, then write the equation for the new graph. She suggested  x² x 2 = y and proceeded to graph it. She realized that it wasn't going to answer Don's question because it had the point (0,0) on it not (0,2). Don said that his question was not as important as seeing what the graph of her equation would look like. Because what Natalie did was just as important as what Don was going for. She graphed a few more points and realized that her graph went up faster than the original; it was skinnier. And the differences were instead of 1, 3, 5, 7, 9,...  were 2x1, 2x3, 2x5, 2x7,... that the odd numbers were still there, but multiplied by 2! Natalie then tried  x² + 2 = y and that moved the original graph up 2 units.

_______________________________________

Day 5: Thursday June 22, 2000 for 1 1/2 hours, an individual session.

Don had Natalie graph the partial sums for the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ... (see chapter 1) and again saw that the series is approaching 1.

Don used a 9x9 square to start Natalie on the infinite series 1/3 + 1/9 + 1/27 + ... She got up to 121/243 and started the graph of the partial sums. She said this was going to 1/2, not what she predicted on day 1.. Then she predicted 1/4 + 1/16 + 1/64 + ... would go the 1/3.

_____________________________

Don started Natalie on changing shapes with matrices ( see sample problems from his book "Changing Shapes With Matrices"- the book which will be published in Japanese early next year!).  He showed her the grocery-store arithmetic to multiply matrices and he plotted a 'dog' on graph paper.  She chose the 2x2 matrix [-1 0 -1 1] as her transformation matrix. Each point on the dog like [2 1] was then multiplied by her transformation to obtain a new point and a transformed dog. They looked at the new dog and Don asked Natalie to describe what happened to the original dog to get the new dog. ----picture to come---

_______________________________________

From the stalk of a sunflower, Don helped Natalie find the number of leaves that grew in a helix to a point directly above the starting one (8). Natalie also counted the number of times the leaves went around the stalk to that point (3). These are both Fibonacci numbers (see chapter 7). He listed the Fibonacci sequence: 1, 1, 2, 3, 5, then asked Natalie to continue. She saw the sequence as adding 2 numbers to get the next one. So she wrote 1,1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377... Then Don asked her to get the ratio (comparing by division) of bigger number to smaller 1/1=1, 2/1=2, 3/2 = 1 1/2 = 1.5, 5/3=1 2/3 = 1.666... (Don did this by long division). Natalie did a couple more and the rest was left for homework.

_______________________________________

Don gave her Mom the special graph paper for building the Golden Rectangle and showed her where to place the compasses to draw the circular arcs and build the new squares (see chapter 7).

Don also gave Mom the special graph paper for building the snowflake curve and showed her where to start it (see chapter 4).

At the end of this session they all hugged and went outside to take some photographs before Natalie and her Mom drove to Chicago's O'Hare airport.  Before leaving, Natalie said, "I will always remember this week with you." Natalie's Mom also signed her up for a month of Don's 'Math By Mail' program, insuring their work together would continue.

========================================

Natalie's Mom wrote in an email to Don after she returned to CA: "I personally believe that you are one of the rarest teachers on earth. Thank you for being."

Natalie wrote to Don after she returned to CA:

"Dear Donald Cohen, What I liked about your class most is your creativity. I had lots of fun because it was hard. (Are there any tornados?) I think kids in my old school would be amazed at what you do. I had a fabulous time with you. I hope every thing is going well. Are you having a good time? I hope you are !! Your student, Natalie"

+++++++++++++++++++++++++++++++++

Email received 6/29/00 from Natalie's Mom

Hi, Don. We did matrices:

-1, 0

 0,-1   got same dog in a square where x<0, y>0

1,0

0,1  got same dog as initially

 1, 0

 0,-1 dog is where x>0, y<0

-1, 0

 0,-1 dog is where x<0, y<0

tried

-1, 1

 0, 0 it turned into a straight line.

Next is from Natalie:

It was so fun looking at the way the dog turned

out,especially -1, 1

                       0, 0

  Having fun teaching math?

I hope you are!

Natalie

Email received 7/12/00

From Natalie & her Mom, Natalya