Jonty (autistic), and George work with Don

George standing, Jonty, and Don

Jonty with Maura (Mom)

# George

He worked on these infinite series    and graphed the partial sums (see

Don's  Ch. 1 sample problems).

-> '   and

He then generalized to

then Don gave him

after guessing and trying different answers, he added these on a calculator and saw the sum was getting close to 0.666 which = 2/3. He then generalized

Don then had George work on the snowflake area and perimeter (ch.4), which he started, then worked on when they got home.

# Jonty

While I was doing the infinite series with George, Maura was doing them with Jonty. After the first hour Jonty was upset with Don, because he didn't get the attention from him as he wanted. Don corrected that right away. Jonty could hear Don very well, didn't look at him, but understood very well what he was asking. Maura drew the pictures for Jonty, as George did , then would hold up the blue board (one with nos., one with letters) or paper and Jonty would respond by pointing to letters, symbols, and numbers for his responses.

George, Don, Jonty and Maura played Guess My Rule (see ch. 6 in Don's worksheet book). Don made up an easy one, like 2x+3=y which both boys figured out quickly. Then George made up this one, Don giving George 0 for the input x. George wasn't sure of his output and said 0 (later changed to infinite), then 8 for 1, 7 1/4 for 2 and 7 1/9 for 3, putting these in the table at the left below. Jonty very quickly, before Don, told George what his rule was!! Then George graphed his rule at the right

They talked about the graph: there was a vertical asymptote at x=0, or the y-axis.. as the |x| got smaller, the graph approaches the y-axis. As |x| gets bigger, the graph approaches 7. 7 is a horizontal asymptote. [One of Don's rule in teaching is if you let the student make up the problem, it will be much harder than the teacher gives!]

Jonty made up the rule x4 + 1 = y and George figured that out.

Don got Jonty graphing  x2=y with Maura's help. He saw the pattern as one starts at (0,0), then moves 1 to the right, 1 up to (1,1), then from (1,1), 1 right and 3 up to (2,4). This pattern continues going up odd numbers and the sum of the first 3 odd numbers 1+3+5= 32 = 9 and Jonty correctly predicted the sum of the first 10 odd numbers  = 102 =100. Then Don had Jonty find the equation of the graph of the parabola that a) moved the original up 2 units (he got  x2 + 2 = y); b) moved the original to the right 3 units (after some work, he got (x - 3)2 = y ); c) move the original parabola right 3 and up 2 (vertex at (3,2)- and he got (x - 3)2 + 2 = y); and d) made the original skinnier (3x2 = y);  and e) reflect  the original in the x-axis ( -x2 = y). These last two were done on the way home!

Don also had Jonty, with Shawna's help (she was the caregiver and great help with Jonty), filled plastic containers with water to show that the volume of a pyramid is 1/3 the volume of a cube with same base and height and the volume of a cone is 1/3 the volume of a cylinder with same base and height.

Great work, everyone!!

Before they left, the boys wanted a picture with Don and his MATHMAN license plate.

The night before they left for home, Don received this email from Maura:

Hi Don:  I want to thank you and Marilyn and your family for being so nice to us while we were here.  I honestly never tasted better Chinese food than that at the Rainbow Gardens- all courtesy of you and Marilyn.   The boys (and Shawna and I !) learned so much in the short time we had and we all enjoyed it so much.  Your work with kids and your book have no doubt changed hundreds if not thousands of people's lives.  If possible, I think both my boys love math even more after having worked with you.

Maura, Jonty, and George