Been busy. Have done hours of tutoring since my last post.
Here is a description of what happened for one of of those hours.
Student 7 years old. Normal to above average intelligence. Slight hyperactivity. Expressed in hyperkenetic activity and extreme talkativeness. He's a regular kid.
First we built tens on down to two's.
Then we talked about subtraction, and with the blocks right there verbally drilled him on facts. 10 take away 4, 8 kids in the room, three leave how many left? Etc. Did 15 different facts and five that were the same. Did not relate subtraction and addition. In other words 10 - 4 = 6 Is exactly the same problem as 6 need what to be ten...when you have the blocks. He had the tray to look at the whole time. So it was easy. Soon he will only have symbols...but first he will be absolutely confortable with blocks and drawing.
Then we built rectangles and labeled the sides and hid the answers underneath. The over the shoulder shot show him labeling 5 eights...
Then we talked about division and multiplication. Here we see symbols only and we have adjusted the symbols to fit the blocks the way we wanted. You don't see that we changed the sevens to show five across 7 down and used 5's because 5 is contained in 35, seven times. We also talked about how to read that expression...4 IS CONTAINED IN 16 four times. He could see that, it was visually obvious.
We then added the arrows for the pattern across and down. and as you can see he got the blocks out to help him. All he had to do was count the down he already knew the one side and how many in the rectangle, I thought he would get out 5's but he got out 4's because he knew the answer. That problem says build 20 with fives. He was in a situation where he could not fail. So he was confident. The important part was he got it right and understood that he had to count the other side. He also understood 4 is contained in 20 five times. Multiplication and division all at once. They are inverse functions after all.
Then we moved on to his favorite topic: ALGEBRA! We moved on to cross teaching or compound teaching concepts. The first "problem" was make as many rectangles as you can with x2+ 7x...
So he built
x2+ 7x + 10
x2+ 7x + 12
x2+ 7x + 6
We talked briefly about the factors (ie the factors of x2+ 7x + 10 are (x + 2)(x + 5) ) but mostly we were concerned with the 10 being made up of 2 fives or 5 twos.
Then we added 3 more x. It was a tiny problem how many more x do we need? We had 7x so we needed 3x to make 10x. Algebra is easy! Now we made as many rectangles as we could with
He started out with x2+ 10x + 25
then did x2+ 10x + 21 then x2+ 10x +16 which you see pictured here. He has 2 on top and 8 on the side. (x + 8) across and (x + 2) up.
x2+ 10x + 16 = (x + 8) (x + 2).
We were more focused on the 2 x 8 part. The algebra is a bonus as it were. (!) It was much faster to make it with two 8's than with eight 2's. This is known as the economy of counting. We discussed this concept only briefly. But it's obvious it's faster to count 8, 16, than 2, 4, 6, 8, 10, 12, 14, 16...we did both of course.
Then x2+ 10x + 24 and lastly x2+ 10x + 9. Note how much math is going on here and why I call it compound teaching. He is learning all the addends for 10, the factors of the algebraic expressions and some the factors of the numbers 25, 21, 16, 24 and 9. We are working on all of this all at once and having fun doing it. Nothing scary or hard about it.
The we continued building the addends to 18...
Then the hour was up. 60 minutes flew by.