Day 1
- Way back at the start of the year, we played with patterns of geometric figures constructed out of toothpicks, using them to construct linear equations. I'd like to refresh our memory on those for a bit -- so please make the equation that matches the following pattern:
x (triangles) | y (toothpicks) |
---|---|
1 | 3 |
2 | 5 |
3 | 7 |
4 | 9 |
5 | ?? |
- Now try for the sequence of squares:
- What we learned before was that all sequences like this could be represented by equations like \(y = mx + b\), where the b represented something about the shared sides between new shapes and the old shapes, the x represented the number of shapes, and the m represented how many toothpicks were added at each level.
- Now I'd like to try a different kind of pattern, made up of dots to represent the numbers, like this:
What is the sequence of numbers that goes with this sequence of dots? What is the equation that matches the sequence? What kind of equation does it have to be?
Day 2
- Now consider the following sequence of dots. The "x" number is the input to some kind of equation, the number of dots is the "y" output of that equation. Figure out what the equation is:
How would you describe how the visual pattern grows?
Can you make a connection between the sequence of square numbers and these numbers?
- Here's another sequence of dots, figure out the equation that matches the sequence. Is there a connection between the length of the sides of these rectangles and the input number x?
Day 3
- Continuing working with sequences of dots. Why is it useful to work with the dot pictures and the equations and the tables of numbers? Sometimes it is difficult or impossible to see a structure unless you look at the same thing in multiple ways. For example, consider the following dot picture. How many dots are in the picture?
Now how about this picture? How many dots are there?
Both images have the same number of dots, but it is much easier to see the number in the second picture, because they are arranged in a structure that facilitates understanding. When we rearrange the dot pictures that match these quadratic equations, we are trying to find a way to make the structure more apparent to facilitate understanding. It is very much like factoring.
Day 4
This was the Health Fair day for the 10th graders, and the Career Fair for 11th.
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