## Tuesday, January 19, 2016

### Day 84: intro to dilations & y=b^x re-intro

Today's estimation was the height of Thomas Jeffersons statue at his memorial. Most students rounded my height up and multiplied by 3 and got pretty close.

I was absent for my 2nd period class to be at my wife's ultrasound appointment. It's going to be a girl!

Students graphed a quadrilateral and multiplied both its coordinates by 2 after predicting what would happen. Most students said it would get bigger but less kids said it would also move right. Some said it would move up.

 Students were demonstrating cutting off a piece of the trapezoid and fitting it to the bottom to make it a rectangle.

Part d asks students to compare the side lengths. They easily saw the sides became twice the size. Some didn't raise hey had to count the side lengths of the squares of the grid paper.

When discussing the word dilation I asked students what happens at an appointment with your eye doctor. They talked about various tests which I said is the eye strength. Eventually one student suggested they got eye drops that dilated their eyes. So, I asked them what happens to your pupils when that happens? They said they get really huge. Well, the original quadrilateral is like your eyes without eye drops. With eye drops, your pupils get bigger, which is like the bigger quadrilateral.

 I showed students how the same type of arrow signals that the pair of sides are parallel to each other.

Once they had to figure out area, only about 1/10 of students in each class knew the name of the quadrilateral. Some were able to see it was made of s rectangle and triangle. They struggled a bit recalling the formula for area of a triangle. Some realized again that it was half of a rectangle if you doubled the triangle.

 This student subdivided the areas then added them together.

Some students remembered the formula for a trapezoid. Most students were familar with the isosceles trapezoid that is from the pattern block set. They remembered it's the red one. They saw the similarities that 1/2bh has to 1/2h(b1+b2). I showed them how the formula was derived, by basically duplicating the trapezoid and flipping it upside down, it formed a rectangle.

 On the right you can see a quick number talk on an alternative way to multiply 19x12. In this class, students asked why it the area quadrupled. I showed an example of a square with side lengths of 3 and 9. It showed that each side was doubled, and when you multiply 2 times 2 you get 4, the factor the area increased by.

In accelerated students were re-introduced to exponential equations, in the form y=b^x. Students remembered that exponential decay was when you repeatedly multiply by a fraction and your line decreases. They then were able to differentiate between bases of 2 and 3, saying that 3 increases faster.

Then I played this awesome Desmos slider by pressing the play button and it would move from -10 to 10. I asked students to notice and wonder as much as they could. Here's what they came up with:

• Students noticed that when b was negative, it disappeared
• Students noticed when the graph appeared it always had a y-intercept of 1
• They wanted to know what it looked like for b=1 and reasoned it must be a horizontal line
• a student asked to slow it down to see the changes better
• students reasoned that when b equals zero that it is a horizontal line from 0 to the right, and that it couldn't be negative because anything divided by zero is undefined
Students asked me to stop it at b=1, but I said no. They had to investigate it. So, students invested when b>1, b=1, b=0, and when 0<b<1. It was a great investigation.

Then I had students re-take a coin problem about systems that had a typo on Friday. Also, they got new seats.

I ordered stickers and a Desmos hat and got some pencils thrown in. I also ordered some tiling turtles for my nieces birthday and an adult coloring book. Christopher Danielson makes the tiles and he is also the author of the book I reviewed on here, Common Core math for parents for dummies. Great book.

 My decagon
 Weird star looking thing.
 Mr Roboto, sort of.