## Wednesday, January 20, 2016

### Day 85: dilations and y=ab^x

Today's estimation was the height of the Jefferson Memorial. A lot of students underestimated. One student anticipated that you couldn't see the dome in the picture, so it was taller than it appeared.

Students started by graphing the green original triangle below. I wrote down the vertices for them to make sure they graphed it accurately. As you can see, this student multiplied all the coordinates by 1/2 to create the blue triangle. Then by multiplying all the coordinates by -1, he saw it rotated 180 degrees.
 I like the color coding. I wish they used a ruler for all the sides of every triangle.
Instead of showing student work on the document camera, I slowly revealed the triangles one by one as students discussed their findings.

 The orange triangle is either a reflection over both the x and y axes, or a 180 degree rotation.
 I like how in my 4th period some international students estimate in centimeters. Eunice thought the memorial was 4 statues tall, and multiplied 580 cm by 4. As you can see the other student Johnny used feet and rounded 19 feet up to 20 feet then multiplied by 4.
In accelerated students investigated y=a*b^x by analyzing the incomplete x/y table on the right. It revisited the introduction of exponentials when students did rebound height with the bouncy balls. They knew that it was decreasing at a different rate each time, so they knew it was using a multiplier. Zoe explained how she got 0.8 because she divided 67.6 by 84.5. Then Michael said he divided 84.5 by 0.8, to get bounce 1, and then divided by 0.8 again to get the initial drop height of 132. Then they wrote y=132*.8^x.

Robert explained that 3 people founded the company in problem b because he substituted x for 0 and 4 to the zero is 1, and 1 times 3 is 3. Students started to see that the a parameter represented your initial value, or y-intercept. The b parameter was the constant multiplier. Finally students analyzed a computer lab infected by virii. They saw the multiplier was 2/3, and that was for uninfected computers. So, they realized that 1/3 of the computers get infected each day.

 Great work and introduction to formal exponential equation: y=a*b^x.