Today's estimation was the length of Queen's "We Will Rock You." Students loved it and laughed when it sped up. Most students accurately estimated that 1/6 of the song was finished, and multiplied 21 seconds by 6. Once again, we discussed place value when it came to minutes and seconds and how 126 seconds was regrouped into 2 groups of 60 seconds with 6 seconds left over.

A common line of thinking. 
I gave students about 5 minutes for these warm up questions. Using their data sheet from yesterday, they investigated whether side lengths of 6, 8, 10 formed a right triangle, squares with areas of 64, 100, and 144 meters squared, and a mixture of 2 squares with side lengths of 5 feet and a square with area of 50 square feet. I made an illustration of the first problem so students could see how we were squaring the side lengths. I also drew a picture after students presented solutions for part C, because many students couldn't visualize what information the problem was giving you. Finally, in part D, they summarized when three squares will form a right triangle.

The warm up questions. 

More of the work. 
Then we drew a right triangle with side lengths of 5 and 12 centimeters. They then used a metric ruler to measure the longest side, which was 13 centimeters. Then we visualized the squares, and they confirmed that 5 squared plus 12 squared equalled 13 squared. In the top right corner we investigated a right triangle with lengths of 9 and 12 cm also.

Using centimeter grid paper to visualize right triangles. 
Then students were formally introduced that the small and medium side lengths of a right triangle are called the legs. The longest side, opposite the 90 degree angle, is called the hypotenuse. This makes up the Pythagorean theorem. They saw that the LEG squared plus the second LEG squared equals the HYPOTENUSE squared. I had students chorally respond when I said hypotenuse as well as Pythagorean. I also pointed out that the theorem is named after a Greek named Pythagoras, but many different cultures discovered the theorem from Arabs, Chinese, Egyptians, and more cultures.
In accelerated I reviewed the difference between x squared times x to the third and x squared to the 3rd. We reestablished that when multiplying power numbers you add the exponents. When you raise a power number to another power, we multiply the exponents. We said that after students told me how to rewrite x squared, to the third power.

Establishing background knowledge. 
Then I gave students more time to complete these problems. They were introduced to 2 ways to solve 4 to the 4th = 16 to the power of x. The methods are in parts A and B, and then they told me how to solve part C.

Solving exponential equations. 
Then volunteers came up and solved problems a through d in 1025.
One part that concerned me was that students thought they were getting rid of the base by dividing both sides by 2. I asked them what exponent does 2 have? (1). So, what happens when you divide it? It actually makes the problem a little more complicated, because the exponents are being subtracted. For example, if you divide both sides of the equation in part A by 2, you would get 2^x+31 = 2^2x1 which doesn't make the problem easier to solve.

A little hard to see, but students were successful here. 
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