Tuesday, March 1, 2016

Day 110: Law of Exponents & Completing the Square

Today's estimation was the length of the Algebra I textbook when placed on top of the Steve Jobs biography. Students tended to underestimate this one. Some of them realized that the paper in the textbook would be thinner than the paper in the biography, so it would have more pages.

The class work started with a great questions, are 3 to the 5th power and 3 * 5 the same? Students were quick to point out that 3 to the fifth was 3 multiplied by itself 5 times. I made sure students contributed the academic vocabulary of the parts of a power number, the small number being called the power or exponent and the large number underneath it being the base number. If students didn't offer it already, I asked how these two expressions were related. A few realized that 3 times 5 was repeated addition while 3 to the 5th was repeated multiplication.

Analyzing the differences.
Then students worked on multiplying power numbers by writing them in factored form, then simplified exponent form. For the first 4, kids were able to generalize that you add the exponents. I stressed the precision of language and they said it is when multiplying power numbers that have the same base, and the base stays the same.

Productive struggle occurred in groups when it came to xy^2. Almost 3/4 of students thought it was xy*xy. Some students weren't convinced and had great conversations. I liked how a student said that xy is x times y so they can be singled out. Another student said that since the x doesn't have an exponent, any number without an exponent can have an exponent of 1. I made sure they shared their ideas to the whole class.

Then they worked on generalizing 20^51 * 20^12, and then analyzed a student mistake where they added the exponents even though they had different bases and multiplied the bases. They then worked on revisiting the commutative property of multiplication.
The main source of productive struggle was the last problem and the 3rd to last problem.
In accelerated we reviewed a problem from friday where a water balloon was launched from the 10 yard line and landed at the 16 yard line. It reached a height of 27 yards. Students said the x coordinate of the vertex was 13, because that was halfway between 10 and 16. Then they wrote y=a(x-10)(x-16). They substituted the coordinates of the vertex to solve for a, which was -3.

Another practice problem gave them a table of values. They saw the vertex and one x intercept. Students reasoned that since the x intercept was 2 and the vertex was (7,25), the highest point, then the other x intercept must be 5 away from the vertex at (12,0). They then repeated the same process to solve for a which was -1.
Classwork review.
Then students worked on a situation where they were shown how to complete the square using algebra tiles. Then they tried it on their own. I had one student write out what was happening algebraically as I manipulated the virtual tiles. As you can see below, 2 unit tiles were needed on the right side to complete the square, so you add 2 to both sides, and write x squared plus 6x plus 9 as (x+3)^2. Then they subtracted 2 from the left side so that y was by itself.
Tile drawing and the algebra.
The virtual algebra tile representation.
Then students were asked to complete the square here. In this situation, there were enough unit tiles so they just had to break up the 11 in to 4 +7, and then rewrite the perfect square trinomial as (x+2)^2. They then could easily see the vertex was (-2,7).

Here you can see the extra tiles not needed to complete the square.
And here is an awesome graph that shows just how amazing Warriors player Steph Curry has been this year when it comes to making 3 point shots:

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