Sunday, April 17, 2016

Day 139: Irrational Numbers Notes & Determing # of solutions to quadratic

Today's estimation was the song Dan California. While some students had no issues multiplying 28 seconds by 7 or 8 or 10, some took that product and did not convert it to minutes. For example. A student said they thought 28 seconds was 1/8 of the song. They multiplied 28 by 8 and got 224. So, they said the song was 2:24 long. I let it stay on the board until students piped into correct it. I forgot to ask students to always thinking about their answer and if it makes sense or not. 2:24 should have seemed awfully low. Luckily, some students rounded 28 up to 30 seconds, multiplied by 8, got 240, and said oh, 240 is 4 groups of 60 seconds, so it's about 4 minutes.

Interpreting what the numbers mean when you divide seconds by 60 seconds.
Students were introduced to irrational numbers as the side lengths of non-perfect squares yesterday. Today we took notes and practiced square rooting numbers on a classwork problem. The notes consisted of the following conversation:

We first discovered squares that had side lengths of 1 through 5. Then to get their areas, students said base times height or square the side length. This got the areas. I said that the squares with these areas were called perfect squares.

I showed that the square root symbol is actually called a radical sign. We will discuss cube root in chapter 10. The number underneath the radical sign is the radicand, and the example was the square root of 16. I said it's a rational number because it can be written as an integer or ratio, 4.

I thought of a clever way to introduce another example of a rational number. I asked what the square root of 1/9 was. At least 3 students knew it was 1/3, because 1/3 times 1/3 is 1/9. I asked students what happens when you write 1/3 as a decimal, and they knew it was 0.3 repeating. Therefore, this proves that repeating decimals are ALSO rational numbers.

The square root of 14 is not irrational. The word for that is "irrational." When you square root it the result goes on forever in a non-repeating pattern. I asked students if they knew a famous number that went on forever in a non-repeating pattern: pi. I reminded them how to write the symbol.

Then I taught them the procedure for approximating square roots. Basically, step 1 is locate the 2 perfect squares the radicand, 14, is between. They knew that would be 9 and 16. So, write an inequality and simplify. Therefore root 14 is between 3 and 4.

Basically, you make a number line and find the difference between the two perfect squares, 16 and 9 which is 7. Then you find the difference between the lower perfect square and 14. 14 minus 9 is 5. Therefore, 14 is 5/7 the distance between 9 and 16, so the square root of 14 is 3 and 5/7. Then you convert 5/7 to decimal form and round to the nearest tenth. The notes are below:

In accelerated students worked on solving quadratics in perfect square trinomial form. They realized that if a perfect square was equal to a negative number, there were no real solutions. If it was equal to zero it had one solution, and if it was greater than zero it had 2 solutions.

I found this photo online for parents but it could definitely be used for teachers too. I like it:

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