Sunday, January 31, 2016

Day 92: missing sides & using substitution to write exponential a*b^x

Today's estimation was a fourth measuring tape. Some students thought it was smaller than the third one. Some thought it was the same size. Some students assumed it would be bigger. Many kids reasoned it had to be a measurement that ended in an even number. This is why I pointed out why 29 feet and 31 feet might be not be the most reasonable estimates.

I wanted to make sure students finished the activity where the used the scale factor from yesterday to find the missing side x on the new figure and missing side y on the new figure. I gave students 7 minutes to finish up, and students who were already done moved on to the next problem. It was tough because some groups were working at different paces.

Here you can see that I brought up the 2 divided by 1/3 problem. Two whole pizzas cut into slices that are 1/3 of a pizza each. How many slices are there total?

Then we had a whole group discussion. I wanted students to tell the class where the scale factor was in the diagram (the arrow with * 5/4 before it). I also wanted them to remind me how they got that scale factor. They said you divide a side on the new figure by its corresponding side on the original figure. Some prefer the decimal 1.25. I asked them if all fractions have nice decimals like that and they said no, and I tried to encourage them to challenge themselves not to use decimals because fractions are never rounded and will always get you the exact answer. Students over rely on calculators a lot.

When students multiplied 6 by 5/4 to get x, through out the day I saw all different misconceptions. Some students put 6/1 and cross multiplied. Some put 6/1, flipped it to 1/6, then multiplied. I made sure that students knew that cross multiplying is a strategy for solving proportions. What I believe they are confusing it with is cross cancelling or cross simplifying.

In each class I asked how do you multiply fractions. The most common explanation was "straight across" which is fine. I asked if students knew a short cut. That's when I labeled the method of dividing 6 in the numerator and 4 in the denominator by 2 as simplifying or cross canceling. Unfortunately, over the years students mix up rules, overgeneralize, then just try what might work.

You can see the multiple ways to multiply fractions here.
To find a missing side length on y, students knew you had to divide instead of multiply. Few students were confident enough to raise their hand to solve 10 divided by 5/4. That's why I drew the picture of the pizzas and asked what is 2 divided by 1/3 the same as? They saw it was the same as 2 times 3. Then they realized it's multiplying by the reciprocal.

A student in 4th period scoffed at this and complained, "this is 6th grade math!!" I stopped the class and addressed the class as a whole that that was not a positive attitude and that there were PLENTY of people in the class that didn't know how to do it and with comments like that it probably lowered their self esteem and self confidence even more. I said that the culture of our class is very important to me and it's OK to not know something, but never think you are too good for it. You can always be a resource for students.

So, in 4th and 5th period, I discussed California schools following social promotion. I don't know if this was a good idea to discuss with the students but I wanted to give them a dose of reality. I said that you could learn NOTHING in 6th grade math and move on to 7th. You could fail every class and you will move on. I said that this is called social promotion. I said that there are schools in the south, and I looked it up, it's Georgia, Florida, and a few other states, that hold students back if they don't pass the state exams. I said that in those states you might have a much older student that should be in high school still in 8th grade. California doesn't believe in that.

I said that while it has its positives, it seems to be a rude awakening for students who didn't succeed in middle school and realize they won't graduate high school without the credits that they need.

In accelerated, students worked on using substitution to write exponential equations from two coordinates. They also worked on what exponent you raise a base to to take the cube root of it. CPM has some creative ways to teach this. They also demonstrate how to simplify fractional exponents by breaking up the exponent into a power of a power problem. We will be working much more on this on Monday because students worked at different paces and we had an assessment too.

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