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.

Friday, January 29, 2016

Day 91: finding scale factor & exponential whiteboard review

Today's estimation was the third measuring tape. A lot of students saw it as a linear pattern and though it may be 20 feet because it the first pair grew by 4 feet. I liked this connection to their previous thinking.

Students devised a way for finding the scale factor between any two similar shapes. It was important that they made the distinction between the original and new shape. For the first problem, students instincts were to say it was scale factor of two. Then students thought and realized the order mattered and that the shape was getting smaller, so it couldn't be 2.

In the second example, they really had to devise a strategy. Some once again reversed it and thought the scale factor was 0.8. I asked students why that didn't make sense. They said if it was larger, the scale factor had to be greater than 1.

So, students came up with the definition: to find scale factor, divide a side on the new shape by it's corresponding side on the original shape. It gives you a ratio, new over original. Rich discussions were had about the equivalence of 5/4, 1.25, and 1 and 1/4. For students struggling with seeing this connection, I said what if the scale factor was x. 4 times x is 5. What equation is that? Now solve it. Ohhhhh...

As you can see I asked students if 6 divided by 3 is the same as 3 divided by 6? Does order matter when dividing? Yes...
Here you can see a discussion about one apostrophe is feet and double apostrophe is inches.
Here I stressed that dividing by a whole number 2, multiplying by a fraction 1/2, and multiplying by 0.5 get you the same result, yet they look so different from each other.
In accelerated, we reviewed how to write an exponential equation from a graph. Then students jigsawed the classwork from yesterday on whiteboards while I was in an IEP meeting and they presented to the class:

Wednesday, January 27, 2016

Day 90 svmi pd sub day: FAL interpreting distance time graphs

Today I was out at a SVMI professional development. We learned about Agency, Identity, and Authority. Basically, all the requirements to have empowered students and great classroom discourse (discussions). We talked about talk moves: wait time, prompting, rephrasing, and relating students ideas to others. We also talked about orchestrating a classroom discussion by anticipating student responses, monitoring their responses, sequencing them presenting, and having students make connections between different students ideas. When I have students do a lesson out of the book while I'm gone, not a lot of work gets done. So, I decided students would work on a FAL: Interpreting Distance-Time graphs and the poster they create would go up on the wall. Here is what I left for my sub, who did a great job:
With 15 minutes to go, students were given the data tables to match up with the graphs and stories.
Students ended up losing all the glue sticks, bummer. But, no students names were written down for bad behavior, YAY! Also, I can't wait to post pictures of how some of the posters turned out. I am quite pleased!

In accelerated, students worked on a lesson out of the book continuing with exponential equations. Tomorrow will be a wrap up day where students will present to the class how they wrote equations from some of the graphs and then go over what they did.

Day 89: rigid T's & dilations. Negative exponents & y=a*b^x from a graph

Today's estimation was a tape measure me that was slightly bigger than yesterday. My favorite reasoning was if was 1 and a half times bigger so we got to do mini number talk on how they multiplied a whole number by a mixed number. 

Here are the success goals for my first class 2nd period. The goals improved as the day progressed. I did not mention reflections as a goal but students still used it as a transformation successfully which pleased me.

I'm taking an online course from west Ed called formative assessment insights and part of the class is picking a lesson and standard and developing building blocks, learning goals and success criteria prior to teaching the lesson. They suggest developing success criteria with the students. It was great. Although it took some time it was time well spent and lots of students wrote it down and participated. 

I like that we had the opportunity to talk about converting meters to centimeters and centimeters to meters in this estimation.
For each learning goal, I had students talk with their elbow partner and group about what the success criteria should be. For translating, I asked what are you SPECIFICALLY doing to the coordinates to move horizontally? What about vertically?

For the second goal, rotating, I said to accurately describe a rotation, there are three requirements. Discuss then tell me. They came up with these 3: how many degrees (90, 180, 270, and we discussed why 360 doesn't matter because the shape ends up right where it started), about what point, and what direction. I asked them to be specific, is there only "one direction?" Pause.... laughter. The students understood the pun and appreciated. They laughed, smiled, as I did.

In this class I got to give them an improvised joke / pun about the direction of a rotation. 5th period got the joke too.
In the last two periods I remembered to include the word scale factor when discussing multiplying both coordinate by the same number, the scale factor.

I had students independently and silently setup their graphs and graph rectangles A and B. Once it was setup they discussed whether the two figures were similar and why. Then they devised steps to transform rectangle A into rectangle B. Halfway through their work I gave them a hint, trying to relate working backwards with equations, with working backwards with their transformation steps. I reminded them that they could start with their end result and undo their dilation. Christian did that here:

Christian undid the dilation by dividing all of B's coordinates by 2. Students were surprised that it ended up overlapped, and not inside the bigger figure.
This group decided to dilate by 2 first, then translate, rotate, and then I believe they reflected across the y-axis. Ethan showed his work here:

I like how the steps are labeled here.
In third period, Mikaela and her group rotated first, then translated to the MIDDLE of the bigger figure, and then dilated. A handful of students understood that it had to be in the middle before dilating anticipating that it would expand out.
Clear diagram with easy to follow instructions.
Here you can see the success criteria and the steps easy to follow.
In accelerated students picked up where they left off. They were making sense of raising to the zero power, to the -1, and finally to the -2. They realized that 1/2 to the -1 was 1/2 divided by 1/2, then divided by 1/2 again. Then they realized that it was the same as taking the reciprocal. Then they reasoned that raising to the negative 2nd power is taking the reciprocal of your base, then raising it to the positive 2nd power.

In the upper right hand corner we discussed their first graph that they had to write an exponential equation of. Students saw that from writing a table of the two points, they could find the multiplier, 7/5, which is your base, or b. The a value, in y=a*b^x is the y intercept, or whatever y is when x is 0. Groups and students within their groups were working at many different paces in this activity.

Examples and generalizing the rules.
The last part of the lesson is a super important learning log. Make up an exponential equation with two coordinates and explain how you find the equation of it. I want students to take pictures of these eventually, blog about it, and then tweet it out so their classmates can see it.
Harrison made a super complicated example, but his explanation seems pretty good. It will be easier to read when I have them type it up in a blog entry.

Monday, January 25, 2016

Day 88: Scale factors & exponential decay (negative exponents)

Today's estimation was Day 81 of Estimation 180, estimating the length of the small tape measurer. I like this sequence because it builds on the previous days. Also, coming up is estimating length of books. I'm especially looking forward to estimating length of songs.

Today I started class by asking students what activity they did with the cut up shapes on Friday. Students remembered we saw which of the figures were similar to the original shape. They remembered that shapes D and G were not similar. So, I asked them to think pair share about what must be true for two shapes to be considered similar after only about 3-5 students raised their hand in each class. They came up with:

I like how one student knew a regular tape measure had about 25 feet, so since this one was small it was about half of it.
Then students found the scale factor between 2 shapes.
They saw that the sides of Q were all multiplied by 2.
Here we had a discussion about a new shape, shape R with a base of 25. They had to figure out the scale factor.

Students talked about how 10 times 2 was 20, and 10 times 1/2 was 5, and 20 and 5 was 25.
In 5th period we had a richer discussion where students found the vertical height of shape R when the vertical height of shape P was 6. They saw that 6 times 2.5 was 15. And they compared it to shape Q and said 3 times 5 is 15.

In 2nd period students had confusion about how to multiply a whole number by a fraction, and I'm sure there were students confused in other classes. Some tried cross multiplying. I discussed how cross multiplying is a strategy for solving proportions. What some of them meant by that was cross canceling or cross simplifying, which is a valid step to have an easier multiplication problem. Unfortunately, some students even confused multiplying fractions with dividing fractions by thinking you had to invert one of the fractions. Nope.

2 strategies for multiplying fractions
The last problem was predicting which type of fraction scale factors would make a figure smaller or bigger. They reasoned that a fraction greater than 1, or when the numerator was greater than the denominator, would make it greater than 1. They said that a fraction less than 1 would make it smaller. I asked them if a scale factor of -1 would make it smaller. Some thought about for a bit, but some students raised their hand and quickly added the fraction is less than 1 but greater than 0. I discussed how precision of language is so important.

I forgot to take a picture before erasing but didn't erase this students reasoning completely. She reasoned a tape measure is about 3 inches around. It would go around about 50 revolutions. 50 times 3 is 150 inches. Then she divided by 12, got 12.5 feet and rounded down to 12 feet.
In accelerated students worked on an exponential decay with half life. Students also worked with finding an exponential function when x=-1. They have some experience with negative exponents, but this is the first time they saw it in a real context. They eventually see that raising a fraction to an exponent of -1 is the same as taking the reciprocal of it.

When the x is zero y is 100. So, 100 is the y-intercept or A value in the exponential. To get y when x is -1, you divide 100 by 1/2 or raise 1/2 to the -1 to get 2, then multiply by 2, technically the same operation.
After a Mathography where a student said she likes writing with her 3D pen, I was curious what she had made with it. I don't know if she made this especially for me, but she gave me this rainbow with clouds. Pretty cool!

Friday, January 22, 2016

Day 87: similar shapes & exponential decay

Today's estimation was a followup to yesterdays. It was how heavy is the money chair. Unfortunately, students were a bit all over the place. I like that some students used their own weight as a reference. I also referenced a bag of cement, which some students knew was around 50 pounds and pretty hard to carry.

Today's lesson is one of my favorite hands on lessons CPM has for similar figures. Basically, students need to compare the lettered shapes to the original shape and justify which are similar and which are not and why. They are prompted to look at the angles and side lengths. They come to the conclusion that there are two requirements for shapes to be similar: congruent corresponding angles and parallel corresponding sides. They also are starting to see and will recall that their side lengths are also proportional to each other by a scale factor. A few thought a scale factor that would make the shape smaller would be a negative figure, while others said it had to be a fraction between 0 and 1 to make it smaller. Here are the card sets:

Students compare by putting the shapes on top or under the original shape to get a closer look comparing.
Some estimation reasoning.
In accelerated we reviewed the lesson from yesterday. I made a T chart on the board and asked volunteers to tell me differences between simple and compound interest along with the equations from yesterday. Here's what they came up with:

I like how this student compared the weight of the chair to their luggage on an airplane trip.
Then students completed trials in the penny lab. They put 100 pennies on their table, spread them out, and then removed the ones with tails side up. They recorded how many were left, shook those ones up, and then removed again.

On Monday we will finish up discussing it's discrete graph and the equation. Then students will work on half-life problems.
Students in action.

Day 86: undoing dilations & compound interest

Today's estimation was how much money is the chair made of. A lot of students underestimated this. Also, some students assumed the coins were quarters, when they were half dollars.

In class students looked at a shape that was dilated by a scale factor of 4, and wrote down how to undo that dilation and plotted the points. Then they plotted points of a shape that had been dilated by multiplying all the coordinates by 1/3. Most realized they had to then divide by 1/3 to get the original shape, and many forgot how to divide fractions. Some students made the jump that if you multiplied by 1/3 to get it, that means you divided by 3. So, the opposite of that is multiplying by 3. Some students did not see the connection between dividing by 1/3 and multiplying by 3 being mathematically equivalent.

During our closure discussion, I talked about 2 whole pizzas that I baked. I was really hungry so I didn't want to make small slices so I cut the pizzas into slices that were 1/3 of the whole. So, 2 divided by 1/3 is literally how many 1/3's are in 2 wholes? They clearly saw that it was 6. I think this illustration helped students make the connection stronger.

Students recalled that dividing by a fraction is like multiplying by the reciprocal.

In accelerated I passed back assessments. Some students struggled with coin problems. Some struggled with writing the equations. Others could not settle on a strategy to use to solve it. I've highlighted some of the strategies I observed below:

This student multiplied his 2nd equation on the left by 4, and then used that to eliminate the q variable, then solve for d.
This student isolated the variable d and then used substitution. With all those decimals, it was a lot of work. Could have been easier to eliminate the decimals first.
This student did a LOT of work to solve the problem. First they used fractions and multiplied by 10, which didn't get rid of the fraction 1/4. Then solved for y. Then used substitution. Eventually, it got him the answer. Was it the easiest way? I think not but it worked.
In class students worked on comparing simple and compound interest. Many didn't get to the graphing part, where it's the first time they graph a piecewise function. I think I am going to show them what that looks like and then move on to the next section today.

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.

Tuesday, January 19, 2016

Day 84: intro to dilations & y=b^x re-intro

Today's estimation was the height of Thomas Jeffersons statue at his memorial. Most students rounded my height up and multiplied by 3 and got pretty close. 

I was absent for my 2nd period class to be at my wife's ultrasound appointment. It's going to be a girl!

Students graphed a quadrilateral and multiplied both its coordinates by 2 after predicting what would happen. Most students said it would get bigger but less kids said it would also move right. Some said it would move up. 

Students were demonstrating cutting off a piece of the trapezoid and fitting it to the bottom to make it a rectangle.

Part d asks students to compare the side lengths. They easily saw the sides became twice the size. Some didn't raise hey had to count the side lengths of the squares of the grid paper.

When discussing the word dilation I asked students what happens at an appointment with your eye doctor. They talked about various tests which I said is the eye strength. Eventually one student suggested they got eye drops that dilated their eyes. So, I asked them what happens to your pupils when that happens? They said they get really huge. Well, the original quadrilateral is like your eyes without eye drops. With eye drops, your pupils get bigger, which is like the bigger quadrilateral.

I showed students how the same type of arrow signals that the pair of sides are parallel to each other.

Once they had to figure out area, only about 1/10 of students in each class knew the name of the quadrilateral. Some were able to see it was made of s rectangle and triangle. They struggled a bit recalling the formula for area of a triangle. Some realized again that it was half of a rectangle if you doubled the triangle.

This student subdivided the areas then added them together.

Some students remembered the formula for a trapezoid. Most students were familar with the isosceles trapezoid that is from the pattern block set. They remembered it's the red one. They saw the similarities that 1/2bh has to 1/2h(b1+b2). I showed them how the formula was derived, by basically duplicating the trapezoid and flipping it upside down, it formed a rectangle.

On the right you can see a quick number talk on an alternative way to multiply 19x12. In this class, students asked why it the area quadrupled. I showed an example of a square with side lengths of 3 and 9. It showed that each side was doubled, and when you multiply 2 times 2 you get 4, the factor the area increased by.

In accelerated students were re-introduced to exponential equations, in the form y=b^x. Students remembered that exponential decay was when you repeatedly multiply by a fraction and your line decreases. They then were able to differentiate between bases of 2 and 3, saying that 3 increases faster.

Then I played this awesome Desmos slider by pressing the play button and it would move from -10 to 10. I asked students to notice and wonder as much as they could. Here's what they came up with:

  • Students noticed that when b was negative, it disappeared
  • Students noticed when the graph appeared it always had a y-intercept of 1
  • They wanted to know what it looked like for b=1 and reasoned it must be a horizontal line
  • a student asked to slow it down to see the changes better
  • students reasoned that when b equals zero that it is a horizontal line from 0 to the right, and that it couldn't be negative because anything divided by zero is undefined
Students asked me to stop it at b=1, but I said no. They had to investigate it. So, students invested when b>1, b=1, b=0, and when 0<b<1. It was a great investigation.

Then I had students re-take a coin problem about systems that had a typo on Friday. Also, they got new seats.

I ordered stickers and a Desmos hat and got some pencils thrown in. I also ordered some tiling turtles for my nieces birthday and an adult coloring book. Christopher Danielson makes the tiles and he is also the author of the book I reviewed on here, Common Core math for parents for dummies. Great book.

My decagon
Weird star looking thing.
Mr Roboto, sort of.