Thursday, May 26, 2016

Day 163: Slope FAL & Pool Draining Task


This was my last day before my paternity leave. I am out Friday through the rest of the school year. The last estimation with myself and not the substitute was the value of a bowl of quarters. Students thought about 8 rolls were in there with each one valued at $10 so it's worth about 80 dollars.

Students will be matching points, equations, and lines on a poster while I am gone. It's a MARS MAP lesson called Points, Slopes and Equations. I passed out the pre-assessment and it was clearly tough for students to engage with finding a missing coordinate of 2 points knowing the slope.

Basically students find missing side lengths using scale factor or proportions between corresponding sides (gotta emphasize that) of similar triangles. Then they figure out if 3 coordinate pairs are all on a straight line. They realized they needed to make slope triangles again and calculate the side lengths. Then they compared the slopes and realized they that one line segment did not have a slope of 3.

Then they find a missing y coordinate when given an x coordinate and another coordinate pair. Students had difficulty connecting writing an equation using the slope formula to undo it and solve for the missing variable.

I'm hoping students recall the vertical change / horizontal change when working with pairs of coordinates. They will need to reason about this when making matches on their posters. The substitute will have the answer key and dialogue highlighted.

Finally, students that didn't finish surveying classmates for their 2 questions for their 2 way frequency table did that. I think that next year I will go into more detail about relative frequency and the percentages within the table. Then when figuring out your hypothesis, you must use conditional frequency, where you make the sample space using one of the marginal frequencies outside the box.
Here are the three different calculations students used to figure out how many hours it would take for the pool to drain. 
In accelerated I used Kyle Pearce's Pool Draining task. As you can see above and below this paragraph, one student plotted the data on a scatterplot, drew a line of best fit, and not pictured, calculated the slope and wrote an equation. He got 57.4 hours. One student plotted after I showed him he could use the Table feature on the Desmos iPhone app. They then typed the linear regressions equation " y1 ~ mx1+b" to make a prediction.

I liked that students used linear and exponential models. I didn't get a picture of the exponential model, but students wrote an equation in y = a*b^x form with the average multiplier which I believe was around 0.975.


Here are the substitute lesson plan binders. I will be detailing what is in these for students and parents to see very soon.

Wednesday, May 18, 2016

Day 161: 2 way frequency tables & measuring the center

Today's estimation was the value of a bowl of nickels. It's a great followup to estimating the bowl of pennies. Students reasoned that nickels are bigger and thicker than pennies so it had to be less than 450 coins. So some said 50 less so 400 nickels than multiplied by 5. 2000 cents 20 dollars. Some students thought of it has instead of 9 rolls of pennies maybe 8 rolls of pennies. I don't want to ruin the answer. Also one student thought of it as layers and I took that opportunity to relate it to finding layers of a circle bases to find the volume of a cylinder. Tomorrow is dimes!!
I started by reviewing the trashketball lesson. I had a student volunteer and closed it out by rewriting it on the board after students broke down the clues and the steps. 

To launch 2 way frequency tables I told students I had a hypothesis. I predict that students who play an instrument are more likely to play a sport. Not many students believed this. Many students play a sport and not an instrument. But in general in most classes the majority of students who played a musical instrument also played a sport except 4th period. 

This task was adapted from Illustrative mathematics. I made half sheets of class rosters. I asked students what 2 questions I had to ask to everyone in the class. I labeled these at the top of 4 columns with yes no yes no below it. Then I had students think of the answer to the first and second question and to be prepared to go down the list. A few hiccups a long the way in one class but we got our data. 



I asked students if it was easy to see if my hypothesis is correct. Some said yes some said no. This is when I introduced the definition of a 2 way frequency table. I showed and will reiterate the differences between joint (inside box) and marginal (outside box) frequencies. Tomorrow we will talk about relative frequencies to further analyze it.

As the standards say below students will need to describe a possible association between the two variables. 


2nd period definitely backed up my hypothesis.
3rd period. 9/12 musical players also play a sport. Pretty good results, looks like an association.
This is 4th period. They were the first class whose data did NOT back up my hypothesis.
5th period. All 16 musical instrument players play a sport! Money results!

Then I asked students to think of 2 questions with yes or no answers to help you figure out an association. Students took their time but came up with some good hypotheses (some commented how it felt like science class). 

People that have a cell phone are more likely to be on social media. 

Students who play video games tend to go to sleep after midnight. 

People that have an iPhone also probably have snapchat. 

People that have Instagram in this class and people who follow me on Instagram. 

People that are girls are still just as likely to play a sport than boys. 

Tomorrow we will revisit and make sure students know what joint, margin, and relative frequencies are.

In accelerated students gave feedback on each other's posters. I am going to post a standings where students post their iqr or interquartile range to see who was the most consistent. (Lower the better because smaller range in 50% of data)

Then they looked at how the median of two city price listings could be the same but their means could be much different. The center can be described in 2 ways and some situations one is more effective than the other. 

Then they looked at 2 medical procedures that detailed response times to an emergency. One was consistently under 15 minutes, while the other was a little more spread out. Students made arguments for each case.

I thought it was funny that the student wrote "units!" with an arrow. I'm pretty sure they didn't want points to be taken off! Standards of mathematical practice: attend to precision.
I found this great link to 2 math board games here http://blog.mindresearch.org/blog/big-list-mathematical-board-games. The last 2 or 3 days of school I will have the substitute make these available to the students. I printed out the first 2, Achi and Dara.
I tutor a 10th grader and these are the topics in Algebra 2. As teachers have been discussing on twitter, it's a disjointed mess of topics. I feel the Common Core standards
Regarding 2 way frequency tables, coincidentally, I saw a few math teachers discussing it. Next year I'd like to incorporate this as my notice and wonder to immerse students in the power of the data display. I do want to know what the situation is about first because I'd hate to leave them without knowing the context.


Monday, May 16, 2016

Day 160: Volume, Trashketball, & Combo Histogram Box Plots

The estimation today was a glass of pennies. It has 3 angles and 1 of the photos shows 1 roll of pennies fitting diagonally at the bottom. I encouraged students to be riskier in their too lows and asked the class why students were putting 50 cents or 1 dollar as their too low. For the most part a majority of students underestimated.

Most students estimated how many rolls would fit in it. A few students talked about layering. During the video they were all guessing how many it would fit and lamenting how low they estimated.

Then I went over the lesson plans for the rest of the week. I then reviewed that they meet in different classrooms Monday, Tuesday, and Thursday of next week according to the schedule on my door. On Wednesday students will meet with the sub in H-1 and do an estimation followed with a lesson on the SBAC topic to make sure they understand the vocabulary of the task. Then Friday they will have a lesson. 

To start class I gave students time to work together on solving the volume for a cylinder and a separate cone. In class discussions I asked students what we discovered with the sand demonstration last week. How are a cylinder and a cone related? At first some students said it was half as big. Then they remembered that the cylinder with the same base and height fit 3 cone fulls of sand. Therefore, students could pretend the cone was a cylinder and then divide their answer by 3 or multiply by 1/3.

Then we launched into Mr. Stadel's Trashketball lesson. EVERY single class a few people said, "Why is he wasting so much paper?" I ended up responding, "for the sake of quality math situations!" Students figured out it was how many trashketballs fit in the trash can. They asked for the diameter and the height of the cylinder. Regarding the sphere, students thought you'd have to find the circumference. Some students realized you could put it on a ruler and measure the diameter. Next year I'm going to have them actually reuse a crumpled ball and measure it themselves.

I like how the task gives students a page of formulas with pi and they have to discern which formula to use. I like how some students pointed out the sphere's volume was the only one with an exponent of 3, so it must be volume. I wonder how they could explain why the cylinder's volume doesn't have an exponent of 3.

They then basically divided the volume of the cylinder by the volume of the sphere. This theoretical answer is off by double digits and makes students think why. In some classes we didn't reveal the answer or have time for a student to present so we will launch class with that tomorrow.

Warmup with cylinder and cone.
Student volunteers dictated to me how they solved. I started with this question: "What advice would you give someone who did not know where to start with this problem?" Most students said you needed to know the volume was the area of the base * the height, and the area formula for a circle.
This is the only student work sample I snapped a photo of. Will take more tomorrow.
In accelerated students finished putting together their combination box plot histogram of their 40 golf shots. I was impressed that without being prompted 1 group made a stem and leaf plot that helped them figure the max, minimum, and median easier.
Steam and leaf plot
A bin width of 20 worked pretty well for this group! The team member asked if the shape was skewed left. Yes! I could instantly see by this and the box plot that their team was pretty consistent overall! Their median was around 40 centimeters away from the hole.

I think this student is making sure he doesn't get points taken off for forgetting the units. SMP: attend to precision!

Day 159: Promotion Field Trip, Subbing

I did not meet with any of my classes. I was going to chaperone the 8th grade promotion field trip to Great America, but I figured it wouldn't be a good idea to be stuck in Mountain View on a bus if my wife went in to labor. So, I substituted Mrs. Fiore's 6th grade Math and science classes all day. They corrected homework, took a test, watched a Bill Nye video on volcanoes, and then worked on a math sheet on finding side lengths of a pen when given 120 feet of fencing. We had some decent discussions.

Thursday, May 12, 2016

Day 158: Notice, Wonder Circle Review & Statistics Posters

This estimation was one of my favorites of the year. Today students estimated the value of the complete cent sign on Day 150 of estimation 180. It's an incomplete cent sign made of pennies. I was impressed when at least one student in a few of the classes looked at the vertical line of pennies was along 5 tiles. They counted 8 tiles in it, and estimated it was 1/5 of the whole line. Therefore 8 * 5 is 40. Then they said there was 20 in the curved top part, and it was 1/3 of the curve. 20 * 3 is 60, so 60 plus 40 is 100. I thought that was awesome thinking. Students also saw it as a reflection so they doubled what they saw and then added more.





Circles are a 7th grade standard and I feel students need practice with it, especially if I expect them to be successful with finding the volume of cylinders and cones. So, here's the plan:
Start by seeing what students notice about the following image (image not mine, attributed to godfathers.com) :


  • **UPDATE**Students noticed so much from this image. They thought it was Al Capone. (It was actually a pizza chain caleld "Godfather's Pizza" I randomly found by Googling pizza sizes
  • Students noticed the number of slices increased by 2 slices as the size increased (!!)
  • all of the numbers are even
  • the circles all touch at the same point on the left
  • the image is half red and half black
  • the inches don't grow at a constant rate. Mini to small is 4 more inches, so is large to jumbo. The other 2 increase by 2 inches
  • a medium is twice the slices and size as a mini
  • concluding that when you think pizza, think circles, and think they are measured by their DIAMETER

  1. Hopefully students notice that the size of the pizza is measured in inches. What part of the pizza is measured in inches? The diameter!
  2. Draw a circle with a diameter of 3 inches. If we know that, what else do we know about the circle?
  3. Solve for circumference, then show this Geogebra circumference visual proof: https://tube.geogebra.org/student/m19655 
  4. Then ask students to find the area of that same circle.
  5. Then show students this area visual and discuss the questions that are asked: http://tube.geogebra.org/m/1845061 
  6. Reveal answers with this visual: http://tube.geogebra.org/m/1844529 
  7. Then find the volume of a cylinder with that same circular base, given a height.
  8. Then mathography, and then assessment.
We didn't get much time to practice finding volume of cylinders but we calculated circumference and saw how it was represented on the coordinate plane in the first Geogebra applet. Then we investigated the 7 questions here http://tube.geogebra.org/m/1845061. Unfortunately it labeled the circumference but students realized it was similar to this first applet as the first outer layer of the circle peeled off. Thank you Kate Nowak for tweeting it out:


Students enjoyed the animation and I feel that triangles are a bit more accessible to students then parallelograms.

My TA's hung up the Wheel of Theodorus projects from yesterday and many students were gazing at them into lunch. I'm glad to see the students put a lot into it and like to see how other people customized it and showed their work. 

In accelerated we discussed types of data displays. Some thought possible a scatterplot with their golf data. Then students realized that was 2 different variables. I did see one student graph the distances with the shot out of 10 as the x axis.

Students reminded me of the 5 important numbers of a box plot. I also asked what other types of ways can we evaluate the data besides the median?
These are the situations we did not discuss yet. Any ideas how to spice of volume for these?

Day 157: Volume of Cone & Pyramid, Penny Golf, Wheel of Theodorus Examples

The estimation was the value of a roll of quarters. I liked how a student related it to a real life experience. She needed to get quarters for laundry so her mom gave her $20. The cashier gave her 2 rolls of quarters, so she reasoned that one roll must be $10. Other students said 10 groups of 4 quarters. I heard a few estimate $7.50 and only a couple over estimated. Students realize it must be a round number in whole dollars.
Students analyzed a small and large container of popcorn. The small was a cone, and the large was a cylinder. The small was $1.50 while the large was $3.50. Some students thought the large was double the size. Then we did a demonstration. Students saw and wrote down that the cone and cylinder had the same height. I asked how we could see if they had the same radius. They said you could put the bases together and see if they match up. They did.

Then I took estimates of how many cones of sand would fit in the cylinder. estimates ranges from 1.5, to 2, 2.5, 3, 3.5, and some even 4. Students were intently watching as I poured and saw it filled it 3 times. So, I said that if you know the volume of this cylinder, do we know the volume of this cone? Students said to divide by 3 or multiply by 1/3. Great.

Then they figured out the "fair price" of the large popcorn, as well as which one was the better deal. Students multiplied the small price by 3 to get $4.50 for a fair price for the large. This is also told them that the large was a better deal because it was a dollar cheaper than 3 smalls. Only a few students reasoned that if you divide 3.50 by 3, you get around $1.17, which is cheaper than the price of the small cone full.

Then they predicted with a prism and a pyramid. Some still thought the prism was twice as big as the prism, though some assumed the relationship would be the same as the cone and the cylinder.
Our sand and the materials I ordered for our department from Amazon.com.
In accelerated we went over the last problem from yesterday on inverse functions. Davin showed how we could solve for the x variable, and then switch the x and y variables at the end. This was a great method after reasoning with it yesterday.

Then we read about the 40 holes of golf task. I set up 8 golf holes with orange duct tape outside that were 200 centimeters away from the cement line. Each team member took turns and 1 threw 10 pennies 1 at a time. Then they worked together to measure their distance from the hole as they picked them up. Then they repeated so all group members went. Tomorrow they will organize their data using box and whisker, bar graph, histogram, or any of their choices. I'm going to push students to go for the ideal which is the combination box plot and histogram. Students will have to convince the rest of the class how consistent their team was and we will decide who the winner is based on how convincing their argument was.
Beautiful day to be outside collecting data.
Students enjoyed themselves.
I graded students Wheel fo Theodorus projects. The requirements were: first 5 triangle calculations, all sides labeled. Also, hypotenuse labeled with an I for irrational or an R for rational. Colorful or designed (1/2 a point). I also deducted 1/2 point if there were multiple triangles that were clearly not right triangles. I walked around the room when they started and emphasized the legs had to be perpendicular to form a right angle if you were using the Pythagorean theorem. Here's some of the work that stood out to me:
My niece would love this one, she's a huge Frozen fan. My brother's family dressed up as them for Halloween.

A patriotic student.
Cool colors.
This is a popular way to shade nowadays.
Taste the rainbow.
A peacock!
Pretty funny!
This one really spiraled around! Cute cat... cheshire?
Very creative... a "cornucopia."
Nice dress.
I like how this student decided to color code the triangles with rational hypotenuses.
A star wars fan!




Tuesday, May 10, 2016

Day 156: SA & Vol of Cylinder & Inverse Functions

Students estimated a roll of dimes. Some students said it was as many pennies because of the thickness so 50 dimes. They multiplied that by 10, to get 500 cents.

The subject matter was surface area of a cylinder as well as the volume. I took the opportunity to review what we discussed on March 14th, pi day. I asked students what we did after they traced a circle and were given a piece of string. They remembered they measured the outside of the circle, the perimeter, also known as the circumference. Then they measured the diameter and compared. They saw then that the circumference was about 3 times the length of the diameter, which ended up being pi or approximately 3.14. Then I asked them what we figured out after folding, cutting the circle, and rearranging it. They said the area formula which was pi times radius squared.
Re-establishing background knowledge and showing how to sketch a cylinder.
Then they worked on the surface area. They had to first interpret the lateral surface area as the shape of a rectangle and multiply 52 by 40 to get 2080. I discussed the mental math strategy of multiplying by 4 which is the same as doubling twice. Then students realized the circumference of the base was 52 inches. Some realized they needed to reverse the circumference formula to find the diameter by dividing 52 by pi or 3.14. This resulted in about 16.6 inches. They then divided by 2 to get the radius, and then found the area of the circle. They doubled this result because there were 2 circular bases and added that to the lateral surface area they found.
Lots of steps and labeling.
I love Darren's idea with his composition book. Fold the page after finishing that page to instantly know what page to flip to the next day. Love this ingenuity.
Had a mostly civil argument with a student about approximating irrational numbers. The student thought that for approximating 51 it was 2 away from 49, and 51 was 13 away from the higher perfect square 64. The 2 somehow meant .2 in the answer.
Through carefully chosen examples, I showed that the square root of 66 would have a large denominator with the wider range between the perfect squares. 66 is 2 away from 64, and 64 is 17 away from 81, so it's 8 2/17. When you convert 2/17 to a decimal it gets you .11 which rounds to .1, not 0.2.
More work.
In accelerated students were introduced to inverse functions. First they examined this function machine and described the pattern in words first. It was multiply the input by 3 and then add for. The equation was f(x)=3x+4. Then in part c they described the inverse of the function, which students said was basically reversing the order of operations and changing to the inverse operation each time. So this was subtract 4 and then divide by 3. They understood the input would be subtracted by 4, and the whole term would then be divided by 3.
Using inverse function notation.
Students came up and showed how they wrote the inverses of each of these functions.
After class a student wanted to review how to graph a quadratic equation as well as solve a system of linear and quadratic functions algebraically.
In periods 2 to 5 I collected the Wheel of Theodorus project. Some of them turned out astounding so I will be posting photos of those tomorrow. These were the requirements of the poster which I used to grade them.