The times when I notice the changes that are happening in our craft over our implementation of the common core standards do not occur with a frequency I would hope for. One of those moments occurred as I am cleaning out my digital closet: I found some notes I had prepared for my algebra students my first or second year of teaching. The standard stated

19.0 Students know the quadratic formula and are familiar with its proof by completing the square.

What I did: Under the guise of direct instruction I wanted to give the kids an “easy” to follow, step-by-step to start with the homogeneous set of a quadratic equation in standard form and through the process of completing the square to arrive at the quadratic formula. So, I created a fill in worksheet that the students and I went through, then I gave them the mathematical equivalent of sentence frames to reproduce the exact steps we had just covered. I would then reward the students for being able to fill in the blanks and follow the brightly illuminated path to solution.

The predictable outcome: The notes and student work look like this

Each step was part of a PPT presentation and the students were to copy those steps and listen to the rationale of those steps. This kind of teaching makes me shutter at this point in my career, the cold detached nature of completely irrelevant math facts, in a complicated manner to find an equation that the students either already knew or could find in a few minutes on their phone. The blank stares, the constant yawning, the glossy eyed students staring back at me as I dutifully taught the lesson in the manner I had been instructed made me think I was doing my job right. At one point, I was even complemented on the lesson and the “good lesson” I was giving on this difficult standard. This problem struck me then, as it does now, how sterile math had become, and this is the example I often think of when “I teach high school math. I sell a product to a market that doesn’t want it, but is forced by law to buy it,” said Dan Meyer.

What is this approach missing: I usually don’t like to repeat the same negativity without offering any solutions, but this one is screaming at me so loud I do not hear new thoughts creating in. The biggest problem I have with this particular piece is the lack of student buy in to the problem. As I stated earlier, the sterility of the presentation of the problem and the lack of student experience to make this a relevant problem are slap you in the face obvious.

What I am missing: The problem I have had since digging this up is creating a relevant scenario that would get the students to buy into how we start with one and end up at the other. The idea I feel coming on is literally a map with the steps are the part key you must solve to get the next piece of the puzzle, and creating a positive interaction to see where kids stand…however, this isn’t a context where the math naturally arises, so it looses a lot of its power. I am coming up blank with a good idea on a way to make this happen, but I will keep looking for ideas.

# Fun with problems

I love a good problem, I love when a problem has the barest essential information and the problem challenges you. I would love to get to the point some day where I am making problems like these to challenge students, but for now I am looking to capture as many problems as I can. This problem posted in a forum and I dug into it for about fifteen minutes, solving it one way and look for alternative solutions…which I’m still looking at.

The goal is to determine the area of the shaded region with four inscribed circles tangent to one another. My solution is in general, I didn’t plug in the unity of the radius, r, but that is a trivial matter.

Solution: The first step seemed obvious which is to write the equation of the shaded region based on the difference of the larger circle with the four inscribed circles. The solution is thus,

Area = pi*(R^2 – 4 r^2)

where R is the radius of the large circle and r is the radius as shown in the figure. The problem may be extended by writing R=R(r), that is R as a function of the radius r. This was accomplished in the straight-forward way of using some geometric facts. Constructing a square from the center of the large circle to the points on tangency with the two smaller circles, we have

where the red line is the larger radius R, and the sides of the square each have unit length r. Since we have a right isosceles triangle, we know the hypotenuse of this triangle (or the diagonal of the square) is sqrt(2)*r. The length of R is the sum of these two lengths, namely

R = sqrt(2) r + r = (sqrt(2)+1) r

Substituting this value into the equation for the shaded region and solving we arrive at the desired solution.

Area = (2*sqrt(2) – 1) pi r^2

I am challenging myself to find at least two other methods to solve the problem, I wonder what I’ll find as the journey is always one of the best parts to a resolution of a problem.

# Dropping the Density Knowledge

What we’re doing: We are having the learners build hot air balloons. We are also teaching the mathematics of density, buoyancy, and basic force analysis.

Where we’re at: We had our second lecture in mathematics today covering the topic of density. I had some trouble coming up with a lesson that would offer the students access to the concepts, and allow them to compute the problems.

The big idea: I wanted the learners to understand that density may be thought of as proportional reasoning. Since our target age groups are 5th to 8th graders, I wanted to ensure that all learners could access the information. Having success at this age is vital for these students to grow in their desire to learn and to push them beyond the classroom experience. The framing of proportional reasoning will (hopefully) lead the students to see the connection with how differing densities create the buoyant force their balloons will need to fly. The computations of this idea with having the students compute how big a volume they would need to displace for flight, and relating these concepts to time of flight may give students a concrete foundation, especially as they build their balloons.

The student responses to the in class questions:

11/9/2013 Lab School – Density Day 2

What are some of your observations?

1. The right rectangle has more dots than left

2. The right one is a solid and the left is a gas.

3. The left one is a gas, and the right is a liquid.

4. There is a lot of dots on the right

5. They look like atoms.

6. Two squares of dots

Which one is denser?

The box on the left is more dense because the particles are closer together.

How did you order your layers with their densities?

1. The different layers are a result of different densities.

2. I organized the layers by largest to smallest density.

3. The layers with 5 and 6 g/cm^3 are close and will mix – resulting black color.

11/9/2013 Lab School – Density Day 2

What are some of your observations?

1. The dots are closer to center and more on right side.

2. More dots on the right than left.

3. On the left one the dots are spread, and the right they’re getting closer together.

4. On left look lighter in color, the right one looks it could be solid and the left could liquid or gas.

5. It means that world could end

6. There are different states of matter, liquid, solid, gas. The one of the right one could be a liquid, the one on the left could be a gas because the molecules.

Why do the liquids separate?

1. You would have to try in many ways to put in the exact order. You would have to know the densities to know how order them.

2. We learned about different liquids, which means different densities, which means the black part separates to different liquids.

Final thoughts: When I ran the lesson past my wife last night I felt the lesson would go quicker than it did, realizing that I was thinking of spending less time in the discovery phase. If I think about the lesson in terms of the MVP cycle, this was a Developing Understanding task, my hope is to get into a Practice Understanding Task by the time they launch their balloons. The lesson in eighth grade was great in the sense this was the most engaged I had seen this class, and the students seemed to really be more interested and trying harder…though their understanding of the material was lacking. The fifth and sixth graders probably walked away with a better conceptual understanding, but I will be very surprised if any of them are able to translate the new concept of the tape diagram and the proportional reasoning together to problem solve. I felt the lesson was better than those I have previously designed and given at the lab school, but students will need more time to truly grasp it. I am also still struggling with talking so much, I want the students to do more of the work, and the amount of talking I am doing seems too much. I am wondering if I could get away with introducing buoyancy and having the students do some computations without talking too much. I am thinking of the lesson I have already prepared and maybe I need to work on removing some of my time and figure out a way for the kids to work through the material…That is my new challenge, I’ll probably post on that this week as a before and after piece as this is a huge part of my desired growth and change.

If you have any ideas, suggestions, or comments please let me know.

# 26.2 X 2

Finished my second marathon yesterday. Overall I was pleased with my performance, but I would loved to have not had a nasty shin splint at mile 21 which slowed me down tremendously for the last six miles. Besides the blisters, the sore muscles, and the incredible realization of how far this is, I am very happy that I was able to finish, especially considering how weak I felt going into this race with all my long runs leading up to this feeling weak. Well, as I sit in the glow of the week long high, I am thinking about how to improve my performance and how excited I will be with winter on its way and those wonderful long runs returning. Cheers.

What we’re doing: We are starting a four-week project in our math and science lab school.

Goal: To use a real life example to frame the learning around a curriculum between mathematics and science. The learning goal is to introduce students to the scientific ideas of density, Archimedes’ Principle, buoyancy, surface area, volume, and the force of gravity. The mathematics is the language and the tool by which we are able to quantify each of these scientific ideas. For example, we will show that proportional reasoning and ratios help us to define and describe density, extending that description to explain concentration gradients and how that relates to buoyancy and Archimedes’ Principle. In addition, we would like to show that relationship between surface area and volume, how they relate, and how we use them in terms of density, buoyancy, and Archimedes’ Principle.

The project: The four-week project will have the students learning about the scientific principles as the students build miniature hot air balloons, which will culminate in the fourth week with a community event in which the students launch their balloons. The students will reinforce their learning and make predictions based upon their calculations as they build and design their balloons.

What we did: In the science part the students began their construction of their balloons immediately due to the short time spread between now and the launch date. The students were introduced to density through the idea of a density column and making predictions about what would happen when common household materials were mixed together. Then the students were given the materials and instructions to start making their hot air balloons.

I haven’t finished constructing the last three mathematics lessons, but I have a general idea for the next three weeks. What I did today was design a lesson that would cache a problem in terms of generating ideas with a general concept and getting specific as we refined our focus on the problem. The first iteration I am attempting to drum up interest in the problem, allowing students a gut level interaction with their imaginations and to fan the flame of their imagination. The next step with today’s lesson was to give them an idea of where we were going, and then have them come up with ideas that would be similar to the result we are aiming at. Today’s lesson involved essentially no direct mathematical concepts or computations, but was more trying to get their creative thinking going as we attempt to problem our situation.

The lesson started off with a quick slide show of cartoons drawing were two guys are sitting on a deserted island, then there is a helicopter and two people rescued. The next slide is a backpack with text that describes the two heroes are looking into their supplies to figure out the main problem situation we will spend the next lab period really digging into which is how to communicate with a helicopter at night, so the helicopter can find us upon its return from dropping off the first two rescued. After viewing the slide show students wrote their own ideas down, then shared out their ideas with the group. A few moments later I called on anyone who had idea in the first group to get more students to take part and have their voices heard (this class is a reluctant group of learners). The second group is famously into the lesson every time, so I followed the same format, but then had each group come up with a consensus of only two ideas – otherwise we would still be their listing ideas. The class results are recorded and displayed below:

***********************************************************

What are some ideas on how to effectively communicate with a helicopter at night?

1. Break glow sticks and write SOS on island. 2
2. Spell help with fire. 5
3. Use flashlight in a strobe light fashion. 6
4. Put a bomb in the water and make it explode. 4
5. Rope and bright things and write the SOS sign. 2
6. Throw rocks at the helicopter. 2
7. Build a sling shot and shoot a fireball. 4
8. Start a bonfire. 15
9. Shoot a flare with the flare gun. 9
10. Build a wood helicopter.  1
11. Build a boat – times two. 6
12. Bang thing to make loud noise. 9
13. Make tree leave wings and fly away. 2
14. Construct a statue of some sort and light on fire.  6
15. Hug a tree.  5
16. Make noise 4
17. Swim away. 3
18. Vocal communication either yelling or walki-talkie. 2
19. Shake the tree.  3
20. Perimeter of the island lined with torches and then climb to the top of the highest tree and light another torch. 2

What was in the video?

That science teacher was here?

He was about to fly. Using gravity (that was Jimmy’s idea)

Seeing which way the wind was blowing to see what direction the current was going.

Signal a helicopter with the big floating bag.

Boy was using force flow of air to make something float.

Ideas?

Signal a helicopter with the big floating bag. (idea too)

Get a leaf blower and act like it’s jet pack.

Current of ocean to sail somewhere.

What can we build with these materials?

Make a Chinese lantern and spell out SOS with the tape.

Construct a little hot air balloon.

Tape trash bags together and make a hot air balloon.

Burn shirts, light with candle, and then reflect fire light with foil.

Build a straw poker and poke the helicopter.

Make a straw and tree pole vault and jump to the nearest helicopter and yeah. No. no yeah. No what. OMG.

Light a candle, cover it and make a strobe light, and breakdance.

What are some ideas on how to effectively communicate with a helicopter at night?

Group 1: 1. Flashlight pointed at sky, 2. Start a fire, boil water, and make beam (like Bat signal)

Shine light beam onto balloon to make a disco ball.

Group 2: 1. Break up into smaller teams with different ideas, 2. Reflect the fire with the ipad

Make a big balloon with coconut inside, light a little piece of wood on fire, and then let it fly.

Group 3: 1. Superman or 2: Or Mr. R’s ugly yellow shoes

(Missing their ideas here)

Group 4: 1. Flying rainbow pooping unicorns 2. Flashlight

Wanted to use leaves, giggle giggle,that’s her like glue bird poop glue. And tree bark that’s what she said. She said that out loud and make they could make the hot air balloon out of leaves and attached the leaves together. No that’s what shae said and then dear gonna gives up food and unicorns go woohoo?

Group 5: 1. Use glass (or smelter) when light shines on the glass and reflects it 2. Using ipad to signal with morse code or use c4 to blow up the island

Make a big balloon and get tree sap in the coconut, it’s a burnable material, and it floats. Draw SOS on balloon.

Group 6: 1. Something on fire to make a signal, 2. Yell really loud and crazy while jumping up and down

Turn balloon into hot air balloon and then fly after running really fast.

Group 7: 1. A bright banana, 2. Light their shirts on fire.

By getting air and filling it up with hot gas.

Group 8: 1. Get coconuts, light them on fire, and then throw them in the air, 2. Start a fire with leaves and glasses.

They make the hot air balloon with bright (giggle giggle) colors girl left in backpack and they need help with they are stranded in the island.

****************************************************************

Next: The next part of the lesson had students watch a short 18 second clip showing a group of eighth graders holding a large balloon made of stitched together trash bags and a student walking around with the bag. I asked the students how we might use this idea with the ideas they had already generated in the same way as the first video (I mean that I ran through the same scenario where the kids wrote first, then shared in the group, and then shared whole class). We listed some of those ideas, and then I showed them my backpack with a slide that listed all the materials that were available. I pulled out all the goodies in the bag, and asked them to create an idea using the ideas they had generated but only using the materials that were in the backpack. We then had them share out as a group, and then whole class.

The lesson finished with having students write a few things they liked and a few things they wondered about. We will take their ideas and have them try to construct their ideas next week in class. We will then teach them the idea of density (described as a ratio) and proportional reasoning as it relates to the idea of hot air and get the kids to predict how that might make a difference in how the balloons will fly. As part of teaching them the idea of density, I am thinking about using hard-boiled eggs and a glass of water. Having students predict if the egg will sink in water, performing the experiment, and then asking them if there is a way we could make the egg float. I feel that the concrete demonstration is important for a better understanding for the kids, but I don’t know if they will become more confused as we are dealing with water and egg materials in one situation and air at different temperatures having different densities though it is the same material…I’ll have to ponder this over the week as I design the last concepts for that lesson.

How the kids did: The student samples from the day showed that a few of them picked up the more general ideas presented in the lesson. The general feedback had the students talking about the story created in the opening slide show. Many students liked the open-ended approach to the lesson, which I was afraid they might be hesitant to attempt, and they liked talking in the groups. One measure of success I feel is that several students made the comment verbally and in writing that they were not sure if this was really a math class. I take this as a compliment because we are trying to design a curriculum that approaches learning through problem solving and application. Another sign of success was several students wrote on their reflection they wondered what would happen next week, so they are already curious and want to come back for more. The last positive news was several students commented how much they enjoyed the class, how fast time goes by in lecture, and how they never get bored which are all great things to hear as a teacher – I’ll not mention the numerous students who wrote comments that I am funny and how much they like me, which is always fun to hear.

What I would like to see more of: I think the lesson is a good attempt at what I am thinking, but I don’t think it’s exactly right yet. I am also struggling with the usual problems of off topic chatter and unrealistic points of view from many of the eighth graders. The eighth graders are students attempting credit recovery, so they come with a lot of negative baggage toward school and are typically the lower performers which is part of the problem. What I am hoping to see is an increase in their engagement and a general increase in their attempts to problem solve. The students are starting to see that this is more gut level and with the absence of right and wrong, the students may come to realize that education is fun and interesting again. I will find it a huge success if I can just get these kids to ask questions and maybe, just maybe, determine some ways to answer their questions – that whole curiosity thing would be amazing. I would also like to increase the rigor and depth of the problems we attack and make them a little more relevant to the kids every day experiences.

Unity for now: What I am excited about is that we have a target for the next four weeks, with a definite outcome that the kids are constructing the product. I like that we have a framework to have some valuable learning go as an outcome and I like that there is a general buzz of good tidings from the students.

I wonder if I’ll be able to get the kids to understand the chain of relationships that different densities create a buoyant force as a result of Archimedes’ Principle which if great enough will overcome gravity. I want the students to understand the relationship between surface area and volume in relation to one another and how these relate to density and Archimedes’ Principle. I wonder if students will be able to compute these ideas as they are quite advanced for fifth to eighth graders, but I wonder if they will be able to scaffold through what they are building to these mathematical principles and their representation in the real world. I wonder if we can make that a reality for these kids.

# 26 x 2

Conveniently this is exactly how I look and feel at this moment with my second marathon almost here. I have to admit I am more nervous for this one than I was my first one…or maybe because my training runs don’t feel as strong as I did last year. In either case, I don’t know if I am ready, but ready or not the clock won’t stop moving forward. I’ll just have to do my best and wish for the best as I slowly grind out another miracle. I don’t think I would have ever believed in high school that I would ever run a marathon and here I am ready for another one. Well here’s to bloody nipples, soar muscles, and that little voice that says you don’t get to take time off. Good luck.

# Exp(x)

A problem a former student asked me recently over a Skype call…apparently I didn’t prepare them well enough for the ravish of college life. Anyway, I thought it was a simple problem to get stuck on, and here’s why.

Here’s the problem

In calculus the number e is sometimes introduced using slopes. In this problem you will explore this idea.

1. Draw the graph of and plot the points (0,1) and (2,4), which are on the graph.
2. Consider the secant line passing through these points. Now, consider the slope of the secant line as the point (2,4) slides along the curve toward the point (0,1). Draw the line that you think will result when the point (2,4) reaches the point (0,1). This is the tangent line to the curve at the point (0,1).
3. Using the tangent line, and the fact that the slope of a line is rise over run, estimate (to the nearest tenth) the slope of the tangent line.

Solution: This problem attempts to develop the conceptual understanding of students by demonstrating the limiting process in a more concrete way. Students draw the connection between the slope of the secant line and how this slope behaves as the displacement along the horizontal direction becomes decreases to zero. The student should see a trace of many secant lines that approaches the tangent line in the limit as the step-size goes to zero.

What I did: There are two software platforms that I am attempting to become more familiar with and figuring out ways to make these software pieces in a lesson if I were a teacher – specifically Desmos and Geogebra. I am far more familiar with Geogebra than I am Desmos, but I chose to use Desmos to answer this question (I learn software through application than just to learn, which is the kinda my point on lesson design).

Part (a) and Part (b) I answered together in terms of the Desmos graph shown below:

Desmos Graphing Calculator” href=”https://www.desmos.com/calculator/j52bc1hvp4″>

The plot of the two points (0,1) and (2,4) on the curve. Next I created a line with a movable point that you can slide along the curve from the ordered pair (2,4) to (0,1) which acts as the secant line through any point in between (0,1) and any point (a,2^a), where “a” is the parameter along the curve.

Part (c):

Here there’s a little work to do, but nothing to bad. Using the idea of the slope of the tangent line is the rate of change at the point, we are using the linear approximation using the slope and a step-size of 0.1. The approximation becomes

m_tan = (2^0.1 – 1) / (0.1-0) ~ 0.71773462536293

If we check this result exactly we have

f’(0.1) = ln(2)*2^0.1 ~ 0.742896753756

The approximation is within the desired limit of one tenth as we should expect with a small step-size the relative error is small, even for an exponential graph, as the rate of increasing isn’t as great near the origin as seen from our results from Part (a)