Geometry

We held our first bowl team session in MTPS today: an friendly, informal competition dividing the class into two groups of roughly equal 'ability.' The bowl competition we attend at the University of Nebraska-Lincoln in November consists of games of 15 questions with 30 seconds to answer each question.

Here is a question we spent some time on in class today.

Original_Cube_Bowl_Question

 

Here is a screenshot from my Promethean board of the work we did as a group with students:

Find_Side_of_Cube_Given_Sum_of_All_Possible_Edges

And here is a screenshot from the Geogebra sheet students constructed in front of the class at my computer terminal to demonstrate what is going on in this problem. If you would prefer to download the Geogebra sheet for your own tinkering, you can find it here.

Cube_Bowl_Question

No one got this question right in the allotted 30 seconds, so we spent some time developing the various parts of the expression. We color coded the edges on the cube (segments AB, AD, and AG) orange. We color coded the diagonals on faces of the cube (segments AC, AH, and AJ) green. We color coded the segment inside the cube (segment AI) black.

The kids had a lot of fun with building the Geogebra sheet and then trying to reconcile between sqrt(2) and the crudely rounded segment length 1.41. Or the sqrt(3) and 1.732.

The pink stuff at the top of Promethean board screen shot is our efforts to generalize this question. I tell the kids we are trying to "hack all possible problems." We use the term invariant and that we are looking to write a question that covers all possible question types. For example, what would happen to the question if s = 2? Or if s=3? Or if s=n? Is it possible to answer the question before the moderator finishes reading it? If so, at what point can we be confident we can buzz in and answer correctly?

Here is a problem I have been working on this afternoon. I worked on building a model in Geogebra to help students understand the situation.

2013 Probe I Problem #3
A cylindrical can with radius of 6 inches is filled with 2 inches of water. When a smaller cylindrical can 4 inches tall is placed inside of the first with its bottom lying on the bottom of the first, the water in the first can rises a further inch to 3 inches. What is the radius of the smaller can?

A. 3   B. sqrt(12)   C. sqrt(15)   D. 4   E. sqrt(18)

Here is a screenshot of the Geogebra model I constructed.

Nested_Cylinders_and_Water

 

Here's a PDF of my typed-up solution.

Nested Cylinders Problem

Here's the Geogebra Sheet (built in the Geogebra 5.0 beta with 3-D graphing) if you'd like to take a look.

Below is a list of games (mostly $0) high school math students will enjoy.

Euclid: The Game

EuclidLevel1

Euclid: The Game, designed by Kasper Peulen, takes the basic compass and straightedge constructions featured in geometry class and makes a game of them. There are 20 levels to complete. A teacher can directly access a particular level by modifying the URL. http://euclidthegame.org/Level1.html (for example, change the "1" to a "7") I would have loved the opportunity to learn geometric constructions in this format when I was in high school.

2048

2048

The premise of 2048 is very simple. Use the arrow keys to slide tiles about the game board. When two tiles with the same number touch, they merge into a new tile. For example, when two tiles with the number 64 on them touch, they merge into a 128 tile. The objective is to make a "2048" tile, which would require two 1024 tiles, and each of those require two 512 tiles, and each of those requires two 256 tiles, and each of those requires two 128 tiles, and each of those requires two 64 tiles, and each of those requires two 32 tiles, and each of those requires two 16 tiles, and each of those requires two 8 tiles, and each of those requires two 4 tiles, and each of those requires two 2 tiles. Each move in the game yields a new number tile on a random space. A user may continue with the game even beyond the 2048 tile. I have personally witnessed one of my students with a 65,536 tile. This game has some uses in the math classroom. The obvious is knowing the positive integer powers of 2 and its parallels to data storage capacity in computer hard drives. We can also use this game to teach an introduction to game theory and optimal strategies. After playing this game for a while, a user tends to see patterns and trends, cycles of values and positions that appear frequently. Students can intuit optimal strategies through trial and error, but this could also allow students to use formal mathematics to move towards establishing optimal strategies systematically.

SET (card game)

Daily_SET_Game_Puzzle

While the card game isn't free, the daily set puzzle is free. My students avoid trying to explain SET to a novice player. It is much easier to play and learn through trial and error than to learn all the rules before doing anything... (sounds a bit like mathematics, in my opinion). Each card has four attributes - shape, color, shading, and number of shapes. A set is a collection of three cards for which each individual attribute is all the same or all different. Here are two examples of possible sets.

TwoSETExamples

 

 

 

 

There are some great combinatorial features associated with this game. A great strategy lends itself incredibly well to mathematical statement: For any two selected cards, there exists a unique third card in the deck which completes the set.

ALICE

Alice

Alice isn't a game, but rather a software package which teaches computer programming as if it were a game. Students can use previously constructed environments and characters to create animated videos. Students can use the interface to make their own games. I have used this software in my class for five years. I have had students make a soccer game that plays a victory sequence for the first player to three goals, a tank rolling through a world that enables collision detection (the default objects can pass through walls; writing code to detect collisions requires some great geometric reasoning), and a first person zombie shooter game *complete* with a zoom-in rifle scope. All my work with students has been with Alice 2.0. I plan to use Alice 3.0 this upcoming school year!

SCRATCH

Scratch

Scratch is like Alice in that it's a programming interface. I have not used it personally with students, but it's another resource I do plan to explore with some of my students this upcoming school year.

KenKen

KenKen

Many are familiar with Sudoku. KenKen is a similar reasoning number puzzle game involving operations in addition to populating the digits.

Alcumus

Alcumus

 

I've mentioned Alcumus previously on my blog, but I can't say enough about the role Alcumus plays in addressing learners' needs. Teachers can now register for a teacher account to monitor students' progress through the self-paced, differentiated curriculum. Students can sign up for free, only an email account is needed. Students can take a pre-assessment to determine areas of strength and deficiency. Quests and experience points provide instant feedback, instant gratification, and a mechanism to keep students engaged and to avoid tedium.

Here is a link to the original post on the broken pipe in a classroom problem.

This is the email I received from a teacher at a different school:

I don't have all the details and may not be able to get them... a pipe in my room burst on Monday and caused water damage in the hallway end and a total of 6 classrooms in full or in part... I have been told varying amounts as to how high the water was in my room and mere estimation by our custodial staff on how much water they themselves eliminated... the plumber said that the little copper pipe was spewing 12 gal per min...

Some of the outcomes of this activity:

  • Students gained a stronger understanding of Excel. Many students chose to write formulas and use the fill handle to brute force the time it would take to fill the various bathtubs.
  • Students deduced the connection between casework, writing out values, finding a trend, and generalizing through trying to write formulas for the volume of each bathtub as time passes.
  • Students discovered finding the "center" of a semicircle is more challenging than they anticipated. (see the half-cylinder tub on the original worksheet) Several used Geogebra to find the center through geometric construction.
  • One of the students made some progress on understanding 3-dimensional graphing with Geogebra 5.0 Beta.

Here are some examples from student work on Excel:

This student (below) chose to write the functions for fill and drain as linear functions in Geogebra:

Here is some work by a student trying to find the center of a semicircle:

A student modeled a theoretical room using Geogebra 5.0. (Apologies in advance for the sound quality.)

The students submitted written reports summarizing their thinking in this activity. I wanted to see a progression in their thinking from specific cases - like the percentage of a tub filled after 10 minutes - to generalized cases.

 

A teacher from another school across the state emailed me last week. She described a terrible facilities issue she has dealt with for days.

I don't have all the details and may not be able to get them... a pipe in my room burst on Monday and caused water damage in the hallway end and a total of 6 classrooms in full or in part... I have been told varying amounts as to how high the water was in my room and mere estimation by our custodial staff on how much water they themselves eliminated... the plumber said that the little copper pipe was spewing 12 gal per min...

We had been working on a different project during MTPS (Math Theory Problem Solving) class that I didn't want students working on without me being present... I had a sub coming in last Friday and decided this would be a great opportunity for students to do some modeling work in the computer lab.

Take a moment to consider how elaborate this situation is. Water accumulates within a room. Many places exist where the water could escape. Some places are obvious, like underneath the door. Other places are not as obvious, such as through electrical outlets or through the cracks in the drywall where the tile meets the wall. There are many potential sources for error. Modeling simple cases is easy, flow in is positive, flow out negative. But this is definitely a problem from the world outside school (I've never been a fan of the term 'real world'... that would imply high school isn't real... and I remember sitting through some interminable classes with a very real feeling of when will this class ever end...)

I worked to write a worksheet that would help students identify some of the potential complications in modeling how water would accumulate in various rooms. I thought about using theoretical "bathtubs" to simplify the computations and help the kids understand some of the complexities they would encounter in modeling a real room.

Bathtub_Images

The students have spent one 46 minute period and one 90 minute period in the computer lab working on modeling the situation. The source worksheet appears below.

The Broken Pipe Problem-1

Students will have another 90 minute period in the computer lab Wednesday, then 30 minutes on Friday before each student gives a 3 minute presentation on Friday about their lab work and findings. I will share some of the students' work on this dilemma later.

P. S. On the worksheet, I use Google images to retrieve pictures of each of the bathtubs. The copy did not turn out as nicely as I had hoped, so I traced over the images and scanned the resulting worksheet. I told the students the half-cylinder tub should be oriented in a way where the curved side is tangent to the floor. (The 3D image on the worksheet makes it look like the tub is tilted, but I did not intend for the half cylinder tub to be tilted). The values for the time in minutes were arbitrary. All of my students have chosen to set up tables of values in Excel so they can address some of the interesting questions (like how many minutes will it take to fill the rectangular prism bathtub?)

 

I am a supporter of the Common Core State Standards for Mathematics (CCSSM). While far from perfect, I think the CCSSM Mathematical Practices are downright beautiful. I see shades of these practices in my classroom from time to time. I am striving to develop these behaviors, like constructing viable arguments and critiquing the reasoning of others, in all of my students.

While reading through the discussion on Algebra (page 62 of the CCSSM document) earlier today, I found the following:

An equation is a statement of equality between two expressions, often viewed as a question asking for which values of the variables the expressions on either side are in fact equal. These values are the solutions to the equation. An identity, in contrast, is true for all values of the variables; identities are often developed by rewriting an expression in an equivalent form...

...The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula for the area of a trapezoid, A = ((b_1+b_2)/2)h, can be solved for h using the same deductive process.

Rewriting an expression can often provide greater understanding of its underlying structure. As a student, I had a rough experience with geometry in high school; however, I did very well in algebra class. Dynamic software provides a learner like me the opportunity to see the connections between algebra and geometry. I can bootstrap my way up to better geometric understanding by seeing the connections between symbolic manipulation and graphs.

Below is an image of a Geogebra sheet I made which demonstrates how we can rewrite the quadratic formula as the axis of symmetry of the parabola plus or minus a distance.

Quadratic_Formula_and_Its_Parts

This sheet demonstrates the graphical connection between the quadratic formula and the roots. Symbolically, we can rewrite the quadratic formula as two fractions with denominator 2a.

The first fraction, the green -b/(2a), gives the axis of symmetry of the parabola. The second orange fraction gives the distance we walk to the left and right of the axis of symmetry. We can think of an x-intercept in the following way: start at the axis of symmetry, then walk the same distance (the second fraction) in the positive or negative direction along x.

If this discriminant b^2-4ac is positive (the radical stuff), there are two distinct real roots.
If this discriminant is zero, there is one repeated real root.
If this discriminant is negative, then there are two imaginary roots. Graphically, there are no real solutions, demonstrated by the Geogebra sheet.

This approach demonstrates visually the utility of the discriminant in determining the number of roots of a parabola written in standard form.

 

I love to see the creativity of my students realized.

Below is our Math Club t-shirt design we will proudly sport at UNL Math Day next week.

Bearcat_Head_with_Image

Exhibit A: Original image of our school mascot, the Scottsbluff Bearcat

Bearcat_Head_Outline

Exhibit B: The piecewise function which graphs the outline of the Bearcat head

Exhibit C: Finished design for the front of the shirt

Exhibit D: Finished design for the back of the shirt

In my opinion, this is one of the greatest lessons I have done in the math classroom. Not only did this activity scream STEM, the kids learned first hand the challenges a small business owner can face and that mathematics is useful in many different disciplines.

Back story: I was at a birthday party for my friend Erik's girlfriend. We got to talking about the new automotive shop he is building. The shop uses a waste oil heater. As they change oil in vehicles, they dump the waste oil into a container. The container has a pump that feeds the waste oil into a long metal tube that burns the oil. The exhaust from the metal tube is piped out of the shop. The air outside the metal tube is heated and blown into the shop area.

Because the shop is close to a municipal water source, the 215 gallon waste oil tank must be inside an enclosure that can contain an equal volume of oil in the event the tank were to rupture and its contents completely empty out. Erik needed to figure out the optimal dimensions of a cinder block enclosure for the desired tank location.

I asked if our MTPS class could serve as consultants and determine the dimensions. I went to the shop building, took photos, and posted them on my school website. Below are the photos I took and showed the students during a Monday class.

TankDimensions

Exhibit A: Dimensions of the tank

Tank Scale Location

Exhibit B: Images to give sense of scale and desired tank location

6 Water Port, Needs 12 inch Clearance

Exhibit C: The water valve, which needs minimum 12 inches clearance

The students began work on modeling the cinder block enclosure needed to contain the oil in the event the tank is compromised. I had many issues in the Monday class trying to answer questions about the constraints. I invited Erik to attend my Tuesday class and describe the tank and the cinder block enclosure requirements.

Erik Nemnich from Nemnich Automotive describes the oil tank storage problem to students during the Tuesday MTPS class. Erik points out in the event of a tank rupture at the bottom of the tank, we can still include the volume of the tank when determining the dimensions of the cinder block oil containment.

The students worked for an approximate total of 100 minutes in class. They had to email their solutions by 8:00 am Wednesday morning. Here are some samples of student solutions.

Oil Tank Solution Samples

Students had to deal with many challenges. Many students wondered how the cinder block would be staggered to provide greater structural integrity. If we simply stack blocks one on top of the other, the resulting structure would be unstable. Since the blocks are 8" x 8" x 16", we can use an interlacing pattern at the corners of the structure to create this desired staggering without cutting block. Designs needed to avoid cutting the cinder block for obvious reasons.

Students had to perform conversions between cubic feet and gallons. Students modeled the position of the structure with Geogebra. Constraints on the placement wall included the location of the water valve, the location of the garage door sensor, the location of a nearby parts washer, and the amount of space the structure would protrude into the shop floor and workspace.

Below is an image of the finished oil tank storage cinder block enclosure.

Finished_Oil_Storage_Tank

 Exhibit D: The finished oil storage tank containment area

Special thanks go to Erik Nemnich at Nemnich Automotive for the opportunity.

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Gesturing is a powerful way for students to learn. I have students gesture with their hands often, particularly when we examine behavior of graphs. One of our teachers was working to help students remember the slope-intercept form of a line. She found this video on Youtube:

The teacher had the students do the gestures to help remember the parts of the slope-intercept form. I wanted to help the students understand the graphical effects of changing the values of m and b on the line y = mx + b. I started with a blank Geogebra sheet and constructed the graph of y = mx + b actively in front of the students.

y=mx+b

Students had no trouble with the effect altering the value of m had on the graph. However, something interesting happened when I asked the class, "What is your prediction for the effect changing b will have on the graph?" One of the students quickly replied, "The black point will move along the x-axis." I think my face betrayed my attempt to remain neutral. I said with my mouth, "You're right, the black point will move along the x-axis. What else will happen?" The students eventually pointed out the relationship between the b-value on the slider and the y-coordinate of the pink point.

But that student's comment has stuck with me since yesterday. I hadn't really considered before how we would go about defining the movement of the black point along the x-axis.

Let's take a second look at the graph.

We can define the x-coordinate at point A using x = (y/m) - (b/m). I defined the text box using the LaTEX editor in Geogebra.

where object i is the y-coordinate of point A. Since point A is the x-intercept, its y-coordinate is always zero. Then the root of y = mx+b is given by x = -1*(b/m).

This result also empowers my Math Bowl participants because they can now rattle off the x-intercept of a line written in slope-intercept form very quickly.

Here is a copy of the Geogebra sheet if you would like to play.

This setting can be used to generate many bowl-type questions where the student is asked to compute the area of the triangle bounded by the y-axis, the x-axis, and a line usually expressed in slope-intercept form or standard form. Below is a screenshot from my additional work on this problem.

Being able to compute the x-intercept immediately gives the student a good chance to solve the area problem very quickly. Since the x-axis and y-axis are perpendicular, the student can quickly find the area of the triangle by taking (1/2)*(b)*(b/m).

Here is a link to the Geogebra sheet displaying the area of the triangle bounded by the x-axis, the y-axis, and the line y = mx+b.

Area of triangle bounded by x-axis, y-axis, line y=mx+b

 

 

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One of the many problems the kids worked on today is below. What I enjoyed most about this problem is that two of my students worked right through frustration and obtained the correct solution.

1998_Problem_19_Four_Angles

 

 

 

 

 

I jumped in while the students were struggling to model how I would try and find the possible boundaries for the value of the sum.

IMG_3682

It was satisfying to work through what the students had done and discovering their work was correct. I was so focused on the work I totally missed the students' guerrilla addition to my meeting announcement. "Be there... or be taken to the second power!"

IMG_3685