As explained in a comment on George Couros blog, when making small talk with people, the question usually comes up about what we do for a living. When they find out that I am a teacher, the next question they always ask is, “What do you teach.”

My typical answer, “Students.” Granted, there is a lot of math that gets discussed, but my focus tries to be on developing the student. After all, other than a high school math class, how often will an average person ever use the “CPCTC” (or “Corresponding Parts of Congruent Triangles are Congruent”) theorem? What about factoring? I think most real-world polynomials of interest cannot be factored.

So why even both having math classes if we are teaching students skills they will never use outside of the classroom? For me, it is because I am trying to teach so much more than math. I am teaching how to follow directions and proceedures. I am teaching logical thinking. I am teaching how to explain their complex thinking to others. I am teaching how to persist, especially in light of difficult problems.

I am also trying to get them to think outside the box. (Also one of the reasons I have participated in Odyssey of the Mind for so long, especially at the regional and state levels.)

I love this student’s answer for this problem. I think the writers of this problem (and consequently the answer key) just assumed that we would assume the pizzas were identical. The purpose of this question was to help students realize that (if the pizzas were the same size), Marty could not have eaten more pizza that Luis, and so the statement that Marty ate more is incorrect.

Marilyn Burns recently wrote a blog post about algorithms and shared the work of a student she called Omar. Part of his work subtracted 98 from 100 and got 88 instead of almost nothing. He also took 120 and subtracted 50 to get an answer of 420 — more than he started with!

Problem like number 8 can hopefully help students learn to self-correct and realize their own errors.

But when I look at this student’s answer, I love how they thought outside the box! They didn’t confine themselves to equally sized pizzas. If the statement made is correct (and why wouldn’t it be; the teacher gave it to us), then there has to be something about the problem that may not be obvious, but allows the given problem to be true. Having pizzas of different sizes would certainly qualify.

If my pre-teen son and I shared the same pizza, we both might eat “half” of the pizza, but I can assure you that when I cut the pizza, the “halves” will not be equal size so I will still eat more of the pizza. I’ve also cut a pizza in quarters and then cut the quarters in half (to make 8ths) or in thirds (to make 12ths) so that my wife and I can have the same number of slices as our children, but still get more in our bellies. If I can get creative serving pizza, why can’t this student?

Granted, my examples were using the same pizza and this problem refers to different pizzas (it must be different pizzas since 4/6 + 5/6 of the same pizza makes more than one pie; also the references are to “his pizza” and not to “the pizza”).

I worked with a Geometry student last week who mathematically got a correct answer, but rationally, she didn’t think the answer made sense. The problem asked her to take a pool (shaped as a quarter sphere) of water, and to find the volume of a model that shrunk each dimension by 6. Although I didn’t know the numbers she used, I knew that the answer would be 216 times smaller than the larger tank. She thought that answer would be too small, so though she made a mistake.

We talked about the reasoning for the difference being so large. As a parting problem, I asked her to calculate the area of pizza next time she ordered it, to see what kind of value she was getting. I gave the example, “Would a 6″ pizza have half the area of a 12″ pizza since 6 is half of 12?” and as a follow-up, “Would we want to charge half as much for a 6″ pizza as the 12″ inch?”

What do you think?

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Since we are an online school with asynchronous courses, we have a lot of talented students who need to be able to “do school” at times it is convenient for them. Athletes and actors are two groups that readily come to mind. Although we don’t list the names of ours students (to protect their privacy), that doesn’t stop some from sharing it themselves.

On the other side of the spectrum, though, are students who struggle in a brick school. For them, our online, asynchronous model makes it possible for them to do school at all, let alone succeed in school. The student I mentioned from the first paragraph came from this group. To protect privacy, I will just refer to her as, “B.”

B has serious vision processing issues. Although I don’t know her full story, I can only imagine the difficulty she would have in a brick school trying to see, and then comprehend, any written learning material presented while there are 29 other kids in the room who are finishing the assignment as she is just finishing reading the directions.

I got involved with the family right before she started her Senior year. Mom reached out to me to discuss options for B to take a 4th year of math her final year of high school. One of the courses that we offered looked like it would be a good fit for her, only the book was changing so the course was not available until all the changes were made and processed.

Since the course wasn’t ready when B wanted to start, she ended up starting the year taking Pre-Calculus. A few weeks in, though, we did have the other course ready. She had to option to stop Pre-Calculus and move to take all of Advanced Applied Mathematics (AAM), essentially doing a few weeks of Pre-Calculus work for nothing, finish the first semester of Pre-Calculus and then switch to the first semester of AAM to finish her year, or stay in Pre-Calculus for the full year.

B decided to switch to AAM immediately and ultimately was glad she did, even though she had to do extra work. Fortunately for me, we have a teacher at our school who also has visual processing issues (he is legally blind); we paired the two of them up for her senior year since they already had something in common.

(You may ask how a blind teacher can work at an online school, especially in math where screen readers don’t yet have the capacity to read math symbols drawn by hand. Sounds like an idea for another post!)

B will be graduating with us in a few month, which is exciting. At our meeting yesterday, though, I learned that she was accepted into her first choice of college! Here is a young lady who desires to succeed despite her obstacles.

We all have obstacles of varying degrees. What are you doing to overcome yours?

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As mentioned before, I teach in an online school. Our courses are designed to be asynchronous, so our students can work at any pace they want. As a result, the content is pre-designed (often by a third party) and there are limits to changes we can make.

Many times, this boils down to preferences. If there is a topic, I might want to teach it one way, but the course teaches it a different way. It isn’t wrong, per se. It just isn’t what I would use. For example, I don’t like the use of the word “cancel” in math because I think students use it too often when they don’t really know what is going on (or why they can do what they are doing). Think of it as “magic.” If I say the word “cancel,” I can do anything I want! (At least Jane Taylor agrees with me in her comment on the same post mentioned above.)

Using the word “cancel,” though is more a preference than an error (although it can be used erroneously). As a result, even though I don’t like it, I cannot request to have it changed in the course. Only errors will get changed.

So in one of the courses I taught, the curriculum used the error listed above. They would say something like √9 = ±3. I know this is wrong, so I asked for it to be corrected. It ended up not being corrected as the curriculum writers saw this as a “preference” as opposed to an “error.”

Eventually a parent called me on it. Of course, I knew he was correct, but there wasn’t anything more I could do about the content. I did let the parent know that I give students credit if they gave the answers of √9 = ±3 or √9 = 3. Unfortunately, that wasn’t good enough for the parent, so we had several conversations through email and phone. What bothered me the most is that I could not do anything about their valid concern. At least not while their student was in the course.

On a positive note, we ended up rewriting the course shortly after that, but mainly because it was an old course. As an old course, we had many, many students go though it and learn it the wrong way. My hope is that they were able to correct their thinking in later courses.

Often when working with a student on radicals, I will give them the following scenario:

If I, as the teacher or as the author of the curriculum, give you a square root symbol, then I will tell you whether I want the positive or negative version. For example, if you see √9, you know I want the positive version. If you see −√9, then you know I want the negative version.

However, if YOU as the student put the square root symbol, then you don’t know if I want the positive or negative version of the square root, so you need to give me both. In that case, a problem like x^{2} = 9 would need to have two solutions since it is the student applying the radical sign.

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Nearly two decades ago, I decided that I wanted to teach at the college level. In looking back at my college experience, I tried to find a common trait that my good professors had–the professors that really stood out and I would qualify as the best in terms of helping me learn and being understanding in the classroom.

With rare exception, there was a common trait: they had all taught high school at some point in their lives.

It was then that I decided I wanted to follow the same path. I wanted to teach at the high school level first before teaching college. As a result, I applied for, and was accepted into, the Teach For America program.

This week, we celebrate all branches of the AmeriCorps program, including Teach For America. Thanks to all the volunteers and those who serve who make our cummunities great.

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But this post isn’t about the video directly. After watching the video, I did a search for “bicycle generator,” which brought me to the instructibles.com website. One of the members, notoriouslev, made the bike and posted a comment to help people try to determine the v-belt length needed for this project.

I think it is obvious that he is not a mathematician, but also obvious the he paid attention in school in math class. His explanation, I think, is perfect for the lay person. It gives some basic informaiton about why it works without getting too technical to scare people, and then gives the steps someone else would need to take to replicate his idea for their project. In his short reply, he references both the circumference of a circle and the Pythagorean Theorem.

Now if a student says, “when am I ever going to learn this,” I can point them to notoriouslev and say, “you might not, but he did!”

He starts by measuring the diameter of his rim to get 26 inches, and converts it to 13 inches as the radius.

Then he says, “The equation to get the circumference of the rim is 2 x pie x radius = 2 x 3.14 x 13 = 81.64″ around.” Although it is pi, not pie, I’ll give him a pass on this step. Alternately, he could have used πd and not had to convert the diameter to a radius.

I don’t know how these belts are measured or bought, but if he is able to get an accuracy of hundredths of an inch, he may want to use a more accurate number for pi. I got 81.68 inches.

Once he has this number, he says to, “Subtract 1/4 of the circumference: 81.64 * 75% = 61.23” I think this is classic! He says to subtract, but then he multiplies instead! Mathematically, it is correct, though. If we take the circumference and subtract 1/4 of the circumference, we can rewrite that as:

C – ¼C = C(1 – ¼) = C•¾ = C•0.75 = C•75%

In his last step, he said to, “Add the diameter: 61.23 + 26 = 87.23″ Long.” At first, I had to look at the diagram to see why he added the diameter. Once I looked, I saw that he was really adding two radaii, which is the same value as the diameter. Essentially, he created a circle with a square drawn on top. The square has a vertex at the center of the circle, and the square’s side length is the same as the radius of the circle. Two of the sides of the square are tangent to the circle.

He does have an optional step 4 listed which isn’t entirely accurate. He says to:

Optional Step 4: You can add extra length to the v belt in case you don’t want the generator sitting just below the rear wheel’s exterior. For instance to place the generator 1 foot behind the rear wheel, you add 12″ for the extra distance behind the wheel, and then you use the

Pythagorean theoremto get the additional v belt required to reach back to the tire: (12 x 12) + (13 x 13) …then get the square root = 17.7″. So to place the generator 1 ft behind the rear wheel of this bike, we add 29.7 inches to the 87.23″ we already had and get 116.93″

This model would assume (or imply) that there is some sort of pully at point D causing the belt to stay on the wheel rim all the way to point D before moving to the generator at point E. According to his description the side labeled d_{1} is 12 inches, the foot behind the read wheel. The side labeled e_{1} is 13 inches, the same as the radius of the wheel.

In actuallity, though, the belt will come off the wheel forming a tangent from the wheel to the generator at Point D.

For his purposes, though, using the hypotenuse might be close enough to get a belt length. In using the tangent line, the belt would leave the rim at point F, so even though the belt length from E to F is longer than the hypotenuse he calculated, the arclength from F to D is no longer traveled, so we lose some length from the circumference (less than 75% used now).

Using Geogebra, I found the measurement of the belt is 53.31 + 13 + 12 +25 = 103.31 inches.

Of note, since the segment BE is also a tangent, it will have the same length as EF (a much easier calculation than using Pythagorean). (Can you show why?)

If we bought a belt using his 116.93 inches calculation, I believe that the generator will be closer to 2 feet behind the back wheel. (About 20.19 inches behind the back wheel.)

What started as a signout video from my email turned into a neat little mathematical tangent. Thanks to notoriouslev for the comments that started this little journey.

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In any event, we were playing a level called “Frost Breath.” There is a 4 by 3 grid of mirror holders. Really, there are 11 mirror holders with the bottom center position as a laser cannon.

The laser cannon can be pointed in 5 different directions — assuming standard position, they are at 0, 45, 90, 135, and 180 degrees. Each of the mirror holders can hold a mirror in 4 positions, but some allow the mirror to face N, S, E, or W. Other holders allow the mirror to face NE, SE, SW, or NW.

The goal is to hit the big ice kitty with the laser. The first shot is easy. However, the kitty gets angry and causes some of the mirror holders to be destroyed. Combine that with the fact that the level is randomized every time you play it, and the player only has 3 mirrors to place, this is a tricky level.

Working with my daughter, we had to figure out which direction to aim the cannon, and how to position the mirrors so that the laser beams bounce the correct way to hit the target. Talk about having fun with angles! It almost reminded me of some of the billiard problems we see in textbooks, but I think much more enjoyable (and more difficult even though the angles were all multiples of 45 degrees).

Although I beat the game years ago, not that I have been refreshed to this puzzle, I am curious if I can work it into a lesson or a review with my students.

Have you ever played Zack & Wiki? Any ideas how to make this academically appropriate?

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I posted many many suggestions, and a few of them were selected, but one suggestion I gave was to organize time. I wrote:

Use a timer to help you organize your time and create a schedule for your courses. For example, set a timer for 50 minutes. Work as hard as you can until the timer goes off, and then take a 5 – 10 minute break from your studies. Your brain needs the rest before learning some more. During your break, do something active – exercise or do some stretches, have a snack or take a quick walk or jog, or play a quick game or read a book to activate your brain in a different way.

For me, it was important to include the part about exercise during a student’s break time. I’m a firm believer that a student needs to take care of their whole body to help them learn. Our parts are too connected between the brain and the rest of the body.

Imagine my delight when I found an article this week by the Washington Post called Is more physical education at school linked to higher student math scores?.

“This finding demonstrates that students’ academic performance improves when there’s a balance between time spent on physical education and time spent on learning,” said Stacey Snelling, dean of American University’s School of Education.

Granted, this study was just looking at a handful of elementary schools, but I don’t see why the results couldn’t be extrapolated to all people of all ages.

Recently my daughter and I started doing “pretend jump rope,” where we move our hands as if we were holding jump ropes, but our hands are empty. We also jump as if we were jump roping. (I started jumping rope after reading a different article by Drs. Oz and Roizen last August.

Can you think of any ways to introduce more activity for your students? Let me know in the comments.

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Chad Lower:Hello Student. How may I help you?

Student:I have a math question.

Student: The following set of coordinates most specifically represents which figure?(-4, 5), (-1, 7), (1, 4), (-2, 2) (5 points) Parallelogram Rectangle Rhombus Square

Student:I know that it is either a square or rhombus, but I don’t know which one

Chad Lower:Student, how do you know that part of it?

Student:I graphed it on graph paper and the figure had 2 pairs of parallel sides and all right angles

Chad Lower:how do you know they are right angles? They may look it, but may not be exactly right angles

Student:Oh I guess I don’t know if they are right angles

Chad Lower:okay, so that is one part we need to check.

Chad Lower:(and checking that will also verify if the opposite sides are parallel)

Chad Lower:the other part is the distance of the sides

Chad Lower:you can have 4 right angles, but then it could be a rectangle or a square

Student:yes

Student:so should i do the distance formula?

Chad Lower:exactly!

Chad Lower:for the length part anyway.

Chad Lower:you will need a different tool to show right angles

Student:they are all equal sides

Chad Lower:oh cool! so then we can narrow down to square or rhombus

Student:yup

Student:what should I use to find the right angles

Chad Lower:so think “perpendicular”

Chad Lower:how can I tell if two lines are perpendicular to each other?

Student:they make a right angle when they intersect

Chad Lower:okay–try this related question — how can you tell if two lines are parallel?

Student:can I use one of the proofs to prove perpendicular lines

Student:they never intersect each other

Chad Lower:so would you draw them forever and ever and see if they ever touched?

Chad Lower:I don’t think a proof will be helpful for this question.

Student:okay

Student:I guess you could

Chad Lower:that would take a long time

Student:but that wouldn’t work out

Chad Lower:and you would never finish — right

Student:hahah right

Chad Lower:so is there another way we could check if they were parallel?

Student:you can find their slopes?

Chad Lower:BINGO!!!!

Chad Lower:so now go back to the previous question. How can you tell if two lines are perpendicular?

Student:the slopes would be opposite reciprocals

Chad Lower:and that is what you need to prove or disprove to determine if it is a square or rhombus

Student:yes thank you!

Chad Lower:(and by finding slopes, you can also tell if opposite sides are parallel)

Chad Lower:but if the side lengths are the same, we get that as a bonus

Chad Lower:without needing to prove it separately

Student:so if they are all right angles, does that mean it is a square?

Chad Lower:correct

Chad Lower:since we know the side lengths are the same

Student:okay thank you!

Chad Lower:My pleasure, Student!

Chad Lower:Anything else today?

Student:nope!

Chad Lower:Sweet! Be sure to tell your teacher you stopped by the Live room. Thank you for your effort

So when I reflect on this question in hindsight, there are some things I like about it. I never did any of the number crunching. In addition, the student came up with the ideas for what to do next; I merely verified if the idea was correct.

If I could do it again, I might not be as blunt about her ideas being correct–maybe ask her to justify why she thought that might help (like the idea of the distance formula). I also didn’t do the problem myself; I only relied on her answers and assumed them to be correct. If she made an error somewhere, then I verified that her incorrect answer was correct. This one doesn’t bother me as much as the first, but I know in the past I *have* caught student mistakes, so there could have been a mistake here as well.

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However, not only is my school online, we are also asynchronous. That means we have students starting classes every day of the year that the office is open. Students can pace as quickly or slowly as they want, as long as they are finished with their course after 10 months.

Because of this dynamic, it is not uncommon for a teacher to have a student submitting their first assessment in a course on the exact same day a different student is submitting their final exam. It would be impossible to make presentations to all of our students on the topic they are at every day (or even every week). As a result, we use a static curriculum that the students can move through on their own.

Of course, sometimes students will need help on a topic they are learning. This fact is true for all subjects we offer, but is especially true for our math courses. To meet this student need, we have developed a Math Lab that we run weekdays, usually 4 times a day, for roughly 37 hours per week. Students can get math help outside of the Math Lab hours if a teacher is available, but these 37 hours, we promise a math teacher will be available.

So when I am talking about Teaching My Lesson, it is in the context of helping a student in one of these Math Lab sessions. The drawback to the Math Lab, since it is drop-in first-come-first-served, we never know what subjects or what questions will show up during Math Lab. I have helped a Pre-Algebra student and a AP Calculus BC student at the same time. All that to say, we cannot do much pre-planning of a lesson as they are all relatively spontaneous.

Wow! I’m glad you got through that. It seems like I need a separate blog post just to introduce this blog post!

So the other day, I was working with a student on subtracting polynomials (Algebra 1). She came in specifically to find out why, when subtracting polynomials, do you multiply the second polynomial by −1 (negative one).

Rather than directly answer, though, I thought it would be helpful to start with elementary school arithmetic and move to integer arithmetic and the distributive property in general before looking at the distributive property with algebraic terms.

We started with the simple 8−5. Anyone can do this problem from 1st or 2nd grade. Then we bumped up the difficulty by reminder her that “subtraction” was the same as “adding a negative.” 8+(−5).

Then back to 8−5, but instead of writing 5, I wrote the equivalent phrase (2+3). And then we explored that 8−(2+3) was the same as taking 8−2−3. (Take the 8, then subtract 2, then subtract 3.) All of these methods yielded 3 as a result.

From here, we went back to the 8−(2+3) line. I rewrote it, and then explained that we can multiply a 1 in front of the (2+3) since 1 times anything is just that anything. Multiplying by 1 doesn’t change the value. From here, I played the “subtracting” is the same as “adding a negative” game we looked at earlier to get 8+ −1(2+3).

From here, we used the distributive property to get 8+ −2+ −3. And then finally “adding a negative” is the same as “subtracting” to get 8−2−3. But we had already seen this line earlier in the lesson! I love it when things work full circle.

Now that we played with numbers (the concrete), I thought it was appropriate to bring in the abstract, really only now answering the question she came here for initially.

I will admit, I was disappointed when she asked that last question, “So what would your final answer be?” I loved the initial question she had. I appreciate students not wanting just “the rule,” but also wanting to know why the rule works. This question is especially fun because it isn’t covered in the course. The student doesn’t have to ask to meet the objectives of the course.

But at this point in Algebra 1, she has practiced combining like terms over the last several lessons. Initially just two terms. Then with adding polynomials the lesson before this. At this point in the course, she should be comfortable with combining like terms (especially before even attempting to subtract polynomials). In hindsight, I should have flipped a clean whiteboard out, and reviewed with her the few lessons leading up to this one.

I will say, this lesson wasn’t atypical, but also wasn’t typical. I get more questions akin to “how do I do this?” With those inquiries, I usually make the student do more writing than me. Math is not a spectator sport. In order to learn the math, you have to do the math.

If I get a student looking for a how, I will see if I can do more screenshots and post again (maybe this week, maybe next). Stay tuned…

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