Zack & Wiki

Tonight, my oldest daughter asked me if I would play Zack & Wiki: Quest for Barbaros’ Treasure with her. Zack & Wiki is a puzzle adventure game for the Wii, released back in 2007, but a great game. (The Smithsonian ranked it as one of the top 80 from the past 40 years, and it won GameSpot’s Best of 2007 Adventure Game Award.)

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.


One possible scenario

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?


Is more physical education at school linked to higher student math scores?

In January, we decided to post tips on my school’s facebook page for students to get organized as part of Get Organized Month.

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.


Teach My Lesson 2

Check out my previous post for an introduction to my school and idea for where this came from. Student name was changed for privacy, and some spelling/punctuation errors were fixed (but I’m sure not all). Also, since we are writing in real time, there are several places where I posted something after the student did that would probably make more sense if it was posted before the student. I did not alter these situations, but if you have a question on anything, please ask.


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: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: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: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?


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.


Teach My Lesson

I feel like I always have to make the disclaimer that I teach for an online school. That in and of itself poses different challenges than a brick school.

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…