Next up with my ninth grade math class: solving rational equations. I’ve been having fun with combined rate problems such as, “How long does it take Timmy and Marsha to mow the grass if it takes Timmy 2 hours on his own and Marsha 1.5 hours on her own?”
I, therefore, with great pride present you my first Vimeo video (Edit: Filling the Sink):
How I’m using it:
- Put it out there as a warmup — how long do you think it’ll take to fill this sink? Show about 10 seconds of the video.
- Put it out there in the lesson — what about when I combine the faucet and pitcher? Teach them combined rate problems as a specific case of rational equations.
- Watch “Trial 1 (faucet only)” and “Trial 2 (pitcher only)”, then compute time to fill the sink with both working together.
- Play “Trial 3” and compare your experimental results to my actual results.
- Close class with this: Assuming your computed time to fill the sink is different from the actual time, discuss what you think is the most significant source of error and how I could fix it in “Filling the Sink, Part II”.
Side note that’s aggravating but true: I’ve struggled with finding great applications of the concepts I teach because they’re rarely as complex as the stuff we’re expected to work with. With rational equations, I’m still looking for something where I’ll need to factor the numerator and denominator, cancel some terms out, and move along. My gut tells me this thing doesn’t exist.
Georgia Performance Standards: MM1A3d Solve simple rational equations.
Electrical circuits is the one area in engineering school I completely sucked at because I had a knack for letting out the magic smoke of many a project. Well, I’m back at it. Hopefully, I’m a bit more knowledgeable now & I’m definitely staying away from capacitors.
I have here a pretty cool project cribbed modified from a task published by the Georgia Department of Education.
The 8 second summary: practice solving for total resistance then assemble several resistors into an equivalent resistor using concepts of series and parallel circuits. Students will be solving rational equations throughout the project.
Below is a circuit that comes early in the project. Students are learning to apply the formula for resistors in parallel.
Here’s a more complex circuit. One of the problems is sufficiently complex that some students will need hints on solving the equation. I’ve split the hints out in a separate file, available to only those students who need it.
After students have a chance to get comfortable with the formulas for resistors in series and parallel, the project puts them in a challenging situation: build a circuit with resistance equivalent to a given value (even though none of the individual resistors have the particular value I’m looking for).
I will have actual resistors and an ohm meter on hand for the final phase of the project.
The files: Student Edition (PDF | Word 2007) & Hints (PDF | Word 2007)
Materials: Copies of the Student Edition, resistors (Ebayed), and an ohm meter (borrowed from the physics department).
Georgia Performance Standards: MM1A3d Solve simple rational equations
I really don’t like going this deep inside the box to think…but my kids have to prepare for major standardized tests. With that caveat, I now present resources I like to prep students for the End of Course Test after Math 1 (9th grade math – algebra, geometry, & statistics).
First up: I like the Georgia DoE’s EOCT Math 1 Study Guide (PDF, free). The 130 pages in this guide has re-teaching alongside practice problems. In my experience, the problems in this book do a great job illustrating how hard the EOCT will be. Here’s a clip from the statistics unit we recently completed:
Second up: the Released Items booklet is helpful for teacher planning. I use it to see sample problems from last year’s test & inform my quiz and test writing. Unfortunately, I realize now that the EOCT relies on at least passing knowledge of the state-authored tasks. Here’s an example:
Fellow math teachers – what EOCT resources do you like?
Get 4 rounds of Probability Trivia (PowerPoint | PDF). Set your Adobe Reader to scroll (View menu > Automatically Scroll or Shift + Ctrl + H). Put on music the kids might actually like. Then roam the room to help kids who struggle.
I’m not much at expounding, but if I had to list the benefits this little game has given me,
- used days before a unit test, I can identify areas the whole class needs to work on
- I have the freedom to work in very small groups with kids who struggle with a particular problem (because the game part kinda runs itself)
- the teamwork aspect means that kids justify their work – I hear the best arguments from kids who know they solved the problem right but their teammates are being meatheads
- the scrolling questions puts the kids under pressure to solve problems quickly
The question set covers counting principles (for number of outcomes), permutations & combinations, mutually exclusive events, dependent events, and conditional probabilities.
Bonus for Georgia teachers: this is Math 1, Unit 4 standards MM1D1 and MM1D2.
I wrote a full howto on Waterfall Trivia back in November.
Recently spotted on the Duarte Blog: Cheating by Charting. An excerpt from Spear’s Practical Charting Techniques. This stuff is genius.
Methinks this could be used in math class. “Hey kids, we’re going to play Corporate Spin today. You’ll hide a disappointing stat in a graph so the public doesn’t realize how much we’re polluting/raising prices/whatever.” (My, I didn’t realize I was feeling so cynical today…)
What time of day was this photo of the Washington Monument taken?
While writing this warm-up question, I came across the Sun or Moon Altitude/Azimuth Table page. Supply a location and a date and this page will tell you the angle of sun in the sky for any time of day. (Interesting side note: This photo of the Washington Monument could not have been taken between November and February because the sun never gets that high in the sky.)
I can already imagine all sorts of interesting extension problems based on this picture but for now I will keep it simple: my students need to practice the inverse trig functions.
(Props to my colleague Annie Sun for the idea for this game)
The goal is to get as close to 21 + 21i without going over.
Georgia Performance Standard MM2N1(b,c)
The math department at the high school where I teach is big into stations. If the concept is new to you, here’s the lowdown: On a station day, several learning centers are set up around the room and students circulate among them. Montgomery County schools in Maryland has published details.
From what I’ve seen, stations are particularly good at providing opportunities for reteaching and practice, in addition to acceleration.
The key here is that stations are useful as a differentiation tool. According to the MCPS folks, you should use assessment data to break students into groups. Not all students will visit all stations and time at the teacher station will differ based on the data.
I’ve applied stations a few times this year and have learned a few things:
- how incredibly important it is to model the concept to the class
- you must give explicit instructions in the stations where students work independently
- I like using assessment data to divide students
- give students their station assignments during warmup
- students need a way to check their work in the stations (I’ve posted solutions on the back of index cards taped to the wall)
- you need an assessment tool or record of students’ work in the stations — they need to turn something in. I’m thinking a culminating question at each station that has no solution posted makes a lot of sense.
Your time spent in planning stations is huge. Not only must you plan approximately three activities but you must also provide differentiation for each group. This could mean upwards of six times the work in advance. This time is so worth it! Do I even need to say it beats the heck out of lecture or drill-and-kill practice?
Long the premise of every lesson on probability, coin tosses are taking a beating (thanks Boing! Boing! for the share). Some interesting research (Diaconis, Holmes, and Montgomery) uncovered these biases:
- If the coin is tossed and caught, it has about a 51% chance of landing on the same face it was launched. (If it starts out as heads, there’s a 51% chance it will end as heads).
- If the coin is spun, rather than tossed, it can have a much-larger-than-50% chance of ending with the heavier side down. Spun coins can exhibit “huge bias” (some spun coins will fall tails-up 80% of the time).
- If the coin is tossed and allowed to clatter to the floor, this probably adds randomness.
- If the coin is tossed and allowed to clatter to the floor where it spins, as will sometimes happen, the above spinning bias probably comes into play.
- A coin will land on its edge around 1 in 6000 throws, creating a flipistic singularity.
- The same initial coin-flipping conditions produce the same coin flip result. That is, there’s a certain amount of determinism to the coin flip.
- A more robust coin toss (more revolutions) decreases the bias.
Always the hacker, I suggest following the link for strategies to winning a coin toss.
Is there a place for this research in my math classes?
Filed under: spherical cows-and-Other-Oversimplifications.
Here’s my first entry in Dan Meyer’s “What Can You Do With This?” meme.
The artifacts: Two photos taken from a tourist’s turn on the London Eye, one from eye-level with Big Ben and the other from well above the famous London landmark.
What meaty, mathy problems can you dream up out of these photos? Tune in next time for the questions my students asked (and answered).