Divide By Zero

February 17, 2010

How To Teach: Will It Hit The Corner?

~~Here’s what I did.~~ I taught this lesson today, and I had mixed results. For the record I’d say that kids in the second group that got the lesson where really into it, whereas the first group gave it a “so-what-else-did-you-cook-up-for-us?” rating. I started this post thinking I’d tell you what I did, but I kept thinking things like “if I’d only done this it would have gone better,” so here’s the if-I-would-have-done-this-then-it-would-have-gone-better version of what I did today. I know, too much hyphenation for one day.

At the end of this lesson, students will be able to

Work backwards to solve a complex problem
Use patterns and inductive reasoning to extrapolate data (logo positions on a large grid)
Apply software to investigate (potentially) algebraic relationships
Construct a rule for the corners problem

Setup: Show kids office clip #1 (see post). Then open up the 5 min. gridded version and play that in the background while acting like you’re trying to give directions. Get the kids to make guesses about whether or not it will hit a corner. Then after they’re done shouting about the epic moment at 1:43 (.333) when it does, get them to predict the next corner. It doesn’t happen. At least, not in the 5 minutes that Dan provided.

Distribute copies of the warm up below

Ws Office Boxcorner Warmups

As kids graduate from this, ask them if they’d like a bigger version (the desired answer being: “YES!”). Then hand them the first page of this sheet.

Ws Office Boxcorner

The answer here being that it doesn’t matter all starting squares result in a corner hit after varying numbers of steps. This can be shown with the software or through a pattern that many kids will notice after they have traced out sufficient numbers of steps. Talk about the patterns, ask them if they have any theories about how this might work on grids of varying dimension.

If you have access to a set of Mac’s you’re in luck. Ask the students if they have developed any general theories about these types of problems. How does the path to the corner depend on the starting location and grid dimensions? Allow students to work with the software to investigate and develop their theories. Use this sheet and have students download office-grid-basic.py, the simulator I designed.¹

Ws Office Boxcorner Pythonlab

I leave off here at this incomplete point, because I’m not sure of a good way to finish this off.

Any ideas? How would you finish this lesson? Would you do anything differently? I’d love to hear, as I might be doing this again with different classes in a week or two.

If you intend to run the program, see the prior post for how-to info. ↩

February 12, 2010

re:WCYDWT – Will It Hit The Corner? – Find Out With Python!

Thanks to Dan for posting a great setup and a problem that turned out interestingly. I worked the problem on paper last night. Then this morning, I came in and re-worked the problem with a few students, and finally read Alex’s algebraic solution. The handout I made from last night is below.

Ws Office Boxcorner

After some time with Alex’s formula I wasn’t getting things to work out (probably a problem of choosing between 63/64 and 79/80). At any rate I really wanted something concrete to test solutions. I bumbled around a bit in python and came up with a program to model different sized grids and logo paths. The program colors in the paths and stops when a corner is hit.

Here’s an overview.

Running the Software on a Mac:

This is easier than a PC because Mac’s come pre-loaded with python already installed. Follow these steps to get the program running.

Download the file office-grid-basic.py and save it to the “Macintosh HD” folder.
Open up a terminal window (hold command and hit spacebar, then type “Terminal” or navigate to your Applications folder from Finder, open Utilities, and double click on Terminal.)
In the Terminal window you’ll have to change directories to get back to the root (which is the Macintosh HD folder where the office-grid file should be)
- To do this, type cd .. then Enter until the directory stops changing.
- Then type ls Enter to view the contents of the current directory. You should see the “office-grid-basic.py” file
Type: python office-grid-basic.py and hit Enter to run the program.
To run the program again, you must first make sure to close the graphic window, then enter python office-grid-basic.py in the Terminal and hit enter.

Running the Software from on a PC

Install python (choose the version that fits your operating system (32 or 64 bit)
Download the file office-grid-basic.py and save it to the root of the “C:\” drive.
Open a command terminal. If you have XP, type cmd Enter in the Run dialog of the Start Menu. See here for info on Vista. On Windows 7 hold Shift and left-click any folder with the mouse and choose “Open Command Prompt Here.”
Type “cd c:\” from the command prompt to change directory to the root of the C-drive.
type “python office-grid-basic.py” Enter

Resources

Worksheet
office-grid.py
python – The program needs python 2.x, version 3 will probably not run the script.
cygwin

February 10, 2010

Error Checking and Basketball Scores

After reading Kate’s Error Finding in Geometry Proofs post which I highly recommend. I had two ideas.

First, that I will be designing some similar activities for my class soon. And, second, that there’s a lot of value in this type of post with the follow up reflections. I’ve recently hit on two activities that I thought were worth sharing so here goes.

Row Games Stolen from Kate at f(t). Basically, pairs of problems with the same answer. Each partner works one of the problems, then they compare answers/results. Lots of self checking, and error analysis, and discussion. Here’s the worksheet I made.

ws-substitution-elimination-pairs

Problem Solving This was supposed to be the warm-up problem for today’s algebra class, but it turned out to be a good problem, and I’m going to use it again tomorrow, with a little more structure to the assignment. Note, I won’t pass out the second page, but I find it helpful to know the correct answer and to have a few different visuals ready before presenting a problem.

ws-systems-inequalities-basketball

December 3, 2009

Structuring a little competition into class.

I teach a “mathletics” class where we do mostly mathcounts problems. I wasn’t happy with the way it was structured last trimester, because kids generally weren’t engaged as much as I would have liked. So this trimester I re-structured things. I keep a spreadsheet of team points, it’s a running total. Today we completed a warm up and two challenges.

The warm up was “how many golf balls does it take to fill up a school bus?” and any reasonably justified answers were accepted. Then we went into mathcounts stuff.

The key is in the structure. Each kid gets the same sheet of problems, and each group of four or five has a single answer key where they write their final answers. At the end of the round, I collect the team answer keys, give the correct answers, and then distribute the next activity. While students are working the next set, I grade the last set and update the team points on the board. To complete 5 challenging problems groups never have more than 10 minutes. To build off the last post, give the most points for the easiest stuff, and only a few points to the hardest material – that will keep students at varying skill levels engaged and allows the most competitive kids to compete without blowing everybody else out of the water.

A few benefits of the structure I noticed today:
1. Because the teams are fairly evenly picked, the scores are close enough to make it a true competition.
2. There’s higher engagement.
3. It’s easy for me to keep the tally accurate because I only have to grade one paper for every five kids in the class.

As we move on I may sprinkle in some tips and tricks, but the vast majority of our time is spent with groups working problems.

November 18, 2009

Easy task high credit, tough task low credit, reasonable results.

If you are assigning scores to questions on a test, I used to make hard questions worth more points than easy questions, because I thought that they should carry a greater reward for correct answers. This caused problems because many students who had a decent level of understanding of material were not getting the questions worth the most points on their assessments.

My current approach is the exact opposite, an easy question should be worth lots of points, and a tough questions should be worth a few. The few points are enough to separate out kids who are good from those who are excellent, but students who do well on a test should not be receiving terrible scores. This tends to agree with our intuition about what a student’s total score should be.

I think I figured this out sometime last year, and I’ve been thankful for it ever since.

November 15, 2009

What day of the week where you born on?

I’ve been reluctant to post lately because I haven’t felt like I have much worth sharing. I’m using BetterLesson, and I’m slowly getting my curriculum updated to it. I’ve spent a lot of time putting together some units, the units are composed of an Overview (a pdf containing lists of skills and links to other pdf worksheets and Smart Board files that relate to the unit), worksheets, notes, and multiple versions of tests with answer keys written in LaTeX and the package eqExam and the AcroTex bundle by DP Story.

I did a little research this afternoon, and thought I’d put it up here. I’ll be giving this the run through tomorrow in my “mathletics” math competition class. Here goes.

Big Question
What day of the week do you think you were born on? Take a guess.

Some Info For you to Consider
1. The number of days in a month can be easily remembered using your knuckles.

Thanks to the Mnemonics Guide at EUDesign

2. If today, November 15th, 2009 is a Sunday, and you want to know what day it was on March 18th, 2009, find the number of days that have elapsed, divide by 7, and use the remainder to determine the day of the week.

Number of days elapsed since March 18th:
March: 18th to 31st is 13 days. 30 days elapsed in April, May is 31, June, 30. July, 31, August, 31, Sep, 30, Oct, 31 and Nov: 15. For a total of 13 + 30 + 31 + 30 + 31 + 31 + 30 + 31 + 15 for a total of 242 days. Since $242/7 = 34\frac{4}{7}$ , March 18th, 2009 was (Sat, Fri, Thus, Wed), 4 days prior in the week relative to today, so it was a Wednesday.

3. 365 divided by 7 has a remainder of 1. So generally, if the a date in year N is Monday, the date in year N + 1 is a Tuesday. But leap years have 366 days, so if a date in year N is Monday, and a February 29th falls within the range of dates in the next year, the date in year N + 1 will be a Wednesday. Note that 2008 was a leap year.

For practice, I’d show how to calculate my own birthday. Here’s a fake example to protect my millions in offshore bank accounts and student loans. Say I was born on July 13th, 1980.

1. Number of Days elapsed since July 13th, 2009, remember today is Sunday, November 15th:
18 + 31 + 30 + 31 + 15 = 125.

2. 125 / 7 = 17 r 6. So 17 and 6/7 weeks have elapsed since July 13th, making July 13th fall six days earlier in the week. July 13th, 2009 must have been a Monday.

3. Years from July 13th, 1980 until July 13th, 2009 = 29. For each year the week days shift forward one, but for leap years they shift by two. Of the 29 years there were 7 leap years. In reverse order they were 2008, 2004, 2000, 1996, 1992, 1988, 1984 (1980 would not count since July 13th would have occurred after February 29th of the leap year). Increment another 7 days from the leap years giving 36 weekdays to account for. 29 + 7 = 36, which has remainder 1. Therefore, July 13, 1980 must have been a Sunday.

Explore
Have each student determine the day of their birthday and the day of a friends.

I’ll be giving them a set of mathcounts problems, that include some day of the week calculation.

October 13, 2009

Consecutive Integers Puzzle from JD2178

This is a perfect extension for some of the problems I’m giving a test over tomorrow, so it caught my eye.

How many ways can 1000 be expressed as the sum of consecutive integers?

My first instinct was to start writing equations to find the sum of a set of consecutive integers with x as the least. I began with $x + x+1= 1000$ . This simplifies to $2x + 1 = 1000$ but the solution is not an integer so there are not two consecutive integers that add to 1000¹.

To test four integers we’d use the equation $4x + 6 = 1000$ . And suddenly infinity starts to dawn on us as we realize there could be a lot checking to be done. So back to equations and formulas. I looked at a set of simplified equations

$x + x + 1 = 1000$ is $2x + 1 = 1000$
$x + x + 1 + x + 2 = 1000$ is $3x + 1 + 2$ or $3x + 3 = 1000$
and $x + \ldots + x + 3 = 1000$ is $4x + 1 + 2 + 3 = 1000$ .

To simplify the second term $1 + 2 + 3$ where $n = 4$ and the total should be 6, I’ll use the formula $(n^2 - n)/2$ . So for $n=4$ , $(4^2 - 4)/2=6$ it checks out.

Knowing that all the equations have the form $nx + (n^2-n)/2 = 1000$ ² is useful. Solving for $x$ we get $x =[1000 - (n^2-n)/2]/n$ . We can now test (n) different consecutive integers to find the starting integer (x). In cases where both x and n are integral we have a solution, count the solutions and the problem is solved.

To count the solutions I wrote up a spreadsheet with three columns, a column for values of n, values of (1000 – (n^2-n)/(2))/n, and a simple column that would take the value of the second column formatted with

if(mod(value,1)=0,”solution”,”")

which holds “solution” only when the value of 1000-(x… is an integer.

I found 6 solutions: 5 consecutive integers (starting with 198: 198, 199, 200, 201, 202), 16 (starting with 55), 25 (starting with 28), 80 (starting with -27), 125 (starting with -54) , and 400 (starting with -197). After checking to see if there are more, I find a 7th solution with 2000 consecutive integers starting with -999 – it checks out $[-999+(-999+1999)]/2 \cdot 2000 = 1000$ .

So I guess I’m starting to wonder whether you couldn’t find an infinite number of solutions. Here’s a similar spreadsheet I cooked up on Google Spreadsheet. On excel I’ve checked up to about 13,000 consecutives and still not found any other solutions…

I also thought I noticed a pattern in the solutions that they were expanding by multiples of 5 so first there were 5 then 16 (doesn’t fit) then 80, then 400, then 2000… so I checked 10,000 consecutives, and found that there is not an integral starting point. So much for the pattern of fives.

My final answer is 7 sets of consecutive integers add up to 1000. And I wonder if there’s an easier way.

499.5 + 500.5 does equal 1000, but 499+500 or 500+501 don’t quite get there ↩
Remember, $n$ is the number of consecutive integers and $x$ is the least of them ↩

October 5, 2009

Asking questions to which you do not know the answer

I try not to do this in class – at least not intentionally. However, today in my math competition class we spent a little time working with formulas for Volume and Surface Area. I wanted to ask a set of tough questions. The final question on the sheet is below

The volume of sphere $_1$ is 100 $\text{u}^3$ larger than sphere $_2$ . How much larger is the radius of sphere $_1$ than sphere $_2$ ?

My thinking is that the solutions could be represented on the graph with x as the initial radius and y as the increment necessary to create a difference of 100. I am getting different outputs from the graphing software on my computer and the output from Wolfram Alpha and I’d like to be able to infer coordinates from the graph.

How would you do this?

September 20, 2009

Year Plans

One of the things I’d like to do this year is engage more students at their level. Perhaps try to have more class time where there are multiple activities/groupings simultaneously. The main idea behind this is that my one class, one-idea, one-option norm is almost guaranteed to miss the sweet spot for low and high learners. Daily I would think it means that someone in the room is spinning their wheels without traction.

So, I read The Differentiated Math Classroom and I’m picking through Differentiation in Practice: : A Resource Guide for Differentiating Curriculum, Grades 5-9. It was strange to go back to the old bound paper method of professional learning – most of mine usually takes place amongst you all on the internets. But, as far as I can see, the best resources for learning about differentiation are still printed, bound, and sold. They’re not yet offered up freely like so much of the great stuff folks are writing and sharing.

The first outcome from this is two simple documents that I drew up to provide a general overview of the units I plan to teach and the time frame in which I hope they’ll take place. The unit planning stage is where much of the thinking behind differentiation actually takes place, but this is a starting point.

Anyway, here are my year plans for Algebra and Geometry.

Plan Alg Year

Plan Geo Year

Scribd bug fix These documents showed up at lousy resolution for a long time. I wrote an email to scribd and had a response back in less than 24 hours (for a free service I’m impressed). Turns out that I had two lines in the latex code that were making the pdf tougher for the scribd reader to read. After deleting

\usepackage{emerald} %a package of free fonts
\usepackage[T1]{fontenc} %font encoding to go along with the free fonts

The pdfs are back to high resolution, easiliy readable, scribd documents. To the other person in the world, if you’re out there, appreciating this solution: you’re welcome.

September 8, 2009

A principle for promoting good learning habits

Unless student exploration and problem solving are the task at hand, give the kids access to correct answers for every activity they complete.

I find that one of the biggest differences between the way that I complete problems and the way most of my students do is that I check the answer using the back of the back, my teacher’s edition, or software¹. Students will complete a problem and go on to the next. Once they’ve written an answer for the last problem, the work is done.

I’ve been trying to change this for a while. Last year I started assigning only problems with answers in the back of the text for homework, and this year I’m requiring all problems that can be checked to be checked.

I also write activities to use in class. Up until last year I almost never had an answer key ready for them when I’d write them. This year I’m trying to be better. After writing a worksheet, I print it out, solve it, and translate my work into the answer key. Not only does this make it much easier for me to hand a reference to students in class, it makes it that much easier for students to verify their process and thinking.

Here’s a sheet I’m going to use tomorrow in Algebra

Ws Solve Props Commut Assoc

This answer key isn’t meant to be the only source of truth, and I’ll try to give student solutions the main stage, but I think providing a complete model (and having it ready) helps students to gain confidence in their own thinking as they work. It’s also a great way to answer the “am I doing this correctly?” when you are needed elsewhere. Very few students will try copy answers, and it’s always clear when they do.

There are times when I would withhold the correct answer/solution for students almost endlessly to force them to think their way through, and times when I want them to practice applying a new skill with answers close by. This is one of those times.

A few new files for equation solving practice are here. Click on a file to download it. The box is friendly and doesn’t add any pop-up junk.

Geogebra or Maxima ↩

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