Circles, Arc Measure, Arc Length

Last Geometry unit I'm teaching before the MCAS: Circles! Thursday and Friday I taught about measures of central angles and arc measures, and introduced arc lengths. As the unit scheduling turned out, this unit is coming right after Similar Figures, so solving proportions are still fresh in everyone's minds for finding arc length (and soon area of sectors) as fractions of the circumference and area of the whole circle. I'm anticipating that they're going to get a kick out of the different properties of arc measures formed by various angles on circles, like central angles versus inscribed angles.

This is the packet I'm using so far for notes and preliminary practice, from TPT Secondary Math Shop: https://www.teacherspayteachers.com/Product/Circles-Geometry-Circles-Arcs-Arc-Addition-Arc-Lengths-Notes-Assignment-1118676

Word of the Day: "Eyebrow" in Spanish is "ceja". After drawing the symbol over the two endpoints to denote an arc, I realized it looks like an eyebrow over the letters, and went with it. Other bonus, I convinced some of my students to salute anytime they use "Major Arc".

Making and Using x-y (Input-Output) Tables

Input-Output tables are an excellent way to introduce students to linear, proportional, and non-linear algebraic reasoning. When set up correctly, students can use the tables to describe relationships between the independent and dependent variables; how the output is changing between each row; and what transformations take place between the input and the output.


Questions to Ask about Input-Output Tables

Real world situations

• Unit price (specials, 2 cans/$5)
• Constant speed (time vs. distance traveled)
• Year vs. age

What is the rule inside the machine?

• Is it a one-step or multi-step transformation?
• Is the rule constant or changing?
• Is the rule describable, either in words as y = f(x)?

What is the pattern?

• As x increases, what happens to y?
• How can you predict the next row on the table?
• If you skip a few rows on the table, what would the 100th row be?

Important numbers

• What happens when you plug in 0?
• What do you have to plug in to output 0?
• By how much do the inputs and outputs increment each time?

Translate table to other forms

• Rows into (x, y), graph on coordinate plane
• Come up with situation or graphic that the table could describe
• Predict what characteristics the graph will have based on the table.

 In this problem, students can make a connection between the term in the sequence and how many squares form that picture. They can see that as the numbers in the Input column increase by 1 each time, the number of squares grows by 2 every time. You can (literally) highlight that change visually by shading the two new squares that are added on to form the new shape. As an extension, you can ask students to analyze other characteristics of the shape by gathering data and making conjectures- for example, finding the perimeter of those shapes, or redrawing it to be a 3-d figure and finding surface area or volume. Or, students may choose to do away with the original independent variable and find the relationship between area and perimeter, or any other two quantities.

The utility of Input-Output tables lies in their flexibility to span grade levels and depths of content, from elementary through calculus and beyond. Students should be able to use tables as a tool to organize and interpret many instances of data and numerical relations.

Solving One-Step Linear Equations, Part 1

This lesson will introduce students to the concept of solving simple equations. The big ideas are:
- The solution to an equation is the number that makes it true.
- You can write a different yet related equation to make the problem simpler.

Step one, bring on the fact families. Remember these?
3+5=8
5+3=8
8-3=5
8-5=3
As long as students have a decent grasp of what addition and subtraction mean, you can use some kind of manipulative to show how the different facts fit together. Using a rectangular array, the same can be done with multiplication and division fact families. When leading students to write out the fact families, ask them to find the patterns about where the largest number is in the addition/multiplication and subtraction/division problems. Is it the part or the whole of the group?

Then, replace one of the numbers in a family with a variable or empty box.
3+[ ]=8
[ ]+3=8
8-3=[ ]
8-[ ]=3
Ask students to fill in the spaces, and then pick out which fact is easiest to "solve" for. They might choose one of the addition facts and solve by counting on, or they'll choose the subtraction fact with all numbers and operations on the left-hand side (LHS) of the equation, and the unknown isolated on the right-hand side (RHS). Whichever one they choose, make sure they check their answer by taking the solution and replacing it in the equation they chose; if it turns out true, then their solution is correct.

Next up, write an equation that has one unknown/blank, such as:
[ ]-6=4
Ask students to write the other three equations that will be in the same fact family (What numbers? What operations? What are the parts and whole?) Circle the equation that has all numbers on one side of the equal sign, and the unknown isolated on the other side. Repeat with similar equations that have the blanks in any of the three spaces until students can identify what different but related equation they would use to solve it.

Resources and Extras:
This worksheet from math-drills.com has algebraic equations with small numbers and the basic operations, with one number blanked out. Have students choose their problems, then write the fact families for them, choose the "easy" equation to solve, then fill in the blank. Other versions of this worksheet have symbols or variables in place of the blank.

Dice are a good tool for getting more tactile with this lesson, especially in the addition/subtraction fact families for lower-skilled students. Roll two dice, place the dice next to each other, and write the addition fact that's shown. Physically change the place of the two dice to show the second addition fact. For example, if they rolled 4 and 5, that gives you 4+5=9 and 5+4=9. Start writing the subtraction fact as 9-[ ]=[ ] and remove one of the dice. Ask, "How many did I take away? How many are leftover? Where do each of those numbers go in the sentence?" Students can practice drawing out each of the problems they do with the dice along with the fact family. Slightly higher-level students can do the same with multiplication/division problems by drawing the array.

Measures of Central Tendency: Range, Mean, Median, and Mode. Halloween Edition!

You and your friends go Trick or Treating, and decide to pool all of the candy and divvy it up evenly when you get back home. You get 23, 27, 18, and 26 pieces of candy.
- How many pieces of candy will each person get?
- What MoCT is this?
- What are the other measures?

Right before you leave the house, your Mom tells you to bring your little sister along with you. Ugh! But, all of the neighbors thought she was so cute that she was raking in the candy. She got 45 pieces of candy!
- What will happen to each of the MoCT? Will they stay the same, increase, or decrease? By a lot or just a little?
- Will each person, including your little sister, end up with more or less candy?

After the first house you visit, you run into a classmate of yours, who tags along for the rest of the night, but doesn't get any candy himself. (All of the pillowcases at his house were in the laundry.) At the end of the night, he asks to get half as much candy as everyone else.
- How do you distribute the candy between all of the people?
- What expression can you write to show this?

I'm one month into my second year of tutoring math at an urban high school, serving almost all ELL students. Now that I have a degree and a year of school-specific experience under my belt, there are some aspects of my tutoring that I want to improve.

1. Spiral Review daily.
   Our official curriculum has built-in review days, but they are designed to target only the skills that we taught that past week. I've been slowly creating review cards with questions from every topic we've studied so far, so I can pull them out and let students choose a random card (skill) to practice.
2. Teach note-taking and note-using skills.
   Even if my students are technically placed in 9th or 10th grade, due to possible interrupted schooling, many of them are not used to "doing school", or what they need to do other than just sit and learn in order to succeed in high school. With some of my younger students, or those with major organizational issues, I write out a set of notes as the lesson goes on, and tell them, "Make your notes look like mine." Then, as we accumulate more lessons in their notebooks, I encourage them to look in their notebooks for answers before they ask me.
3. Create cumulative vocabulary sheets.
   I'm still undecided whether it'd be better to create a long list of words at the back of their notebooks (or on a separate sheet), or to make an outline before each section with the words and phrases written in, with space for them to write definitions and translations, and show examples. The latter worked excellently last year for the Algebra I unit on linear equations, so I might try that again this year, even if it requires more upfront work on my part.
4. ELL support, even for English speakers.
   By the nature of this particular high school, every student in the academy is enrolled in an ELL class along with their content classes. Even if the students are comfortable with spoken, social English, all of them need help practicing the academic language structures that will let them be successful in their content classes, and eventually be able to transfer into one of the other academies (that isn't specifically designed for newcomers and ELLs). These techniques could include writing sentence stems into their notes; encouraging students to discuss problems with each other in English or Spanish and responding to them in English; and pre-teaching the vocabulary needed for an upcoming section.

As the year goes on, I also intend on keeping detailed notes on how I teach all of the different topics, and what did/didn't work for various students. Even though I can remember most of the techniques for myself the next time I see that topic, my notes could possibly help some of the new tutors this year and in the future.

This isn't directly related to teaching, but I bet I can find something relevant to say about it.
As of the past two weeks, I've been taking a free, online college course called "Principles of Economics for Scientists." I'd somehow managed to never take any sort of Econ course before, and this one is a great response to my hesitation about the subject relying too much on discussion of human nature and the sociology aspect of things. (While I do respect both of those approaches as valid, neither of them would have helped me make intuitive sense of this new subject.) As it happens, the "for Scientists" part of the course title roughly translates to, "for people who know and understand Calculus", which squarely includes me.
(Ooh, I figured out the connection to teaching and learning!)
In this class (at least so far), the professor is using Calculus as the lens through which he is teaching the fundamentals of economics. Therefore, for the intended population of students, there is a basic level of math needed in order to comprehend the material of the course. We are not doing math just for math's sake; we are using it as a tool for something else. Without a conceptual and practical understanding of calculus, the professor would have to take a more convoluted approach to teaching the exact same topics, probably resulting in students appreciating the material less.
This is equivalent to the pedagogical idea of "learning to read vs. reading to learn," the transition that all Language Arts teachers encounter in their respective grade levels. Once students can physically read a text out loud, what do they get out of it? How will the skill of reading translate into a truly functional literacy?
These same questions can be asked of math: "learning to do math vs. doing math to learn." Many secondary math classes spend all of their time teaching and learning sequences of math skills that build upon each other, with the promise that students "will need to know this later on." But for students that have no desire to continue in math or science classes, that later time never comes. I see myself as a perpetual student of math, but I've realized that while some math is universally necessary in participating in adult life, and other math is elegant and interesting in its own right, some topics are useful not just for the sake of doing math, but rather as a tool for understanding other topics, spread far and wide across the human quest for knowledge.
Every math teacher gets asked the question, "Why do we have to know this? When am I ever going to use this?" In response, think further than just the upcoming test, and even further than the next few courses in the school's sequence. Math is pervasive, and we need to convince our students that there is a bigger picture to the subject, that stretches far beyond the classroom.

Life update, since the last post:
- Completed my Math major.
- Started and finished my student teaching (MS and HS), and my Secondary Education major.
- Graduated from college =)
- Became a certified Secondary Mathematics teacher in two states so far, working on the third.
- Taught my two summer classes for the 2nd year, and got asked back for my 3rd 2013 summer.
- Currently working as a full-time math tutor at a HS in urban Massachusetts, and job-searching for next year.

Hoo boy!

What's been on my mind lately:
- Do I want to start my classroom teaching career in an urban, more difficult-to-manage school, or in an "easier" suburban environment?
- How will I measure the success of my students mid-year and at year-end?
- What can I do to increase my students' retention of math content from week to week, other than just in basic skills?
- What else can I do this school year to improve myself as an educator and thinker, and make myself a more marketable teacher candidate?

One day down, only 18 more to go!

Note to self: Leave enough time to get coffee before classes. Also, bring water and wear comfy shoes.
So, my first day of classes was rather surprising. The younger class, which I was most worried about, is remarkably easy to manage, even though the kids are antsy at the end of the day. The older class, which I thought could be a little more open, is a mash-up of all different kids and grades and personalities and motivations (or lack thereof). This will be a challenge.
What I imagined for this first class: All of the kids show up within two or three minutes of the official start time. There is paper and a pencil ready on each desk. I have a 3x3 magic square drawn on the board, and ask the kids if anyone knows what's special about it. Some kids get it quickly, and after a minute I give a hint to the remaining kids. After a few more minutes, everyone had gotten it. I hand out a homemade worksheet with partially-filled magic squares, answer a few classwide questions, and let kids work on it in their small groups. Much inter-student discussion ensues. Once most have finished, I draw a table on the board to flesh out some patterns that showed up in the squares. For the last part of class, I hand out mini challenges to each group ("Make your own magic square that...") and let them work together on it. If someone finishes a challenge and doesn't want to start a new one right before class ends, I give them a puzzle photocopied from MathMania. Done.
What actually happened: Kids show up after lunch. One of the first girls in is not actually on the attendance list, so she sits and mopes until our counselor gets here and can take her to the office to see what class she's actually signed up for. I show the class the magic square on the board, and ask if they can tell what's special about it. Some kids know already. I give the hint and some scaffolding, and a few more kids get it. I, along with student volunteers, flesh out what the trick is, and a few more kids say they get it. Our last student to show up walks in, sits down at the very back table, and I tell her to move up to one of the tables with other people at it. Since there are about 12 kids who've already mostly figured out what's going on, I hand out the worksheet to everyone. The girl who came in late has no idea what's going on with the math, so I ask the girls at her table to try explaining how to fill them out. After a few minutes, I realize that one of the problems I put down doesn't actually work, so I tell everyone to cross out that square. After looking at a few finished papers, I clarify that each sub-square must have a different number, that zero isn't one of them, and that squares can have different magic numbers. A few more kids fix and finish their worksheets, or just get frustrated after trying for a while. There's about 10 minutes left in class, so I give kids the options of either trying one of the challenges, or doing the other puzzle. Most kids chose the puzzle, but a few attempt the challenges and come up with some good work. Kids say they don't get what's the goal in the puzzle, even though there are directions written on the page. Class ends.
What I could do differently:

• Hand out the photocopied puzzles at the start of class to keep students occupied while everyone gets to class, and to suggest some math techniques. (The puzzles will be related thematically to the actual lesson.)
• Have all necessary supplies on the desks before students come in, along with extra paper. 
• Up on the board or individually, give a set time for students to share some techniques and tips for the day's activity, to use as reference at all of the tables.
• Designate a bathroom/water break 15 minutes into class. (It is right after lunch, after all.) 
• Have one calculator for each small group, so that students who are slower at arithmetic can still get something out of it. (This is for elementary and not secondary students, so I'm not totally contradicting previous opinions >_<.)
• Check and double-check any papers I hand out.
• Use the worksheets as scaffolding, with directions, hints, and prompts included. 
• Figure out how to integrate the kids that are signed up for my class because their parents don't want them to fall behind over the summer, which may or may not be the same kids that show a lot of learned helplessness. Yeah, Differentiated Instruction!
• Commandeer another counselor (already done for tomorrow). 

I'm going to try to implement as many of these changes as I can for tomorrow's lesson. Hopefully, I'll be able to convince my kids that I do actually know what I'm doing, and that this will be a positive environment for exploring math. First days are always rough.

Strangest shopping list ever.

Things I have bought for my classes so far:

• Posterboard (for cutting up into DIY rulers)
• Small plastic cups
• Pack of plastic rulers
• Dry-erase markers
• Snack-size Ziploc baggies
• Wagon-wheel pasta
• Silver star stickers
• Mini post-its 

Also, my mom discovered about a dozen issues of Puzzlemania and Mathmania books (anyone remember those?) in our attic, along with a number of other math puzzle books. So many things...

How I (Will Have) Spent My Summer Vacation

The next few months are going to be very different from what I've gotten used to as a college student. Just as soon as I got the hang of being a decent student, I end up teaching instead. It's all for the better, anyway; according to my professors, I haven't actually learned any useful teaching skills yet. I have to get those on the job. Well, here's that there job.
For the next four weeks, I will be teaching two elementary school-aged math classes at a community summer program (where I previously worked as a counselor, and was a camper before that >_< ). My first class each day is "Math Fun and Games," where I'll try to convince students entering 3rd-6th grade that there's more to math than worksheets and drills by encouraging them to look for patterns and think creatively. The class for kids entering 1st-3rd grades is called "Number Sense," and I hope to give the younger students an idea of what different physical quantities look like by learning how to measure and record.
Classes start on Monday, and I honestly have no idea what to expect:

• The kids are outside of my typical age range for teaching or tutoring.
• This is my first time in charge of an actual class.
• I've come up with the lesson plans myself, and don't know how good they are.
• I don't yet know how old the kids that signed up are, or why they're signing up.

I'm pretty sure I find out part of that last one tomorrow. It's only really an issue for the younger class, because there's such a wide range in what I can do with a 5 y.o. fresh out of kindergarten, and a 7 y.o. who just finished 2nd grade. (All made more annoying by the fact that NJ doesn't have statewide math standards, nor does my school district across schools.) There will be a lot of modifying to do after the first day or two of class, but I feel like I'll manage to not screw up my campers too badly the first time I teach this. 

Fun- and preparation-filled weekend ahead. I'll keep all y'all posted. 

(Looking at past posts, I'm good about writing at the start of a "semester". I'm going to break that habit this summer.)

What my semester looks like

Sorry it's been quite a while since my last post. Classes started at the end of August, and life has been pretty crazy.

On that same note, my schedule will give me a good number of future post topics; I presume that's the occupational hazard of completing a college (double) major in what the blog is supposed to be about. My classes for this semester are as follows:

1. Multi-Variable Calculus
2. Linear Algebra (both for my Math major)
3. Adolescent Development
4. Computers, Teaching, and Math Visualization
5. Secondary Education Practicum
6. Jewish Science Fiction.

I know the last course seems a bit out of place, but it's a break from classes that I specifically need for my majors. Anyway...

The first five courses are fitting together surprisingly well. The straight-up math courses certainly teach the stuff they're supposed to, but it gives me some opportunity to play with ideas from the other three classes. And then topics covered in one of the education classes give me something to focus on from the other two.

An example of what's been on my mind:

In Adolescent Development, the 2nd chapter we covered was cognitive development. I forget which researcher it was, but someone talked about the stage at which kids are able to make leaps from the concrete to the abstract. In terms of implications on math education, that would be when students are able to understand the concept of a variable. So, depending on whether the student is an early- or late-maturer, he may get lost in class if the teacher introduces variables before he can really get it.

I witnessed a possible instance of this first-hand as I observed a classroom through my practicum: 7th grade Advanced Pre-Algebra. The lesson of the day was the Properties of Addition (Associative, Commutative, Distributive, etc.). The teacher talked about grouping like terms, and why students couldn't simplify the term "x + x^2". He asked a student why it wasn't possible, and the student replied that it was because they "don't know what x is." This sounds right, but it could be that the student thinks that "x" has a different value at both places in the the expression, which it doesn't. From a practical viewpoint, is there a good way of diagnosing cognitive misconceptions, or whether or not the student has reached the level of understanding in order to get what a variable is in this context?

So today in my Math Technologies class, we got into a discussion about whether or not technology should be used as a crutch to help students that are lagging still get something out of a lesson that may be over their head. The consensus was that if a student has a hard time carrying out the algorithm to compute a problem, they may still be able to benefit by viewing the problem more conceptually using a graphing calculator or Geometer's Sketchpad. In the above example, is there some way that a certain technology could be used effectively by the teacher to help the student visualize the logic of "grouping like terms" before they understand how variables are used?

And then when I don't feel like thinking about (relatively) little kids anymore, I can apply the same mindset of a teacher to my own math classes. Even once we reach college, the same things still apply. I find it interesting to contrast the teaching styles of both of my current math professors, mostly to find some justification in really not liking, at all, how one of them teaches. (The fact that I can't understand his accent half the time doesn't help, either.) A review book will be desperately needed for that class.

On that note, I should be studying. Any feedback would be enjoyed.

How's about those positive role models?

My friend showed me this video of Patricia Heaton at Guest Week on "Who Wants to Be a Millionaire". (Skip to about 3:00.)

It isn't often that I yell at videos, but this is ridiculous. As soon as anything with numbers showed up on the screen, she completely shut down even before looking at the question. This is a grown woman freaking out about middle school math. A kid would look at the video and say, "She can't do math and got to be famous. I can do that too!" Never mind the fact that when Regis walked her through it, she was perfectly capable of solving the problem.

A defeatist attitude is lethal to learning math or any other subject. Unfortunately, mathematics often gets the short end of the stick in situations like these; that's what credit cards are for, right? It's a lot easier to get away with never doing simple, mental calculations than to get through life unable to write a cohesive paragraph. Who cares that you can't balance your own checkbook, as long as you can write thank-you notes to your generous donors. 

Does anyone know of a celebrity who likes doing math in public, and is proud of it? Who knows, maybe Patricia is just stuck in a junior high mindset and doesn't want to be called out as a nerd. 

Welcome to MathEd Out!

“For Today’s Graduate, Just One Word: Statistics”  (NYT 8/6/09)

“I keep saying that the sexy job in the next 10 years will be statisticians. And I’m not kidding.” -Hal Varian, chief economist at Google

Math stole the front page of the New York Times today. In short, it claimed that a good number of the careers that will pop up in the next decade are going to be based on a math-heavy education, namely the field of statistics. With the growing number of technologies born to process huge quantities of information, we need more human intelligence to do something with all of it, and to make those technologies more efficient.

One example:

A while back, MIT experimented by giving 100 students Blackberries that tracked their movements and actions on campus. One small step for Big Brother, one large step for statisticians. It’s one thing to have a massive amount of data on record about your student body; the hard part is focusing on a specific trend, and extrapolating from it. Humans come into play by deciding how to organize the information into something useful, or at least interesting. 

The headline for the general public, or least for school-age students, should be shouted loud and clear: Math is everywhere, not just in your textbooks. You don’t have to be a typical “nerd” to find something that catches your attention in the field of mathematical applications. Most people take advantage of forms of advanced mathematics every day, and anyone with the right kind of analytical mind could contribute to the field in ways yet unimagined. 

For once, math education is being glorified based on something other than only salary! (Though, going by the jobs mentioned in the article, that part doesn’t sound too shabby either.) Soon, the secret will be revealed that math can, in fact, be cool to study. And that it involves far more than learning techniques and regurgitating them on worksheets. You could land a gig delving into technologies we take for granted, and making them even better and more user-friendly. Who wouldn’t want to take part in that? I don’t know how many pre-college students are going to read this article, but I would encourage them to do so, before they make up their minds to dismiss math completely.

“You’re Leaving a Digital Trail. What About Privacy?”  (NYT 11/30/08)