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?