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.

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.

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.)