Systems of Equations: Substitution

Keep in mind that we have already done systems by graphing and reviewed finding solutions of equations. I started class with this:

Next, I presented them with this slide and asked them how to solve:

They easily knew what to do and how to write it as an ordered pair. Next I presented them with this slide: 

I think maybe one or two students kind of knew what to do but for the most part, they were stumped. Now that is where the index cards come in. So we took one half (I just had them cut them up so I wouldn't have to use as many) and on one side they wrote y. On the other side, they wrote x - 1 because that is what y equals. Next, I passed out legal size white paper. I had them write the second equation down. Instead of writing the y though, I told them to draw parentheses big enough that your index card would fit inside it. We practiced looking at the equation with the index card showing the y and then flipping it over to see the x - 1. I think this was a good 'kinesthetic' activity because we were literally replacing the variable in the equation with an expression. So now students knew we had to do distributive property and then combine like terms.

From there we did one or two more examples and then they figured it out with out having to use the index card.

The next day, we practiced again.

This time, I had students get a colored pencil to write the substituted parts to make them stand out better. So we started out with a system. I told the students to make a cloud around the equation where the variable is alone. In the example below, we clouded 5y - 1. I asked the students what that equaled and when they answered x, I showed them to then circle the x in the second equation. From there we plugged in (with the colored pencil) the cloud where the x had been. We went through the steps of solving and found that y = 2. Then, with colored pencil, make a cloud around that. Draw an arrow from one cloud to the other. I asked them what happens when two clouds hit? (You know where I'm going right?) Make it rain! Last year my students always forgot to find the second coordinate of the ordered pair but now I know they will always remember to make it rain (if nothing else!).

I found this joke worksheet last year and I really like it because students can self-check using the answers on the right. But beware, students can figure out the joke. Specifically tell them you will be checking their work, not just the answers.

I assigned #1-4 for homework at first because you don't have to solve for a variable to start out.

One student told me she cried because she couldn't figure it out. I think what frustrated them about this was that they knew what to do, they knew that they could do it, but they were just making mistakes. Some students have actually told me that they like this and one girl said it was the easiest thing we've done all year. I've tried to stress to them that this is a new concept but a familiar process. As they practice, they improve. But what I have been impressed by is that they don't give up. They know they can do it.

Right before lunch one girl said I was making her hungry with all of this thinking and hard work. Success!

Those of you that have been reading and responding to my tweets have probably noticed my moody-bitter-I-hate-everybody-what-is-the-point-of-teaching diatribe. I still feel that way. I feel like even though my students are learning at this point, they won't know in two weeks how to do it. Or two months. And definitely not in two years. We have data to back that up.

I feel like my job is pointless. They aren't retaining anything and I don't know how to make them retain it. We've identified that students aren't really doing any critical thinking but I don't know how to fix that. Or teach it better. Or make it stick.

So. I will keep going on but yesterday, I wanted to quit. This job is impossible. If the people before me couldn't do it, what makes me think I can? I think a mark of a good teacher is when you can make the lower level students learn. I'm not doing that. Anyone can teach the top students. How do you reach the rest? 

Even then, our smart kids are not retaining anything. 

What. Is. The. Point.



  1. It is not the case that "Anyone can teach the top students." Teaching is difficult, and you won't be able to reach everyone. Work on expanding the number you reach.

    Students do tend to learn and forget, because we set up the game that way for them. Try to set up your curriculum so that skills get continually re-used rather than learned and forgotten.

  2. Any suggestions on how to accomplish that? I've been trying to review old material through bell ringers at the beginning of class and throwing an old question on a quiz every now and then. That's about the best I've got.

  3. You know Ms. C, I was going out the door to dinner with a friend and only had time to read the first half of your post. The friend’s an educator who works with a similar population and I mentioned your “draw the parentheses wide enough to fit the index card in” approach, even taking out a piece of paper and a business card to show what it looks like, and we were both impressed as heck with the brilliance of that idea. (READ: We’re both stealing it.)

    And then when I got home, I read down to see your comment of “I feel like my job is pointless” and I’m thinking, “WTH?!” You have a real gift as a teacher, Elissa, whether you realize it or not. And you can make a meaningful difference for your students, even if it isn’t as much as you want or think you should be able to.

    You make a difference when a student passes algebra and graduates with a high school diploma, instead of dropping out because they couldn’t pass math.

    You make a difference when a student gets a C and decides to give community college a try.

    You make a difference when a student remembers enough math on the ASVAB to become a tank mechanic when they join the marines.

    You make a difference when three years from now one of your former students tells you that they finally got their act together and wants to thank you for pushing her when she was in your class.

    There is a point to what you do. Believe it.

    Teaching honors kids is “easier” in the sense that they have the intrinsic or extrinsic motivation to do the homework consistently, which means that learning is more likely to stick and/or you don’t have to re-teach as much. Beyond that, I agree with gasstationwithoutpumps that it’s just different with different challenges.

    In the average honors class, daily homework gets done by 75% or more of the class. In the average non-honors class, the percentage can go as low as 30 to 40%. And that’s even when homework counts towards the final grade.

    Getting more of your students to do homework consistently or getting all of your class to do homework on the same day once a week would be a meaningful goal to shoot for. Seriously, if bribes work and you can pull it off, the first time the entire class has homework done on a Thursday, I would allow them to watch “Donald in Mathmagic Land” on that Friday.

    Also, Henry Piccioto, of “Eliminating the Textbook” fame, has an interesting practice of assigning homework the day AFTER he teaches the lesson, i.e. the homework for the day 1 lesson isn’t assigned until day 2. It gives the students the opportunity to review their notes and to ask questions on day 2 before the homework is assigned. In addition, if he realizes that he didn’t cover an important point, the students need an extra example, etc., he can take a few minutes on day 2 to provide the additional instruction or clarification. You may want to give that approach a try. (It is how I teach most of my classes.)

    Hang in there, Ms. C!

    Paul Hawking
    The Challenge of Teaching Math
    Latest post:
    I get what you’re going through

  4. Paul,
    Thanks for your encouragement. The problem is students aren't remembering enough math to pass the ASVAB or community college. Yes, I know that I encourage and motivate students. They like me. They'll remember funny stuff I said and that I wore cute clothes and made them laugh. They might even remember some fun activities we did. But! And it's a huge but!! They aren't remembering any math. They aren't learning how to be responsible, how to learn, how to be organized, how to study, how to work in teams. That is the problem. Yes, I will make a difference. I could have done that by being a good friend or mentor or club sponsor or neighbor. But my job is to teach math. And that's what's not happening.

  5. Hi Elissa, thanks for your honesty, here and on Twitter. I think Paul is right, everytime I visit your blog I find an idea I want to use--I'm impressed by the thought you put into your work.

    Have you been able to talk to your head of department? What does s/he say? Or is there another teacher who can be a listening ear?

    About re-using skills, as gasstationwithoutpumps mentioned, I think you are on the right track. Lesson starters that have five questions different topics from earlier in the year are one way (I assume this is what you mean by bell ringers). The first time you ask "what is the area and circumference of a circle with radius 4 cm?" then the next day there are five more questions, and this time one says "what is the area and circumference of a circle with radius 7 cm?". After a week of this, ask something different, then come back to area and circumference every third time.

    Also, I often budget a lesson just for a review worksheet or activities, about once a fortnight or once every three weeks.

    Peer marking and feedback helps them remember, I think. For example, the "Tear and Share" review worksheet. The page is devided into four quadrants. Each quadrant has a single review question, such as "solve this system of equations..". Each student completes the four questions, then tear into four pieces. Collect the A, B, C, and D pieces, and assign a small group to analyse one question. (Depending on the class size, you might have two or three small groups analysing question A, just divide the A responses between them.) Then the groups feedback to the class about what was done well and how to improve for each question.

    I hope things look up and you regain a feeling of purpose. Keep up the ranting and raving--it's helpful to hear.

  6. I know that it stinks that you can’t help every student in your care to succeed. And that it’s an incredibly frustrating feeling if you care about your students, which you obviously do. Surgeons sometimes lose patients on the operating table, in spite of their best efforts, but they don’t give up because doing so would cause all of the patients to be lost. You do what you can, the best that you can, believing that you can make a difference, even if you don’t know how right now. For now, you just have to have faith.

    I know your school is under the gun to improve overnight, so anything short of 100% improvement can look like failure. From a perspective viewpoint, how many of your students ARE getting and retaining the math? I read into the above post that ALL of your students were failing the class and I wanted to get a fuller picture of things.

    I had a few more ideas below that you could try. None of them is a magic bullet, or will work every time: teaching is more heuristics than algorithms. But you can’t lose sight of the incremental gains that you can make if you keep on trying.

    Paul Hawking

    Hand out index cards and have your students spend 10 minutes answering two questions: “What else can I do to help you succeed in this class?” and “What else can you do to help you succeed in this class?” You will learn a lot from what they have to say.

    After a lesson, ask one of your struggling learners to show you their notes and take a few minutes to help correct them, if they miscopied an example problem, and to add any additional details to their notes that they missed.

    Assign readings from a trade book that’s fun to read. Chapter 10 Systems of Equations from “Algebra The Easy Way” by Douglas Downing is written as a medieval fairy tale with dragons and kings and wizards.

    Give your students a “capstone project” that pulls together everything they’ve learned so far this chapter. I just put up on my blog such a project that I use with my regular algebra students when we get to solving systems of linear equations word problems:

    Teaching word problems (systems of linear equations)

  7. 1) I'm so annoyed that I've never thought of using index cards as a way to physically move around parts of equations

    2) This is a really good lesson, and your students will remember it. In a year, they'll be able to get up to proficiency in a week, rather than a month. You're not just teaching them some math, you're re-wiring their brains.

    3) Ask your students what they need to learn better, in a survey. Or better yet, have them write a math autobiography. Then proceed to give them lots of practice with lots of feedback and when they've finally got it on their work, give them a graded assessment.

  8. Hi - Just came across your blog and comments about students not retaining anything. I feel the same way - and this goes even for the AP Stats kids that I teach. Some ways that I try to improve retention is (a) all exams are cumulative - 20% of the exam is "old stuff"; (b) I go at it at a fast pace so that before the AP Exam we have a month of review. In a non-AP context that would mean before the final or before the state tests have about 10 days of mixed problems and perhaps a cumulative test.
    I have reached the conclusion that as a math teacher and former engineer I know better than most what concepts the students need for their next courses or careers. So I cut down on some non-essential stuff and go more in depth, repeat more and assess more cumulatively (if that is a word).
    Best of luck.

  9. I think your students' relationship to math is kind of like my relationship to working out. I dislike working out, and I always think, "What's the point? I'm going to work out for an hour and then eat, and I will get those calories right back. Versus if I don't work out for a while, it doesn't seem to have an immediate negative effect on my body."

    But, the reality is that if I don't work out at least sometimes, it prevents me from being in good-enough shape to do things like climbing a volcano or exploring a jungle when I go on a really great vacation somewhere. (For your students, this could be equivalent to them not being able to get the job that they want in the future.) So, working out, for me, is a necessary evil even though I don't always see its benefits right away (or, ever).

    To make working out bearable for me, I do it in a way that minimizes the boredom -- I get with a friend and do yoga once a week. (I know -- it's REALLY minimal. But I'm trying to illustrate the point to you.) I am not in great shape, and it's sometimes a hassle to make time to meet up with my friend to do yoga during the week, but I always feel good afterwards. And the result is that when I do go on vacations or go on an 8-hour hike somewhere, I have enough physical strength to pull myself through. (That's equivalent to your students in the future remembering HOW to study for math and remembering that they CAN get it with hard work, even if the specifics of the algebra has escaped them.)

    That's just my two cents, anyway. Year 2 is deceptive, because you think everything should have gotten easier, but the truth is that you start to be more self-critical as well. Good luck hanging in there!

  10. Oh, just re-read part of it. Maybe it's not totally clear why your teaching math is valuable, in relation to my analogy: I work out sometimes (yoga and walking) just to keep myself at a minimum fitness level. What you're doing is keeping your students at a minimum relationship to math. Of course it'd be great if you can improve that relationship with your creative lesson ideas, but even if you can't, you're at least making sure that they're staying in equilibrium and not getting further away from math, in case they do need to play "catch up" at some later point. (Just imagine the alternative -- what if NO ONE teaches them anymore math starting tomorrow? What chances would they have to graduate from high school and to attempt college?)

  11. mathfeedsback,

    I am the head of department. Lol-small school. Lately my bell ringers have been more homework review or else preview questions. I had been doing some ACT prep questions but I've fell away from that too. I should take a day to review every couple weeks- I have a lot of review game ideas. But for this year, I am too far behind already. I LOVE the tear and share idea! Do you use that with review questions or current material? What happens after the class shares the feedback? Do students have a chance to try again?

    Not all students are failing but only 15% of students are meeting on state tests. We have data to back up the fact that students aren't retaining from 6th-11th.

    I've asked them many times for ideas on how to help them but most if the time they don't know enough about the way they learn to help themselves. Some want direct instruction and notes every day but to me, they are just memorizing and manipulating. They don't particularly love my activities that make them think or struggle.

    What is a trade book? Would that book be appropriate for high school? I do want to incorporate more reading and writing.

    I have not taught any word problems yet so I am looking forward to using your investigation after I teach the elimination method. Thanks for suggesting it. I do want to do more projects that will tie things together.

    I don't think many students have any idea how they learn, let alone how I could help them learn better. Do you have any suggestions for how/when to give them feedback?

    Definitely support a cumulative exam idea. I tried that at the beginning of the year but I was terrible at writing assessments so I will definitely be trying that in some way next year.

    What you said about year 2 is a perfect summary of my feelings! Do you teach students how to study for math? I need to but I don't know how to myself. It is true that I am keeping them at a minimum relationship with math but I still feel like it's pointless because next year they will have lost the minimal relationship and start over again.

  12. @misscalcul8:

    Textbook vs a trade book

    A math textbook is on average written at a reading level two grade levels above the students. A trade book is a guide to math you pick up at a book store and is written to be read and understood by the student. "Kiss My Math" by Danica McKellar is an example. "Algebra The Easy Way" is very readable and covers much of the material in a high school algebra 1 class. If you can get a copy from your curriculum library of this guide,

    Teaching Reading in Mathematics, 2nd Edition (A Supplement to Teaching Reading in the Content Areas) (from McREL.org)

    you'll get some good ideas to ponder and possibly use should you try to incorporate reading and "writing to learn" into your classroom.

  13. Paul,

    Thanks for the suggestion. Do you use trade books? How do you envision using these type of books in the math classroom?

  14. Short answer is that I don't.

    Longer answer and a possible lead to someone who knows how to . . .

    The Teaching Reading in Mathematics guide has some good suggestions that I have used. One of them, for example, is that you need to show more than one representation of a geometric figure. If you only draw right triangles with the right angle in the lower left corner, and not in the lower right corner, upper right corner, etc., students may not connect with the idea that these are also right triangles.

    My stumbling block with using selections from trade books is that some of my regular math students would rather do ANYTHING else but read. For innumerable reasons, reading can be difficult and frustrating for those students. I think math class is one of the few classes where they didn't have to deal with their reading difficulties and my concern is that I will do more harm than good.

    I've never had the benefit of training from a literacy coach who's actually used reading in a math class, but one of your readers who commented above has and has blogged about it at little. I'm hoping hillbilly will share some more how to suggestions on her blog, as her two posts on the subject were more geared to how she used reading in her physics classes (HINT: that was a pleading request for a post on this, hillbilly :-)

  15. Paul,
    I have that same fear but then again some students would love the reading part. I also want to integrate some art projects too. But there again, some students would love that and some people would hate it.

  16. With reading, my concern is for the students whose reading difficulty makes it difficult for them to pull meaning from what they've read, so it's more than just love vs hate.

    With art projects, I agree with the research in "Why Don't Students Like School?" that more time is spent on putting the art together than on reinforcing the math concepts, but if doing such a project motivates your students to become more involved in class because it's fun then I say go for it. Amber at "Maximizing Learning" posted this week on a geometry project that I could see students having fun with.

  17. Paul,
    My goal is to have a project of some kind (reading, writing, or art) to motivate students to get done with their work so they can have project time. Or for the top students, a form of extension, and for lower students, maybe a way to help boost their grade/participation/motivation if they struggle with math.

    I checked out Amber's project and it is something I did last year. But, I hadn't heard of Amber's blog before so thanks for mentioning it.

  18. I just wanted to let you know that I used the index card idea to teach substitution this week and it was absolutely brilliant. I ended up cutting card stock into small pieces (maybe 1.5" squares)to save supplies, and my students even asked if they could use more squares on their tests. The card flipping built on prior knowledge (we talk about the sides of an equation all the time -- referring to sides of an index card is a natural extension) and really helped them see what they were doing. Thank you for sharing that awesome idea!

    It sounds like you have a tough school. It's hard not to get swept up in the glowing math teacher blogs, but success doesn't always look like the each student achieving the deep conceptual understanding of math we all want them to have. For some students, every day that you help them enjoy or feel successful in your class is a huge victory because it encourages them to come back to school the next day. I know that the more chaos my students have in their lives, the less they like activities and investigations and the more they like direct instruction and a very predictable class structure. You already seem to be making your students grow so much, and the fact that they are working so hard is something that you should give yourself a lot of credit for!

  19. Emily,
    Thanks so much for your positive and encouraging comments! We both know feedback is so important. I'm glad the idea worked out well for you. I'm wondering if you could/have blogged about teaching equations or the sides of the equation that you mentioned. I'd be interested to know how you do that.

    I also never thought about the fact that the more chaos there is, the more structure my instruction should have. Thank you. That is insight that I didn't have and gives me something to chew on.

  20. Love the index card substitution idea. I plan to use that one in the future.

  21. What does break out of jail mean?

    Love this...phenomenal work.