I have always been frustrated trying to teach students how to turn the slope-intercept form of a line into the standard form. I cannot count how many times I have pointed out that, to get rid of the denominator, students can multiply the equation by the denominator. Why wasn't this working? Because I was the one saying it. As the saying goes, "The person doing the talking is the person doing the thinking". I have heard this for years, but I didn't know how to utilize it until recently. I changed my lesson drastically, and this time, the students did the thinking! I started the lesson by showing them the following three equations. I asked them what they noticed and annotated what they said on the board with the equations. I then had them type all three equations on Desmos and tell me what they noticed about the graphs. They realized that the three equations graphed the same line. We went back to the equations and talked about slope-intercept form versus standard form of a line. The last two equations are in standard form and we concluded that standard form has no fractions and that x and y are on the same side of the equation, with the constant on the other side. Next, I randomly grouped the students into groups of 3 and sent them to the whiteboards. (One marker per group, the person doing the writing is not deciding what is being written). I challenged them to use algebra to change the slope-intercept form of the line from above into standard form. They worked for several minutes, trying different things. (I did not give them any ideas or tips at this point) It was time to move them forward and I decided to try something new at this point, workshops. (I read about workshops in Geoff Krall's book Necessary Conditions. I highly recommend the book! ) I paused the class and asked one person from each group to come over to me and the rest keep on working. I wasn't sure how this moment would go, but they did exactly what I wanted, one person came over to me and the rest turned back to their boards. It took a matter of 5 seconds. When the small group of students got to me, I wrote down the equation in slope-intercept form and asked the students what they have done so far. They gave me a couple of answers. One of their answers was that they moved the term with the x over to the other side by subtracting it on both sides. They also talked about how they wanted to get rid of the fraction, but weren't sure exactly how. I asked them what they had tried so far and I found out that they knew what to do, they were just insecure. I told them that they had great ideas and should try them. They went back to their groups to report what we did. Some groups went on with what they were doing and some went back with new ideas to try. When groups finished, they were able to check their answers against the standard forms on the board. I then gave them a new equation in slope-intercept form to change into standard form. As they finished, they wanted me to confirm if they were correct. I decided instead to have them check using Desmos. But, instead of telling them exactly what to do, I asked them what happened when we put our original equations into Desmos. This was enough to spark the idea of how to use Desmos to check their answers. I wanted the lesson to progress, so the next problem was a little bit different. I gave them the following problem: A line goes through the point (2,7) and is parallel to y=3/4x+5. Write the equation of the line in standard form. At first, the students were convinced that they couldn't do this problem. My first response was to tell them that I wouldn't give them something they couldn't do. About half of the groups went on to figure it out themselves. For the rest, I asked the following questions:
Students jumped back into the problem at various questions. This is the fun part as a teacher using the thinking classroom structure. Students are motivated to solve problems, they are invested in it and want to keep going! Once again, students used Desmos to make sure they got the correct answer. I visited each group to check how Desmos helped them. I wanted to make sure that they plotted the original parallel line, the point that went through the new line, the equation in slope-intercept form and their standard form. It turned out to be an important conversation with the groups. All the groups made it to the last problem (above), but not all of them made it to the next one. This is to be expected, given that the groups work at their own pace. I was happy as long as they made it to the last one. If they hadn't, I would have continued the next day. The next problem was intended to push students who were ready for a push. One of my favorite tasks for increasing Depth of Knowledge is from openmiddle.com. I finished this lesson with the following problem: It was fun to watch groups tackle this problem. None finished during class, but many came back the next day having worked on it outside of class. Of course, we used Desmos to check.
We followed the lesson up with practicing changing from slope-intercept form to standard form independently. Note: This was my first independent lesson in the thinking classroom. Seeing my students work independently and with self-motivation sold me on the teaching method. Now, I often stop and just soak in the math discourse and learning going on in my classroom. My students are now the ones doing the talking, and therefore doing the learning. I have recently realized that my classroom is now truly student-centered. Although my involvement with questioning and helping is important, when they leave and I reflect on the class, my thoughts are about how hard they worked and how well they did, not about how well I did presenting the lesson.
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I wanted to teach a lesson on using systems of linear equations to solve an interesting problem instead of the typical gym membership problems. I turned to the MTBoS search engine, but struck out (I was being really picky). I did, however, run across Christopher Danielson's "Oreo Manifesto" and found my inspiration. Then I found a picture at residentevents.com that helped inspire the rest of the problem. To spark interest and start the problem solving process, I started the lesson by showing the picture of the oreo stack on my smartboard. I had the students notice and wonder while I wrote what they said on the board. I then shared two things that I wondered: 1) How many calories would this oreo stack be? 2) How many grams would it be? In order to answer these questions, we brainstormed some things we might need to know. We also had a conversation about if the stufs in the oreo stack were from regular oreos or double stuf oreos. We decided to say that they were from regular oreos, but some were convinced that they were from double stuf. I followed up by giving the students the following image. Note: At this point, the students have brainstormed, but I have not told them what they need or how to answer the questions (in reality, I have not told them anything). I used the public brainstorming as a way to let them share ideas, but didn't want to do the thinking for them.
I randomly grouped the students together with cards and sent them to the whiteboards. We decided to work on the first question, "How many calories is the stack?" I let the students play around with the problem for a while. They struggled, which I anticipated. It became the headache and they wanted the aspirin (see Dan Meyer). As they worked, I visited each group to hear their ideas and thoughts. After a few minutes, I called the class together to one of the white boards. I explained that something that helps get a problem going is figuring out what the variables are. "What don't you know that you want to know?" We wanted to know how many calories the stack is. But what do we need to know more specifically? 1) How many calories is a wafer 2) How many calories is a stuf I sent the students back to their boards with the mission to now create equations. Once again, they struggled, but kept trying. I walked around to the groups and visited with them about what they were trying. (When talking with the students, it is my goal to learn what they understand and then move them forward with careful questioning. My rule is that I cannot tell them anything, I must ask a question) After a while, I called the class back together to talk about how to build the equations (aspirin to their headache again). They went back to their boards, wrote their equations and found how many calories were in the stack. (They have already learned how to solve systems). When each group answered the first question, I asked them to work on the second equation, "How many grams is the stack?" They jumped in with enthusiasm and worked until the end of the hour. Some finished, some did not. Either way, we all ate oreos before leaving for the day! To follow up, the next day, we worked on the typical word problems: from the situations, we constructed equations and solved the systems. I have always been frustrated with teaching zero exponents and negative exponents. It is difficult to really get the students to understand. But I think I have finally found a good lesson! Of course, it includes visuals that students can experiment with. When we were done with it, I was pleasantly surprised when, in answer to the question, "What is a^0?", both classes answered, "the division by itself." (Which was not a phrase I used in class, but a phrase they created from their exploration).
I struggled at the beginning of the lesson to get them to forget about the exponent rules they memorized and really understand what we were exploring. But, through the process, they forgot about the memorized rules, and embraced the math we were doing. I count this lesson a success! You can get the lesson here. First handout for Stand and Talk Collection of various patterns for groups to work with |
AuthorI teach mathematics for grades 7-12. Teaching mathematics is my passion. Archives
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