Goals: 1) Students understand what they are doing when they solve equations, beyond memorized steps 2) Students see/understand the efficiency of getting rid of the constants first 3) Students know they can multiply/divide first, but also understand that they have to do it to the entire side, not just the term with the variable 4) Students continue to keep their equations balanced 5) Students continue to make connections between solving equations abstractly and solving with Algeblocks Lesson: Day 1: I began this lesson with a number talk. The idea is to give them one that they have to do in a couple steps and then relate the lesson back to it at the end of the hour. Number Talk: (See picture above) I have six tables. I told the kids that I had 20 pencils that I wanted to put in the resource boxes on the tables. If I put the same number of pencils in each box, how many did I put in each box? The students shared about 5 different solutions and we went on to the lesson. (I wish I had taken a picture of these. I thought I did, but cannot find it any where on my phone). Next I wanted the students to explore solving a two-step equation. I decided to use the balance scales because I wanted the students to see it being balanced or unbalanced after their choice of operation. (We have been talking a lot about how the equal sign in an equation means that it is balanced). The equation I used was 2x+3=9. My paraprofessional and I set up the balance scales at each table. Before we started solving, I asked the students what they noticed about the scales. This step is something I was missing in the past. It is important to take the time to make sure students have studied what is in front of them before asking them to make decisions on how to solve it. They noticed: "There are two canisters" (We wondered and confirmed that they were the same weight and worth) "There are 3 white tokens with the canisters" "There are 9 white tokens on the other side" "The two sides are balanced, so they weigh the same" "The equation is 2x+3=9" At this point I told the students that I wanted them to figure out how much ONE canister weighs, but it is important that they "have a math reason for what they are doing". (There are probably better ways of saying this, but my ninth graders nodded their heads and got busy, so it must have made sense to them). There was great discussion within the groups on how to solve the equation. They talked about making sure they did the same thing on both sides and that they needed to keep it balanced. They pretty quickly decided to remove the three tokens from each side. Dividing by 2 was a bit trickier with the balance scales, for two reasons. First, when we worked with Algeblocks to do this with one-step equations, you could easily split the pieces into two groups. But the pans of the balance scale are not big enough to do this. So the students had to remove one of the two groups. Second, because of the actual balancing of the scales, they could "cheat" to figure out how many tokens to leave on the right. They would just remove one token at a time until it was balanced, instead of splitting the tokens in half and removing one of those groups. At this point I had to have them put the pieces back on and ask them to talk me through their steps. To wrap up this problem, I had the students set up their scales again with the same equation and I wrote the equation on the board. As a whole class, they talked me through their steps and then we discussed how we could write the same thing with the equation. See our work below. I next wanted to make a point about the efficiency of removing the constants first, but also point out that you could divide/multiply first (as long as you do it correctly). I chose to have the students solve the equation 2x+4=10 on the balance scales. I wrote the equation on the board and had them set it up on their scales. Note: I made sure the value of x stayed the same so that they could just use the canisters as they were. I also chose even numbers so that they could physically divide them all by 2. My instructions to the students were: "I want you all to solve this equation on the scales. When you are done, I want you to put it back on and solve it a different way the second time. Make sure what you do to one side, you do the exact same thing to the other and make sure you keep it balanced". The students quickly solved it the first way by removing the constants (extra tokens) first and then dividing into two groups, just like the first problem. When they went to solve it a second way, one of three things happened: 1) They didn't know what to do and sat somewhat quietly 2) They tried different things and maybe ended up at the answer but with sketchy math steps (see previous trick from first problem) 3) They did well with dividing both sides by 2 first and figuring out that you also have to divide the four tokens. Once again, I had all the groups put the equation back on their scales. As a class, we walked through the first solution while they did the scales and I did the algebra on the board. Then I had them put the equation back on the scales so that we could talk about the second step. Knowing that very few figured it out, I decided to try to lead the groups to the second method. One of my favorite questions to ask during these solving equations lessons is "What is our goal?". I find students need to be reminded that they are trying to solve for x, which means they are trying to get x by itself. So I asked that and then followed with, "What are the two things we need to get rid of?" Students knew they wanted to get rid of the 2 and the 4. They also realized at this point, that if they got rid of the 4 first last time, they will want to get rid of the 2. To purposely mislead them, I showed dividing the 2x by 2 and the 10 by 2 on the board (I did NOT divide the 4 by 2). I wanted them to see the unbalanced result on the balance scale. It was interesting to see the reactions of students. Some eagerly went to the scales to do it, but there were a few that started to protest. I quietly nodded in agreement to them and they quickly accepted my choice (I am thankful that they have gotten to know me well enough by now to trust my teaching moves). The students pretty quickly realized the mistake of not dividing the four. It was great to hear their conversations of why they were not balanced and how to fix it. We reconvened on the board and discussed as a class what happened. They did a great job justifying why we need to divide EVERYTHING, not just the 2x and 10. You can see our work below. But there was an important discussion to have still. Which method should we use? When should we use each method? Is there a more efficient method? So, we went back to our first problem, 2x+3=9. I had the students set it back up and then told them to solve it by dividing first. I barely started to walk around the room before the students started protesting. I challenged them to keep working it out, but it didn't last long before many were giving up. We gathered together as a class again to discuss what their issues were. Although they knew what they wanted to do, it was impossible to split a token in half, therefor they could not solve this problem this way on the balance scales. We did talk about how we could do it on paper, but it would be more efficient to do it by getting rid of the constant first. One of my favorite parts of this lesson is bringing it back to the number talk from the beginning. I asked the class to look at the solutions, which I purposely left on the board. I asked, "Which one of the solutions took away the constant first?". The surprised reactions of the students was fun to listen to as they realized that they did ALGEBRA during their NUMBER talk and that they naturally do algebra, even without writing a variable. At this point, the class period was done. I really wanted to get out the Algeblock mats to have them do some equations with those. I wanted each student to be active with physically manipulating the equations. However, being out of time, I just gave them 6 problems to solve on paper as homework. Day 2: I wasn't quite sure how this day was going to go. Because I didn't get to have the students work with the Algeblocks the day before, I was prepared to do that today. But first I wanted to check how they were doing. I decided to start the class with the activity My Favorite No. Each student got an index card and I wrote the problem 8=3x+2 on the board, purposely writing the x on the right. The kids finished somewhat quickly and I collected the cards. I was pleased to see that most of the solutions were done correctly. There were just a couple that had subtracting mistakes, but nothing incorrect algebraically. So, I decided to take the activity in a little different direction. I pretended that the following was an incorrect solution from someone in the class: 8=3x+2 8=5x 8/5 = x Instead of asking what was right and then analyzing the mistake, I had them visit within their table groups to answer the following questions: Pretend you are the teacher, 1) Figure out why they did what they did 2) What would you say to help them understand why they shouldn't do that? 3) How can you use Algeblocks to help? The students did a great job talking through the task. When we discussed it together as a class, I played the role of the confused student, asking questions that I have been asked myself. For example, one student said that "2 isn't an x, so you cannot add it to the 3". I said, "I know 2 isn't an x, but 3 isn't either, so why can't I add them together?". Soap box: When someone understands algebra, 3x and 2 are obviously different things, but when a student just sees algebra as just numbers and letters, it is completely logical to add 3 and 2, they are both numbers. That is what I love about Algeblocks. They force us to analyze what 3x means (3 groups of x, which would be 3 long, yellow blocks) and that 2 is 2 units or 2 green centimeter cubes. When they can make a visual representation of the expression, they can actually SEE that they are not the same thing. However, this is not automatic either. It takes time to establish this, and it may take longer for some students than others. Be patient and willing to put in the time. The rewards are worth it. And this is how our conversation went. They were able to tell me that 3x means 3 groups of x, not "3 green guys" (which is how we refer to the green centimeter cubes now). So we have 3 yellow blocks and 2 small green blocks. This cannot be simplified any further, so they cannot be written as 5x (which would be 5 yellow blocks). I also wanted this lesson to include talk about simplifying a side before solving. Not that you have to, but we needed to make sure they knew how. I posed the following mistake: 2x + 3 + x = 9 -x -x x + 3 = 9 (basically, the student subtracted x twice on the same side) Class discussion included talk about keeping the equation balanced and that you cannot subtract x on the same side twice. We also looked at the Algeblocks and discussed how we could just combine the x's since they were on the same side. At this point I was please with where the students were at and wanted them to practice. So we spent the rest of the class period practicing problems as an assignment.
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I was so excited when I discovered the usefulness of Algeblocks, I jumped right into using them. My enthusiasm carried me through, but I soon learned that I had more to learn. Luckily, it was time for me to do the research part of my Master's, and I had just discovered my topic: How to use Manipulatives in the High School Math Classroom. It was also the year of snow storms and I managed to do 80% of my research on snow days! It was meant to be. It is a long paper, but if you really want to read it, I attached the link. Years ago I had an interesting conversation with one of the elementary teachers in my building. She said she uses manipulatives extensively, but she doesn't know how to get the students to make the connections between the blocks and the math they are supposed to do on paper. I realized then that I didn't either. In fact, I still look at that as being the toughest part of teaching with manipulatives, and how you make the connection can be different depending on the student. So, what does research say? First, I will apologize for not citing my sources here. Please reference my paper in the link above for official-ness. Now, here comes the good stuff. There are fundamental steps to follow, steps that of course have variations through the years, but I will stick with the basics. They are the Concrete, Representational, and Abstract stages. I will refer to them as CRA. Concrete is the use of manipulatives ONLY. There is no paper involved. Abstract is the use of paper only, this is where students are using the algorithms we want them to know in the end. Representational is the way you get from concrete to abstract. Representational has many different looks. Concrete Stage This stage will take more time than you want, but it is worth it. Make sure you pick a manipulative that does a good job representing the math you want them to learn. This is important! I use Algeblocks pretty extensively when teaching Algebra, but I have been known to use Cheerios for units and envelopes for x when in a bind. Also, before you get down to business, let the kids play for a few minutes. It helps to keep them from playing when you are trying to do math. To begin, you need to define the manipulatives. It is often the case that the teachers "see" the connection much easier than the students. Therefore, you need to be clear about what you want the students to "see". For example, in Algeblocks, the x is a yellow stick that is a little over 3 green cubes long. Many students want to give the yellow stick the value 3. (I tease them that if they are going to do that, then Pi would be more accurate). But then we have the conversation that it would be physically impossible to create a stick that doesn't line up with some number of green cubes. So we will have to accept that the yellow stick can be worth 5 or -10 or whatever else x is worth. Another thing that might be annoying at first, but turn into some great conversations and "aha" moments, is that you may need to start with earlier concepts to build to what you want to do. For example, before I have students solving equations, we have to understand how to add, subtract, multiply and divide integers. We have to understand why a negative times a negative is a positive, which has nothing to do with bad things happening to bad people! We also need to understand what opposite value means. When we get to actually solving equations, we also have to discuss what the symbol "=" means (Hint: it does not mean "here comes an answer"). The main part of the concrete stage is getting the students to see the physical representation of the math concept you are teaching. Create several examples for them to struggle through. Make problems that will challenge what they already know and think, and make sure you cover several variations so they don't think it all works the same way all the time. Make sure to do them yourself so that you can anticipate the challenges. Above everything else, give them lots of TIME. Do not rush this part of the process. You are trying to get them to internalize the concept. That doesn't happen in five minutes. During this stage, you need to be circulating and challenging students' solutions. Get them talking and explaining. That is one of the cool things about using objects, no matter their level, they can still talk about green cubes and yellow sticks, even if they don't have the vocabulary. I often make them tell me like I am a 5 year old, don't use math words at first, just tell me what you are doing physically. Representational Stage There are several ways to attack this stage. It all depends on your resources, your topics, and your students. 2D version of the 3D manipulative: You can have students use a virtual manipulative via the internet or an app. (Brainingcamp has some great apps for this). One of the cool parts of virtual manipulatives is that they show the abstract along with the manipulation, so it will often "catch" them if they make a mistake. I often just have students draw their work out, see the picture below. This can often be a tool some students use to help them if they are not ready for abstract when everyone else is, which is perfectly fine. Let them be where they are comfortable and understand what they are doing. Although I haven't done much work with it, I would imagine this is the stage for the clothesline math. Simultaneous work with concrete and abstract: When we are nearing the end of the concrete stage, I start introducing the way we write the same math abstractly. For example, I will ask the students to show me 2x + 3 = 5. I write the equation on the board and walk around to make sure they have two yellow sticks and not 2 green guys in front of a yellow stick. I will then ask them to tell me what their first step will be (remember we have already done several problems, so they have already done their exploring and are pretty ready to do it right away). I have them do their step and then I ask them how I could show the same thing on the board. We compare what I have on the board to what they have on their desks. I also have one of the students demonstrate the manipulative work via the document camera while I do the abstract on the board. This allows me to model how I want them to work together. Once I have demonstrated that, I have the students pair up. One will do the manipulative and the other will write. After so many problems, they switch roles. I also insist they talk to each other about what they are doing, they need to do each step together and double check each other's work.
I also enjoy doing this in groups of four. Student 1 does the manipulative work, Student 2 does the abstract work on paper, Student 3 announces the work of Students 1 and 2 (think sports announcer, most enjoy making a show of it, some just talk normal), and if there is a Student 4, they act as referee and throw the flag if someone isn't doing their job. Abstract Stage The part everyone wants to get to. This is what you normally teach: the algorithms of the concept. No more physical objects, no more drawings; just numbers and symbols. But here is the cool part, you can still talk about those other things. I still reference the "green guys" and zero pairs. Let's say I have a struggling student stuck on solving a two-step equation. Most of the time, I just have to ask if they have any green guys, and they will forget I exist and solve the equation. The other cool part: 2x + 3 is not 5x anymore. In the last 5 years of teaching this, I have seen maybe three kids make that mistake. Why so few? They "see" 2x and 3 as different things, just like we want them to. And those three kids? They are asked to show me 2x + 3 using Algeblocks and explain to me why they shouldn't get 5x. When I was a student in high school, I did a lot of peer tutoring. It was when I figured out I wanted to be a math teacher. I was struggling to get a peer to understand how to solve a one step equation. I used the method I saw my teachers use, basically just talking about inverse operations, but it wasn't working; no matter how many times I said it. I asked my teacher at the time if there was another way, but was disappointed when I was told there wasn't. I was also skeptical of this answer. How can there not be another way? Fast forward to my early teaching career. I found Algebra Tiles in the cupboard, where they were stashed away from the previous teacher, never used. I tried to make sense of them, but got confused and stashed them back in the cupboard. Fast forward another couple of years. I went to a workshop by Nancy Berkas and Cyntha Pattison in my northwest corner of Minnesota. They introduced Algeblocks (the 3D version of Algebra Tiles). As they explained their use, a lightbulb burst on and lit the path for the rest of my career. I found the "other" way to teach solving equations and more. I found a way that all students can grasp an abstract concept and succeed. I also found a way to get my smarty pants students to think differently and challenge their tendencies to take the easy road of memorizing steps. So, here is the idea, give students something physical that they can manipulate to make sense of the operation/concept you are teaching. They do it frequently in the elementary with base ten blocks. I have yet to meet a math teacher that would condone teaching without base ten blocks in the elementary, but it is difficult to find math teachers that have the same philosophy for the Algebra class. And yet, we complain when students make 2x + 3 into 5x. They do this because they don't understand what 2x is and what 3 is. They need to understand what they are in order to do the operation correctly. Before I continue, let me describe Algeblocks. (I have nothing against Algebra Tiles, they work the same). Constants are represented just like you would with base ten blocks. The kit comes with green centimeter cubes. The number 3 is represented by getting out 3 green cubes, we call them the "Green Guys". X is represented with a yellow stick, it is 1 centimeter by 1 centimeter by a little over 3 centimeters. There are also blocks that stand for y, xy, x^2, y^2, x^3, y^3, xy^x, and yx^2, I think that covers them all. It is always interesting to start a lesson asking students to show me 2x. If this is the first time they have used Algeblocks, I often see two green cubes with a yellow stick to the right of them. If you have Algeblocks or Algebra tiles, try it (after introducing and defining them). It is a huge eye opener that they don't understand what 2x means! It is also interesting to represent x/2, they often put a yellow stick above and 2 green cubes below it. Sometimes they even use another yellow stick in between to represent the fraction bar. I do many things with manipulatives. This post is to just introduce manipulatives in general. I will have future posts to discuss specific uses and lessons. I love how using manipulatives gives me a place to go with students who are struggling, but also gives me a way to challenge students' thought processes when I feel they are just memorizing steps. I use many different manipulatives, not just Algeblocks. I use Base Ten Blocks, Balance Scales, XY Axis Pegboards, Geosolids, Linking Cubes, 1in Square tiles, and Anglegs; all in the high school classroom. I also use digital manipulatives, my favorite being Demos. One of the most difficult parts of using manipulatives is giving up the time to do them. Lots of time is a key factor to teaching with them! But, I promise, teaching with manipulatives saves time in the end. You don't have to take time to reteach misconceptions. Plus, your students actually UNDERSTAND what they are doing, which is always a positive. But be warned, they are not magical. There is a process you need to use (see next post). I often fear when I see an article or blog criticizing the use of manipulatives. But when I read them, I agree with their arguments. You need to know what you are doing. But when done right, they can open doors to understanding! I wouldn't teach without them now that I have experienced what they can do. |
AuthorI teach mathematics for grades 7-12. Teaching mathematics is my passion. Archives
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