It is the age old frustration for anyone teaching math. How do we get kids to be better at solving word problems? There are many answers out there; many opinions. But let's be honest, many of us are hoping there is an easy answer that won't require a new curriculum, hours of prep and completely revamping our teaching methods. Although I think that all three things are often needed, it is not reality. About a year ago, I discovered I had effectively isolated my professional life to within the four walls of my classroom since 2009. I think it was a combination of raising children and wearing too many hats within my school-district. Luckily, I broke free! And one of the first things I learned was about the simple, but impactful, question "What do you Notice?". Really Quick Rant: How is this not the cornerstone of preservice teacher's education? In case you are like I was a year ago, "What do you Notice?" is a question you ask students whenever you want them to notice things, expand their thinking, or take time before they dive into a question. Basically, don't notice/say anything that the kids can notice/say. If you haven't seen it, watch Annie Fetter's ignite video
Applying this to word problems has completely changed my classroom.
Whenever we are going to work on a word problem, I REMOVE THE QUESTION. How many times have you asked or been asked, "How do I get kids to spend more time on word problems before just 'doing' the math they think it wants?"? It is simple, remove the question. Next: Ask the kids what they notice. When you are first doing this, it is important to MODEL, MODEL, MODEL. Have the students popcorn out what they notice and write it all on the board. And I mean ALL. Give each and every thing that is noticed validation by writing it on the board. You are letting EVERY student know that what they notice is valuable and encouraging them to keep noticing. When you have exhausted everything that can be noticed, ask the kids to come up with the question that can be asked and then solve it. Because they just spent so much time noticing, they are typically ready to do the math without the support they would have required in the past. You can/should extend this to be done individually also. Take a piece of paper and put the problem in the middle with a circle around it (no question again). Have the students write everything they can notice around the circle in a web-format (concept map). Then at the bottom ask them to come up with a question and solve it. What can you expect:
In reflection, it is embarrassing that this was not an intuitive thing to do with my students. Instead, I was always reading the problem outloud, then jumping to the question and asking the students what information they need. I would then try to get them to dissect the information, but they were too eager to just do math with the numbers given and get the problem done. By establishing this model, I am hoping students will be more apt to do this on their own. And I am hoping to see a change in how they attack their state test in the spring.
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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. |
AuthorI teach mathematics for grades 7-12. Teaching mathematics is my passion. Archives
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