Multiplying/Factoring with Algebra Tiles and the Box Method

Transcription

1 Multiplying/Factoring 1 Multiplying/Factoring with Algebra Tiles and the Box Method I. Content: This first lesson in my unit will focus on both reviewing old skills and teaching new skills. The students have been adding and subtracting since elementary school. They learned how to combine like terms in middle school. However, reviewing all of these old skills with algebra tiles will help when we introduce multiplying both monomials and binomials with the same tiles. Therefore, the end content is multiplying monomials and binomials with algebra tiles. They will see the problem done out with multiple methods algebra tiles, box method and written out in a formal algebraic expression. II. Learning Goal(s): Students will know and be able to: - Use physical models (algebra tiles) to perform operations with polynomials - Multiply and divide monomials - Add and subtract polynomials - Multiply a binomial by a monomial or a binomial - Use the box method to perform operations with polynomials - Translate the physical models or box method solution into an algebraic expression III. Rationale: This unit is really going to test whether or not the students are ready for the abstract. Multiplying and factoring polynomials is an incredibly abstract concept. Of course, there are some real life scenarios or story problems I could have given them to demonstrate these concepts. However, as I am learning, there is a time and place in the Algebra I curriculum where my students need to be able to simply do the algebra. In addition, I am scaffolding it by reviewing old concepts first, things that they are comfortable with and using algebra tiles so the ideas are not completely abstract. The tiles are still physical models the students can use to represent the problems we are doing. This first lesson will introduce the big ideas of adding and subtracting polynomials (combining like terms) and multiplying binomials and monomials. We are using the concrete algebra tiles here instead of a gimmicky story problem but I believe the students can handle it. In addition, I am introducing the box method way of multiplying binomials and monomials as well for those students that do prefer to look at things that way. But this is an introduction to these concepts; the lessons after this will solidify what is introduced here. IV. Assessment: As an assessment here, I am going to utilize entrance and exit slips as well as classwork. The entrance slips will be called PODs (Problem of the Day) as they have been in my class all year. However, the concept is the same. The POD on the first day will be on an index card and I will quickly sort them upon collecting them to see if the students answered a specific question I was looking for correctly (question 1). The exit slip on the first day will assess how the students have grasped the concepts we got through so far there will be a question on how to represent 4 x 5 with algebra tiles, what 2(2x + 5) is and if we get to multiplying binomials: (x+1)(x+2). On the second day, I varied a bit from what I had originally planned. I had time, due to a snow day, to reevaluate my lesson plan and change it based on what I saw during Day 1. Therefore, instead of beginning with a POD, I started class with a discussion then got right into working. This day their assessment will be based on the completion of their classwork. The worksheet they will be doing has multiple columns the factors, the algebra tiles, the box method and the algebraic expression. I am requiring that they do all of the columns to get practice with each method. After they 1

2 have worked through multiplying binomials with addition, subtraction and perfect squares; they will answer questions about which method they prefer to use and why. If I need to do an exit slip on day 2, I have one ready but I am thinking that the repetition of the classwork will be enough to evaluate whether or not they get it. By the end of the second day, both the students and I will be able to identify this based on how they complete their classwork. V. Personalization: This lesson is incredibly scaffolded for those students that need it and all students, whether or not they choose to admit it. We begin with adding and subtracting with algebra tiles, move onto combining like terms (which they ALL struggled with when I did my pre-assessment) and then into multiplying monomials and binomials. Adding and subtracting polynomials with the algebra tiles reinforces the idea that x 2 and 3x CANNOT be combined to make 3x 3 or 4x 3 or 4x or 4x 2 yes, they came up with all of those answers during the pre-assessment. While I accept the algebra tiles as an AWESOME method to teaching these concepts I am aiming towards, I also know that some students will prefer a more straightforward, algebraic method. This is why we will be doing the box method side-by-side with the algebra tiles. This gives students practice with both methods and the option (after concluding practice with both) to choose which they prefer to work with going forward. It gives them multiple tools in their tool box though and reinforces the idea that there is always more than one way to solve a problem (I love that!). I know that in middle school the students used algebra tiles to demonstrate combining like terms (I checked in with their eighth grade teacher). However, they have not used the tiles to multiply monomials/binomials or for factoring. Regardless, provided that the students fly through what I have planned for Day 1 rather quickly, we will be able to get the two days done in one. Also, as we are working through with the tiles, if the problems are too easy I will introduce problems that cannot be solved with the tiles as well. These problems will be a bit more complicated (which mostly just means they have bigger numbers to work with). In addition, to engage all students in the class, and to ensure every student is doing the work while we are doing whole class demonstrations I will be asking for volunteers to do problems at the ELMO. I will also be walking around the class during the lesson, during the work time, to check that each student has an attempt at a solution on his/her desk. VI. Activity description and agenda: Grouping: Students will be arranged into an inner/outer semicircle where they can all work individually at their desks and still see the board to follow along. Materials: Algebra tiles, ELMO, worksheets, index cards Day 1 Time What Student Does What Teacher Does 0-3 Enters room, finds seat, responds to POD. Get out your problem statement for Ms. M to check off. 2 Projects POD about combining like terms/distributive property 5x(2x + x +1) OR 2x 2 + 7x x 4x Hands in index card Sorts index cards, yes, no. Pick my favorite no. If there isn t a wrong answer, pick my favorite yes. One that shows out their steps and you can clearly see what they did 5-10 Points out good things about favorite no. they distributed the outside to both terms, etc Prompts students to point out the things this student did write

3 Points out the mistake probably that they didn t square the x Students will receive tiles and without a doubt play with them and make designs and whatever else Students will find each tile in their bags and follow along. If they have any questions, they will ask them now Students will arrange their tiles overhead Students will arrange their tiles Students will arrange their tiles 3 Prompts students to point out what the mistake is Hand out algebra tiles to students. Give students 2 minutes to build whatever patterns they re going to with the tiles before we begin Project the tiles on the overhead Little green square = 1 Little red square = -1 Green rectangle = x Red rectangle = -x Green square = x 2 Red square = -x 2 After introducing each we will go over it again with the students telling me what each means. I will stress that green means positive and red means negative **As we are doing these on the elmo I will be showing the tiles, labeling the tiles over them and writing the algebraic expression off to the right to make it consistent so when we get to harder problems the students will be comfortable with expressing these algebraically as well I will introduce adding the green tiles = 7, = 14, do 5 examples of addition. Have students come up to the overhead and answer them for me. Participation points! Then we ll do subtraction with the green tiles 8-3 = 5, 9-2=7, etc Introduce adding positive and negative = -2. Use zero pairs or cancelling. Then do 3-8 = -5. The rule is you change subtracting a bigger positive number to adding a negative = = 2. Change the second orange tiles to adding a positive with the green

4 >>> If students are catching on quickly, instead of just using the 1s. We can add and subtract just using Xs and X 2 s Students will arrange their tiles Students will arrange their tiles Students will arrange their tiles Exit Slip Students will answer the exit slip questions on index cards. They will then put their algebra tiles back in the bag and hand in both the tiles and the index card. Using all the tiles do addition and subtraction. Do a few examples of that = combining like terms. They struggled with that on the scavenger hunt. Set up multiplication in the t- chart or in a box (closer together) Multiply just the ones tiles. 4 x 3 = 12 To help students remember whether answer is positive/negative When colors of factors match = good = green When colors don t match = bad = red Multiplying monomial by binomial 3(2x + 2) -3 (2x +2) -3 (-2x -2) (x 2 + 3x + 2) + (x 2 2x 1) =? Represent using algebra tiles: 4 x 5 =? IF we get to this 2 (3x 2) =? Original Plan for Day 2 Time What Student Does What Teacher Does 0-3 Enters class, sits, begins POD POD about combing like terms or 4

5 multiplying a monomial with a binomial 3-5 Hands in index card Sorts index cards, yes, no. Pick my favorite no. If there isn t a wrong answer, pick my favorite yes. One that shows out their steps and you can clearly see what they did 5-10 Points out good things about favorite no. they distributed the outside to both terms, etc Points out the mistake probably that they didn t square the x 5 Prompts students to point out the things this student did write Prompts students to point out what the mistake is Hand out tiles Hand out tiles. Refresher of what each tile means quickly. Begin where we left off yesterday (2x + 7) -4(2x + 7) -4(2x 7) (x + 1)(x + 2) Do it out with algebra tiles. Write what each tile means on white board. Write algebraic expression next to the picture. Arrange tiles and add like terms Students will do each problem at their seats. Volunteers will come up and show their work at the elmo. (x + 3)(x + 1) (x + 2)(x + 3) (x + 4)(x + 1) (x-3)(x + 2) Do it out with algebra tiles. Write out what each tile means on white board. Write algebraic expression next to the picture Students will do each problem at their seats. Volunteers will come up and show their work at the elmo. (x 4)(x + 1) (x + 2)(x 4) (x + 3)(x 1) (x 3)(x 2) Do it out with algebra tiles. Write what each tile means on white board. Write algebraic expression next to the picture (x 1)(x 4) (x 3)(x 1) (x 2)(x 4) (x 3)(x + 3) Difference of squares Introduce factoring

6 EXT If students move through activities faster than expected, give problems that cannot be solved with the algebra tiles. (x + 57)(x +1) etc Exit Slip (x + 1)(x + 2) (x + 1)(x 2) (x 1)(x 2) Give answer and one factor, have students find the other factor Do a few examples NEW Plan for Day Students will enter the classroom and sit in rows. They will wait for Ms. M to begin class Students will look over the worksheet as I explain the directions. They will offer answers to the first one as we do it as an example. They will be drawing and writing as I am on the board Students will work individually at their seats. They will draw the algebra tiles, complete the box method and write the algebraic expression As students wrap up their multiplying worksheet, they will move onto the 6 I will begin class with a discussion about the previous class. I think the algebra tiles are a super cool, useful way to do the stuff we re doing. And based on the exit slip that I collected from them all on Monday, none of them got the answers right. Therefore, they should not have been acting the way they were during class. They should have been paying attention, doing it out at their seat and learning what we were learning that day. Etc. HOWEVER, I acknowledge that some students said during class Monday that they would learn better from doing it on their own with a worksheet rather than following along with the whole class. Therefore Pass out the worksheet. Pass out algebra tiles. Go over the directions. Students will draw the algebra tiles representation, do out the box method and write the algebraic expression. They will be doing this individually; however, we will do the first one together as an example. I will circulate around the classroom. I assume that once the students get to the problems with negatives, we will do another example on the board as a whole class. However, if they do not need this, I want to see how they do on their own before suggesting it. The worksheet has the algebra tiles pictures and the algebraic expressions

7 factoring one. Again, they will work individually at their seats Students will complete the exit slip on the provided index cards. They will turn in their algebra tiles and the index card. Ext. If students finish with the multiplying worksheet, factoring worksheet and exit slip they can work on their homework that is due tomorrow. for the students to find the missing factor. This may be difficult for some students. We may have to do an example at the board. However, I am hoping that with all the multiplication practice they have just done, they will be able to do it in reverse here. (x+1)(x+2) (x-1)(x+2) (x-1)(x-2) They have a POW due that is a lot of work and I know a lot of them have not even started it. VII. List the Massachusetts Learning Standards this lesson addresses. N.Q.2 - Define appropriate quantities for the purpose of descriptive modeling. A.SSE.1A - Interpret expressions that represent a quantity in terms of its context. Interpret parts of an expression, such as terms, factors, and coefficients. A.SSE.2 - Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). A.APR.1 - Perform arithmetic operations on polynomials. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. VIII. Resources: IX. Reflection This is a complicated reflection to write. I originally had planned out Day 1 and Day 2 (both of which are included here), however, after Day 1, I went back and totally re-evaluated how I wanted to structure Day 2 (I also included that here). Against my better judgment, as per Shannon s advice, I planned both days as full class discoveries. I should have known better. The students do not do well with full class lessons for that long. Day 1 in 9A, honestly, some of the students sat there and did absolutely nothing. In my opinion, they thought the algebra tiles were beneath them. That is the way they were acting, they were answering questions like they already knew how to do this and that they didn t need the lesson at all. Surprisingly, Lilly was one of them. She told me at the very beginning of class that she thought the tiles were stupid and she didn t need them. By the end of the class period I was frustrated to say the least, I was also somewhat mad. Therefore, I gave an exit slip different than the one I had originally planned to. It was (x-1)(x-2) and (x-2)(x+1). I gave a very brief but annoyed speech on how if they thought they could do it without the lesson, if they thought the tiles were beneath them and that I was wasting both their time and mine, they needed to prove to me that they could do it. Of course, not one single student in the entire class got both problems correct. Duoc got one right and Terry got one right, no real surprise there. However, no other student in the class even came close. I got a lot of answers with only two terms, some with only one term. Clearly, they did not know what they were doing. However, they showed me that I was right in my original way of thinking. While some parts of this lesson needed to be done as a full class (such as the 7

8 introduction of the tiles and what they represent), I should have created a worksheet or a packet for them to follow along with at their seats. They would have been drawing the tiles and writing the expressions themselves creating a sense of accountability and making them actually practice. In 9B, the lesson went a bit better but I still didn t feel that the students got out of it what I wanted. A lot of them seemed to check out, much in the way 9A did, though not quite as obviously. So at the end of class, I took the same approach with them, gave them the same exit slip and got the same results. Because while some students were very engaged in the lesson, volunteering and appreciating the tiles, most of the class was not following along with the tiles at their seats as I was asking them to do. However, again, I should have held them more accountable and I should have hyped the tiles up more. I learned from my mistake. For Day 2, I set up a worksheet with the factors, a space for algebra tiles, a space for the box method I had introduced in addition to the tiles, and a space for the algebraic expression. The students worked individually at their seats after we did a whole class example with the first one. I cannot even begin to explain how much more effective this was. The students were actually engaged and working the entire period. Lilly even came up to me at the end of class, when turning in her worksheet where she had to identify which method she preferred guess what she chose the tiles! Success! Of course, also, like I anticipated, a lot of them do prefer the box method. They think it s easier, doesn t require as much work and makes more sense to them. That is perfectly fine with me. I wanted to give them multiple methods to work with; it is up to them ultimately what method they are most comfortable using. However, Ada during this lesson was another success, she was so excited after completing the worksheet she said she actually gets it! For her to have confidence in herself is absolutely awesome. In addition, I also started the class by explaining how awesome and cool AND useful, I found the tiles to be. I explained to them how when I was in school we learned FOIL and that was it. You remembered it and used it or you didn t. We didn t get these awesome manipulatives and multiple ways to solve things. I learned to use the algebra tiles this summer and I thought it was the best thing ever. I wish that I learned this way when I was in school. It made me understand FOIL and multiplying binomials all the better. AND I even explained and showed it to a few MATs because I was so excited and they thought it was awesome as well. So while, yes, some people may prefer the box method or double distribution (FOIL), we are going to learn all the ways and appreciate them before choosing one we like. I will probably always be partial to FOIL because I learned it that way, but I may have thought differently if I learned all these other ways. Adding personal stories makes it more relatable to them and makes them appreciate it more. Why do I forget this sometimes? I should always add as much of myself into things as I can. I guess I just thought that algebra tiles were abstract and unrelatable while still useful and cool. But even that small story about myself made them open up to the idea more and accept it as a viable method. Another cool thing, originally, when drawing the algebra tiles I was thinking the students would label g, b, r, y (to represent the color of the tiles so they d know whether it was positive or negative). On their own, they each came up with the idea of shading the negatives to tell the difference. Sometimes, when students are left to their own devices, they come up with better ideas than we do. If I were to do this lesson again in the future I would definitely structure more of it to be individual with examples to follow. I think that the students can handle that and I think they would benefit more from it. It is incredibly evident in the way these two days were. While they claim to hate worksheets, they do actually learn a lot from them. And surprisingly, none of them complained about this worksheet. The only complaints I got were from those students which right away knew they preferred the box method, they didn t want to have to draw out all the algebra tile pictures, but they survived. In the end, I think I salvaged the lesson adequately. After Day 2, each student was able to multiply monomials and binomials. Unfortunately, only one student (of course, Duoc) got to the factoring, but it was worth it for them to fully grasp the multiplying concepts. In this lesson I also saw students advocating for their own learning. They were very vocal about when they needed extra help; some students that never ask for help. I know that I am trying to move them 8

9 towards being more independent and relying on each other versus always asking me for help but this lesson was hard for them to do individually, I know that, hence why I am okay that they asked for help from me. I also really love that about an even balance of students said they preferred the algebra tiles versus the box method. It s interesting to see who prefers which method and why. 9