Multiplying and Factoring Notes


 Julius Williamson
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1 Multiplying/Factoring 3 Multiplying and Factoring Notes I. Content: This lesson is going to focus on wrapping up and solidifying concepts that we have been discovering and working with. The students have been practicing how to multiply monomials and binomials with algebra tiles, the box method and double distribution. We have also been working towards the rules for factoring; they have some ideas on how that works. However, at some point we do need to put those thoughts into concrete rules and ideas in their notes. They need to have those notes to look back on in the future. II. Learning Goal(s): Students will know and be able to:  Multiply binomials  Identify and define the difference between a monomial, binomial, trinomial and polynomial  Factor trinomials  Identify and define the quadratic, linear and constant terms in a monomial, binomial, trinomial and polynomial  See patterns between the factors and the binomial/polynomial answer, such as: o When multiplying two binomials they add to the middle (linear) term and multiply to the last (constant) term o If there are two addition signs in the polynomial answer, the factors must have addition signs between both their terms o If there are two subtraction signs in the polynomial answer, the factors must have two different signs between their terms and the greater will be negative o If there is an addition sign first and a subtraction sign second in the polynomial answer, the factors must have two different signs between their terms and the greater will be positive o If there is a subtraction sign first and an addition sign second in the polynomial answer, the factors must have subtraction signs between both their terms III. Rationale: We will have just completed two lengthy activities delving into multiplying and factoring polynomials (Algebra Tiles and Tic Tac Times). Therefore, it is time to discuss the concept they uncovered. In this grid format we will approach the concept from all angles, different learning styles and tie the ideas together. I want them to be able to see multiplying binomials using the algebra tiles, box method and algebraic formats. Then, I want them to have concrete examples of each type of polynomial and how it is factored. I know taking notes will not completely teach them this, they will not automatically absorb it from this lesson; it will set them in the right direction. After these notes we will delve deeper into factoring and continue doing more skills practice on both concepts. This is just tying together all their thoughts thus far. IV. Assessment: Notes are a difficult thing to assess. However, the graphic organizer I give them will be mostly empty; therefore, they will be responsible for filling it in as we go along in class. This ensures they are paying attention and are responsible for their learning. In addition, I will keep track of who is contributing to the class discussion in forming the notes and give points for that. Of course, this will require me setting up the class in this manner. I will tell them at the beginning on class in order to hear from everyone, I will be keeping a list and giving participation points. Also, I do periodic notebook checks where they are graded on how complete their notebooks are (table of contents, includes all notes, everything is filled in, etc). This means that these notes will come up later down the line, they will keep reappearing. V. Personalization:
2 While I will give them a graphic organizer to glue into their notebooks, they will be in charge of filling it in as we go along in class. They need structure and organization which is why I have created the organizer for them. A lot of my students are on 504s or IEPs, struggling with organization, use of organizers when possible is recommended. Also, by giving them this form that they simply need to fill as we go, ideally it will save time. I KNOW for a fact that the second I tell them to divide their pages into these boxes immediately I ll hear cries of Miss, I need a ruler! Yes, 100% predictable. They love their lines to be straight; it also, of course, wastes class time. This way, their lines will be straight, the grid is organized and I will save that class time. Also, there is a lot of vocabulary being covered in this unit and this lesson specifically. I will try to tie each word back to something that they are familiar with (for example, the linear term is the x term where x is raised to the first power, like when they learned about linear equations, x was only raised to the first power). Relating the terms to their everyday lives (for example: bicycle and tricycle, tricolor, triathlon, biannual, monorail, monochrome, monogamy, polygon, etc) may help them remember more. We need to break down the words into parts they understand. We will also be filling in examples next to the words to solidify together the words, definitions and algebraic examples. VI. Activity description and agenda: Groups: The class will be set up in an inner/outer semicircle (the usual format when we do notes) so that every student is able to see the board Materials: graphic organizers (multiplying AND factoring ones), glue sticks, video, lyrics, ELMO Time What Students Do What Teacher Does 07 Students will enter the classroom, find their POD book, which will be set out for them and begin to work on the problems. The POD will have students identifying expressions as monomial, binomial, trinomial or polynomial. It will also have them labeling each term as quadratic, linear or constant Students will volunteer to share a few answers quickly One student will pass out the glue sticks and students will glue the first graphic They will then update their table of contents as I write it on the board Students will follow along and fill in the definitions and examples as we do them as a class. The definitions and examples will come from the students; they will raise their hands and volunteer as I ask for them Students will glue the second graphic Students will volunteer to fill in the graphic organizer in my notes rather than just me doing it. *Get notebooks out* We will go over A FEW of the examples as a whole class. However, the notes following this discussion will be solidifying these concepts as well. Update table of Contents: 3/26/13 Polynomial Vocab 3/26/13 Multiplying Binomials 3/26/13 Factoring Quadratics We will fill in the definitions for each vocabulary word in the table. We will also fill in examples as we go. I will relate each of these words to real life things they can relate to: bicycle and tricycle, tricolor, triathlon, biannual, monorail, monochrome, monogamy, polygon, etc. We will fill in the graphic organizer for multiplying binomials. Most of the answers will come from the students. I will ask students to come up and draw the algebra tiles, the box
3 5260 Students will complete the exit slip method and the algebraic expression. 1.Give an example of Give one example of a polynomial expression. 2. Multiply (x+2)(x2) Students will be following along with the lyrics and underlining important math rules or things that stand out to them Students will glue the last graphic Students will volunteer answers when I ask for them. They should be filling in their organizers as we are doing it as a class and as I am on the board Students will complete the exit slip Pass out the lyrics for Teach me how to factor Show the video We will factor the quadratic equations in the graphic organizers as a class. We will look for common factors that multiply to the last term and add to the middle term. Then we will pick factors that work. 1.Give an example of Give one example of a polynomial expression. 2. Multiply (x+2)(x2) Day Students will enter the room and sit in the seats marked by their POD books. They will begin to work on the POD that is on the board Students will answer my questions by raising their hands as I ask them With the first table, students will volunteer answers and we will fill it in together. Then they will move onto working with their partners Students will be following along with the lyrics and underlining important math rules or things that stand out to them Students will glue the last graphic Students will volunteer answers when I ask for them. They should be filling in their organizers as we are doing it as a class and as I am on the board. 1.What pairs of numbers sum to 10? 2. What pairs of numbers multiply to 16? 3. How would you factor x x + 16? 4.What pairs of numbers sum to 9? 5.What pairs of numbers multiply to 20? 6. How would you factor x 2 9x + 20? We will go over the POD. We will make a chart for numbers 1 and 2 then find the common factors to answer 3. We will do the same for 4 and 5 to solve 6. We will go over how to fill in the first table together. Then I will circulate to see how students are doing. If need be I will call the class back to a full class discussion. Pass out the lyrics for Teach me how to factor Show the video We will factor the quadratic equations in the graphic organizers as a class. We will look for common factors that multiply to the last term and add to
4 5260 Students will complete the exit slip the middle term. Then we will pick factors that work. 1. Factor x 2 5x Factor x 2 + 6x + 8 VII. List the Massachusetts Learning Standards this lesson addresses. 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.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. A.SSE.3A  Factor a quadratic expression to reveal the zeros of the function it defines. 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. A.REI.4B  Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions 1 and write them as a ± bi for real numbers a and b. VIII. Resources: IX. Reflection In 9A the notes took a lot longer than expected. We got through the vocabulary and multiplying binomials, however, I added additional practice factoring at the last minute (after their exit slips yesterday) and we didn t even get to that let alone the notes. That means tomorrow will be the additional factoring practice before we solidify Thursday with notes and a check in quiz. In 9B, it was much the same. They are struggling even more than 9A is with factoring. However, I would say 9B s multiplying skills are superior. I am not sure what has caused these differences in comfort levels but it is an interesting divide. In general, the class was engaged during the polynomial vocabulary. All students were writing and answering questions. However, during the multiplying binomials notes some students went ahead and filled in all their boxes very quickly. This meant that while some students were struggling, others had already flown through it. I think in the future I could avoid this by doing notes on multiplying binomials before this point. They were here because I realized we had never put them in our notes and probably should. Also, I could have just let the students fill in the boxes on their own and move onto another task. However, I wanted to go over the boxes as a class to make sure everyone had the correct answers in their notes. At the end, students were engaged while going over them. I had various students are the board at the same time filling in my notes for me with others double checking in their seats. The factoring practice I added in here was most definitely needed. Even after this additional day, students were still confused. I was anticipating that though. Factoring is a hard thing to discover on your own. They had some ideas about it and that was what I wanted going into notes. Even after the notes, factoring is still a lot of guessing and checking for some students. With the factoring practice, students were confused on how to use the box method to factor. I myself
5 am not completely comfortable with it and in showing them; most of them decided they did not want to use that method anyways. It is unfortunate because a lot of them prefer to use the box method to multiply. Then again, the box method to factor is kind of strange in my opinion. Finally, we arrived at Factoring Notes Thursday for 9A and Tuesday for 9B. It was supposed to be Monday for 9B; however, a parent meeting came up so I needed to leave an activity they could do without me. The factoring notes were not an activity they could do without me. Therefore, they worked on What good are factors and we did the notes Tuesday. They LOVED Teach me how to factor! Ruthie even said she wished our school was that cool and made videos like this one. I don t know why they underestimate UPCS so much. Unfortunately, we don t have time to do that in this unit, otherwise I would. I also think that after seeing this video on factoring it would be too hard for them to come up with their own and not think about the way they did it. Even after the factoring notes I think the students are still very iffy on the rules or patterns. They are still going to need a lot of practice. Luckily, the zero product property also utilizes factoring so they will get practice there as well. The Check In Quiz that ended up with the Factoring Notes in both classes produced surprising results. The students struggled a lot more with the vocabulary than I expected. Especially since it was known that they could use their notes like the notes we took on the vocabulary which would have given them all the answers. Also, on the quiz I had them multiply, add and subtract binomials. I stressed during the quiz that they should pay attention to the operation symbols, even circling them on the board a lot of them still tried to multiply all of them. This proves they do not pay attention to detail and they do not listen when I give directions, a little disheartening. Other than that, it showed me about what I expected skill wise in terms of all my students. They were about where I expected. I knew who was struggling and this confirmed it.
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