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Polynomials 9th Grade Algebra Text:
Algebra 1 by Holliday, Cuevas, Moore-Harris, Carter, Marks,
Casey, Day, & Hayek. |
Strategies: |
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| Extra Materials: Note cards (one for each student) | ||||
Motivation: --Upon learning the definitions of binomial, trinomial, and polynomial, 9 th grade Algebra 1 students will be able to recognize the difference between a monomial and a polynomial and be able to identify whether an expression is a monomial, binomial, trinomial, or other polynomial. --After today's lesson, 9 th grade Algebra 1 students will be able to determine the degree of a polynomial and be able to arrange polynomials in ascending or descending order in terms of one variable. --In addition, students will read Section 8.4 from the text using an Anticipation Guide. --Students will complete story problems using the SQRQCQ strategy. --Students will review what they have learned using the Find the Fake strategy. |
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The text: Before reading the text, students will complete the anticipation guide individually. Hand this out, then say the following:
Good morning class! I've heard from more than one of you that you're having trouble reading the textbook, so we're going to try a few new strategies to help you guys out. The first strategy we're going to use is called an Anticipation Guide. This is the worksheet you have in front of you. The directions are on the top of the page. When you finish marking the "Before Reading" column then (and only then) you may start reading your textbook. I want you to read pages 432 through the top of page 434. [write that on the board] Make sure to read all the tables, examples, and side notes as well. When you finish reading, fill out the "After Reading" column, also making notes in the "Text" column. I realize this is a lot of information, so if you have questions on anything, just raise your hand. I'll give you all a half-hour to do this. I think that should be enough time.
When the half-hour is up or the majority of students are finished with the Anticipation Guide, bring the class back together and go through the answers, calling on a different student to try and answer each one. If a question is answered incorrectly, kindly inform them of the fact and call on another student. Have the students correct the false statements. The correct answers are: • True - p. 432 • False - p. 433 (The degree of a monomial is the sum of the exponents of all its variables. OR The degree of a polynomial is the greatest degree of any term in the polynomial .) • True - p. 432 • False - p. 432 (5x + 2 - 6x is a binomial because it can be simplified as 2 - x) • False - p. 432 (A polynomial is a monomial or a sum of monomials . It may have four terms, it may have more, or it may have less.) • True - p. 434 • False - p. 433 (4x^2 - 6x + 1 is written in descending order . OR 1 - 6x + 4x^2 is written in ascending order.) |
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DURING LESSON The Lesson: --After reading through the text, you all should have a good idea as to the kind of material we're going to be covering in class. We're going to start by talking about the different types of polynomials. [write 'polynomial' on the board] You should remember from class earlier in the week what a monomial is. [write a couple monomials on the board] Can anybody tell me? (a number, a variable, or the product of a number and one or more variables) Yes. Well, a monomial is an example of a polynomial. [write 'monomial' on the board] Now, if we add two monomials together [put a plus sign between two of the monomials written on the board] , we get what's called a binomial. A binomial is simply the sum of two monomials. [write 'binomial' on the board] A binomial is another example of a polynomial. Finally, if we add together three monomials [put a plus sign between three monomials on the board] , can anybody guess what that would be called? (a trinomial) That's right, a trinomial . [write 'trinomial' on the board] Again, a trinomial is a type of polynomial. Now polynomials can have one, two, or three terms, but they can also have four, five, or 726. But only the first three have special names. Everything else is simply referred to as a polynomial. --Now remember, before you can decide which type of polynomial you are dealing with, don't forget to combine like terms. [write 3x + 7 - 2x on the board] What type of polynomial does this look like? (students will probably say trinomial) Right, because it has three terms. But look, we can add together 3x and -2x, so this is actually x + 7. [write x + 7 on the board] --Of course, it's also important to remember that the rules of monomials we talked about earlier in the week still apply here. We still cannot have negative exponents or variables in the denominator. --The next thing we're going to talk about is degree. The degree of a monomial is simply the sum of the exponents of all its variables. [write a few examples of monomials on the board: 3x 5 , a 3 b 2 c 4 , 5x 4 yz 2 , 32] The first, 3x 5 , has degree 5, the second, a 3 b 2 c 4 , has degree 9, the third, 5x 4 yz 2 , has degree 7 (because any variable without an explicit exponent is considered to have an exponent of 1), and the fourth, 32 , has degree 0, because only the exponents of variables are considered. --The degree of a polynomial, on the other hand, is the greatest degree of any term in the polynomial. [write a few polynomials on the board: 8x 3 y, 3m 2 n + 5mn + 7, 15a 5 b 2 c - 8a 4 bc + 4a 2 b + 5a + 9] The first polynomial, 8x 3 y , has degree 4, because it only has one term, and its degree is 4. The second polynomial, 3m 2 n + 5mn + 7 , has degree 3, because the degrees of each term are 3, 2, and 0, respectively, and 3 is the greatest. The third polynomial, 15a 5 b 2 c - 8a 4 bc + 4a 2 b + 5a + 9 , has degree 8, because the degrees of each term are 8, 6, 3, 1, and 0, respectively, and 8 is the greatest. --Now it's your turn to try a few. I'm going to write some monomials and other polynomials on the board, and I want you to raise your hand and tell me what their degree is. Okay? [choose from the following ( adapted from Chapter 8 Resource Masters); answers are in parentheses:]
--Finally, we're going to take a look at how to put polynomials in ascending and descending order. Polynomials are put in either ascending or descending order in terms of the powers of one variable. Therefore, even if a polynomial contains both x and y, the directions may tell you to put it in order in terms of the powers of x. This means you can ignore the powers of y. --Ascending order would look like this. [write x + x^2 + x^3 on the board] You can see that the exponents count up: 1, 2, 3. Descending order would look like this. [write x^3 + x^2 + x on the board] You can see that the exponents count down: 3, 2, 1. Does everybody see how that works. Let's try a couple. I'll write a few polynomials on the board, and I want you to raise your hand and tell me what order they should be put in. Let's start with ascending. [choose from the following ( adapted from Chapter 8 Resource Masters); answers are in parentheses:]
And now let's try descending. [choose from the following ( adapted from Chapter 8 Resource Masters); answers are in parentheses:]
--Finally, let's try doing a story problem from this section using SQRQCQ. I'm sure you remember by now what all those letters stand for, but let's just review. • Survey: Read the problem rapidly, skimming to determine its nature. • Question: Decide what is being asked-in other words, what the problem is. • Read: Read for details and interrelationships. • Question: Decide which processes and strategies should be used to address the problem. • Compute: Carry out the necessary computations. • Question: Ask whether the answer seems correct. Check computations against the facts presented in the problem against basic arithmetic facts. (From: Leu, D. J., & Kinzer, C. K. (1991). Effective Reading Instruction, K-8 Second Edition. New York: Macmillan Publishing Company. Cited on http://www. galesburg205.org/churchill/sqrqcqmath.htm (Retrieved March 20, 2007)) --Turn to page 435. We're going to look at problem number 55: [work this out with students]
PACKAGING A convenience store sells milkshakes in cups with semispherical lids. The volume of a cylinder is the product of pi, the square of the radius r , and the height h . The volume of a sphere is the product of 4/3, pi, and the cube of the radius. Write a polynomial that represents the volume of the container. (Answer: pi r^2 h + 2/3 pi r^3 ) Survey: This question is about the volume of a milkshake container. Question: We're being asked to find an equation to determine the volume of the milkshake container. Read: The volume of a cylinder is the product of pi, the square of the radius r , and the height h . The volume of a sphere is the product of 4/3, pi, and the cube of the radius. Question: Well, we have half a sphere on top of a cylinder, so maybe we can just add the cylinder volume to half the volume of a sphere. Compute: pi r^2 h + (4/3 pi r^3 )/2 = pi r^2 h + 2/3 pi r^3 Question: Does this answer make sense? Yes it does! |
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Recap after Lesson: Use the "Find the Fake" strategy. Put students in groups of four (or let them find their own groups if they can do so responsibly). Give each student a note card. On it, the students should write three statements about the day's lesson. Two of the statements should be true, and one of the statements should be false. In their groups, students should take turns reading their statements, with the group trying to determine which of the three statements is false. When everyone has gotten a chance to read their card in the group, students should choose one member of their group to read their card and challenge the entire class. From: opsu.edu/education/Reading Strategies Applied to Math Presentation.pdf Give students the following directions: Okay. Now we're going to see what you all have learned today. You guys need to get in groups of four. I'm going to give each one of you a note card. On the note card, you should write three statements about today's lesson. Two of the statements should be true, and one should be false. In your groups, you need to take turns reading your statements (all 3 at once) and the other group members need to decide which of the three statements is false. When you're all finished, choose one member of your group to read his or her card to the rest of the class. Make sense? Go to it! |
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