Student Probe Dividing Mononomial Expressions Perform the following division: 4x6÷ x3 . Answer: Students who answer have the misconception that, since the indicated operation is division, the exponents should be divided as well as the coefficients. Lesson Description This is a guided discovery lesson which uses expanded form of exponents and simplifying fractions to help students understand how monomial expressions are divided. Students will divide various types of monomials and form their own conjectures, with teacher guidance, about the rules for dividing these expressions. Rationale The ability to divide monomial expressions efficiently is an important skill for success in algebra and serves as a foundation for dividing and simplifying more complex expressions. An understanding of how monomial expressions are divided will provide a basis for understanding negative exponents and a justification of . Rather than memorizing a list of rules for dividing these expressions, students are encouraged to use their knowledge of numerical expressions and the properties of real numbers to deepen their understanding of dealing with algebraic expressions. At a Glance What: Dividing monomial algebraic expressions Common Core State Standard: CC.9-­‐
12.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. Matched Arkansas Standard: AR.9-­‐
12.LA.AI.1.5 (LA.1.AI.5) Perform polynomial operations (addition, subtraction, multiplication) with and without manipulatives Mathematical Practices: Look for and make use of structure. Look for and express regularity in repeated reasoning. Who: Students who incorrectly divide monomial algebraic expressions Grade Level: Algebra 1 Prerequisite Vocabulary: coefficient, base, exponents, expanded form Prerequisite Skills: write numerical and algebraic expressions as a product of their factors (expanded form) Delivery Format: individual, small group, large group Lesson Length: 15-­‐30 minutes Materials, Resources, Technology: none Student Worksheets: Dividing Monomial Expressions Preparation Prepare a display so that all the problems and their solutions can be seen as students work and think about the common patterns in the answers. Run copies of the worksheet or additional practice problems, if necessary. Lesson The teacher says or does… Expect students to say or do… 1. What does mean? This is called “expanded form”. 2. Now write in expanded form. 3. We are going to divide some algebraic expressions and I want you to think about what patterns you see as we work. (Keep the solution to each example posted as students work.) 4. What happens when we divide: ? First let’s rewrite the problem using a fraction bar. This will make the problem easier to understand. 5. Now let’s write both the numerator and denominator in expanded form. 6. Since this is a division problem, let’s divide out the common factors. or or If students do not, then the teacher says or does… Review exponent notation. Prompt students as they work. Prompt students as they work. This is the same way we divided out common factors when we simplified fractions. Prompt students as they work. The teacher says or does… Expect students to say or do… If students do not, then the teacher says or does… Prompt students as they work. 7. Now, what do we have? 8. Now, write our answer using exponents. Prompt students as they work. 9. Now divide Prompt students as they work. ? The teacher says or does… Expect students to say or do… 10. Students should work additional examples until you see that they understand the concept. Then say: This is a lot of writing! Does anyone see a shortcut we can use to make this easier? Show me your shortcut on this problem: . Explain your shortcut. 11. Would someone summarize what we have learned? I know that is 3 x’s. . is 4 x’s and If students do not, then the teacher says or does… Prompt students as they work. If some students’ thinking seems unclear, work through the problem using expanded notation, counting the x’s together, if necessary. so that is 1 x. When we divide monomial expressions, we divide (or simplfy) the coefficients and we subtract the exponents. What do we do with the coefficients? What do we do with the exponents? Teacher Notes: 1. Students should not be taught a rule or procedure (i.e., Laws of Exponents) prior to this lesson. 2. The intent of this lesson is for students to use the commutative and associative properties for multiplication of real numbers and simplifying fractions to make sense of the division of monomial algebraic expressions. This will provide them with a strategy to use in case they “forget” the rule or procedure. 3. Allow students to forego the use of expanded notation once they ascertain the shortcut. However, encourage students to use the expanded form whenever they are not confident in their answers. 4. Using fraction bar notation rather than the division symbol will help students visualize the division more readily. The fraction bar notation is more frequently used in algebra. 5. Use the term “divide out” rather than “cancel” when explaining the division to students. “Cancel” frequently leads students to the misconception that division is simply crossing out terms. The operation of division should be stressed. 6. If all terms “divide out” ( as in Step 8 ), make sure that students understand that a 1 remains rather than 0. 7. Do not require students to use parentheses when writing the expressions in expanded form unless you feel that it helps clarify the monomials for the student. 8. Frequently students become confused when the exponents or coefficients are understood to be 1. If that occurs, write the 1 in the expression. Example: Formative Assessment Perform the following multiplication: Answer: or References
Paulsen, K., & the IRIS Center. (n.d.). Algebra (part 2): Applying learning strategies to intermediate algebra. Retrieved on 3 10, 2011, from http://iris.peabody.vanderbilt.edu/case_studies/ICS-­‐010.pdf . Dividing Monomial Expressions 1.
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