Multiplying Mononomial Expressions Student Probe At a Glance 2
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Perform the following multiplication: 4x ·∙ x . Answer: . Students frequently answer . They have the misconception that since the indicated operation is multiplication, the exponents should be multiplied as well as the coefficients. Lesson Description This is a guided discovery lesson which uses expanded form of exponents and the associative and commutative properties of multiplication to help students understand how monomial expressions are multiplied. Students will multiply various types of monomials and form their own conjectures, with teacher guidance, about the rules for multiplying these expressions. Rationale The ability to multiply monomial expressions efficiently is an important skill for success in algebra and serves as a foundation for multiplying and simplifying more complex expressions. Rather than memorizing a list of rules for multiplying 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. Preparation Prepare a display for the problems so that all the problems and their solutions can be seen as students work and think about the common patterns in the answers. Prepare copies of the worksheet or additional practice problems, if necessary. What: Multiplying 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 multiply 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: Multiplying Monomial Expressions Lesson The teacher says or does… 1. Let’s write 12 as a product of its prime factors. 2. Does it matter in which order we write the factors? What property of numbers tells us that? 3. We can write the product using exponents. Who can tell me what that would be? 4. What does mean? 5. This is called “expanded form”. 6. Now write in expanded form. 7. We are going to multiply 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.) 8. What happens when we multiply ? Expect students to say or do… No Commutative Property of Multiplication Explain that is a shortcut meaning . 9. Now let’s try x3 ·∙ x4. 10. What about ? If students do not, then the teacher says or does… Assist students as they find the prime factors, such as using a factor tree. Demonstrate that multiplying the factors in any order results in a product of 12. Prompt students as they work. Prompt students as they work. Prompt students as they work. The teacher says or does… Expect students to say or do… 11. Multiply 3x5 ·∙ 5x . 12. 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. 13. Now let’s work a slightly different kind of problem: . Who would like to show us how to think about this problem? 14. Would someone summarize what we have learned? If students do not, then the teacher says or does… Prompt students as they work. Prompt students as they work. If some students’ thinking seems unclear, work through I know that . is 4 x’s the problem using expanded and is 5 x’s. so that notation, counting the x’s together, if necessary. is 9 x’s all together or . If some students’ thinking seems unclear, work through the problem using expanded notation, counting the x’s and the y’s , if necessary. When we multiply monomial expressions, we multiply the coefficients and we add 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 to make sense of the multiplication 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. 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. 5. 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: References
Paulsen, K., & the IRIS Center. (n.d.). Algebra (part 2): Applying learning strategies to intermediate algebra. Retrieved on [month day, year,] from http://iris.peabody.vanderbilt.edu/case_studies/ICS-­‐010.pdf Multiplying Monomial Expressions 1. x2 ·∙ x5 2. x ·∙ x4 3. d6 ·∙ d4 4. 2x4 ·∙ x 5. 3y2 ·∙ 5y3 6. 5m3 ·∙ 5m 7. d4 ·∙ 6d4 8. 2x ·∙ 5x 9. 3x ·∙ x 10. 2a ·∙ 4b 11. 3x ·∙ 5y 7m ·∙ 8n 12.