Simplifying Numerical Expressions Grade Five
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1 Ohio Standards Connection Number, Number Sense and Operations Benchmark E Use order of operations, including use of parenthesis and exponents to solve multi-step problems, and verify and interpret the results. Indicator 9 Use order of operations, including use of parentheses, to simplify numerical expressions. Related Benchmark F. Apply number system properties when performing computations. Lesson Summary: In this lesson, students learn about the order of operations and how to use the order of operations to simplify expressions. First, students learn the parenthesis signifies which operation is executed first when simplifying expressions. Then, students analyze simplified expressions, write the steps needed to simplify the expression and then generalize the order for which operations are executed. In journals, mathematical terminology and summaries of understanding are recorded. Estimated Duration: Two hours Commentary: Students learning about the order of operations are typically presented the pneumonic, Please Excuse My Dear Aunt Sally. However, this technique has caused misconceptions and confusion for many students. In this lesson, students discover the order through analysis of simplified expressions. They can make and test conjectures that will engage them in the understanding and use of the order of operations. Mathematical Processes Benchmarks F. Use inductive thinking to generalize a pattern of observations for particular cases, make conjectures, and provide supporting arguments for conjectures. H. Use representations to organize and communicate mathematical thinking and problem solutions. Pre-Assessment: Assess strategies for solving multiple step problems and writing equations to represent the situation. Pose the following situation to the students. The teacher ordered gifts for the 20 students in the classroom. The mini notebooks cost $1.50 and the dazzling pens cost $2.50. Direct students to determine the cost for the gifts and write an expression to represent the situation. Select students to share their determined cost and expressions. Ask students questions to explain the methods used. List the expressions created by students on the board. Scoring Guidelines: Assess students problem solving skills and understanding of the operations. The process should include an acceptable way to determine the cost for all of the gifts. Note students who multiply each price by 20 and then add, and those that multiply the sum of the items by 20. 1
2 Ohio Standards Connection Post-Assessment: Have students simplify a variety of expressions and write the steps for simplifying. Distribute Simplifying Numerical Expressions Post- Assessment, Attachment A. Scoring Guidelines: Use the rubric to determine the level of students understanding and progress. Attachment B contains the simplification and steps. Score Description Meeting Simplifies all expressions accurately Expectations Provides the appropriate steps for simplifying Progressing Simplifies expressions with minor flaws Adequately Intervention Needed Instruction required Provides appropriate steps for simplifying Provides evidence of partial understanding Executes the same minor errors throughout all simplifications. Uses a random order to simplify each expression or simplifies from left to right regardless of operation. Instructional Procedures: Part One 1. Direct students to look at the equations on the board. 20 $ $2.50 = $80 $ $ = $80 a. Describe the problem situation from the pre-assessment and relate it to the first equation, Twenty students multiplied by the cost of the mini notebook and the dazzling pen. b. Describe the situation as it relates to the second equation, The cost of the mini notebook and the dazzling pen multiplied by 20. c. Have students check the equations for accuracy. As written, in arithmetic thinking, the answer to the first equation is $ d. Explain to the students that, in mathematics, parentheses are used to signify an order for which to simplify the expression. e. Rewrite the first equation. Tell students that in this equation the parenthesis are used to signify the first step in solving the expression. 20 ($ $2.50) 2
3 f. Ask students how they would change the second equation to signify which operation to complete first. 2. Pose the following situation: The seven cheerleaders found sweaters priced at $15 on sale. The cost of each sweater was then reduced by $3. Determine the cost of the seven sweaters and write an expression using parenthesis to represent the situation. 3. Observe students as they complete the task and assist as needed. Note students who begin to use the parenthesis appropriately and those who multiply seven and the difference of the regular price and the three dollar reduction ($12). 4. Select students to share their responses and equations. Write the expression on the board and reinforce the significance of the parenthesis. 7 ($15 - $3) = 5. Present number expressions for students to simplify for practice. 5 + (3 7) (26) 21 (12 6) (19) 56 (7 6) (14) 6. Introduce vocabulary to students. expression (any combination of numbers and symbols, excluding the equality or inequality symbols) simplify (to perform all operations within an expression, like solving an equation) a. Have students write the terms in their mathematics journal. Have students discuss each term. b. Ask questions such as, What is it? What is it like? or not like? Can you give an example? 7. Have students write statements, in a mathematics journal, about the significance of parenthesis in simplifying expressions. Provide exercises to practice for homework. Part Two 8. Review the use of parenthesis in simplifying expressions. 9. Present Finding the Order of Operations, Attachment B, to students and place them into groups of four. The groups analyze several expressions and determine the steps used to simplify the expression. Students write the steps for each expression in the boxes. See Attachment C, Finding the Order of Operations Answer Key. Attachment B contains two pages of expressions. Page one contains expressions using four to five operations, while page two offers less complex expressions. Choose the expressions which best suit the needs of the students. 10. Review the steps used to simplify each expression with the students and allow them to make corrections as needed. Ask students questions about the steps and the order in which the operations seem to appear. If applicable, which always occurs first? (Operations in parenthesis). Do you notice any order to the operations as they are listed in the steps? (They always appear after the parenthesis, where applicable. It appears that division and multiplication are used before addition and subtraction.) 3
4 Do you notice any instances in which addition and subtraction occur before multiplication and division? (Yes, but only when addition and subtraction occur within parenthesis.) Does it appear that there is an order for division and multiplication? (No, they are interchangeable. Yes, you complete them from left to right in the expression.) Does it appear that there is an order to addition and subtraction? (No, they are interchangeable. Yes, you complete them from left to right in the expression.) 11. Introduce vocabulary to students. order of operations a. Have students write the terms in their mathematics journal. Have students discuss each term. b. Ask questions such as, What is it? What is it like? or not like? Can you give an example? 12. Have the groups write statements, in a mathematics journal, about the order of operations and how they are used to simplify expressions. Include a graphic organizer or visual. 13. Assign expressions for students to practice simplifying using the order of operations. Differentiated Instructional Support: Instruction differentiated according to learner needs, to help all learners either meet the intent of the specified indicator(s) or, if the indicator is already met, to advance beyond the specified indicator(s). Use calculators, number lines, and factor charts as tools for students who are learning basic facts. Present expressions with fractions, decimals and percents. Have students write expressions to represent different situations they may face in daily life. Extensions: Have the students create expressions for stories, and then simplify using order of operations. Using 2, 4, 6, 8, 9 create an expression using all four operations that will obtain the largest possible value with the given set of numbers. Create an expression that will obtain the least possible value with the given set of numbers. Write and simplify expressions that contain a variable. 4
5 Home Connections: Discuss with parent and ask what has happens to a recipe when the measurements of an ingredient have been changed, such as only making half of a recipe or when you double a recipe. Materials and Resources: The inclusion of a specific resource in any lesson formulated by the Ohio Department of Education should not be interpreted as an endorsement of that particular resource, or any of its contents, by the Ohio Department of Education. The Ohio Department of Education does not endorse any particular resource. The Web addresses listed are for a given site s main page, therefore, it may be necessary to search within that site to find the specific information required for a given lesson. Please note that information published on the Internet changes over time, therefore the links provided may no longer contain the specific information related to a given lesson. Teachers are advised to preview all sites before using them with students. For the students: Calculators, journals Vocabulary: expression order of operations simplify Technology Connections: Identify the difference between Algebraic and Arithmetic calculators. Algebraic calculators follow the order of operations, where Arithmetic calculators do not follow the order of operations. Have students use spreadsheets when looking at different sequences. Have students use mathematics software that would enhance the lesson on use of order of operations Research Connections: "BSCS Science: An Inquiry Approach." BSCS Biological Sciences Curriculum Study. 23 Dec < Marzano, Robert J., Jane E. Pollock and Debra Pickering. Classroom Instruction that Works: Research-Based Strategies for Increasing Student Achievement, Alexandria, VA: Association for Supervision and Curriculum Development, 2001 Attachments: Attachment A, Simplifying Numerical Expressions Post-Assessment Attachment B, Simplifying Numerical Expressions Post-Assessment Answer Key Attachment C, Finding the Order of Operations Cards Attachment D, Finding the Order of Operations Cards Answer Key 5
6 Attachment A Simplifying Numerical Expressions Post-Assessment Name Date Directions: Simplify the expressions and write the steps used in the process. 1. Simplify the expression: (4 + 7) 3 2. Simplify the expression: Simplify the expression: 40 + (6 5) 3 4. Simplify the expression: (9 4) 6
7 Attachment B Simplifying Numerical Expressions Post-Assessment Answer Key 1. Simplify the expression: (4 + 7) Add Multiply 11 and 3 2. Simplify the expression: Divide, Multiply 5 and 8 3. Add 4 and Simplify the expression: 40 + (6 5) Multiply 6 and 5 2. Divide 30 by 3 3. Add 40 and Simplify the expression: (9 4) Subtract Divide 36 by 6 3. Multiply 6 and 3 4. Add 18 and 5 7
8 Attachment C Finding the Order of Operations Cards 8
9 Attachment C (continued) Finding the Order of Operations Cards 9
10 Attachment D Finding the Order of Operations Answer Key 10
11 Attachment D (continued) Finding the Order of Operations Answer Key 11
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