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Algebra and Operational Thinking In Grade 5 Overview 5th Grade Content is preparation for Expressions and Equations Students begin working more formally with expressions (5.OA.1 and 5.OA.2) Exploratory rather than for attaining mastery Write expressions Evaluate and Interpret Expressions Should be no more complex than expressions using associative and distributive properties Students prepare for studying proportional relationships and functions in middle school (5.OA.3) Progression – 5.OA.1 and 5.OA.2 4th 5th 6th •4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35=5x7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. •5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. •5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. •6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true Progression- 5.OA.3 4th 5th 6th •4.OA.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. •5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. •6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. 5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. Instructional Strategies start with expressions that do not involve any grouping symbols and have two different operations Ex: 4 X 5 + 7 switch the operations around and discuss why the solutions are different Ex: 4 X 5 + 7 and 4 + 5 x 7 PEMDAS Introduce the rules that must be followed, noting that multiplication and division, as well as addition and subtraction should be solved left to right PEMDAS P = Parenthesis E= exponents MD= multiplication and division (whichever is first, from left to right) AS= addition and subtraction (whichever is first, from left to right) http://www.amathsdictionaryforkids.com/dictionary.html Strategies Have students place parentheses around the multiplication or division part in the expression and discuss the similarities and differences After students have solved multiple expressions without grouping symbols begin presenting problems with parentheses, then with brackets and/or braces Give students an expression and solution and they must fill in the appropriate operations in order to get the given solution. Ex: 7 _ 8 _ 3 _ 2=17 More complex you could ask them to insert parentheses, brackets, or braces Give students a solution and they must come up with the expression Ex: I wrote an equation using parentheses and all four operations with an answer of 25. What might the equation be? Write a matching story to fit the expression. This will provide insight to whether or not they fully understand the order of operations. Have students solve expressions using a calculator and have them decide what operation the calculator did first in order to get the same answer Common Misconceptions with 5.OA.1 Students may believe the order in which a problem with mixed operations is written is the order to solve the problem. Allow students to use calculators to determine the value of the expression, and then discuss the order the calculator used to evaluate the expression. Do this with four-function and scientific calculators Misconceptions about PE MD AS These mnemonics Do not replace the need to understand the meaning of the order. Students continue to do poorly on order of operations items on high-stakes assessments, and this is due to a lack of understanding. What part of the order of operations is due to convention, it is largely due to the meaning of the operations. Because Multiplication represents repeated addition, It must be figured first before adding on more. Because exponents represent repeated multiplication these multiplications must be considered before multiplying or adding. Misconceptions about PE MD AS Continued A common misconception with exponents is to think of the two values as factors so 5 to the 3rd is thought of as 5×3 rather than correct equivalent expression of 5×5×5 this is further problematic when students hear things like it is 5 three times since the word times indicates Multiplication. Avoid confusing language, and spend significant time having students state and write Equivalent expressions. When experiencing difficulty with exponents, students should write or include parentheses to indicate explicit groupings. 5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. Student Thinking Students will understand that the word “then” implies one operation happens after another and parentheses are used to indicate the order of operations. Example: “Add 8 and 7, then multiply by 2” can be written as (8 + 7) x 2. http://www.youtube.com/watch?v=swHuC9o JVZo Students will understand how to write a realworld problem as an expression. Real World Application Students will generate expressions for word problems. Edwin buys school supplies for the beginning of the school year. On his first trip to the store he purchases 10 pencils. Edwin realizes he needs to make a second trip to the store to purchase 20 more. Every year for he last 6 years he has followed this pattern. Write an expression that matches Edwin’s story. Student Thinking Cont. Students will recognize that 3 × (18,932 + 921) is three times as large as the sum of 18,932 + 921, without having to solve. Students will make the connection that 3(18,932 + 921) is the same thing as 3 x (18, 932 + 921). Teaching Approaches Visual Models 4 x (9 +2) Set model Area model Teaching Approaches Creating problem context for a given expression Your turn! Heads together, create a word problem that would match the following expression; 8 x (3 + 5) Misconceptions The need of grouping symbols Expression vs. Equation Literature Connections Alexander, Sunday Who Used to Be Rich Last Judith Viorst The Grapes of Math Greg Tang Games http://illuminations.nctm.org/ActivityDetail.aspx?ID=173 5.OA.3 Analyze patterns and relationships CCSS.Math.Content.5.OA.B.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so. Standards for Mathematical Practices (MP) to be emphasized: MP.2. Reason abstractly and quantitatively. MP.7. Look for and make use of structure. Patterns, Functions, Algebra Patterns are key factors in understanding mathematical concepts. The ability to create, recognize, and extend patterns is essential for making generations, seeing relationships, and the order/logic of mathematics. Students investigate numerical and geometric patterns; describing them verbally; representing them in tables and graphically. Students can make predictions, generalizations, and explore properties of our number system, eventually learning about various uses of variables and how to solve equations. Then students should be able to understand the three goals in functions, tables, formulas, and graphs. Turning on the Common Core Students extend their Grade 4 pattern work by working briefly with two numerical patterns that can be related and examining these relationships within sequences of ordered pairs and in the graphs in the first quadrant of the coordinate plane. 5.OA.3 This work prepares students for studying proportional relationships and functions in middle school. An example of a pattern might be: Students use a table for recording the terms of the pattern and the number of stars, and extend the pattern. Number of Stars 2 7 12 17 22 27 Examples Suppose you fold a piece of paper in half, and then in half again, and again, until you make six folds. When you open it up, how many sections will there be? Suppose you draw ten dots on a circle. If you draw lines connecting every dot to every other dot, how many lines will you draw? Function Tables 1. ‘The Fly on the Ceiling’ draw a simple picture that can be formed with straight lines connecting points on a coordinate grid. Use at least 8 points but no more than 10 points. 2.Record the ordered pairs you plotted in the order in which you connected them. 3. Next, double each number of the original pair and plot the ordered number pairs on a second grid. Connect the points in the same order that you plot them. Challenge: What would happen if you: -doubled only the first number of each original ordered pair? - doubled only the second number of each original ordered pair? Misconceptions Engage in pattern work without developing any algebraic thinking. Students often reverse the points when plotting them on a coordinate plane. In graphing a function, the function rule does not need to be fully understood. In generating a number pattern with 2 rules, stop after the first rule. http://learnzillion.com/lessons/797-generate-a-pattern-sequence-using-a-tchart Teaching Considerations Functions can be represented in many ways. Generalization of patterns should be realized by students. Context helps student make sense of what changes in a function. Example: Brian is trying to make money by selling hot dogs from a cart during ball games. He pays the cart owner $35 each time he uses the cart. He sells hot dogs for $1.25 each. His costs for the hot dogs and condiments etc. are about 60 cents per hot dog on average. The profit from a single hot dog is 65 cents. Verbal Description is the functional language. Symbols are used to express a function as an equation. Tables provide a concise way to look at recursive and explicit rules. Graphical representation allows one to see “at a glance” relationships and adds understanding to context. Algebraic Vocabulary for Communicating Mathematically Independent variable is the input or whatever value is used to find another value. Dependent variable is the number of objects needed—the output or whatever value one gets from using the independent variable. Discrete relates to graphical representations and whether the points plotted on a graph should be connected or not. When isolated or selected values are the only ones appropriate for the context, the function is discrete. Continuous relates to the connected points on a graph. Domain of a function comprises the possible values for the independent variable. Range is the corresponding possible values for the dependent variable. References Source: Utah Education Network http://www.uen.org/core/math/downloads/5OA 2.pdf Marilyn Burns, About Teaching Mathematics NCTM K-5 Teaching Resources Turning on the Common Core University of Arizona Progression documents Zimba chart Van De Walle , Elementary and Middle School Mathematics Teaching Developmentally Feedback What part of the lesson were you most engaged in? Would you have sequenced the lesson the same or different? Is there anything you would have included that we didn’t? Lesson Agenda Read Aloud Discussion Introductory expressions Discuss solutions Order of Operations PEMDAS (Graphic Organizer) Order of Operations song/TPR Hopscotch Journal page Exit Ticket Trailer http://www.wbrschools.net/technology/c trailers/orderofoperations%20g7gle3%20g 8gle35%20g9%20gle8.wmv Guiding Question How does the punctuation affect the meaning? Lesson Objective: Students will be able to explore the order of operations by a read aloud, class discussion, and engaging activities. Solve these two problems: 5x3+6= 5+3x6= Discuss Does why the values are different? the order of operations effect the solution? Exit Ticket Three students evaluated the numerical expression 7 + (8-3) X 2. Tom said the answer was 24. Nicole said the answer was 17. Sam said the answer was 19 Who was correct? Why? Explain your thinking.
