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Writing and solving equations can be abstract and confusing for students. Learn nonconventional ways to encourage flexible thinking and develop a deeper understanding of inverse relationships, fact families, and variables representation. Walk away with three easy-to-use activities to expand students' toolkit for solving equations.

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- 1. Speaker: Shephali Chokshi-FoxCo-Speaker: Victoria Miles1

2. AgendaRationale for using non-conventional methodsMenu Math What is a variable?Flowcharts and Backtracking How to use inverse operations.Fact Families What is the relationship between the terms?Wrap-up / Questions2 3. Menu MathCommon Core Standard Addressed6.EE.2 Write, read, and evaluate expressions in which letters stand fornumbers.a.Write expressions that record operations with numbers and with lettersstanding for numbersb. Identify parts of an expression using mathematical terms (sum, term,product, factor, quotient, coefficient); view one or more parts of anexpression as a single entity.c. Evaluate expressions at specific values of their variables. Includeexpressions that arise from formulas used in real-world problems. Performarithmetic operations, including those involving whole-number exponents,in the conventional order when there are no parentheses to specify aparticular order (Order of Operations).3 4. 4Lets look at this menu 5. Menu Mathh + f =1.85 + 2.15 = $4.00c+ f + s =$ 4.157f =$ 7.353h + c + f + 3x =$ 14.904c + 3f + s + m + l =$15.503c + 3d = $11.10 What does d = ?$1.55 (large soda)5 6. Write what each customer ordered and calculatehow much was paid for each order:63h + 3f =3h + f =3(h + f)=3(1.85) + 3(1.05) = $8.703(1.85) + 1.05 = $6.603(1.85 + 1.05) = $8.70Which two customers ordered the same food and paidthe same price? 7. Different members of the same family placed thefollowing orders. Simply the orders:7 8. 83 (1.05) + 6 (1.85) + 5(?) = $24.50? = $2.05 (extra large) 9. Hamburger = $3Fries = $19Can you find the price of a hamburger and of anorder of fries at each of these restaurants?At Restaurant A, how much does a singlehamburger and a single order of fries cost?(WITHOUT using symbolic algebra) 10. 10Menu Math Wrap-Up What concepts are students learningintuitively? What skills are they building? How does real-world application provide acontext for connecting prior knowledge tomore abstract learning? 11. Solving Equations using a FlowchartCommon Core Standard Addressed6.EE.5 Understand solving an equation or inequality as a process of answeringa question: which values from a specified set, if any, make the equation orinequality true? Use substitution to determine whether a given number in aspecified set makes an equation or inequality true.7.EE.1. Apply properties of operations as strategies to add, subtract, factor, andexpand linear expressions with rational coefficients.8.EE.7b Solve linear equations with rational number coefficients, includingequations whose solutions require expanding expressions using thedistributive property and collecting like terms.11 12. 12Number puzzleIm thinking of a number when I add 5 to the number my answer is 18.What number am I thinking of?x + 5 = 18To solve we must find the value of x using inverseoperations. 13. Using a flowchart to introduce inverseoperationsExample of a Flowchart:8 2 + 4 4 316 20 5 2 2 4 4 + 3Backtracking is working backwards bycarrying out the inverse operation.13 14. 14Solving Equations Using a Flowchart:Working Backwardsn 50n=10 4 + 1040 4 10 15. 15Solving Equations Using a Flowchart: 3x 2x = 18 36 4+ 4 16. 16FlowchartWork Boards:Scaffold andSupportKinestheticLearners 17. 17Solving Equations Using a Flowchart:^2777+37.16739.84*0.017.77-1.056.723.2n=2.1 18. 18Solving Equations Using a Flowchart:-24 (-3)n=8 (-3) 12 19. 19Solving Equations Using a Flowchart:-6 5n=-30 +(-14) 20. 20Solving Equations Using a Flowchart:(-5) + 6 2-50 6-44x=10 5-22 2 21. 21Solving Equations Using a Flowchart: 22. 22BacktrackingBenefitsIntuitively builds anunderstanding of undoing(focus on inverse operation)Flowchart provides a visualpromptBuilds understanding of thestructure of algebraicexpressionsWorks well with quadraticequationsLimitations Only applies to equationswith ONE Unknown (cant beused to with x+1=2x) Focus on numbers does notassist students in moving tomore algebraic approach 23. Solving equations using a Fact-Family Approach23Common Core Standard AddressedUse properties of operations to generate equivalentexpressions.6.EE.3 Apply the properties of operations to generateequivalent expressions .7.EE.1 Apply properties of operations as strategies to add,subtract, factor, and expand linear expressions with rationalcoefficients. 24. 24Using Fact Family Triangles94 5213 74 + 5 = 95 + 4 = 99 4 = 59 5 = 43 7 = 217 3 = 2121 3 = 721 7 = 3 25. 25Using Fact Family Trianglesto Solve Equationsn3 -10-30n 5n 3 = -10n (-10) = 33 + (-10) = n-30 n = 5-30 5 = nn 5 = -30 26. Summary Points Learning Non-Conventional approaches Allows students to understand the underlying concepts Bridges students prior knowledge of number theory to moreabstract concepts of algebraic thinking Supports seeing the interconnectedness of strands of Algebra Take-Away: In order for students gain a deeperunderstanding of abstract concepts, they need opportunities toexplore using hands-on examples and visual models.26

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