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Non Conventional Methods for Solving Equations
The document discusses non-conventional methods for teaching math, specifically focusing on concepts like menu math, flowcharts, and fact families, aligned with Common Core standards. It emphasizes the importance of real-world applications and visual aids in helping students grasp abstract algebraic concepts and develop problem-solving skills. The wrap-up points highlight the benefits of engaging with hands-on examples to deepen understanding of mathematical principles.
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Non Conventional Methods for Solving Equations
- 1. Speaker: Shephali Chokshi-Fox Co-Speaker: Victoria Miles 1
- 2. Agenda Rationale forusing non-conventional methods Menu Math – What is a variable? Flowcharts and Backtracking – How to use inverse operations. Fact Families – What is the relationship between the terms? Wrap-up / Questions 2
- 3. Menu Math CommonCore Standard Addressed 6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers. a.Write expressions that record operations with numbers and with letters standing for numbers b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). 3
- 4. 4 Let’s lookat this menu…
- 5. Menu Math h+ f = 1.85 + 2.15 = $4.00 c+ f + s = $ 4.15 7f = $ 7.35 3h + c + f + 3x = $ 14.90 4c + 3f + s + m + l = $15.50 3c + 3d = $11.10 What does d = ? $1.55 (large soda) 5
- 6. Write what eachcustomer ordered and calculate how much was paid for each order: 6 3h + 3f = 3h + f = 3(h + f)= 3(1.85) + 3(1.05) = $8.70 3(1.85) + 1.05 = $6.60 3(1.85 + 1.05) = $8.70 Which two customers ordered the same food and paid the same price?
- 7. Different members ofthe same family placed the following orders. Simply the orders: 7
- 8. 8 3 (1.05)+ 6 (1.85) + 5(?) = $24.50 ? = $2.05 (extra large)
- 9. Hamburger = $3 Fries = $1 9 Can you find the price of a hamburger and of an order of fries at each of these restaurants? At Restaurant A, how much does a single hamburger and a single order of fries cost? (WITHOUT using symbolic algebra)
- 10. 10 Menu MathWrap-Up What concepts are students learning intuitively? What skills are they building? How does real-world application provide a context for connecting prior knowledge to more abstract learning?
- 11. Solving Equations usinga Flowchart Common Core Standard Addressed 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. 7.EE.1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. 8.EE.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 11
- 12. 12 Number puzzle… I’m thinking of a number – when I add 5 to the number my answer is 18. What number am I thinking of? x + 5 = 18 To solve we must find the value of x using inverse operations.
- 13. Using a flowchartto introduce inverse operations Example of a Flowchart: 8 × 2 + 4 ÷ 4 – 3 16 20 5 2 ÷ 2 – 4 × 4 + 3 Backtracking is working backwards by carrying out the inverse operation. 13
- 14. 14 Solving EquationsUsing a Flowchart: Working Backwards n 50 n=10 × 4 + 10 40 ÷ 4 – 10
- 15. 15 Solving EquationsUsing a Flowchart: ÷ 3 x 2 x = 18 × 3 6 – 4 + 4
- 16. 16 Flowchart WorkBoards: Scaffold and Support Kinesthetic Learners
- 17. 17 Solving EquationsUsing a Flowchart: ^2 777 +37.16 739.84 *0.01 7.77 -1.05 6.72 ÷3.2 n=2.1
- 18. 18 Solving EquationsUsing a Flowchart: -24 × (-3) n=8 ÷ (-3) –12
- 19. 19 Solving EquationsUsing a Flowchart: -6 × 5 n=-30 +(-14)
- 20. 20 Solving EquationsUsing a Flowchart: ×(-5) + 6 ÷ 2 -50 – 6 -44 x=10 ÷5 -22 × 2
- 21. 21 Solving EquationsUsing a Flowchart:
- 22. 22 Backtracking Benefits Intuitively builds an understanding of “undoing” (focus on inverse operation) Flowchart provides a visual prompt Builds understanding of the structure of algebraic expressions Works well with quadratic equations Limitations Only applies to equations with ONE Unknown (can’t be used to with x+1=2x) Focus on numbers does not assist students in moving to more algebraic approach
- 23. Solving equations usinga Fact- Family Approach 23 Common Core Standard Addressed Use properties of operations to generate equivalent expressions. 6.EE.3 Apply the properties of operations to generate equivalent expressions . 7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
- 24. 24 Using FactFamily Triangles 9 4 5 21 3 7 4 + 5 = 9 5 + 4 = 9 9 – 4 = 5 9 – 5 = 4 3 × 7 = 21 7 × 3 = 21 21 ÷ 3 = 7 21 ÷ 7 = 3
- 25. 25 Using FactFamily Triangles to Solve Equations n 3 -10 -30 n 5 n – 3 = -10 n – (-10) = 3 3 + (-10) = n -30 ÷ n = 5 -30 ÷ 5 = n n × 5 = -30
- 26. Summary Points Learning Non-Conventional approaches… Allows students to understand the underlying concepts Bridges students’ prior knowledge of number theory to more abstract concepts of algebraic thinking Supports seeing the interconnectedness of strands of Algebra Take-Away: In order for students gain a deeper understanding of abstract concepts, they need opportunities to explore using hands-on examples and visual models. 26
Editor's Notes
- #23 “Do the same to both sides” does not have these limitations.