Non Conventional Methods for Solving Equations

Speaker: Shephali Chokshi-Fox Co-Speaker: Victoria Miles

Agenda Rationale for using 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

Menu MathCommon Core 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).

Let’s look at this menu…

Menu Mathh + f =c+ f + s =7f =3h + c + f + 3x = 4c + 3f + s + m + l =3c + 3d = $11.10 What does d = ?

1.85 + 2.15 = $4.00

$ 4.15

$ 7.35

$ 14.90

$15.50

$1.55 (large soda)

Write what each customer ordered and calculate how much was paid for each order:

3h + 3f =

3h + f =

3(h + f)=

Which two customers ordered the same food and paid the same price?

3(1.85) + 3(1.05) = $8.70

3(1.85) + 1.05 = $6.60

3(1.85 + 1.05) = $8.70

Different members of the same family placed the following orders. Simply the orders:

3 (1.05) + 6 (1.85) + 5(?) = $24.50? = $2.05 (extra large)

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)

Hamburger = $3Fries = $1

Menu Math Wrap-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?

Solving Equations using a 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.

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.

Using a flowchart to introduce inverse operationsExample of a Flowchart:

8

Backtracking is working backwards by carrying out the inverse operation.

× 2 + 4 ÷ 4 – 3

+ 3 × 4 – 4 ÷ 2

16 20 5 2

Solving Equations Using a Flowchart: Working Backwards

n=10

40

÷ 4

× 4 + 10

– 10

50n

Solving Equations Using a Flowchart:

x = 18 × 3

÷ 3

6

– 4

+ 4

2x

Flowchart Work Boards:

Scaffold and Support Kinesthetic Learners

Solving Equations Using a Flowchart:

^2

777

+37.16

739.84

*0.01

7.77

-1.05

6.72

÷3.2n=2.1

Solving Equations Using a Flowchart:

n=8 –12÷ (-3)

-24

× (-3)

Solving Equations Using a Flowchart:

n=-30 +(-14)

-6

× 5

Solving Equations Using a Flowchart:

-50

– 6

-44

÷5x=10

×(-5) + 6 ÷ 2-22

× 2

Solving Equations Using a Flowchart:

BacktrackingBenefits

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

Solving equations using a Fact-Family Approach

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.

Using Fact Family Triangles

4 5

9

3 7

21

4 + 5 = 95 + 4 = 99 – 4 = 59 – 5 = 4

3 × 7 = 217 × 3 = 2121 ÷ 3 = 721 ÷ 7 = 3

Using Fact Family Triangles to Solve Equations

3 -10

n

n 5

-30

n – 3 = -10n – (-10) = 33 + (-10) = n

-30 ÷ n = 5-30 ÷ 5 = nn × 5 = -30

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.