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.

1. Speaker: Shephali Chokshi-Fox
Co-Speaker: Victoria Miles
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2. 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
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3. Menu Math
Common 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).
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4. 4
Let’s look at 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)
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6. Write what each customer ordered and calculate
how much was paid for each order:
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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 of the same family placed the
following orders. Simply the orders:
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8. 8
3 (1.05) + 6 (1.85) + 5(?) = $24.50
? = $2.05 (extra large)

9. Hamburger = $3
Fries = $1
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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 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?

11. 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.
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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 flowchart to introduce inverse
operations
Example of a Flowchart:
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× 2 + 4 ÷ 4 – 3
16 20 5 2
÷ 2 – 4 × 4 + 3
Backtracking is working backwards by
carrying out the inverse operation.
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14. 14
Solving Equations Using a Flowchart:
Working Backwards
n 50
n=10
× 4 + 10
40
÷ 4
– 10

15. 15
Solving Equations Using a Flowchart:
÷ 3
x 2
x = 18 × 3
6
– 4
+ 4

16. 16
Flowchart
Work Boards:
Scaffold and
Support
Kinesthetic
Learners

17. 17
Solving Equations Using a Flowchart:
^2
777
+37.16
739.84
*0.01
7.77
-1.05
6.72
÷3.2
n=2.1

18. 18
Solving Equations Using a Flowchart:
-24
× (-3)
n=8 ÷ (-3) –12

19. 19
Solving Equations Using a Flowchart:
-6
× 5
n=-30 +(-14)

20. 20
Solving Equations Using a Flowchart:
×(-5) + 6 ÷ 2
-50
– 6
-44
x=10 ÷5
-22
× 2

21. 21
Solving Equations Using 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 using a Fact-
Family Approach
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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 Fact Family 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 Fact Family 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.
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