Region 11: Math & Science Teacher Center
Solving Equations
Where we’ve been…
Equality
Ratio &
Proportion
Pattern
Generalization
Patterns Review - Justify It!
3rd Grade MCA
Practice Problem
1
4
7
10
Patterns Review - Justify It!
3rd Grade MCA
Practice Problem
1
4
7
10
Patterns Review
With a partner, talk about how a third
grader would talk about this problem:
• Draw the next figure in the pattern
• How many dots will be in the next
figure?
• Describe how make the pattern
Adult Challenge: How many dots
would be in the nth figure?
Patterns Review
Share in Grade Level Groups:
• What strategies did you see kids use?
• What did the students find most
challenging?
• What growth did you see?
• What surprised you?
Patterns Review
Watch video and record:
Good questions
Not-so-good questions
Patterns Review
Discuss video at your tables:
• What questions helped most to get your
students to explain the explicit rule for
different patterns?
• What did you learn about scaffolding
questions?
•How do you know when you are done
asking questions?
Solving Equations
Goals:
• Identify a developmental sequence for
solving equations
• Structure “math talk “about expressions &
equations
• Emphasize equivalence and how we
communicate this with students
• Discuss properties of equations
Work with a partner to solve problems on page 1
Solve.
3
x

5

17

x
Find all values that make the statement true.
Solve.
b

b

b

7

b

1
Find all values that make the statement true.
Solve.
2
x 9
Find all values that make the statement true.
Solve.
23

2
x

7
Find all values that make the statement true.
Solve.
8
m

2

2
m

5
Find all values that make the statement true.
Solve.
x

y

5
Find all values that make the statement true.
What does solve mean?
Break
Think/Pair/Share…
• Define expression
• Define equation
Big Idea!
• We solve equations because we can
make them true or false.
• We don’t solve expressions because
we can’t make them either true or
false.
Expression vs. Equation
Share and discuss in your group as you
work on page 2:
• How would you write the directions?
• How would you want/expect students
to show their work?…What would you
write on the board?
Expression vs. Equation
5

4
a

13
Expression vs. Equation
3
x

5

7
x

1
Expression vs. Equation
3
a

6

3
(
a

2
)
Expression vs. Equation
1 2  3
Expression vs. Equation
5x
Expression vs. Equation
5 x
40

8
x
Expression vs. Equation
• Directions make a big difference!
• Directions depend on context and where
you are in the curriculum
• Expressions are not equations
• No one “right way” to show work
• Most middle school textbook authors have
thought carefully about what strategies to
use to solve equations
CGI Algebra Video Clips
Benchmarks in Student Thinking
About The Equal Sign
Description
1
Children are asked to be specific
about what they think the equal sign
means
2
Children accept as true some
number sentence that is not of the
form a + b = c
3
4
Children recognize that the equal
sign represents a relation between
two equal numbers, particularly
through calculation (relational
understanding of the equal sign)
Children are able to compare the
mathematical expressions using
relational thinking without carrying
out the calculations (relational
understanding across the equal sign)
Note: These benchmarks are a guide, not a firm sequence
CGI Algebra Video Clips
Solving the Equation
b

b

b

20

1
(4th grader, 2 minutes, 42 seconds)
CGI Algebra Video Clips
Solving the Equation
k

k

13

k

2
(4th grader, 1 minutes, 51 seconds)
CGI Algebra Video Clips
Solving the Equation
e
e
e

e


2
(4th grader, 39 seconds)
CGI Algebra Video Clips
Solving the Equation
4

82

p

p
(4th grader, 51 seconds)
Benchmarks in Student Thinking
About The Equal Sign
Description
1
Children are asked to be specific
about what they think the equal sign
means
2
Children accept as true some
number sentence that is not of the
form a + b = c
3
4
Children recognize that the equal
sign represents a relation between
two equal numbers, particularly
through calculation (relational
understanding of the equal sign)
Children are able to compare the
mathematical expressions using
relational thinking without carrying
out the calculations (relational
understanding across the equal sign)
Note: These benchmarks are a guide, not a firm sequence
Lunch
Methods for solving linear equations of
the form ax ± b = cx ± d
Traditional Approach
vs.
Functions Approach
Traditional approach for solving linear
equations of the form ax ± b = cx ± d
1) Use of number facts
(solve mentally)
Example:
3+x=7
Not-so-good for:
3x + 7 = 5x – 14
Traditional approach for solving linear
equations of the form ax ± b = cx ± d
2) Generate and evaluate
(“guess and check” or “trial and error substitution”)
Example:
2x + 3 = 4x – 7
Not-so-good for:
3x – 7 = 10 – 4x
Traditional approach for solving linear
equations of the form ax ± b = cx ± d
3) a. Undoing
(or working backwards)
Example:
20 = 3x – 4
24 = 3 • x
8=x
Not-so-good for:
3x + 7 = 5x – 14
Traditional approach for solving linear
equations of the form ax ± b = cx ± d
3) b. Undoing
17 = 3p – 1
x3
p
6
-1
3p
18
÷3
3p – 1
17
+1
Traditional approach for solving linear
equations of the form ax ± b = cx ± d
4) Cover-up
Example:
k + k + 13 = k + 20
k + k + 13 = k + 13 + 7
k=7
Not-so-good for:
3x + 7 = 25 – 5x
Traditional approach for solving linear
equations of the form ax ± b = cx ± d
5) Transposing (change side-change sign)
Example:
3x – 7 +7=2x
+
5x = 15
x=3
Not-so-good for:
8 – 2x
2
x  20
3
Traditional approach for solving linear
equations of the form ax ± b = cx ± d
6) Equivalent equations
(performing the same operation on both sides)
Example:
17 = 3x – 7
17 + 7 = 3x – 7 + 7
24 = 3x
24/3 = 3x/3
8=x
Traditional approach for solving linear
equations of the form ax ± b = cx ± d
Group Task #1
• Break into six table groups
• Write one or two good problems for each
method on the table
• When the bell rings, move to the next table
Traditional approach for solving linear
equations of the form ax ± b = cx ± d
Group Task #2
• Move to the table where you started
• Work the problems using that particular
method
• Star any problems that can’t be solved
easily with the method
• Determine the minimum benchmark level
needed to solve these problems
• Be ready to report out
Functions approach for solving linear
equations of the form ax ± b = cx ± d
1) Table using graphing calculator
(similar to guess and check)
Example:
3x – 4 = x + 6
Y1
Y2
x=5
Functions approach for solving linear
equations of the form ax ± b = cx ± d
2) Graphing
Example:
3x – 4 = x + 6
Y1
Y2
x=5
Why standards?
“By viewing algebra as a strand in the
curriculum from prekindergarten on, teachers
can help students build a solid foundation of
understanding and experience as a
preparation for more-sophisticated work in
algebra in the middle grades and high
school.”
- NCTM, 2000, p.37
True or False
Your textbook determines the algebra
concepts and skills that you should cover
at a particular grade level.
TRUE
FALSE

don’t know
False: In a standards-based system, the
focus is shifted from what is TAUGHT to
what is LEARNED.
The standards tell us what students should
know and be able to do.
True or False
Algebra content has been shifted down
and now starts in the middle grades.
TRUE
FALSE

don’t know
False: Algebra and algebraic thinking are
integrated across K-11 in the state
standards. Every teacher has to do his/her
part to give students the opportunity to
learn the grade-level content.
Sort the Standards Involving
Solving Equations
In your groups,
• Look through individual standards
• Classify for grades 1 - 8
• Ask for an answer key when finished
• Take time to reflect on standards
with your group members
BIG IDEA
TRADITIONAL
ALGEBRA I
BIG IDEA
TRADITIONAL
ALGEBRA I
8th GRADE
STANDARDS
Break
Equivalence - Expressions
On page 7:
• State directions for each problem
• Simplify each expression or
equation one operation at a time,
leaving a trail down of equivalent
expressions or equivalent
equations.
Equivalence - Expressions
Directions:
7

3
(
2

5

3
)
Equivalence - Expressions
Directions:
3

2
(
3
x

7

5
x
)

2
Equivalence - Expressions
Directions:
3
m

14

2
(
m

5
)

5
m
Check:
Equivalence - Expressions
Expressions are equivalent…
…if every line has the same value for
the same value of x
Equivalence - Expressions
Equations are equivalent…
…if they have the same solution set
(but each line has a different value
for a given value of x).
Equivalence - Expressions
Can I add 5? … 10?
3
x
x
13

8

7
STRETCH
Balance metaphor for equations
Same weight on both sides
page 8 of the handout
Balance metaphor for equations
Build a balance to solve:
4 x + 4 = 12
Balance metaphor for equations
Build a balance to solve:
x+y=7
Balance metaphor for equations
Build a balance to solve:
2x+4=2x+9
Problems with balance metaphor
Negatives
Problems with balance metaphors
Subtraction
3
x

5
x

6

x

1
Metaphors for equations –
Balance (same weight on both sides)
1. Open Number sentences (CGI)
Metaphors for equations –
Balance (same weight on both sides)
ii. Algebra Tiles (McDougal Littell) or
Bags of Gold (CMP)
x
=
x
x
Algebra Tiles – McDougal Littell (Course 2)
CMP – Moving Straight Ahead (7th grade)
Metaphors for equations –
Balance (same weight on both sides)
3. Equation Mat (CPM – Algebra
Connections)
+
+
x
x
x
-
x
= +1
x
-
= -1
2
x

1


3

3
x
STRETCH
Baseline Assessment
Baseline Assessment
1. Find the value of m that makes
the number sentence below true.
12 = 4 • m
Baseline Assessment
2. Find the value of b that makes the
number sentence below true.
15 + 3b = 42
Baseline Assessment
3. Find the value of x that makes the
number sentence below true.
12x - 10 = 6x + 32
Baseline Assessment
4. Find the value of n that makes the
number sentence below true.
Show your steps to demonstrate
how you solved the problem.
4 + n - 2 + 5 = 11 + 3 + 5
Baseline Assessment
5. Balance A is balanced. The amount on
the left side of Balance A is tripled for
Balance B. Draw in what should appear
on the right side of Balance B to be
?
balanced.
Balance A
Balance B
Explain why your answer works.
Baseline Assessment
6. Determine which of the equations
below are equivalent to: 3b – 4 = b + 6
Circle yes, no or do not know for
each part.
Equivalent to 3b Π 4 = b + 6
a) 3b + 4 = b Π 6
yes
no
do not know
b) 3b Π 4 + 7 = b + 6 + 7
yes
no
do not know
c) 2b Π 4 = 6
yes
no
do not know
d) 4b Π 3 = 6 + b
yes
no
do not know
Properties of Equations
Addition Property of Equality
Words
Adding the same number to each side
of an equation produces an equivalent
equation
Algebra
from McDougal Littell Math Course 2
Properties of Equations
What other properties maintain
equivalence?
Words
Algebra
Big Ideas
We want to:
• Learn the different methods that
children use to solve linear equations
• Learn how to select examples to push kids
from less sophisticated methods (i.e.
guess and check) to more algebraic
methods
• Learn different metaphors for helping
students solve equations
Evaluation
On an index card, please record:
P: one positive from today’s work
M: one ‘minus’ or concern from
today’s work
I: something that you found interesting or
intriguing from today’s work.
Thanks for your feedback.