COMMON CORE STATE
STANDARDS
BUILDING MEANING OF
INTEGERS, EXPRESSIONS,
AND EQUATIONS
Q
MICHELE DOUGLASS, PH.D.
MD SCHOOL SOLUTIONS, INC.
May 2014
5/6/2014
SBAC EXAMPLE: GRADE 5
5/6/2014
SBAC EXAMPLE: GRADE 6
 Connect each situation to an equation that matches that situation. t h th t it ti
5/6/2014
SBAC EXAMPLE: GRADE 6
 Partner with a Person at your Table
at e
t a e so at you ab e
 Partner A Construct the following
How would you represent the account pictorially you y
p
p
yy
add $5 to some amount of money in the bank?
 How would you represent this same amount if it is tripled?


How would you write each of these symbolically?
How would you write each of these symbolically?
5/6/2014
LET’S BUILD AN EXPRESSION
 Partner B Construct the following:
g

How would you represent any amount of money in an account? 
How would you represent the account if the amount in it was tripled?

How would you represent the account pictorially if the amount in it was tripled and you added $5?
the amount in it was tripled and you added $5? 
How would you write each of these symbolically?
5/6/2014
LET’S BUILD AN EXPRESSION
CONCEPT OF INTEGERS
 Three Key Concepts
Three Key Concepts

Opposites are an equal distance from 0 on the number line

Opposites added together make zero (cancel each other out). We return to 0.

A negative means “the opposite”. The effect with A
i
“h
i ” Th ff
ih
the algebra tiles means flipping the tile to make a value the opposite of what it was.
value the opposite of what it was.
CONCEPT OF INTEGERS
 Beginning with symmetry and opposites on a Beginning with symmetry and opposites on a
number line (absolute value)


The opposite of 3 is ‐3
The
opposite of 3 is 3
The opposite of ‐3 is 3
-5 -4 -3
-2 -1
0
1
These are equal
distances from zero in
both directions (think
absolute value).
2
3
4
5
CONCEPT OF INTEGERS
= 3
= ‐3
What do we get when we put opposites
together?
In other words, the sum of two opposites is
the same as what number?
CONCEPT OF INTEGERS
 Describe the length of the red line and the length of g
g
the black line.
S
Symmetry
from
f
0
|‐ 5| = |5|
 The lines are the same length because of absolute g
value being the way we talk about a distance.
 Use your algebra tiles to represent each of Use your algebra tiles to represent each of
the following situations.



You earned $8 for taking out the trash.
You
earned $8 for taking out the trash
You lost $5 on your way to school
You owe your brother $4
You owe your brother $4.
 What words indicated whether the number Wh t
d i di t d h th th
b
was positive or negative?
5/6/2014
REPRESENTING SITUATIONS
OPERATIONS FOR INTEGERS: USING
ALGEBRA TILES
 In order to add integers, it is important to return to In order to add integers, it is important to return to
addition of whole number.
 Addition is combining sets or an augmentation to a set.
3 + 2 = ???
“Adding Quantities to a Set” ADDITION OF INTEGERS
-3
3
-1
1
-1
+ -2
2
-1
1
-1
1
-1
5
= -5
ADDITION OF INTEGERS
-3
3
-1
1
+ 1
-1
1
=
-2
2
1
-1
Notice that we have opposites:
Opposites make zero
There is 1 zero pair.
ADDITION OF INTEGERS
 Let’s try a couple more together.
et s t y a coup e o e toget e .
7 + ‐3 =
‐5 + 6 =
‐3 + ‐7 =
8 + ‐2 = 29 + ‐35 = How did you represent the positive?
H
How
did you represent the
h negative?
i ?
How many zero pairs exist?
What is the value of the expression?
SUBTRACTION OF INTEGERS
 Subtraction of Integers: This is where the Subtraction of Integers: This is where the
challenge begins for students.
3 2 = 1
3 –
2 1
 The fundamental concept of subtraction is the “removal from a set”
SUBTRACTION OF INTEGERS
-3 – (-1 ) = -2
-1
-1
-1
Subtraction is about
removing from a set. This
problem asks to remove a
negative
i one from
f
the
h set.
This is typically a hard problem for students due to the symbols
involved. However, when you think about this concept, you are
removing negative from a set of negatives.
SUBTRACTION OF INTEGERS
3 – (-2) =
1
1
1
There are no negatives
g
to
remove from the set.
SUBTRACTION OF INTEGERS
5 – (-3) =
1
1
1
1
1
There are no negatives
g
to
remove from the set.
SUBTRACTION OF INTEGERS
-5 – (2) =
What about if we start with a negative amount?
How do we remove a positive 2 from this set?
SUBTRACTION OF INTEGERS
 Eventually, we want students to remember Eventually we want students to remember
the “add the opposite” rule for subtraction. •
•
•
Notice the pattern
Notice
the pattern
Provide opportunity with representations
Periodically come back to the representation
Periodically, come back to the representation 3 – (-2) is the same as 3 + 2
SUBTRACTION OF INTEGERS
 Let
Let’ss try a couple more together.
try a couple more together.
7 – (‐3) =
‐5 – 6 =
‐3 – ‐7 =
‐8
8 – ‐2
2 = How did you represent your set to
begin the problem?
Do you need any zero pairs?
What is the value of the expression?
problem did you
y also
What addition p
represent?
 What would be a problem whose sum or What would be a problem whose sum or
difference is ‐7? 5/6/2014
FORMATIVE ASSESSMENT: REVERSIBILITY
5/6/2014
FORMATIVE ASSESSMENT: GENERALIZING
YOUR THINKING
5/6/2014
SBAC EXAMPLE: GRADE 6
5/6/2014
SBAC EXAMPLE: GRADE 7
MULTIPLICATION OF INTEGERS
 Order makes a difference when using visuals
Order makes a difference when using visuals
3  ‐2
3 2 is not the same as is not the same as ‐2
2  3
 The idea of The idea of “negative
negative number of groups
number of groups” does not make sense to student


We can’t have ‐2 groups of 3
g p
Critical to understand the negative sign as “the opposite of”
MULTIPLICATION OF INTEGERS
3  ‐2
3 2 means 3 groups of means 3 groups of ‐2
2 in each group
in each group
-1
-1
-1
-1
-1
-1
3  -2
2 = -6
6
MULTIPLICATION OF INTEGERS
- 32
Perform
multiplication
first and then
find the
opposite (flip)
1
1
1
1
1
1
The problem is asking
about “the opposite” of
3 groups of
f 2.
32
Now you are ready
to think about the
opposite of 3 x 2.
MULTIPLICATION OF INTEGERS
‐3
3  ‐2
2 means we have means we have ‐3
3 groups of groups of ‐2
2 in in
each group
 We can also read this as We can also read this as “the
the opposite
opposite” of 3 of 3
groups of ‐2
-1
-1
groups off -2
THE PROPERTY RELATED TO INTEGERS
What This Process Looks Like with Symbols
What
This Process Looks Like with Symbols
1  ‐1 = ‐1 the Multiplicative Property of ‐1
‐1 
1 1 = ‐1
1 1
‐1  ‐1 = 1
‐1  a = ‐a the Multiplicative Property of ‐1
‐1 
1 5 = ‐5
5 5
‐1  ‐5 = ‐(‐5) or 5
THE PROPERTY RELATED TO INTEGERS
‐3
3  2 = 2=
‐3 is the same as ‐1 
3 i th
1 3
‐3  2 = ‐1  3  2
‐3  2 = ‐1  (3  2)
‐3  2 = ‐1  6
‐3  2 = ‐ 6
DIVISION OF INTEGERS
3)) 6
1
1
1
1
1
1
• In order to divide integers, it is important to return g ,
p
to division of whole number.
• The operation of division asks, “How many groups?” or “How many are in each group?”
DIVISION OF INTEGERS
3) -6
-1
-1
-1
1
-1
1
-1
1
-1
• We still are going to partition this set into 3 sets. g g p
Now our set is beginning with negative values. DIVISION OF INTEGERS
-2
3) -6
-1
-1
-1
1
-1
1
• There are two negatives in each set.
g
-6
6
= -2
3
-1
1
-1
1
DIVISION OF INTEGERS
The opposite of this set will be 2
will be ‐2. - 3) 6
The
negative
flips the
tiles.
-1
-1
-1
-1
-1
-1
EXPRESSIONS
LIKE AND UNLIKE TERMS
 We tell students “you can’t combine unlike e te stude ts you ca t co b e u e
terms” (3 + a)
And then we show them this:
3a + 5
So we eventually get this:
3(3 + a) = 9a
EXPRESSIONS
NUMBERS VS. VARIABLES
directly apply to variables.
apply to variables
 Number Number “rules”
rules don
don’tt all all
43 = 40 + 3 but 43
40 3
b t
35 ≠ 3  5 but 2(6 + 1) = 14 but
ab
b ≠ a + b
b
a  b = ab
2(a + 1) = 2a + 2
This is the source of many student misconceptions
that aren’t explained
p
visually
y during
g instruction.
EXPRESSIONS: CONFUSION OVER
PROPERTIES

The Associative Property The
Associative Property – How are terms How are terms
grouped? Simplify 3x + x + 2 + ‐2x + 5 is the same as 2x + 7
But why? And how can you explain this beyond the
symbols?
y
Use your tiles to represent each of these situations.
Use
your tiles to represent each of these situations
 A student has $3 and some money in the bank.
 A bank has $10 on the counter and two drawers filled of cash to begin the day. (Each drawer filled
of cash to begin the day (Each drawer
contains the same amount to begin the day.)
 The other bank branch has 3 drawers of cash and The other bank branch has 3 drawers of cash and
needs to pay out $25 to its first customer.
5/6/2014
REPRESENTING SITUATIONS
EXPRESSIONS - VOCABULARY
 Coefficients and Expressions
Coefficients and Expressions
Variables
x
x
x
Numbers
1
1
1
1
1
= 3x + 5
• 3 is called a coefficient
• 5 is called a constant
• 3x + 5 is called an expression
EXPRESSIONS - REPRESENTATION
 Your Turn: How would you represent each of Your Turn: How would you represent each of
these expressions?

3x + 5

‐4 + (‐3x)

‐1 – 3x + 4
5/6/2014
SBAC EXAMPLE: GRADE 6
The locations of six
buildings need to be
added to the
coordinate grid.






Bank (-8, 5)
S h l ((-8,
School
8 -6)
6)
Park (4, 5)
Post Office (-9, 5)
Store ((-9,
9 -6)
6)
Library (?, ?)
5/6/2014
SBAC EXAMPLE: GRADE 6
EXPRESSIONS - USING PROPERTIES
Example 1
Simplify:
p y -3 + 3x + x + -2x
-1
-1
-1
x
x
x
x
-x
-x
EXPRESSIONS - USING PROPERTIES
Example 2
Simplify:
p y -3 + 2x + 1 + x
-1
-1
-1
x
x
1
x
EXPRESSIONS - USING PROPERTIES
Example 3
Simplify: 2x – 1 – 3x + 2
How do we represent this expression?
EXPRESSIONS - USING PROPERTIES
Example 1
Simplify:
p y 3(x
( + 2))
Remember what multiplication
p
means.
3g
groups
p of ((x + 2))
EXPRESSIONS - USING PROPERTIES
Example 2
Simplify:
p y -2(2x
(
+ -1))
Remember what multiplication
p
means.
“the opposite
pp
of 2 g
groups
p of ((2x + -1))
EXPRESSIONS - USING PROPERTIES
Distributive Property
Undoing or Factoring
Factor: 2x + 6
x
x
1
1
1
1
1
1
Two expressions are shown below.
p
P: 2(3x − 9)
Q: 6x −9
Part A
 Apply the distributive property to write an expression that is equivalent to expression P
expression that is equivalent to expression P.
Part B
Part
B
 Explain whether or not expressions P and Q are equivalent for any value of x.
5/6/2014
SBAC EXAMPLE: GRADE 6
5/6/2014
SBAC EXAMPLE: GRADE 6
EXPRESSIONS - EVALUATING
THINK: x =
Example 1
Simplify 2x +2 where x = 3
x
1
x
1
1
1
1
EXPRESSIONS - EVALUATING
Example 3
Simplify 2x + 4 where x = -2
2
x
1
1
x
1
1
EXPRESSIONS - EVALUATING
Example 4
Simplify -x
x + 4 where x = 2
-x
1
1
1
1
….and we need
to know the
value for -x
Notice we
are given the
value for x….
x
EXPRESSIONS - EVALUATING
 Your Turn: Use algebra tiles to evaluate these Your Turn: Use algebra tiles to evaluate these
two expressions.

Example 4: 3x + 5 where x = ‐2

Example 6: ‐1 – 3x + 4 where x = 2
Part A
Ana is saving to buy a bicycle that costs $135. She
Ana is saving to buy a bicycle that costs $135. She has saved $98 and wants to know how much more money she needs to buy the bicycle. The equation 135 = x + 98 models this situation, where x represents the additional amount of money Ana needs to buy the bicycle
needs to buy the bicycle.
 When substituting for x, which value(s), if any, from the set {0 37 98 135 233} will make the
from the set {0, 37, 98, 135, 233} will make the equation true?
 Explain what this means in terms of the amount p
of money needed and the cost of the bicycle.
5/6/2014
SBAC EXAMPLE: GRADE 6
5/6/2014
SBAC EXAMPLE: GRADE 6
5/6/2014
SBAC EXAMPLE: GRADE 7
UNDERSTANDING EQUALITY TO SOLVE
EQUATIONS
 Key Emphasis with Equality and Equations
Key Emphasis with Equality and Equations



The equal sign (=) separates two parts, and each part or expression represent the same value
part or expression represent the same value.
To maintain equality, what you do to one side of an equation, you have to do to the other side of
an equation, you have to do to the other side of the equation (Properties of Equality).
Solving equations means finding the value the g q
g
make the equation true (connects to lower grade problems in generalizing).
Example 1: How do we solve this equation?
 I ordered 6 items. It was delivered in 3 cases. I ordered 6 items It was delivered in 3 cases
How many items were in each case?


How do you model this problem?
Wh t
What symbols can you use to model the b l
t
d l th
problem?
5/6/2014
UNDERSTANDING EQUALITY TO SOLVE
EQUATIONS
UNDERSTANDING EQUALITY TO SOLVE
EQUATIONS
Example 1: How do we solve this equation?
3
3x
=
6
=
II ordered 6 items. It was delivered in 3 cases. ordered 6 items It was delivered in 3 cases
How many items were in each case?
Example 2: How do we solve this equation?
II ordered 3 cases and 2 additional individual ordered 3 cases and 2 additional individual
items. The company delivered 1 case and 4 individual items How many items are in a
individual items. How many items are in a case?


How do you model this problem?
Wh t
What symbols can you use to model the b l
t
d l th
problem?
5/6/2014
UNDERSTANDING EQUALITY TO SOLVE
EQUATIONS
UNDERSTANDING EQUALITY TO SOLVE
EQUATIONS
Example 2
3 +2
3x
= x+4
I ordered 3 cases and 2 additional individual items. The company delivered 1 case and 4 individual items. How many items are in a case?
SOLVING EQUATIONS WITH ALGEBRA
TILES
Example 3
-2x
2 + -2
2
=
=
x+4
SOLVING EQUATIONS WITH ALGEBRA
TILES
Example 4
2 (3x
(3 – 2)
How does the
di ib i
distributive
property work?
= 5 ( x – 1)
Remember:
Subtraction is the
same as adding an
opposite.
SOLVING EQUATIONS WITH ALGEBRA
TILES
Example 4 – The Problem REWRITTEN would look like this:
2 (3x
(3 + -2)
2)
= 5 ( x + -1)
1)
 What would be an equation whose solution is What would be an equation whose solution is
‐7?  Is there another one? How do you know?
Is there another one? How do you know?
5/6/2014
FORMATIVE ASSESSMENT: REVERSIBILITY
5/6/2014
SBAC EXAMPLE: GRADE 6
All books in a store are being discounted by 30%. Part A
 Let x represent the regular price of any book in the L t
t th
l
i
f
b k i th
store. Write an expression that can be used to find the sale price of any book
Part B
 Jerome bought a book on sale at the store. The sale Jerome bought a book on sale at the store The sale
price of the book was $8.96. Write and solve an equation to determine the regular price of the book to the nearest cent
the nearest cent.
5/6/2014
SBAC EXAMPLE: GRADE 7
Renee, Susan, and Martha will share the cost to rent a vacation h
house for a week.
f
k
• Renee will pay 40% of the cost.
• Susan will pay 0.35 of the cost.
• Martha will pay the remainder of the cost
• Martha will pay the remainder of the cost.
Part B
P
tB
 The cost to rent a vacation house for a week is $850. How much will Renee, Susan, and Martha each pay to rent this house for a week?



Renee will pay $ __________
Susan will pay $ ___________
Martha will pay $ p y $ __________
5/6/2014
SBAC EXAMPLE: GRADE 7
5/6/2014
SBAC EXAMPLE: GRADE 8
In the following equation, a and b are both t e o o g equat o , a a d b a e bot
integers.
a(3x – 8) = b – 18x
 What is the value of a?  What is the value of b?
5/6/2014
SBAC EXAMPLE: GRADE 7
5/6/2014
SBAC EXAMPLE: GRADE 8
EQ
QUATIONS AND MULTIPLE METHODS
 Mathematical Practices call for students to Mathematical Practices call for students to
“Construct viable arguments and critique the reasoning of others”
reasoning of others
 Most problems can be solved differently.
M t
bl
b
l d diff
tl
3(4x + 2) = 6x + 18
COMPARING STRATEGIES
Solution 1
3(4x + 2) = 6x + 18
12x + 6 = 6x + 18
12x + 6 + ‐6 = 6x + 18 + ‐6
12x = 6x 12x 6x + 12
12
12x + ‐6x = 6x + ‐6x + 12
6x = 12
1/6 6x = 1/6 
6x 1/6  12
x = 2 Solution 2
3(4x + 2) = 6x + 18
3(4x
+ 2) = 6x + 18
3(4x + 2) = 3(2x + 6) 1/3  3(4x + 2) = 1/3  3(2x + 6) 1(4x + 2) = 1(2x + 6) 4x + 2 = 2x + 6
4x + 2x + 2 = 2x + 2x + 6
4x + ‐2x + 2 = 2x + ‐2x + 6
2x + 2 = 6
2x + 2 + ‐2 = 6 + ‐2
2x = 4
½ 2x = ½ 4
x = 2
5/6/2014
SBAC EXAMPLE: GRADE 8
work correct? If so how do you
so, how do you know? If not, where is it
where is it incorrect and correct it
correct it.
5/6/2014
CRITIQUING
Q
THE WORK OF OTHERS
Solve for x:  Is this student
Is this student’ss 3x 9 = 12x
3x ‐
9 = 12x
 What types of experiences will your students What types of experiences will your students
have with problem solving?
 How will they be allowed/expected to build models?
 How will you connect the models to How will you connect the models to
equations/expressions and the manipulation that is done to solve/simplify?
5/6/2014
VISUAL RESIDUE
 What are your three key things to Remember What are your three key things to Remember
and do? (go to Agenda)
 With at least one other person at your grade llevel, discuss what types of learning l di
h tt
fl
i
experiences students will need to participate i t b
in to be successful with this content?
f l ith thi
t t?
5/6/2014
IMPLEMENTATION
What is different in your thinking from
when we began with the quick write?
 The negative symbol represents the opposite of a value where as the subtraction symbol is an operation that means y
p
to remove or reduce or compare.
 Operations imply an action including absolute value.
 Context is extremely helpful to modeling and solving problems.
 Modeling is important to the process of learning how to manipulate an equation.
 While algebra tiles model whole values, they support While algebra tiles model whole values they support
students in learning the process to apply to any set of numbers.
5/6/2014
KEY IDEAS FOR INTEGERS, EXPRESSIONS,
AND EQUATIONS