When I am dissatisfied and would like to go back to my youth, I
think of algebra. a retired lawyer
Slide 3
Every time I see a math word problem it looks like this: If I
have 10 ice cubes and you have 11 apples, how many pancakes will
fit on the roof? Answer: Purple, because aliens dont wear
hats.
Slide 4
Students must learn mathematics with understanding, actively
building new knowledge from experience and prior knowledge.
NCTM
Slide 5
The Math Things Mingle
http://www.insidemathematics.org/index.php/classro
om-video-visits/public-lesson-equations-inequalities-
properties/243-equations-inequalities-a-properties-
introduction-part-a?
http://www.insidemathematics.org/index.php/classro
om-video-visits/public-lesson-equations-inequalities-
properties/243-equations-inequalities-a-properties-
introduction-part-a?
Slide 6
ExpressionsVocabulary Look at the terms in each box. Write what
you know about each one.
Slide 7
Labeling an Expression Cut out the vocabulary terms. Glue them
next to the appropriate part of the expression. (You may need to
draw arrows to get them to fit.) Check with your partner. Do you
agree?
Slide 8
Vocabulary Match Match each term with the appropriate
representation. Check with your partner. Do you agree?
Slide 9
Vocabulary Assessment Label each part of the expression with
the correct vocabulary.
Slide 10
Common Core State Standards 6.EE.2b 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. For example, describe the expression 2(8 + 7) as a
product of two factors; view (8 + 7) as both a single entity and a
sum of two terms.
Slide 11
Students need ample opportunities to work on translating words
and contexts into symbols.
Slide 12
Grounding early algebraic experiences in familiar contexts can
help students to see the relevance of algebra to everyday
life.
Slide 13
Expressions Numerical expressiona combination of numbers and
operations 16 + 32 25 50 24/4 Algebraic expressiona combination of
variables, numbers, and at least one operation 6x + 2 x x + y
Slide 14
Writing Expressions with Bar Diagrams Kevin has 6 more baseball
cards than Eli. Step 1: Eli has an unknown number of baseball cards
c. Use a bar diagram to show Elis cards. Eli c cards Step 2: Kevin
has 6 more baseball cards than Eli. Complete the bar diagram below
to show how many baseball cards Kevin has. Kevin c cards 6 cards
So, Kevin has c + 6 base ball cards.
Slide 15
Sam sent 10 fewer messages in July than in August. Step 1: Sam
sent and unknown number of messages, m, in August. Label the bar
diagram to represent the messages Sam sent in August. August m
messages Step 2: Sam sent 10 fewer messages in July. Label the bar
diagram to show the messages Sam sent in July. July m messages 10
fewer So, Sam sent m 10 messages in July.
Slide 16
A bottlenose dolphin can swim d miles per hour. Humans swim one
third as fast as dolphins. Step 1: Dolphins can swim is an unknown
number of miles per hour, d. Use a bar diagram to represent the
speed a dolphin swims. dolphins d miles per hour Step 2: Humans
swim one third as fast as dolphins. Divide and shade a second bar
diagram to represent the speed humans can swim. dolphins d miles
per hour humans So, humans can swim d 3 miles per hour.
Slide 17
Clarks dog weighs 5 times as much as his cat. Step 1: The
weight of Clarks cat is an unknown number. Label the bar diagram to
represent the weight of Clarks cat. cat w cat Step 2: Clarks dog
weighs 5 times as much as his cat. Add four additional bars to
represent the dog as five times the length of the cat. dog So, the
weight of the dog is 5w.
Slide 18
With a partner, complete the table. Draw a bar diagram Write an
algebraic expression for the situation
Slide 19
Common Core State Standards 6.EE.1 Write and evaluate numerical
expressions involving whole-number exponents. 6.EE.2a Write
expressions that record operations with numbers and the letters
standing for numbers. For example, express the calculation subtract
y from 5 as 5 y.
Slide 20
Common Core State Standards 6.EE.2c 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). For example, use the
formulas V= s and A = 6s to find the volume and surface area of a
cube with sides of length s = .
Slide 21
Distributive Property Three friends are going to a concert at
the fair. They each want admission to the fair, which is $6.00 and
admission to the concert, which is $22.00. What is the total that
the three friends will spend? Step 1: Write an expression to
represent the amount spent in dollars. 3(6 + 22) friends fair
admission concert
Slide 22
Step 2: Use area models to evaluate the expression. Method 1:
Add the lengths. Then multiply. 6 22 3 3 (6 + 22) = 3(28) = 84
Method 2: Find each area. Then add. 6 22 3 3 (3 6) + (3 22) = 18 +
66 = 84
Slide 23
Since both expressions are equal to 84, they are equivalent.
So, 3(6 + 22) = (3 6) + (3 22)
Slide 24
With a partner, complete the table. Rewrite the expression
Evaluate the expression Check with your partner. Do you agree?
Slide 25
Prime Factorization The prime factorization of an algebraic
expression contains both the prime factors and any variable
factors. For example, the prime factorization of 6x is 2 3 x.
Slide 26
Factoring an Expression Factor 12 + 8 Write the prime
factorization of 12 and 8. 12 = 2 2 3 8 = 2 2 2 The GCF of 12 and 8
is 2 2 or 4. Write each term as a product of the GCF and its
remaining factor. Then use the Distributive Property to factor out
the GCF. 12 + 8 = (4 3) + (4 2) = 4(3 + 2) So, 12 + 8 = 4(3 +
2)
Slide 27
Factor 3x + 15 Write the prime factorization of 3x and 15. 3x =
3 x 15 = 3 5 The GCF of 3x and 15 is 3. Write each term as a
product of the GCF and its remaining factor. Then use the
Distributive Property to factor out the GCF. 3x + 15 = (3 x) + (3
5) = 3(x + 5) So, 3x + 15 = 3(x + 5)
Slide 28
With a partner, complete the table. Write the prime
factorization Rewrite the term using the GCF Use the Distributive
Property Check with your partner. Do you agree?
Slide 29
Common Core State Standards 6.EE.3 Apply the properties of
operations to generate equivalent expressions. For example, apply
the distributive property to the expression 3(2 + x) to produce the
equivalent expression 6 + 3x; apply the distributive property to
the expression 24x + 18y to produce the equivalent expression 6(4x
+ 3y); apply properties of operations to y + y + y to produce the
equivalent expression 3y.
Slide 30
6.EE.4 Identify when two expressions are equivalent (i.e., when
the two expressions name the same number regardless of which value
is substituted into them). For example, the expression y + y + y
and 3y are equivalent because they name the same number regardless
of which number y stands for.
Slide 31
Equation An equation is a mathematical sentence showing two
expressions are equal.
Slide 32
When you replace a variable with a value that results in a true
sentence, you solve the equation. The value for the variable is the
solution of the equation.
Slide 33
A solution to a problem in algebra might be 5x + 2, which
students might interpret as a lack of closure because their
previous experiences have taught them that an answer to a
mathematics problem should consist of a single term or number.
Slide 34
Hands-on-Equations Hands-on-Equations is a concrete and
pictorial model to help build Algebra understanding.
Slide 35
Common Core State Standards 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.
Slide 36
6.EE.6 Use variables to represent numbers and write expressions
when solving a real-world or mathematical problem; understand that
a variable can represent an unknown number, or, depending on the
purpose at hand, any number in a specified set.
Slide 37
6.EE.7 Solve real-world and mathematical problems by writing
and solving equations of the form x + p = q for cases in which p,
q, and x are all non-negative rational numbers.
Slide 38
Writing Inequalities You must be older than 13 to play in the
basketball league. a > 13 l l l l l l l l l l l l 10 11 12 13 14
15 16 17 18 19 20 21
Slide 39
You must be at least 18 years old to vote. a 18 l l l l l l l l
l l l l l l l l 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
25
Slide 40
Jason has less than 65 pages of his book to read. p < 65 l l
l l l l l l l l l l l 58 59 60 61 62 63 64 65 66 67 68 69 70
Slide 41
The movie will be no more than 90 minutes in length. m 90 l l l
l l l l l l l l l l l l 84 85 86 87 88 89 90 91 92 93 94 95 96 97
98
Slide 42
Write an inequality and draw a number line diagram for each
situation
Slide 43
Solving Inequalities Regina sent x text messages before lunch.
She sent another 4 text messages after lunch. She sent less than 7
text messages today. x + 4 < 7 x + (4 4) < 7 4 x < 3 l l l
l l l l l l l l 0 1 2 3 4 5 6 7 8 9 10
Slide 44
Write an inequality Solve it Draw a number line diagram
Slide 45
Common Core State Standards 6.EE.8 Write an inequality of the
form x > c or x c or x < c have infinitely many solutions;
represent solutions of such inequalities on number line
diagrams.
Slide 46
Effective mathematics teaching requires understanding what
students know and need to learn and then challenging and supporting
them to learn it well. NCTM
