6 th grade big idea 3 teacher quality grant. big idea 3: write, interpret, and use mathematical...

66thth Grade Big Idea Grade Big Idea 33

Teacher Quality GrantTeacher Quality Grant

Big Idea 3: Big Idea 3: Write, Write, interpret, and use mathematical interpret, and use mathematical expressions and equations. expressions and equations.

MA.6.A.3.1: Write and evaluate mathematical MA.6.A.3.1: Write and evaluate mathematical expressions that correspond to given expressions that correspond to given situations.situations.

MA.6.A.3.2: Write, solve, and graph one- and MA.6.A.3.2: Write, solve, and graph one- and two- step linear equations and inequalities. two- step linear equations and inequalities.

MA.6.A.3.5 Apply the Commutative, MA.6.A.3.5 Apply the Commutative, Associative, and Distributive Properties to Associative, and Distributive Properties to show that two expressions are equivalent. show that two expressions are equivalent.

MA.6.A.3.6 Construct and analyze tables, MA.6.A.3.6 Construct and analyze tables, graphs, and equations to describe linear graphs, and equations to describe linear functions and other simple relations using functions and other simple relations using both common language and algebraic both common language and algebraic notation. notation.

Big idea 3: Big idea 3: assessed assessed with Benchmarkswith Benchmarks

Assessed with means the benchmark is present Assessed with means the benchmark is present on the FCAT, but it will not be assessed in on the FCAT, but it will not be assessed in isolation and will follow the content limits of the isolation and will follow the content limits of the benchmark it is assessed with. benchmark it is assessed with.

MA.6.A.3.3 Work backward with two-step MA.6.A.3.3 Work backward with two-step function rules to undo expressions. (Assessed function rules to undo expressions. (Assessed with MA.6.A.3.1.) with MA.6.A.3.1.)

MA.6.A.3.4 Solve problems given a formula. MA.6.A.3.4 Solve problems given a formula. (Assessed with MA.6.A.3.2, MA.6.G.4.1, (Assessed with MA.6.A.3.2, MA.6.G.4.1, MA.6.G.4.2, and MA.6.G.4.3.) MA.6.G.4.2, and MA.6.G.4.3.)

Big idea 3: Big idea 3: Benchmark Item Benchmark Item SpecificationsSpecifications

Big Idea 3: Big Idea 3: Prerequisite Prerequisite knowledgeknowledge

Order of OperationsOrder of Operations

Fractions and ratiosFractions and ratios

DecimalsDecimals

PercentPercent

Big idea 3: Big idea 3: Variable videoVariable video

Writing Algebraic Expressions

Be able to write an algebraic Be able to write an algebraic expression for a word phrase or write a expression for a word phrase or write a

word phrase for an expression.word phrase for an expression.

Although they are closely related, a Great Dane weighs about 40 times as much as a Chihuahua.

When solving real-world problems, you will need to translate words, or verbal expressions, into algebraic expressions.

Since we do not know the can weight of the Chihuahua we can represent it with the variable c

=c So then we can write the Great Dane as

40c or 40(c).

Notes

•In order to translate a word phrase into an algebraic expression, we must first know some key word phrases for the basic operations.

On the back of your notes:

AdditionAddition SubtractionSubtraction

MultiplicationMultiplication Division Division

Addition Phrases:• More than

• Increase by

• Greater than

• Add

• Total

• Plus

• Sum

Subtraction Phrases:

• Decreased by

• Difference between

• Take Away

• Less

• Subtract

• Less than*

• Subtract from*

Multiplication Phrases:

•Product

•Times

•Multiply

•Of

•Twice or double

•Triple

Division Phrases:

•Quotient

•Divide

•Divided by

•Split equally

Notes

•Multiplication expressions should be written in side-by-side form, with the number always in front of the variable.

•3a 2t 1.5c 0.4f

Notes

Division expressions should be written using the fraction bar instead of the traditional division sign.

€

c

4,x

24,15

y

Modeling a verbal expression

• First identify the unknown value (the variable)

• Represent it with an algebra tile

• Identify the operation or operations

• Identify the known values and represent with more tiles


Example: Lula read 10 books. Kelly read 4 more books then Lula.

• There is no unknown value

• More means addition

• The known values are: Lula 10 books and Kelly 4 more

10 books4

books10 +4

Singapore math introduction

• Level 1

• Level 2

• Level 3

• Enrichment

• Links for other strategies

http://www.thesingaporemaths.com/p6math1f.swf




http://www.thesingaporemaths.com/stratf.html

Examples• Addition phrases:

• 3 more than x

• the sum of 10 and a number c

• a number n increased by 4.5

Examples• Subtraction phrases:

• a number t decreased by 4

• the difference between 10 and a number y

• 6 less than a number z

Examples

• Multiplication phrases:• the product of 3 and a

number t

• twice the number x

• 4.2 times a number e

Examples

• Division phrases:• the quotient of 25 and a number

b

• the number y divided by 2

• 2.5 divide g

Example games

• Snow man game

• Millionaire game

http://www.harcourtschool.com/activity/operation_snowman/

http://www.math-play.com/Algebraic-Expressions-Millionaire/algebraic-expressions-millionaire.html

Examples• converting f feet into inches

• a car travels at 75 mph for h hours

• the area of a rectangle with a length

of 10 and a width of w

12f

75h

10w

Examples• converting i inches into feet

• the cost for tickets if you purchase 5 adult

tickets at x dollars each

• the cost for tickets if you purchase 3

children’s tickets at y dollars each

5x

i

12

3y

Examples

• the total cost for 5 adult tickets and 3 children’s tickets using the dollar amounts from the previous two problems

5x + 3y = Total Cost

Example

Great challenge problems are located on the website bellow:

Challenges

http://www.learner.org/workshops/math/puzzlers.html

PROBLEM SOLVING

What is the role of the teacher?

“Through problem solving, students can experience the power and utility of mathematics. Problem solving is central to inquiry and application and should be interwoven throughout the mathematics curriculum to provide a context for learning and applying mathematical ideas.”

NCTM 2000, p. 256

Instructional programs from prekindergarten through grade 12 should enable all students to-

•build new mathematical knowledge through problem solving;•solve problems that arise in mathematics and in other contexts;•apply and adapt a variety of appropriate strategies to solve problems;•monitor and reflect on the process of mathematical problem solving.

Teachers play an important role in developing students' problem-solving dispositions.

1.They must choose problems that engage students. 2.They need to create an environment that encourages students to explore, take risks, share failures and successes, and question one another.

In such supportive environments, students develop the confidence they need to explore problems and the ability to make adjustments in their problem-solving strategies.

• Three Question Types

– Procedural

– Conceptual

– Application

• Procedural questions require students to:

– Select and apply correct operations or procedures

– Modify procedures when needed

– Read and interpret graphs, charts, and tables

– Round, estimate, and order numbers

– Use formulas

• Sample Procedural Test Question

A company’s shipping department is receiving a shipment of 3,144 printers that were packed in boxes of 12 printers each. How many boxes should the department receive?

• Conceptual questions require students to:– Recognize basic mathematical concepts

– Identify and apply concepts and principles of mathematics

– Compare, contrast, and integrate concepts and principles

– Interpret and apply signs, symbols, and mathematical terms

– Demonstrate understanding of relationships among numbers, concepts, and principles

• Sample Conceptual Test QuestionA salesperson earns a weekly salary of $225 plus $3 for every pair of shoes she sells. If she earns a total of $336 in one week, in which of the following equations does n represent the number of shoes she sold that week?

(1) 3n + 225 = 336

(2) 3n + 225 + 3 = 336

(3) n + 225 = 336

(4) 3n = 336

(5) 3n + 3 = 336

• Application/Modeling/Problem Solving questions require students to:– Identify the type of problem represented

– Decide whether there is sufficient information

– Select only pertinent information

– Apply the appropriate problem-solving strategy

– Adapt strategies or procedures

– Determine whether an answer is reasonable

• Sample Application/Modeling/Problem Solving Test Question

Jane, who works at Marine Engineering, can make electronic widgets at the rate of 27 per hour. She begins her day at 9:30 a.m. and takes a 45 minute lunch break at 12:00 noon. At what time will Jane have made 135 electronic widgets?

(1) 1:45 p.m.

(2) 2:15 p.m.

(3) 2:30 p.m.

(4) 3:15 p.m.

(5) 5:15 p.m.

According to Michael E. Martinez There is no formula for problem solving

How people solve problems varies

Mistakes are inevitable

Problem solvers need to be aware of the total process

Flexibility is essential

Error and uncertainty should be expected

Uncertainty should be embraced at least temporarily

What steps should we take when solving a word problem?

1. Understand the problem

2. Devise a plan

3. Carry out the plan.

4. Look back

Reads the problem carefully Defines the type of answer that is required Identifies key words Accesses background knowledge regarding

a similar situation Eliminates extraneous information Uses a graphic organizer Sets up the problem correctly Uses mental math and estimation Checks the answer for reasonableness

K W E SWhat do you KNOW from the word problem?

What does the question WANT you to find?

Is there an EQUATION or model to solve the problem?

What steps did you use the SOLVE the problem?

•What am I asked to find or show? •What type of answer do I expect? •What units will be used in the answer? •Can I give an estimate? •What information is given?•If there enough or too little information given? •Can I restate the problem in your own words?

UNDERSTAND THE PROBLEM

Ask yourself….





Pattern: 1, 3, 6, 10, 15, …

What are the next 4 numbers?

11+2=33+3=66+4=1010+5=15

The amount being added increases by 1 each time so:15+6=2121+7=2828+8=3636+9=45





Number of chips:3 green4 blue1 red8 total chips

What fraction of the total chips is green?

€

What you want

Total

€

GreenChips

Total Chips

€

3

8

Solve problems out loud Explain your thinking process Allow students to explain their thinking

process Use the language of math and require

students to do so as well Model strategy selection Make time for discussion of strategies Build time for communication Ask open-ended questions Create lessons that actively engage learners

Jennifer Cromley, Learning to Think, Learning to Learn

This is simply checking all steps and calculations. Do not assume the problem is complete once a solution has been found. Instead, examine the problem to ensure that the solution makes sense.

LOOK BACK

Hierarchical diagramming

Sequence charts

Compare and contrast charts

Algebra

Calculus Trigonometry

Geometry

MATH

Category

What is it?Illustration/Example

What are some examples?

Properties/Attributes

What is it like?

Subcategory

Irregular set

Compare and Contrast

Positive Integers

Numbers

What is it?Illustration/Example

What are some examples?

Properties/Attributes

What is it like?

Fractions

Compare and Contrast - example

Whole Numbers Negative Integers

Zero

-3, -8, -4000

6, 17, 25, 100

0

Prime Numbers

5

7 11 13

Even Numbers

4 6 8 10

Multiples of 3

9 15 21

32

6

TRI-ANGLES

Right Equiangular

Acute Obtuse

3 sides

3 angles

1 angle = 90°

3 sides

3 angles

3 angles < 90°

3 sides

3 angles

3 angles = 60°

3 sides

3 angles

1 angle > 90°

Word = Category + Attribute

= +

Definitions: ______________________________________________________________________________________

Word = Category + Attribute

= +

Definition: A four-sided figure with four equal sides and four right angles.

Square Quadrilateral 4 equal sides & 4 equal angles (90°)

1. Word: 2. Example:

3. Non-example:4. Definition

1. Word: semicircle 2. Example:

3. Non-example:4. Definition

A semicircle is half of a circle.

Divide into groups Match the problem sets with the appropriate

graphic organizer

Which graphic organizer would be most suitable for showing these relationships?

Why did you choose this type? Are there alternative choices?

Parallelogram RhombusSquare QuadrilateralPolygon KiteIrregular polygon TrapezoidIsosceles Trapezoid Rectangle

Counting Numbers: 1, 2, 3, 4, 5, 6, . . .Whole Numbers: 0, 1, 2, 3, 4, . . .Integers: . . . -3, -2, -1, 0, 1, 2, 3, 4. . .Rationals: 0, …1/10, …1/5, …1/4, ... 33, …1/2, …1Reals: all numbersIrrationals: π, non-repeating decimal

Addition Multiplication a + b a times b a plus b a x b sum of a and b a(b)

ab

Subtraction Divisiona – b a/ba minus b a divided by ba less b

a ÷ b

Use the following words to organize into categories and subcategories of Mathematics:NUMBERS, OPERATIONS, Postulates, RULE, Triangles, GEOMETRIC FIGURES, SYMBOLS, corollaries, squares, rational, prime, Integers, addition, hexagon, irrational, {1, 2, 3…}, multiplication, composite, m || n, whole, quadrilateral, subtraction, division.

POLYGON

Parallelogram:has 2 pairs ofparallel sides

Kite

Square, rectangle,rhombus

Kite: has 0 sets ofparallel sides

Irregular: 4 sidesw/irregular shape

Quadrilateral Trapezoid: has 1 set of parallel sides

Trapezoid, isosceles trapezoid

Kite

REAL NUMBERS

Operations

Subtraction

MultiplicationDivision

Addition

____a + b____

___a plus b___

Sum of a and b

____a - b_____

__a minus b___

___a less b____

____a / b_____

_a divided by b_

_____a b_____

___a times b_______a x b__________a(b)__________ab______

Numbers Operations Rules Symbols

GeometricFigures

Mathematics

Triangle

Quadrilateral

HexagonInteger

Prime

Rational

Irrational

Whole

Composite

Addition

Subtraction

Multiplication

Division

Corollary

Postulate m║n

√4

{1,2,3…}

Mike, Juliana, Diane, and Dakota are entered in a 4-person relay race. In how many orders can they run the relay, if Mike must run list? List them.

Mrs. Stevens earns $18.00 an hour at her job. She had $171.00 after paying $9.00 for subway fare. Find how many hours Mrs. Stevens worked.Try solving this problem by working backwards.

Use the work backwards strategy to solve this problem.

A number is multiplied by -3. Then 6 is subtracted from the product. After adding -7, the result is -25. What is the number?

Big Idea 3: Big Idea 3: Patterns and Patterns and EquationsEquations

Analyzing patterns and sequences (lesson ENLVM)

http://enlvm.usu.edu/ma/nav/toc.jsp?sid=__shared&cid=emready@patterns_and_sequences&bb=course

Properties ofProperties ofAddition & Addition &

MultiplicatioMultiplicationn

Why do we need rules or properties in math?

Lets see what can happen if we didn’t have rules.

Before We Begin…Before We Begin…• What is a VARIABLE?

A variable is an unknown amount in a number sentence represented as a letter:

5 + n 8x 6(g) t + d = s

Before We Begin…Before We Begin…• What do these symbols mean?

( ) = multiply: 6(a) or group: (6 + a)

* = multiply

· = multiply

÷ = divide

/ = divide

Algebra tiles and Algebra tiles and counterscounters

• Represent the following expressions with algebra tiles or counters:

1.3 + 4 and 4 + 3

2.3 - 4 and 4 – 3

3. and

€

3⋅4

€

3 ÷ 4

Algebra tiles and Algebra tiles and counterscounters

• Represent the following expressions with algebra tiles or counters:

1.9x+ 2 and 2 + 9x

1.9x - 2 and 2 – 9x

Commutative PropertyCommutative Property• To COMMUTECOMMUTE something is to change it

• The COMMUTATIVECOMMUTATIVE property says that the order of numbers in a number sentence can be changed

• Addition & multiplication have COMMUTATIVECOMMUTATIVE properties

Commutative PropertyCommutative Property• One way you can remember this is when you commute you don’t move out of you community.

Commutative PropertyCommutative Property

Examples: (a + b = b + a)

7 + 5 = 5 + 7

9 x 3 = 3 x 9Note: subtraction & division DO NOT have commutative properties!

a b

b a

As you can see, when you have two lengths a and b, you get the same length whether you put a first or b first.

a

b a

bThe commutative property of multiplication says that you may multiply quantities in any order and you will get the same result. When computing the area of a rectangle it doesn’t matter which side you consider the width, you will get the same area either way.

Commutative PropertyCommutative PropertyPractice: Show the

commutative property of each number sentence.

1. 13 + 18 =

2. 42 x 77 =

3. 5 + y =

4. 7(b) =

Commutative PropertyCommutative PropertyPractice: Show the

commutative property of each number sentence.

1. 13 + 18 = 18 + 13

2. 42 x 77 = 77 x 42

3. 5 + y = y + 5

4. 7(b) = b(7) or (b)7

+ +toYou can change

You can change to

And the result will not change

Keep in mind the and do not have to be numbers.

They can be expressions that evaluate to a number.

Lets see why subtraction and division are NOT commutative.

The commutative property: a + b = b + a and a * b = b * a

7 + 3 = 3 + 7 and 7 * 3 = 3 * 7

Try this subtraction: 8 – 4 = 4 – 8 8 ÷ 4 = 4 ÷ 8and division

10 = 10 21= 21

4 ≠ -4 2 ≠ 0.5

Associative PropertyAssociative PropertyPractice: Show the associative

property of each number sentence.

• (7 + 2) + 5 = 7 + (2 + 5)

• 4 x (8 x 3) = (4 x 8) x 3

• 5 + (y + 2) = (5 + y) + 2

• 7(b x 4) = (7b) x 4 or (7 x b)4

Identity propertyIdentity propertyMultiplication:

1. 4 x 1 = 4

2. why is

Division:

1. €

2

3=

6

9

€

10 ÷1=10

Distributive PropertyDistributive Property• To DISTRIBUTEDISTRIBUTE something is give it out or share it.

• The DISTRIBUTIVEDISTRIBUTIVE property says that we can distribute a multiplier out to each number in a group to make it easier to solve

• The DISTRIBUTIVEDISTRIBUTIVE property uses MULTIPLICATIONMULTIPLICATION and ADDITIONADDITION!

Distributive PropertyDistributive Property

Examples: a(b + c) = a(b) + a(c)

2 x (3 + 4) = (2 x 3) + (2 x 4)

5(3 + 7) = 5(3) + 5(7)Note: Do you see that the 2 and the 5 were shared (distributed) with the other numbers in the group?

Distributive PropertyDistributive PropertyPractice: Show the distributive

property of each number sentence.

1. 8 x (5 + 6) =

2. 4(8 + 3) =

3. 5 x (y + 2) =

4. 7(4 + b) =

(8 x 5) + (8 x 6)

4(8) + 4(3)

(5y) + (5 x 2)

7(4) + 7b

Ella sold 37 necklaces for $20.00 each at the craft fair. She is going to donate half the money she earned to charity. Use the Commutative Property to mentally find how much money she will donate. Explain the steps you used.

Use the Associative Property to write two equivalent expressions for the perimeter of the triangle

€

6€

41

2

€

51

2

Six Friends are going to the state fair. The cost of one admission is $9.50, and the cost for one ride on the Ferris wheel is $1.50. Write two equivalent expressions and then find the total cost.

Identity and Inverse Properties

Identity Property of Addition

The Identity Property of Addition states that for any number x, x + 0 = x

5 + 0 = 5 27 + 0 = 27

4.68 + 0 = ¾ + 0 = ¾

Identity Property of Multiplication

The Identity Property of Multiplication states that for any number x, x (1) = x

Remember the number 1 can be in ANY form.

€

2

3 ⎛ ⎝

⎞ ⎠

3

3 ⎛ ⎝

⎞ ⎠=

6

9 ⎛ ⎝

⎞ ⎠=

2

3 ⎛ ⎝

⎞ ⎠

The number 1 can be in ANY form. In this case 3/3 is the same as 1.

same

Inverse Property of AdditionThe inverse property of addition states that for every number x, x + (-x) = 0

4 and -4 are considered opposites.

4 + -4 = 0

+4

-4

What number can be added to 15 so that the result will be zero?

-15

What number can be added to -22 so that the result will be zero?

22

Inverse Property of Multiplication

The Inverse Property of Multiplication states for every non-zero number n, n (1/n) = 1

The non-zero part is important or else we would be dividing by zero and we CANNOT do that.

Properties of Equality

In all of the following properties

Let a, b, and c be real numbers

Properties of EqualityAddition property:

If a = b, then a + c = b + c

Subtraction property:

If a = b, then a - c = b – c

Multiplication property:

If a = b, then ca = cb

Division property:

If a = b, then for c ≠ 0c

b

c

a

Addition PropertyThis is the property that allows you to add the same number to both sides of an equation.

STATEMENT REASON

x = y given

x + 3 = y + 3 Addition property of equality

Subtraction PropertyThis is the property that allows you to subtract the same number to both sides of an equation.

STATEMENT REASON

a = b given

a - 2 = b - 2 Subtraction property of equality

Multiplication Property

STATEMENT REASON

x = y given

3x = 3y Multiplication property of equality

This is the property that allows you to multiply the same number to both sides of an equation.

Division Property

STATEMENT REASON

x = y given

x/3 = y/3 Division property of equality

This is the property that allows you to divide the same number to both sides of an equation.

More Properties of Equality

Reflexive Property:

a = a

Symmetric Property:

If a = b, then b = a

Transitive Property:

If a = b, and b = c, then a = c

Substitution Property of Equality

If a = b, then a may be substituted for b in any equationor expression.

You have used this many times in algebra.STATEMEN

TREASON

x = 53 + x = y

givengiven

3 + 5 = y substitution property of

equality

Solving One-Step Equations

Definitions

Term: a number, variable or the product or quotient of a number and a variable.

examples:12 z 2w c

6

Terms are separated by addition (+) or subtraction (-) signs.

3a – ¾b + 7x – 4z + 52How many Terms do you see?

5

Definitions

Constant: a term that is a number.

Coefficient: the number value in front of a variable in a term.

3x – 6y + 18 = 0

What are the coefficients?

What is the constant?

3 , -6

18

Solving One-Step Equations

A one-step equation means you only have to perform 1 mathematical operation to solve it.

You can add, subtract, multiply or divide to solve a one-step equation.

The object is to have the variable by itself on one side of the equation.

Example 1: Solving an addition equation

t + 7 = 21

To eliminate the 7 add its opposite to both sides of the equation.

t + 7 = 21

t + 7 -7 = 21 - 7

t = 14

t + 0 = 21 - 7

Example 2:

Solving a subtraction equation

x – 6 = 40

To eliminate the 6 add its opposite to both sides of the equation.

x – 6 = 40x – 6 + 6 = 40 + 6

x = 46

Example 3:

Solving a multiplication equation

8n = 32

To eliminate the 8 divide both sides of the equation by 8. Here we “undo” multiplication by doing the opposite – division.

8n = 32 8 8

n = 4

Example 4:

Solving a division equation

To eliminate the 9 multiply both sides of the equation by 9. Here we “undo” division by doing the opposite – multiplication.€

x

9=11

€

x

9=11

€

9( )x

9= (11)(9)

€

€

x = 99

€

Identify operations

Undo operations

Balance equation

Repeat steps

Solve for variable

Check solution

Identify Operations

€

x

2− 3 = 8

Fraction bar means division

Minus sign means subtraction

Use Opposite Operations or “undo” Operations

Addition is opposite of subtraction (addition undoes subtraction)

Subtraction is opposite of addition (subtraction undoes addition)

Multiplication is opposite of division (multiplication undoes division)

Division is opposite of multiplication (division undoes multiplication)

Keep Equation Balanced

What ever you do to one side of the equation you do to the other side of the equation.

Repeat these steps until the equation is solved. 1-step equations

2-step equations

7x + 15 = 85

7x +15 – 15 = 85 - 15

7x = 707 7

x = 10

Example:

€

2

3x − 6 = 28

€

2

3x − 6 + 6 = 28 + 6

€

2

3x = 28

€

3

2 ⎛ ⎝

⎞ ⎠2

3x = 28

3

2 ⎛ ⎝

⎞ ⎠

€

x = 42

Example:

When graphing the solution to a linear equation with one-variable on a number line you would put a dot (point) on the answer.

x – 3 = -7x – 3 + 3 = -7 + 3 x = -4

Graphing a Linear Equation

6 th grade big idea 3 teacher quality grant. big idea 3: write, interpret, and use mathematical...

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