6th Grade Big Idea 3 Teacher Quality Grant.

6th Grade Big Idea 3 Teacher Quality Grant

Big Idea 3: Write, interpret, and use mathematical expressions and equations.
MA.6.A.3.1: Write and evaluate mathematical expressions that correspond to given situations. MA.6.A.3.2: Write, solve, and graph one- and two- step linear equations and inequalities. MA.6.A.3.5 Apply the Commutative, Associative, and Distributive Properties to show that two expressions are equivalent. MA.6.A.3.6 Construct and analyze tables, graphs, and equations to describe linear functions and other simple relations using both common language and algebraic notation.

Big idea 3: assessed with Benchmarks
Assessed with means the benchmark is present on the FCAT, but it will not be assessed in isolation and will follow the content limits of the benchmark it is assessed with. MA.6.A.3.3 Work backward with two-step function rules to undo expressions. (Assessed with MA.6.A.3.1.) MA.6.A.3.4 Solve problems given a formula. (Assessed with MA.6.A.3.2, MA.6.G.4.1, MA.6.G.4.2, and MA.6.G.4.3.)

Big idea 3: Benchmark Item Specifications

Big Idea 3: Prerequisite knowledge
Order of Operations Fractions and ratios Decimals Percent

Big idea 3: Variable video

Writing Algebraic Expressions
Be able to write an algebraic expression for a word phrase or write a 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:
Addition Subtraction Multiplication Division Have the students fill in this chart first. Then go through the next slides to see what needs to be added.

Addition Phrases: More than Increase by Greater than Add Total Plus
Sum Ask the participants to add any additional phrases they feel would mean addition.

Subtraction Phrases: Decreased by Difference between Take Away Less
Less than* Subtract from* Ask the participants to add any additional phrases they feel would mean subtraction.

Multiplication Phrases:
Product Times Multiply Of Twice or double Triple Ask the participants to add any additional phrases they feel would mean multiplication.

Division Phrases: Quotient Divide Divided by Split equally
Ask the participants to add any additional phrases they feel would mean division.

Notes Multiplication expressions should be written in side-by-side form, with the number always in front of the variable. 3a t c f Introduce the vocabulary word coefficient, meaning the number in front of the variable in a multiplication problem.

Notes Division expressions should be written using the fraction bar instead of the traditional division sign. Discuss the other symbols that are used for division.

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 Utilizing models for this is a great bridge from using counters to using algebra tiles. This is the first example in the explore lesson from chapter 6. 10 +4 10 books 4 books

Singapore math introduction
Level 1 Level 2 Level 3 Enrichment Links for other strategies These links lead to power presentations on creating models to solve word problems. The last link includes some other strategies for solving word problems.

The next 2 slides continue with the explore lesson and practice how to represent the verbal expressions as models.

Examples Addition phrases: 3 more than x the sum of 10 and a number c
a number n increased by 4.5 Anytime you give examples ask the participants to come up with some on their own.

Examples Subtraction phrases: a number t decreased by 4
the difference between 10 and a number y 6 less than a number z Anytime you give examples ask the participants to come up with some on their own.

Examples Multiplication phrases: the product of 3 and a number t
twice the number x 4.2 times a number e Anytime you give examples ask the participants to come up with some on their own.

Examples Division phrases: the number y divided by 2 2.5 divide g
the quotient of 25 and a number b the number y divided by 2 2.5 divide g Anytime you give examples ask the participants to come up with some on their own.

Example games Snow man game Millionaire game
These two sites provide more examples in the format of games for students. Play some of the games in the workshop.

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 When working with word problems it is often easier to assign variables that relate back to the word problem. If you were trying to find how many dollars you could use d as the variable. 10w

Examples i 12 5x 3y 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 Ask the participants to explain when they would know to multiply or divide. 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 Use these challenge problems for activities depending on your time choose the ones you want. Put the teachers in groups and have them discuss not only the problems but how they can use them in class for higher order thinking.

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.
They must choose problems that engage students. 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 Procedural questions are those typical types of questions that we see on so many different tests. Procedural questions ask students to select and apply a specific operation or procedure to solve the question.

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? The problem requires that the student select and apply the correct operation, division, and compute the answer. The content area for this problem is number operations and number sense.

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 Sometimes these questions are in the format of a “set-up” question. They require that students recognize and manipulate math concepts. This high-order type of question requires that students have an understanding of relationships among numbers, as well as concepts and principles.

Sample Conceptual Test Question
A 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 = 336 (2) 3n = 336 (3) n = 336 (4) 3n = 336 (5) 3n + 3 = 336 ” This sample question requires that students use algebraic thinking skills in order to “set-up” the correct equation to solve the problem. Notice that this question does not require that students solve the problem.

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 This question type uses real-world scenarios where students use their problem-solving skills to solve the problem.

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? 1:45 p.m. 2:15 p.m. 2:30 p.m. 3:15 p.m. 5:15 p.m. This is an example of a sample test question that assesses basic problem solving skills. The question requires that the student identify the problem, select the information needed, the strategy to use, and then solve the problem.

What is Problem Solving?
Problem Solving and Mathematical Reasoning What is Problem Solving? 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 For more detail on this article by Michael E. Martinez as published in Phi Delta Kappan on April 1998, please refer to Chapter 7, Problem Solving and Mathematical Reasoning.

What steps should we take when solving a word problem?
1. Understand the problem 2. Devise a plan 3. Carry out the plan. George Polya has had an important influence on problem solving in mathematics education. He noted that good problem solvers tend to forget the details and focus on the structure of the problem, while poor problem solvers do the opposite. 4. Look back

An Effective Problem Solver
Problem Solving and Mathematical Reasoning An Effective Problem Solver 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 We have all been faced with problems, whether in our personal lives or in education. In fact, being an effective math problem solver is similar to being an effective problem solver in real-life situations. In real-life situations, we generally explore the problem with which we are faced by identifying what it is and carefully defining what it will take to solve it. In fact, we often access knowledge based on similar situations we have faced. Sometimes, we get overwhelmed with all of the extra things that occur, but we try to persevere and set up a solution that we can try. Sometimes we use our kinesthetic/tactile skills to picture the solution. If our problem-solving efforts do not work, we check things over and try again. This is very similar to the problem-solving process that we encourage students to use – discovery and exploration. [Note: Review the different ideas that support being an effective problem solver. You may wish to share a story or an event that shows each of the effective problem-solving skills. Many of the current investigative television series use effective problem-solving strategies. You may wish to correlate this same type of process to the discovery and exploration of a mathematical problem.]

K W E S What 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?

UNDERSTAND THE PROBLEM
Ask yourself…. •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? Ask the participants if there are other questions they should ask themselves about the problem.

George has written a number pattern that begins with 1, 3, 6, 10, 15
George has written a number pattern that begins with 1, 3, 6, 10, 15. If he continues this pattern, what are the next four numbers in his pattern? K W E S What 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? Pattern: 1, 3, 6, 10, 15, … What are the next 4 numbers? 1 1+2=3 3+3=6 6+4=10 10+5=15 The amount being added increases by 1 each time so: 15+6=21 21+7=28 28+8=36 36+9=45

Karen has 3 green chips, 4 blue chips and 1 red chip in her bag of chips. What fractional part of the bag of chips is green? K W E S What 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? Number of chips: 3 green 4 blue 1 red 8 total chips What fraction of the total chips is green?

Improving Problem-Solving Skills
Problem Solving and Mathematical Reasoning Improving Problem-Solving Skills 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 How can you assist your learners to be better problem solvers? There are many different techniques available to assist students in becoming better problem solvers. Jennifer Cromley’s work on Learning to Think, Learning to Learn (Cromley, J. (2000). Learning to Think, Learning to Learn: What the Science of Thinking and Learning Has to Offer Adult Education. Washington, DC: National Institute for Literacy) provides good information about improving problem-solving skills. She has completed work on relating research to practice. Ideas in her publications include: Some problem-solving strategies use lots of working memory – such as looking at the question and finding a formula that includes a variable. Have instructors consider giving questions with open-ended answers; solving problems involves using mental models. Cromley also supports that active learning is more effective than lectures. Good problem solvers have more and better developed mental models than poor problem solvers. If understanding depends on mental models, then students must actively engage in learning, since it is their mental models that lead to understanding, not the understanding of the teacher. Teachers need to demonstrate or model for students the process of solving a problem in a particular area. The best way for students to learn to think is to watch teachers solve problems out loud and explain their thinking process, practice their thinking process, and receiving feedback on it.

LOOK BACK 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. Stress it is always important to check your answer. Does the answer make sense? Is it mathematically correct? Does it answer the question?

Types of Graphic Organizers
Hierarchical diagramming Sequence charts Compare and contrast charts

A Simple Hierarchical Graphic Organizer

A Simple Hierarchical Graphic Organizer - example
Geometry Algebra MATH Have the participants make up another simple organizer the topic Mathematical Operations. Trigonometry Calculus

Compare and Contrast Category What is it? Illustration/Example
Properties/Attributes Subcategory Irregular set What are some examples? What is it like?

Compare and Contrast - example
Numbers What is it? Illustration/Example Properties/Attributes 6, 17, 25, 100 Positive Integers Whole Numbers -3, -8, -4000 Negative Integers Zero Fractions What are some examples? What is it like?

Venn Diagram

Venn Diagram - example Prime Numbers 5 7 11 13 2 3 Even Numbers 4 6
5 7 Even Numbers 8 10 Multiples of 3 3 2 6 Have the participants make up a Venn Diagram using divisibility rules.

Multiple Meanings

Multiple Meanings – example
Right Equiangular 3 sides 3 angles 1 angle = 90° 3 sides 3 angles 3 angles = 60° TRI- ANGLES Acute Obtuse 3 sides 3 angles 3 angles < 90° 3 sides 3 angles 1 angle > 90°

Series of Definitions Word = Category + Attribute = +
= Definitions: ______________________ ________________________________ Example 4 can be considered a sequential organizer showing the process of defining a word by combining the category and attribute.

Series of Definitions – example
Word = Category + Attribute = + Definition: A four-sided figure with four equal sides and four right angles. 4 equal sides & 4 equal angles (90°) Square Quadrilateral Example 4 can be considered a sequential organizer showing the process of defining a word by combining the category and attribute.

Four-Square Graphic Organizer
1. Word: 2. Example: 4. Definition 3. Non-example: These could be redone as some type of foldable.

Four-Square Graphic Organizer – example
1. Word: semicircle 2. Example: 4. Definition 3. Non-example: A semicircle is half of a circle.

Matching Activity Divide into groups
Match the problem sets with the appropriate graphic organizer Divide the participants into groups. Have each group decide which graphic organizer would be best to use with the given information. After have each group present their organizer to the entire class. Discussions will follow as groups may come up with different results. After you can walk them through the answers given here.

Matching Activity Which graphic organizer would be most suitable for showing these relationships? Why did you choose this type? Are there alternative choices?

Problem Set 1 Parallelogram Rhombus Square Quadrilateral Polygon Kite
Irregular polygon Trapezoid Isosceles Trapezoid Rectangle

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

Problem Set 3 a ÷ b Addition Multiplication a + b a times b
a plus b a x b sum of a and b a(b) ab Subtraction Division a – b a/b a minus b a divided by b a less b a ÷ b

Problem Set 4 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.

Possible Solution to PS #1
POLYGON Square, rectangle, rhombus Parallelogram: has 2 pairs of parallel sides Quadrilateral Trapezoid, isosceles trapezoid Trapezoid: has 1 set of parallel sides Kite Kite: has 0 sets of parallel sides Kite Irregular: 4 sides w/irregular shape

REAL NUMBERS Irrational Numbers Rational Numbers Integers Counting Numbers Whole Numbers

Possible Solution PS #3 Addition Subtraction Operations Multiplication
____a + b____ ___a plus b___ Sum of a and b ____a - b_____ __a minus b___ ___a less b____ Operations Multiplication Division ___a times b___ ____a x b_____ _____a(b)_____ _____ab______ ____a / b_____ _a divided by b_ _____a  b_____

Mathematics Geometric Figures Numbers Operations Rules Symbols Rational Addition Postulate m║n Triangle Prime Subtraction √4 Corollary Hexagon Integer Multiplication Irrational Division Quadrilateral Whole Composite {1,2,3…}

Mike, Juliana, Diane, and Dakota are entered in a 4-person relay race
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. Math Connects Plus 6 6-1 Pg #4

Try solving this problem by working backwards.
Mrs. Stevens earns $18.00 an hour at her job. She had $ after paying $9.00 for subway fare. Find how many hours Mrs. Stevens worked. Try solving this problem by working backwards. Math Connects Plus 6 7-3 Pg. 398 #7

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 What is the number? Math Connects Plus 7 Pg #5

Big Idea 3: Patterns and Equations
Analyzing patterns and sequences (lesson ENLVM) The ENLVM lesson is great for students to investigate patterns and sequences involving shapes and numbers. Go through the lesson and have teachers come up with a unique way of implementing the lesson.

Properties of Addition & Multiplication

Why do we need rules or properties in math?
Lets see what can happen if we didn’t have rules.

Before We Begin… What is a VARIABLE?
A variable is an unknown amount in a number sentence represented as a letter: 5 + n x (g) t + d = s

Before We Begin… What do these symbols mean?
( ) = multiply: 6(a) or group: (6 + a) * = multiply · = multiply ÷ = divide / = divide

Algebra tiles and counters
Represent the following expressions with algebra tiles or counters: and 4 + 3 3 - 4 and 4 – 3 and Discuss if it is the same result as they make each of the pairs and why? This is again a good way to keep using manipulative and make the students tell you if these are the same.

Algebra tiles and counters
Represent the following expressions with algebra tiles or counters: 9x+ 2 and 2 + 9x 9x - 2 and 2 – 9x Discuss if it is the same result as they make each of the pairs and why?

Commutative Property To COMMUTE something is to change it
The COMMUTATIVE property says that the order of numbers in a number sentence can be changed Addition & multiplication have COMMUTATIVE properties

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

Commutative Property 7 + 5 = 5 + 7 9 x 3 = 3 x 9
Examples: (a + b = b + a) 7 + 5 = 5 + 7 9 x 3 = 3 x 9 Note: subtraction & division DO NOT have commutative properties!

As you can see, when you have two lengths
b 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. b a

b a a b The 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 Property Practice: Show the commutative property of each number sentence. = 42 x 77 = 5 + y = 7(b) =

Commutative Property Practice: Show the commutative property of each number sentence. = 42 x 77 = 77 x 42 5 + y = y + 5 7(b) = b(7) or (b)7

+ + to   to And the result will not change You can 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
10 = = 21 Try this subtraction: 8 – 4 = 4 – ÷ 4 = 4 ÷ 8 and division 4 ≠ ≠ 0.5

Associative Property Practice: 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 property Multiplication: 4 x 1 = 4 why is Division:

Distributive Property
To DISTRIBUTE something is give it out or share it. The DISTRIBUTIVE property says that we can distribute a multiplier out to each number in a group to make it easier to solve The DISTRIBUTIVE property uses MULTIPLICATION and ADDITION!

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? As you do this have teachers and or students build with algebra tiles and counters as we did before. First show 2 times 3 with counters then try the 2 times (3+4). (use different colors for the 3 and 4 so that then you can lead in to 2 times (3x+4)

Practice: Show the distributive property of each number sentence. 8 x (5 + 6) = 4(8 + 3) = 5 x (y + 2) = 7(4 + b) = (8 x 5) + (8 x 6) 4(8) + 4(3) (5y) + (5 x 2) 7(4) + 7b

Practice: Show the distributive property of each number sentence. 8 x (5 + 6) = (8 x 5) + (8 x 6) 4(8 + 3) = 4(8) + 4(3) 5 x (y + 2) = (5y) + (5 x 2) 7(4 + b) = 7(4) + 7b

Practice: Show the distributive property of each number sentence. 4(3x+2) = 3(6x+2y) = = Use the algebra tiles and it is important to discuss the last 2 and if there are differences.

Ella sold 37 necklaces for $20. 00 each at the craft fair
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. Math Connects Plus 6 Pg #22

Use the Associative Property to write two equivalent expressions for the perimeter of the triangle
Math Connects Plus 6 Pg #19

Six Friends are going to the state fair
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 $ Write two equivalent expressions and then find the total cost. Math Connects Plus6 Pg #4

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 Reinforce the idea of – Identity….Identical answer = ¾ + 0 = ¾

Identity Property of Multiplication
The Identity Property of Multiplication states that for any number x, x (1) = x Important at this point to stress that the number 1 can be in any form. (ex. Fraction 3/3) Remember the number 1 can be in ANY form.

The number 1 can be in ANY form. In this case 3/3 is the same as 1.
When we simplify the fraction 6/9 we get 2/3 the number we started with. We multiplied by 3/3 which when simplified is 1. same

Inverse Property of Addition
The inverse property of addition states that for every number x, x + (-x) = 0 4 and -4 are considered opposites. Have participants give examples of other other opposites.

= 0 -4 +4 Have participants use a number line to show how adding opposites will always result in zero.

-15 22 What number can be added to 15 so that the result will be zero?

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. Review the concept of reciprocal. Have participants understand that a number multiplied by its reciprocal is 1. Zero does not have a reciprocal which is a reason we cannot divide by zero.

Properties of Equality
In all of the following properties Let a, b, and c be real numbers

Properties of Equality
Addition 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 ≠ 0 Reinforce the idea that to keep an equation balanced what ever you did to one side of the equation you must do to the other side.

Addition property of equality
This 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 This property is applied when solving equations. For example, if you have x – 3 = 7, it can be solved by adding 3 to both sides of the equation: X – 3 = 7 X – = 7 + 3 X + 0 = 10 X = 10

Subtraction property of equality
This 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 This property is applied when solving equations. For example, if you have x + 4 = 6, it can be solved by subtracting 4 from both sides of the equation. This is the same as the Addition property if you consider adding a negative 4 to each side.

Multiplication Property
This is the property that allows you to multiply the same number to both sides of an equation. STATEMENT REASON x = y given 3x = 3y Multiplication property of equality This property is applied when solving equations. For example, if you have x/2 = 6, it can be solved by multiplying both sides of the equation by 2.

Division property of equality
This is the property that allows you to divide the same number to both sides of an equation. STATEMENT REASON x = y given x/3 = y/3 Division property of equality This property is applied when solving equations. For example, if you have 2x= 6, it can be solved by dividing both sides of the equation by 2.

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 equation or expression. You have used this many times in algebra. STATEMENT REASON x = 5 3 + x = y given 3 + 5 = y substitution property of equality Explain since x equals 5 the x can be replaced by 5.

Solving One-Step Equations

Definitions Term: a number, variable or the product or quotient of a number and a variable. examples: z w c 6

Terms are separated by addition (+) or subtraction (-) signs.
3a – ¾b + 7x – 4z + 52 How 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 is the constant? 3 , -6 18
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: x + 7 = -3 We want the x by itself.

Example 1: Solving an addition equation
To eliminate the 7 add its opposite to both sides of the equation. t = Adding the opposite to both sides keeps the equation balanced. t + (-7) is the same as t - 7 By adding the opposite the t + 7 becomes t + 0 which becomes t t + 0 = t = 14

Solving a subtraction equation x – 6 = 40
Example 2: Solving a subtraction equation x – 6 = 40 To eliminate the 6 add its opposite to both sides of the equation. x – 6 is the same as x + (-6) and the opposite of -6 is 6. Again reinforce that equals 0 and that x + 0 is x. This shows why we do this and how we are using the properties. x – = x = 46

Solving a multiplication equation 8n = 32
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. Dividing the 8n by 8 results in 1n and (1)(n) is n. 8n = 32 n = 4

Solving a division equation
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. Show: (9/1)(x/9)= 9x/9 9/9 = 1 and 1x = x

How do you solve Two-step Equations
Identify operations Undo operations Balance equation Repeat steps Solve for variable Check solution

Identify Operations Minus sign means subtraction
Fraction bar means division

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) Talk about “undo” What is the “undo” of taking 2 steps forward? Go through more examples.

Keep Equation Balanced
Stress this. And use the power points linked on the next slide to do this. What ever you do to one side of the equation you do to the other side of the equation.

Repeat repeat repeat Repeat these steps until the equation is solved.
1-step equations 2-step equations Use the lesson plan included with Intro to expressions and equations (Lesson).doc

Example: 7x + 15 = 85 7x +15 – 15 = 85 - 15 7x = 70 7 7 x = 10
Start with the constant. Want to move it to the other side of the equation so do the opposite. Subtract 15 from both sides. 15 – 15 = x + 0 = 7x x = 10

Example: Walk the participants carefully through the steps. ⅔x divided by ⅔ is the same as multiply by the reciprocal. A number multiplied by its reciprocal is equal to 1. 1(x) = x. As you go through the steps always state the properties you are using. (Ex. Multiplicative Identity)

Graphing a Linear Equation
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 = -7 x – = x = -4

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