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# Algebra

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## Casey Hughes

on 24 April 2013

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#### Transcript of Algebra

Elementary School NCTM Standards describe, extend, and make generalizations about geometric and numeric patterns
represent and analyze patterns and functions, using words, tables, and graphs Middle School NCTM Standards represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules
relate and compare different forms of representation for a relationship
identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations. Understand patterns, relations, and functions Represent and analyze mathematical situations and structures using algebraic symbols develop an initial conceptual understanding of different uses of variables
explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope
use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships
recognize and generate equivalent forms for simple algebraic expressions and solve linear equations Use mathematical models to represent and understand quantitative relationships model and solve contextualized problems using various representations, such as graphs, tables, and equations. Analyze change in various contexts use graphs to analyze the nature of changes in quantities in linear relationships. Understand patterns, relations, and functions Represent and analyze mathematical situations and structures using algebraic symbols identify such properties as commutativity, associativity, and distributivity and use them to compute with whole numbers
represent the idea of a variable as an unknown quantity using a letter or a symbol
express mathematical relationships using equations Use mathematical models to represent and understand quantitative relationships model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions Analyze change in various contexts investigate how a change in one variable relates to a change in a second variable
identify and describe situations with constant or varying rates of change and compare them SC Standards Grade 3 Standard 3-3: Through the process standards students will demonstrate an understanding of numeric patterns, the use of symbols to represent an unknown quantity, and increasing
change over time.

3-3.1 Create numeric (whole number operations) patterns.
3-3.2 Apply procedures to find missing numbers in a given numeric (whole number operations) pattern.
3-3.3 Use symbols to represent an unknown quantity in a simple addition, subtraction, or multiplication equation.
3-3.4 Illustrate situations that show change over time (increasing). SC Standards Grade 4 Standard 4-3: Through the process standards students will demonstrate an understanding of numeric and nonnumeric patterns, representing simple mathematical relationships, and applying procedures to find the value of an unknown.
4-3.1 Analyze numeric (all operations), nonnumeric, and repeating patterns (including decimal patterns through hundredths)
4-3.2 Generalize a rule for numeric (all operations), nonnumeric, and repeating patterns.
4-3.3 Use a rule to complete a sequence or a table.
4-3.4 Translate among, letters, symbols, and words to represent quantities in a simple mathematical expression or equation.
4-3.5 Apply procedures to find the value of an unknown letter or symbol in a whole number equation.
4-3.6 Illustrate situations that show change over time as increasing, decreasing, and varying. SC Standards Grade 5 Standard 5-3: Through the process standards students will demonstrate an understanding of the use of patterns, relations, functions, models, structures, and algebraic symbols to represent quantitative relationships and analyze change in various contexts.
5-3.1 Represent numeric, algebraic, and geometric patterns in words, symbols, algebraic expressions, or algebraic equations. 5-3.2 Analyze patterns and functions with words, tables, and graphs. 5-3.3 Match tables, graphs, expressions, equations, and or verbal descriptions of the same problem situation. 5-3.4 Identify applications of commutative, associative, and distributive properties with whole numbers. 5-3.5 Analyze situations that show change over time. SC Standards Grade 6 Standard 6-3: Through the process standards students will demonstrate an understanding of writing, interpreting and using mathematical expressions, equations, and inequalities.

6-3.1 Analyze numeric and algebraic patterns and pattern relationships.
6-3.2 Apply order of operations to simplify whole number expressions.
6-3.3 Represent algebraic relationships with variables in expressions, simple equations, and simple inequalities.
6-3.4 Use the commutative, associative, and distributive properties to show that two expressions are equivalent.
6-3.5 Use inverse operations to solve one-step equations (whole number coefficients and solutions). SC Standards Grade 7 Standard 7-3: Through the process standards students will demonstrate an understanding of proportional relationships.
7-3.1 Analyze geometric patterns and pattern relationships. 7-3.2 Analyze tables and graphs to describe the rate of change between quantities. 7-3.3 Understand slope as a constant rate of change. 7-3.4 Use inverse operations to solve two-step equations and two-step inequalities. 7-3.5 Represent on a number line the solution of a two-step inequality. 7-3.6 Represent proportional relationships with graphs, tables, and equations. 7-3.7 Classify relationships as directly proportional, inversely proportional, or non-proportional. SC Standards Grade 8 Standard 8-3: Through the process standards students will demonstrate an understanding of equations, inequalities, and linear functions.
CCSS.Math.Content3.OA.A.2 Interpret whole-number quotients of whole numbers
CCSS.Math.Content.3.OA.A.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities
CCSS.Math.Content.3.OA.A.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers Understand properties of multiplication and the relationship between multiplication and division CCSS.Math.Content.3.OA.B.5 Apply properties of operations as strategies to multiply and divide
CCSS.Math.Content.3.OA.B.6 Understand division as an unknown-factor problem Multiply and divide within 100 CCSS.Math.Content.3.OA.C.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division Solve Problems involving the four operations, and identify and explain patterns in arithmetic CCSS.Math.Content.3.OA.C.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
CCSS.Math.Content.3.OA.C.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table). Use the four operations with whole numbers to solve problems CCSS.Math.Content.4.OA.A.1 Interpret a multiplication equation as a comparison.
CCSS.Math.Content.4.OA.A.2 Multiply or divide to solve word problems involving multiplicative comparison.
CCSS.Math.Content.4.OA.A.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding Gain familiarity with factors and multiples CCSS.Math.Content.4.OA.B.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite. Generate and analyze patterns CCSS.Math.Content.4.OA.C.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. Write and interpret numerical expressions CCSS.Math.Content.5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
CCSS.Math.Content.5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. Analyze patterns and relationships CCSS.Math.Content.5.OA.B.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. Activity 1: Two of Everything Materials:
Two of Everything story
"Magic Pot"
2 Sponges
6 Apples
2 "babies"
2 "men"
2 "women"
Index cards for students Standards Addressed SC State:
Standard 3-3: The student will demonstrate through the mathematical processes an understanding of numeric patterns, symbols as representations of unknown quantity, and situations showing increase over time.
3-3.1 Create numeric patterns that involve whole-number operations
3-3.2 Apply procedures to find missing numbers in numeric patterns that involve whole number operations. Common Core:
Solve problems involving the four operations, and identify and explain patterns in arithmetic
•CCSS.Math.Content.3.OA.D.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding
•CCSS.Math.Content.3.OA.D.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations NCTM:
Understand patterns, relations, and functions
•Describe, extend, and make generalizations about geometric and numeric patterns As story is read, students will recognize the pattern that relates the relationship of things going into the pot with the things coming out. Then the class will do an activity together to determine a new pattern. Common Core Standards Grade 6 Common Core Standards Grade 8 Apply and extend previous understandings of arithmetic to algebraic expressions. CCSS.Math.Content.6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents.CCSS.Math.Content.6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers.CCSS.Math.Content.6.EE.A.2a Write expressions that record operations with numbers and with letters standing for numbers.
CCSS.Math.Content.6.EE.A.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.
CCSS.Math.Content.6.EE.A.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).
CCSS.Math.Content.6.EE.A.3 Apply the properties of operations to generate equivalent expressions.
CCSS.Math.Content.6.EE.A.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). Reason about and solve one-variable equations and inequalities CCSS.Math.Content.6.EE.B.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.
CCSS.Math.Content.6.EE.B.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.
CCSS.Math.Content.6.EE.B.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
CCSS.Math.Content.6.EE.B.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. Represent and analyze quantitative relationships between
dependent and independent variables CCSS.Math.Content.6.EE.C.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. Common Core Standards Grade 7 Use properties of operations to generate equations CCSS.Math.Content.7.EE.A.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
CCSS.Math.Content.7.EE.A.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related Solve real-life and mathematical problems using numerical and algebraic expressions and equations CCSS.Math.Content.7.EE.B.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.
CCSS.Math.Content.7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities
CCSS.Math.Content.7.EE.B.4a Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.
CCSS.Math.Content.7.EE.B.4b Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. Expressions and Equations Work with radicals and integer exponents CCSS.Math.Content.8.EE.A.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions.
CCSS.Math.Content.8.EE.A.2 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
CCSS.Math.Content.8.EE.A.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.
CCSS.Math.Content.8.EE.A.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology Understand the connections between proportional relationships, lines, and linear equations CCSS.Math.Content.8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
CCSS.Math.Content.8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Analyze and solve linear equations and pairs of simultaneous linear equations CCSS.Math.Content.8.EE.C.7 Solve linear equations in one variable.
CCSS.Math.Content.8.EE.C.7a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
CCSS.Math.Content.8.EE.C.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
CCSS.Math.Content.8.EE.C.8 Analyze and solve pairs of simultaneous linear equations.
CCSS.Math.Content.8.EE.C.8a Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
CCSS.Math.Content.8.EE.C.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
CCSS.Math.Content.8.EE.C.8c Solve real-world and mathematical problems leading to two linear equations in two variables. Define, evaluate, and compare functions CCSS.Math.Content.8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1
CCSS.Math.Content.8.F.A.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
CCSS.Math.Content.8.F.A.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear Use functions to model relationships between quantities CCSS.Math.Content.8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
CCSS.Math.Content.8.F.B.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally Activity 2: Integers in between Materials
Deck of playing cards
Score card for each student Procedure

First, the class is divided into groups, and each group designates a dealer. Each student is dealt 3 cards, with one facing down. Now, each student must guess whether the face down card is between the other two cards.
If they are correct, they keep the cards. If they are incorrect, cards are returned to the dealer. There is a new dealer for each round until the deck is gone. This is when the game is over. Standards Addressed SC State:
Standard 4-3: The student will demonstrate through the mathematical processes an understanding of numeric and nonnumeric patterns, the representation of simple mathematical relationships, and the application of procedures to find the value of the unknown.
Indicator 4-3.1 Analyze numeric, nonnumeric, and repeating patterns involving all operations and decimal patterns through hundredths.
Indicator 4-3.4 Translate among letters, symbols and words to represent quantities in simple mathematical expressions or equations NCTM Elementary Standards: Understand patterns, relations, and functions Common Core:
Use the four operations with whole numbers to solve problems.CCSS.Math.Content.4.OA.A.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. South Carolina Standards Students build upon pattern knowledge that was essential in 2nd grade. Now they will be advancing to patterns with whole numbers (not just 1-10) using simple addition, subtraction, and multiplication.
Procedures will be applied (using addition, subtraction, and multiplication) to find the missing number in a given pattern. This missing number could be within the pattern or located at the end.
Students will also begin to use symbols to represent an unknown number in a 3-number problem. They will understand that the unknown amount could be in the place of any of the three numbers in a calculation.
They will become familiar with situations that increase over time. They will also recognize times of no change. Common Core Standards Students learn to use representations (including grouping), properties, missing variables, and practice to become familiar with multiplication and division.
All of the practices are limited to whole numbers up to 100. NCTM Standards Summary This requires a variety of practices utilized in the elementary period.
Students become more familiar with patterns through practice and learn to find an unknown number.
They understand the relations between whole number patterns and functions.
Includes utilization of the process standard, representation to analyze methods of multiplication and division including diagrams, tables, charts, and math models. South Carolina Standards Patterns knowledge from 3rd grade expanded to include non-numeric patterns
Pattern recognition taken a step further to rules.
Identifying rules of major importance
number operations now include decimals up to the hundredths place.
Students continue to use symbols, but add in letters and words to represent mathematical values (Ties in process standard)
In addition to situations that increase with time, students will become familiar with situations that decrease, or vary over time. Common Core Standards Common Core Focuses around the four operations, and students abilities to manipulate them in order to represent the different properties associated with the operations.
The ability of students to solve multi-step word problems.
The ability to write expressions with letters representing missing numbers. (Once again, ties in process standard)
Students have to gain familiarity with concepts of factors and multiples
Move past recognizing patterns into generating and analyzing them. 5th Grade Summary South Carolina Standards Introduction of functions, algebraic symbols, models, and relations
Specifically, how they can represent quantitative data (Process Standard)
Writing equations and functions using algebraic symbols
Analyzing said functions and expressions with visual aids like charts tables and graphs
Be able to match correct function to correct graph
Finally learn the different properties of of operations (3rd grade common core)
Expand knowledge of situations showing change over time from recognition to analysis. Common Core Common Core all about Patterns Relationships and Expressions
Order of operations (parentheses!)
Patterns! Again! Do you see one?
Relationships between patterns ( what do these numbers have in common?)
5x7=5x(4+3) <---understand this without having to do it. Sixth Grade Summary South Carolina Standards
SC Sixth grade standards consists of kids showing knowledge of math expressions, equations, and inequalities
Analyze numeric and algebraic patterns
Use order of operations to simplify whole number expressions.
Use variables in expressions, simple equations, and simple inequalities to represent relationships.
Use the commutative, associative, and distributive properties to show that two expressions are equivalent.
Use inverse operations to solve one-step equations. Common Core Standards Apply and extend previous understandings of arithmetic to algebraic expressions.
Reason about and solve one-variable equations and inequalities
Represent and analyze quantitative relationships between dependent and independent variables Seventh Grade Summary South Carolina Standards The South Carolina standards are about using the process standards to demonstrate an understanding of proportional relationships.Represent and discuss geometric patterns and pattern relationships
Use and interpret tables and graphs to describe the rate of change between quantities
Know slope as a constant rate of change.
Use inverse operations to solve two-step equations and two-step inequalities.
Use a number line to show the solution of a two-step inequality.
Represent proportional relationships with graphs, tables, and equations.
Classify relationships as directly, inversely, or non-proportional. Common Core Standards Use properties of operations to generate equations
Solve real-life and mathematical problems using numerical and algebraic expressions and equations Activity 3: Predicting Future Stats Materials
pencil
paper Procedure
Students will be split into three groups. Each group will be assigned a sport with a data set. Members of the group will draw a graph on given paper. The members of the group will then work together to create a physical model with their bodies Standards Addressed Summary of NCTM South Carolina:
7-3.2 Analyze tables and graphs to describe the rate of change between quantities.
7-3.6 Represent proportional relationships with graphs, tables, and equations. Common Core:
CCSS.Math.Content.7.EE.B.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. Wants students to be able to analyze information given from equations, graphs and tables and be able to compare the information
Be able to develop a form of understanding to manipulate variables
Be able to use the algebraic symbols to represent situations (Be able to write out a word problem)
Use mathematical models to represent information from equations
Analyze changes in graphs and tables and nature of change in linear relationships NCTM:
Understand patterns, relations, and functions
Use mathematical models to represent and understand quantitative relationships
Analyze change in various contexts The students are now building upon their knowledge of equations, inequalities, and linear functions
Learning how to set up and read linear equations (y = mx + b) 1st time being introduced to this
The problems now have multiple steps, where previously they had only one
Learning how to use properties to manipulate problems to solve them
Are now finding slopes of the lines and x and y intercepts SC State Standards Common Core Standards cover linear equations more in depth
Not just solving linear equations, but also using manipulatives, like the triangle, in order to prove the slope of lines
Solving simultaneous linear equations and seeing if they cross given coordinates
Study relationships of two linear equations through points of intersection Algebra
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