Algebra Internet Sites for further study
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ASE MA 1: Algebraic Concepts and Expressions |
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MA.1.1 Number and Quantity: The Real Number System and Quantities. |
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Objectives |
What Learner Should Know, Understand, and Be Able to Do |
Teaching Notes and Examples |
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MA.1.1.1 Rewrite expressions involving radicals and rational exponents using the properties of exponents. For example: The expression
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Convert from radical representation to using rational exponents and vise versa.
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Examples https://www.illustrativemathematics.org/HSN-RN
Rewrite Expressions Involving Radicals and Rational Exponents
Radical to Rational Expressions |
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MA.1.1.2 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
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Interpret units in the context of the problem. When solving a multi-step problem, use units to evaluate the appropriateness of the solution. Choose the appropriate units for a specific formula and interpret the meaning of the unit in that context. Choose and interpret both the scale and the origin in graphs and data displays |
For example, Speed = Distance/Time. That is why the units of measurement of speed are in miles (distance) / (per) time (hour)
Use Units as a Way to Understand Problems https://learnzillion.com/lessonsets/397-use-units-as-a-way-to-understand-and-solve-problems
https://www.virtualnerd.com/common-core/hsn-number-quantity/HSN-Q-quantities/A/1 |
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MA.1.1.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. |
Determine the accuracy of values based on their limitations in the context of the situation. |
Choosing Appropriate Accuracy |
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MA.1.2 Algebra: Seeing Structure in Expressions |
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Objectives |
What Learner Should Know, Understand, and Be Able to Do |
Teaching Notes and Examples |
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MA.1.2.1 Interpret expressions that represent a quantity in terms of its context. Interpret parts of an expression, such as terms, factors, and coefficients
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Identify the different parts of the expression and explain their meaning within the context of a problem. For example, interpret P (1+r)n as the product of P and a factor not depending on P.
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Interpreting Expressions: What is a Variable/Coefficient/Constant https://www.virtualnerd.com/common-core/hsa-algebra/HSA-SSE-expressions-seeing-structure/A/1
Interpret Real World Expressions |
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MA.1.2.2 Use the structure of an expression to identify ways to rewrite it. |
Rewrite algebraic expressions in different equivalent forms:
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For example, see x4 – y4 as (x2) 2 – (y2) 2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
Use the Structure of an Expression to Identify Ways to Rewrite It https://www.virtualnerd.com/common-core/hsa-algebra/HSA-SSE-expressions-seeing-structure/A/2
Using Different Factoring Techniques https://learnzillion.com/lessonsets/718 Factoring Using Common Factors |
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MA.1.2.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Factor a quadratic expression to reveal the zeros of the function it defines.
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Write expressions in equivalent forms by factoring to find the zeros of a quadratic function and explain the meaning of the zeros.
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Choose and Produce Equivalent Forms Examples https://www.illustrativemathematics.org/HSA
Factor Quadratics to Reveal Zeros https://www.virtualnerd.com/common-core/hsa-algebra/HSA-SSE-expressions-seeing-structure/B/3/3a |
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MA.1.3 Algebra: Arithmetic with Polynomials and Rational Expressions |
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Objectives |
What Learner Should Know, Understand, and Be Able to Do |
Teaching Notes and Examples |
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MA.1.3.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
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Emphasis should be on operations with polynomials. Understand the definition of a polynomial. Understand the concepts of combining like terms and closure. Add, subtract, and multiply polynomials and understand how closure applies under these operations. |
Simplify Polynomials
Adding Polynomials https://www.virtualnerd.com/algebra-1/polynomials-and-factoring/addition-example.php
Subtracting Polynomials
Multiplying Polynomials |
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MA.1.3.2 Rewrite rational expressions.
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Rewrite simple rational expressions in different forms using inspection, or, for more complicated examples, a computer algebra system.
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Simplifying Rational Expressions video
Using Wolfram Alpha to Rewrite Rational Expressions https://www.wolframalpha.com/input/?i=factorize+%285x6+–+4x2+–+5x%29+++%286x6+–+7x2+++2x%29+ |
can be re-written as; ( 5a4b12)1/2 which can also be re-written as: 51/2.(a4)1/2.(b12)1/2
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ASE MA 2: Equations and Inequalities |
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MA.2.1 Algebra: Creating Equations |
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Objectives |
What Learner Should Know, Understand, and Be Able to Do |
Teaching Notes and Examples |
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MA.2.1.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions and simple rational and exponential functions.
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Create linear, quadratic, rational and exponential equations and inequalities in one variable and use them in a contextual situation to solve real world problems. Example: A contractor is purchasing some tiles for a new patio. Each tile costs $3 and he wants to spend less than $1000. The size of each tile is 1 square foot. Write an inequality that represents the number of tiles he can purchase with a $1000 limit then figure out how large the patio can be. |
Create Equations and Inequalities in One Variable
Inequalities Video https://www.khanacademy.org/math/algebra/linear_inequalities/inequalities/v/inequalities |
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MA.2.1.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. |
Create equations in two or more variables to represent relationships between quantities. Graph equations in two variables on a coordinate plane and label the axes and scales. |
Create Equations in Two or More Variables |
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MA.2.1.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context.
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Write and use a system of equations and/or inequalities to solve a real world problem. Recognize that the equations and inequalities represent the constraints of the problem. Use the Objective Equation and the Corner Principle to determine the solution to the problem. (Linear Programming)
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For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
Modeling Equations or Inequalities
Example Problems |
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MA.2.1.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. |
Solve multi-variable formulas or literal equations, for a specific variable.
Make a variable the subject of a given formula or algebraic expression. When rearranging a formula or algebraic equation, all operations on the right hand side must be repeated on the left hand side of the equal sign. |
Solve a Formula for a Variable https://www.virtualnerd.com/common-core/hsa-algebra/HSA-CED-/A/4/isolate-variable-from-formula
Example: How long will it take David to cover a distance of 26 miles if he was running at 7mph? Re-arrange the equation Distance = Speed x Time to highlight Time and solve the problem. Dividing both sides by Speed gives: Distance/Speed = Time à26/7 = 3.7 Hours |
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MA.2.2 Algebra: Reasoning with Equations and Inequalities |
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Objectives |
What Learner Should Know, Understand, and Be Able to Do |
Teaching Notes and Examples |
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MA.2.2.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. |
Assuming an equation has a solution, construct a convincing argument that justifies each step in the solution process. Justifications may include the associative, commutative, and division properties, combining like terms, multiplication by 1, etc. |
Justify Steps in Solving Equations Using Properties of Equations
https://www.cpm.org/pdfs/state_supplements/Justification_for_Solving_Equations.pdf
https://mathbitsnotebook.com/Algebra1/LinearEquations/LEjustify.html |
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MA.2.2.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
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Solve simple rational and radical equations in one variable and provide examples of how extraneous solutions arise. |
Solve Simple Rational and Radical Equations https://learnzillion.com/lessonsets/280-solve-simple-rational-and-radical-equations-in-one-variable
Extraneous Solutions to Radical Expressions |
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MA.2.2.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. |
Solve linear equations in one variable, including coefficients represented by letters.
Solve linear inequalities in one variable, including coefficients represented by letters. |
One Step Inequalities https://www.khanacademy.org/math/algebra/linear_inequalities/inequalities/e/one_step_inequalities
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MA.2.2.4 Solve quadratic equations with one variable.
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Solve quadratic equations in one variable by simple inspection, taking the square root, factoring, and completing the square. |
Using Different Methods to Solve Quadratic Equations https://www.illustrativemathematics.org/illustrations/618
Solve Quadratic Equations by Inspection https://learnzillion.com/lessons/743-solve-a-quadratic-equation-by-inspection
Solve Quadratic Equations by Taking the Square Root
Solve Quadratic Equations by Factoring https://www.virtualnerd.com/algebra-1/quadratic-equations-functions/solve-by-factoring.php
Solve Quadratic Equations by Completing the Square |
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MA.2.2.5 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
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Solve systems of equations using graphs. |
Solve a System of Equations by Graphing
Solve a System of Equations by Graphing |
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MA.2.2.6 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
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Understand that all solutions to an equation in two variables are contained on the graph of that equation.
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Represent and Solve Equations and Inequalities Graphically https://www.virtualnerd.com/common-core/hsa-algebra/HSA-REI-equations-inequalities-reasoning/D
Using Technology to Solve Equations: Wolfram Alpha Input x2+5x+6 into Wolfram Alpha to see the graph of the equation: |
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ASE MA 3: Algebraic Functions and Modeling |
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MA.3.1 Interpreting and Modeling Algebraic Functions: Understand the concept of a function and use function notation. |
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Objectives |
What Learner Should Know, Understand, and Be Able to Do |
Teaching Notes and Examples |
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MA.3.1.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
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A function occurs when each input (x) has only one output (y). Shown an equation, table, or graph, students can determine whether it is a function. Students understand that the domain is the set of x values and the range is the set of y values. In a function, f(x) is used instead of y. |
A function defines the relationship between algebraic variables. For a function f(x); x is used an input into the function to produce a set or series of outputs depending on the numerical value of x. So if f(x) = x+5; if x=0 then f(0) = 5; if x=1 then f(1) = 6; if x=2 then f(2) = 7. The values of x (0, 1, 2) is called the domain while the outputs (5, 6, 7) are called the range. The input is the domain, the output is the range.
What is a Function?
Functions: Domain and Range |
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MA.3.1.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. |
Students recognize f(x) function notation. Students can evaluate function for different inputs.
f(x) = 2x + 5 What is f(4)? f(4) = 2(4) + 5 f(4) = 8 + 5 f(4) = 13 |
Evaluating with Function Notation
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MA.3.1.3 Interpret functions that arise in application in terms of the context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. |
For example, for a quadratic function modeling a projectile in motion, interpret the intercepts and the vertex of the function in the context of the problem. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. |
Examples from Illustrative Math https://www.illustrativemathematics.org/HSF
Algebra Functions and Modeling Handout https://abspd.appstate.edu/teaching-resources
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MA.3.1.4 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. |
Students identify an appropriate domain from a graph based on context. Students also identify the meaning of a point in terms of context.
Example: Jennifer’s cell phone plan charges her $20 each month for the phone and 10 cents for each minute she’s on the phone. What domain would describe this relationship? Describe the meaning of the point (10,21).
Solution: The domain is the set of positive integers since there cannot be a negative number of minutes and parts of minutes are not charged. (10,21) means the charge for 10 minutes of service would be $21. |
Relate the Domain of a Function to its Graph https://learnzillion.com/lessonsets/679-relate-the-domain-of-a-function-to-its-graph
Examples from Illustrative Mathematics https://www.illustrativemathematics.org/HSF
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MA.3.1.5 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
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Estimate the average rate of change during an interval from a function’s graph. In the example below, between hours 1 and 2, a person drove 50 miles so the average rate of change is 50.
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Average Rate of Change |
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MA.3.1.6 Analyze functions in different representations. Graph functions expressed symbolically and show key features (properties described above) of the graph, by hand in simple cases and using technology for more complicated cases. |
Given the function y = 2x – 6, students can provide a written description (y is equal to two times a number minus six) and show the function in a table or graph.
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Words, Equations, Tables, and Graphs https://www.youtube.com/watch?v=apktS70tYPo
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MA.3.1.7 Use properties of exponents to interpret expressions for exponential functions.
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For example, identify percent rate of change in an exponential function and then classify it as representing exponential growth or decay. |
Introduction to Exponential Functions https://www.youtube.com/watch?v=PEtIQqvIoGU
Examples from Illustrative Math |
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MA.3.1.8 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
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Compare the following functions to determine which has the greater rate of change. Function 1: y = 2x + 4 Function 2:
Solution: The rate of change for function 1 is 2; the rate of change for function 2 is 3. Function 2 has the greater rate of change. |
Understanding and Comparing Functions https://www.youtube.com/watch?v=Mq6iePhLQGM
https://www.youtube.com/watch?v=6AjBsO4qsww
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MA.3.2 Build a function that models a relationship between two quantities. |
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Objectives |
What Learner Should Know, Understand, and Be Able to Do |
Teaching Notes and Examples |
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MA.3.2.1 Write a function that describes a relationship between two quantities.
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Understand how to interpret words into independent and dependent variables. Understand how to map the variables into numerical values. Understand how to identify the relationship between the variables by mapping the generated values into a graph by hand or using computational methods for complex relationships. |
Introduction to Linear Functions |
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MA.3.3 Construct and compare linear, quadratic, and exponential functions models and solve problems. Interpret expressions for functions in terms of the situation they model. |
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Objectives |
What Learner Should Know, Understand, and Be Able to Do |
Teaching Notes and Examples |
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MA.3.3.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. |
Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. |
Example: Given a function that contains the following points: (1,11); (2,14); (3,19); (4,26); (5,35). Determine whether the function is linear or non-linear.
Understanding Linear and Exponential Models |
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MA.3.3.2 Interpret the parameters in a linear or exponential function in terms of a context.
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Based on the context of a situation, explain the meaning of the coefficients, factors, exponents, and/or intercepts in a linear function. Example 1: Given a linear function y=mx+b; the coefficient m is the slope of the line, and b is the y intercept. X is the independent variable, and y is the dependent variable. Example 2: Given an exponential decay function A=Aoe-kt; Ao is the starting point, k is a constant, t is the time (an independent variable) and A is the dependent variable. |
Exploring Linear Relationships
Exponential Growth and Decay Word Problems |
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https://abspd.appstate.edu/node/377 accessed 1/4/2015