Chapter 1 Equations and Inequalities



ALGEBRA 1 OBJECTIVES

Unit 1 Properties of Real Numbers

1. Perform operations on real numbers. (N2D.2) (4%)

. Add and subtract real numbers, including positives and negatives.

(Adding and subtracting negative numbers) (Adding negative numbers)

b. Multiply real numbers, including positives and negatives. (Multiplying and dividing negative numbers)

c. Divide real numbers, including positives and negatives.

(negative number word problems)

2. Compare and order real numbers and find their approximate location on a number line.

(N1A *compare and order rational and irrational numbers, including finding their approximate locations on a number line DOK1 MA5 1.10) (4%)

d. Classify whole numbers, integers, rational numbers, irrational numbers, and real numbers.

e. Graph numbers on number lines. (number line 2) (number line 3) (decimals on the number line 1 and 2)

(fractions on the number line 1)

f. Compare real numbers using inequality signs (, ≤, ≥) and write them in order. (ordering negative numbers)

Unit 2 Expressions, Equations and Functions

3. Write an algebraic expression, which may include an exponent, and evaluate an expression using the order of operations. (5%)

(N2D *apply operations to real numbers, using mental computation or paper-and-pencil calculations for simple cases and technology for more complicated cases DOK2 MA1 1.10)

a. Evaluate expressions (Find the value, plug in for variables and get an answer). (evaluating expressions 1 and 2)

b. Evaluate using the order of operations. (order of operations)

c. Simplify expressions using the distributive property. (distributive property with variables)

4. Use unit analysis to verify that a variable expression will give a reasonable calculation based on the units in the answer. (4%)

(M2E *use unit analysis to solve problems DOK2 MA4 1.6)

d. Find the unit rate.

e. Given an expression which models a situation, analyze the units included in the terms in the expression to verify that the expression will give a reasonable calculation based on the units in the answer.

5. Write and use a linear equation or inequality to model and solve a problem.

(A2A *use symbolic algebra to represent and solve problems that involve linear and quadratic relationships including equations and inequalities DOK3 MA4 3.3) (5%)

e. Translate phrases into expressions/equations (writing expressions 1 and 2)

f. Determine whether a given number is a solution of an equation or inequality.

6. Write a rule for a function or make a graph given the inputs and outputs, or write the pairing given the rule or a graph. (4%)

(A1B *generalize patterns using explicitly or recursively defined functions DOK2 MA4 1.6)

g. List domain and range of a function. (domain and range 0.5)

h. Write the rule of a function given the input and the output.

i. Write a rule for a function when given the graph of the function.

Unit 3 Solving Linear Equations

7. Write a linear equation to model a real-life situation and solve linear equations in one variable. (10%)

(A2C *use and solve equivalent forms of equations (linear, absolute value, and quadratic) DOK2 MA4 3.2)

(A2A.2)

a. Solve one-step equations. (One step equations 0.5) (One step equations) (Arithmetic Word Problems) (Linear Equations 1)

b. Solve equations with 2 or more steps. (Linear Equations 2)

c. Solve equations with a variable on both sides. (Linear Equations 3 and 4)

d. Solve absolute value equations. (Absolute value Equations)

e. Given a verbal description of a real-life problem, translate it into an algebraic equation. (Age Word Problems)

f. Use the solution of the equation to help you solve the problem.

8. Solve problems involving proportions. (5%)

(N3E *solve problems involving proportions DOK2 MA1 3.2)

(M2D *describe the effects of operations, such as multiplication, division, and computing powers and roots on magnitudes of quantities and effects of computation on precision which include the judging of reasonableness of numerical computations and their results DOK2 MA2 1.7)

g. Write a ratio in simplest form; write a proportion.

h. Solve a proportion using a property of equality or the Cross Products Property. (Proportions 1 and 2)

i. Write a proportion to model a situation, including problems of scale drawings or percents, and use the solution of the proportion to help solve the problem. (Writing Proportions) (Percentage Word Problems 1)

9. Given an equation with two or more variables (literal equation), solve the equation for a specified variable. (A2C.2) (4%)

j. Rewrite an equation with one variable in terms of the others. (Solving for a variable)

k. Rewrite an equation or formula, then evaluate the new formula.

Unit 4 Graphing Linear Equations and Functions

10. Graph a linear equation or function. (A1B.2) (10%)

a. Graph the line of an equation using a table of points. (Plugging in values)

b. Graph horizontal and vertical lines from their equations.

c. Find the x & y intercepts of the graph of a linear equation. (Solving for the x-intercept)

d. Graph the line of an equation using x & y intercepts.

e. Graph the line of an equation using slope-intercept form. (Graphing linear equations)

f. Solve and graph linear equations in function notation [f(x)]. **

g. Graph absolute value equations.

11. Compare linear equations by their slopes and y-intercepts. (6%)

(A1E *describe the effects of parameter changes on linear, exponential growth/decay and quadratic functions including intercepts DOK2 MA4 1.6)

(A4A *analyze linear and quadratic functions by investigating rates of change, intercepts and zeros DOK3 MA4 1.6)

h. Using slope (rate of change), classify lines with positive slope, negative slope, zero slope, or undefined slope, and recognize the effect slope has on a line. (Interpreting linear equations)

i. Find the slope (rate of change) of a line using the slope formula. (Slope of a line)

j. Find the slope and y-intercept of a line using the slope-intercept form.

k. Compare linear equations using their y-intercepts.

Unit 5 Writing Linear Equations

12. Write a linear equation to model a real-life situation and use the equation to help solve a problem. (A2A.3) (10%)

(A1C *compare and contrast various forms of representations of patterns DOK3 MA4 1.6)

(A1D *understand and compare the properties of linear and nonlinear functions DOK2 MA4 1.6)

a. Identify the slope and a point from point-slope form. (A2A/A1D)

b. Write linear equations in slope-intercept form. (Slope Intercept form)

c. Write linear equations in point-slope form. (Point slope form) [Converting between point slope and slope intercept)

d. Write linear equations in standard form. (Converting between slope intercept and standard)

e. Write the equation of a line taken from a graph. (Equation of a line)

f. Write the equations of lines that are parallel and/or perpendicular to a given line.

g. Use a linear equation to solve a problem.

h. Relate arithmetic sequences to linear functions (A1C)

i. Find the inverse of a linear function.

13. Construct a scatter plot, determine an equation for a line of best fit, make predictions based on a set of data, make conjectures about possible relationships between two characteristics of a sample based on a set of data. (5%)

(D2C *given a scatter plot, determine an equation for a line of best fit DOK2 MA3 1.6) a&b

(D3A *make conjectures about possible relationships between 2 characteristics of a sample on the basis of scatter plots of the data DOK3 MA3 3.5) c&d

j. Find a line of best fit for a set of data.

k. Make a scatter plot.

l. Write an equation for line of best fit, include the rate of change.

l. Make a prediction based on a scatter plot.

m. Determine the correlation between two characteristics. (positive/negative)

Unit 6 Solving and Graphing Linear Inequalities

14. Solve a linear inequality in one variable. (local) (5%)

a. Solve and graph linear inequalities [, ≤, ≥] in one variable. (Inequalities on a number line) (One step inequalities)

b. Solve and graph compound inequalities (and/or). (Linear inequalities)

c. Solve and graph compound inequalities (and/or). For example: -3 < x + 2 < 7 (Compound Inequalities)

d. Solve an inequality that is true for all real numbers or determine if an inequality has no solutions.

15. Write a linear inequality to model a real-life situation and use the solution of the inequality to help solve a problem. (A2A.4) (5%)

e. Given a verbal description of a real-life problem, translate it into an algebraic inequality in one variable, or in two variables.

f. Use the solution of the inequality to help solve the problem.

16. Solve/Graph an absolute value inequality in one variable. (A2C.3) (4%)

g. Solve absolute value inequalities.

h. Graph absolute value inequalities.

i. Determine when an absolute value inequality has no solution or infinite solutions.

.

Unit 7 Systems of Equations and Inequalities

17. Use and solve systems of linear equations with 2 variables. (6%)

(A2D *use and solve systems of linear equations or inequalities with 2 variables DOK2 MA4 1.6)

for sections a-c (Systems of equations word problems)

a. Solve system of equations by graphing. (Graphing systems of equations)

b. Solve system of equations by substitution. (Systems of equations with substitutions)

c. Solve system of equations by elimination. (Systems of equations with simple elimination)

(Systems of equations with elimination)

d. Determine the number of solutions of a system of linear equations. (Solutions to systems of equations)

e. Determine whether or not an ordered pair is a solution of a system.

18. Use and solve systems of linear inequalities with 2 variables. (A2D.2) (5%)

f. Graph a linear inequality on a coordinate plane. (Graphing linear inequalities)

g. Solve of system of linear inequalities by graphing. (Graphing systems of inequalities)

h. Use linear programming to find maximums and minimums.

Unit 8 Exponents and Exponential Functions

19. Use properties and definitions of exponents to simplify expressions. (5%)

(A2B *describe and use algebraic manipulations, including factoring and rules of integer exponents and apply properties of exponents (including order of operations) to simplify expressions DOK2 MA4 3.2)

a. Simplify expressions using multiplication & division properties of exponents. (Exponent rules)

b. Simplify expressions using zero & negative properties of exponents. (Negative exponents) (Positive and zero exponents)

c. Simplify expressions with fractional exponents. (Fractional exponents) (Simplifying expressions with exponents)

20. Use scientific notation to represent very large and very small numbers. (5%)

(N1C *use a variety of representations to demonstrate an understanding of very large and very small numbers DOK2 MA5 1.6)

(N2B *describe the effects of operations, such as multiplication, division, and computer powers and roots on the magnitude of quantities DOK2 MA1 1.10)

(N3D *judge the reasonableness of numerical computations and their results DOK3 MA1 3.2)

(M2D.2)

d. Convert numbers between scientific notation and standard form. (Scientific Notation) (Scientific notation intuition)

e. Multiply and divide using scientific notation. (Multiplying and dividing scientific notation)

f. Judge reasonableness of scientific notation computations. (N3D)*

21. Write and graph exponential growth and exponential decay functions. (5%)

(A1B.3) (A1D.2)

(A3A *identify quantitative relationships and determine the type(s) of functions that might model the situation to solve the problem DOK2 MA4 1.6)

g. Write a rule for an exponential growth/decay function when given a table of ordered pairs.

h. Graph exponential growth & decay functions.

i. Solve exponential growth & decay problems.

j. Write a geometric sequence.

Unit 9 Probability and Data Analysis

22. Find the probability of an event(s). (5%)

(D1A *formulate questions and collect data about a characteristic which include sample spaces and distributions DOK3 MA3 1.2)

a. List the sample space of an experiment.

b. Find simple probability.

c. Find probability of compound events.

d. Use combinations/permutations to count possibilities.

23. Calculate statistical measures of a set of data. (6%)

(D2A *apply statistical measures of center to solve problems DOK2 MA3 1.10)

d. Find mean.

e. Find median.

f. Find mode.

g. Determine the best measure of central tendency for a given situation.

24. Represent and analyze data distributions. (5%)

(D1C *select and use appropriate graphical representation of data and given one-variable quantitative data, display the distribution and describe its shape DOK3 MA6 1.8)

h. Find range and inter-quartile ranges.

i. Analyze surveys and samples, identify bias in a survey.

j. Construct stem-and-leaf plots.

k. Construct box-and-whisker graphs.

l. Construct histograms.

m. Construct a line plot.

Unit 10 Polynomials and Factoring

25. Add, subtract, and multiply polynomials. (A2B.2) (6%)

a. Identify the coefficients, including leading, of a polynomial.

b. State the degree of polynomial.

c. Add & Subtract polynomials.

d. Multiply by a monomial.

e. Multiply 2 binomials.

f. Multiply polynomials with many terms.

g. Multiply special product polynomials.

26. Factor polynomials. (A2B.3) (6%)

h. Factor out a common factor.

i. Factor trinomials into two binomials.

j. Factor the difference of two perfect squares.

k. Factor by grouping.

Unit 11 Quadratic Equations and Functions

27. Graph quadratics. (A4A.2) (A1D.3) (6%)

a. Graph simple parabolas in the form of ([pic]).

b. Graph parabolas with a “b“ term ([pic]).

c. Identify/analyze functions (linear, quadratic, exponential, absolute value, piecewise, and step)

28. Solve quadratic equations and functions. (A2C.4) (5%)

d. Solve simple quadratic equations.

e. Solve quadratic equations by graphing.

f. Solve quadratic equations by factoring.

g. Solve quadratic equations by completing the square.

h. Solve quadratic equations by using the quadratic formula.

i. Use the discriminant to find the number of solutions to a quadratic.

Unit 12 Radicals and Geometry

29. Simplify radicals. (N2D.3) (5%)

a. Simplify radicals (break them down--factor trees).

b. Rationalize denominators (don‘t get caught with your radical down).

c. Add & Subtract radicals.

d. Multiply radicals.

.

30. Solve and graph radical equations. (A2B.4) (5%)

e. Graph radical equations.

f. Solve simple radical equations.

g. Identify extraneous solutions.

h. Solve equations with two radicals.

31. Use radicals to solve geometry problems. (5%)

(N1B *use real numbers and various models, drawings, etc. to solve problems DOK3 MA5 3.3)

(G4B *draw or use visual models to represent and solve problems DOK3 MA2 3.3)****

g. Use Pythagorean theorem to find missing sides of a right triangle.

h. Find the distance between two points.

i. Find the midpoint between two points.

Unit 13 Rational Equations and Functions

32. Simplify rational expressions and solve rational equations. (N2D.3)

a. Simplify rational expressions.

b. Multiply & divide rational expressions.

c. Simplify complex fractions.

d. Solve rational equations.

e. Dividing polynomials.

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