|
School City of Hobart Algebra I In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives - operating computer equipment, planning timelines and schedules, reading and interpreting data, comparing prices, managing personal finances, and completing other problem-solving tasks. What they learn in mathematics and how they learn it will provide an excellent preparation for a challenging and ever-changing future. |
| Algebraic Concepts |
|
Operations with Real Numbers
The learner will be able to simplify and compare expressions. They use rational exponents, and simplify square roots.
|
|
A1.1.1
The learner will be able to compare real number expressions.
|
|
A1.1.2
The learner will be able to simplify square roots using factors.
|
|
A1.1.3
The learner will be able to understand and use the distributive, associative, and commutative properties. Example: Simplify (6x2 - 5x+1) - 2(x2 + 3x - 4) by removing the parentheses and rearranging. Explain why you can carry out each step. .
|
|
A1.1.4
The learner will be able to use the laws of exponents for rational exponents.
|
|
A1.1.5
The learner will be able to use dimensional (unit) analysis to organize conversions and computations. Example: Convert 5 miles per hour to feet per second. .
|
|
Linear Equations and Inequalities
The learner will be able to solve linear equations and inequalities in one variable. They solve word problems that involve linear equations, inequalities, or formulas.
|
|
A1.2.1
The learner will be able to solve equations and formulas for a specified variable. Example: Solve the equation q = 4p - 11 for p. .
|
|
A1.2.3
The learner will be able to find solution sets of linear inequalities when possible numbers are given for the variable. Example: Solve the inequality 6x - 3 > 10 for x in the set {0, 1, 2, 3, 4}. .
|
|
A1.2.4
The learner will be able to solve linear inequalities using properties of order. Example: Solve the inequality 8x - 7 <= 2x + 5, explaining each step in your solution. .
|
|
A1.2.5
The learner will be able to solve combined linear inequalities. Example: Solve the inequalities -7 < 3x + 5 < 11. .
|
|
A1.2.6
The learner will be able to solve word problems that involve linear equations, formulas, and inequalities. Example: You are selling tickets for a play that cost $3 each. You want to sell at least $50 worth. Write and solve an inequality for the number of tickets you must sell. .
|
| Functions |
|
Relations and Functions
The learner will be able to sketch and interpret graphs representing given situations. They understand the concept of a function and analyze the graphs of functions.
|
|
A1.3.1
The learner will be able to sketch a reasonable graph for a given relationship. Example: Sketch a reasonable graph for a person's height from age 0 to 25. .
|
|
A1.3.2
The learner will be able to interpret a graph representing a given situation. .
|
|
A1.3.3
The learner will be able to understand the concept of a function, decide if a given relation is a function, and link equations to functions. Example: Use either paper or a spreadsheet to generate a list of values for x and y in y = x2. Based on your data, make a conjecture about whether or not this relation is a function. Explain your reasoning. .
|
|
A1.3.4
The learner will be able to find the domain and range of a relation. Example: Based on the list of values from the last example, what do the domain and range of y = x2 appear to be? How can you decide whether you are correct? .
|
| Algebraic Concepts |
|
Graphing Linear Equations, Inequalities
The learner will be able to graph linear equations and inequalities in two variables. They write equations of lines and find and use the slope and y-intercept of lines. They use linear equations to model real data.
|
|
A1.4.1
The learner will be able to graph a linear equation. Example: Draw the graph of the line with slope 3 and y-intercept -2. .
|
|
A1.4.2
The learner will be able to find the slope, x-intercept and y-intercept of a line given its graph, its equation, or two points on the line. Example: Find the slope and y-intercept of the line 4x + 6y = 12. .
|
|
A1.4.3
The learner will be able to write the equation of a line in slope-intercept form. Understand how the slope and y-intercept of the graph are related to the equation. Example: Write the equation of the line 4x + 6y = 12 in slope-intercept form. What is the slope of this line? Explain your answer. .
|
|
A1.4.4
The learner will be able to write the equation of a line given appropriate information. Example: Find an equation of the line through the points (1, 4) and (3, 10), then find an equation of the line through the point (1, 4) perpendicular to the first line. .
|
|
A1.4.5
The learner will be able to write the equation of a line that models a data set and use the equation (or the graph of the equation) to make predictions. Describe the slope of the line in terms of the data, recognizing that the slope is the rate of change. Example: As your family is traveling along an interstate, you note the distance traveled every 5 minutes. A graph of time and distance shows that the relation is approximately linear. Write the equation of the line that fits your data. Predict the time for a journey of 50 miles. What does the slope represent? .
|
|
A1.4.6
The learner will be able to graph a linear inequality in two variables. Example: Draw the graph of the inequality 6x + 8y >= 24. .
|
|
Pairs of Linear Equations, Inequalities
The learner will be able to solve pairs of linear equations using graphs and using algebra. They solve pairs of linear inequalities using graphs. They solve word problems involving pairs of linear equations.
|
|
A1.5.1
The learner will be able to use a graph to estimate the solution of a pair of linear equations in two variables. Example: Graph the equations 3y - x = 0 and 2x + 4y = 15 to find where the lines intersect. .
|
|
A1.5.2
The learner will be able to use a graph to find the solution set of a pair of linear inequalities in two variables. Example: Graph the inequalities y <= 4 and x + y <= 5. Shade the region where both inequalities are true. .
|
|
A1.5.3
The learner will be able to understand and use the substitution method to solve a pair of linear equations in two variables. Example: Solve the equations y = 2x and 2x + 3y = 12 by substitution. .
|
|
A1.5.4
The learner will be able to understand and use the addition or subtraction method to solve a pair of linear equations in two variables. Example: Use subtraction to solve the equations: 3x + 4y = 11, 3x + 2y = 7. .
|
|
A1.5.5
The learner will be able to understand and use multiplication with the addition or subtraction method to solve a pair of linear equations in two variables. Example: Use multiplication with the subtraction method to solve the equations: x + 4y = 16, 3x - 2y = -3. .
|
|
A1.5.6
The learner will be able to use pairs of linear equations to solve word problems. Example: The income a company makes from a certain product can be represented by the equation y = 10.5x and the expenses for that product can be represented by the equation y = 5.25x + 10,000, where x is the amount of the product sold and y is the number of dollars. How much of the product must be sold for the company to reach the break-even point? .
|
|
Polynomials
The learner will be able to add, subtract, multiply, and divide polynomials. They factor quadratics.
|
|
A1.6.1
The learner will be able to add and subtract polynomials. .
|
|
A1.6.2
The learner will be able to multiply and divide monomials.
|
|
A1.6.3
The learner will be able to find powers and roots of monomials (only when the answer has an integer exponent).
|
|
A1.6.4
The learner will be able to multiply polynomials. Example: Multiply (n + 2)(4n - 5). .
|
|
A1.6.5
The learner will be able to divide polynomials by monomials.
|
|
A1.6.6
The learner will be able to find a common monomial factor in a polynomial.
|
|
A1.6.7
The learner will be able to factor the difference of two squares and other quadratics.
|
|
A1.6.8
The learner will be able to understand and describe the relationships among the solutions of an equation, the zeros of a function, the x-intercepts of a graph, and the factors of a polynomial expression.
|
|
Algebraic Fractions
The learner will be able to simplify algebraic ratios and solve algebraic proportions.
|
|
A1.7.1
The learner will be able to simplify algebraic ratios.
|
|
A1.7.2
The learner will be able to solve algebraic proportions.
|
|
Quadratic, Cubic, and Radical Equations
The learner will be able to graph and solve quadratic and radical equations. They graph cubic equations.
|
|
A1.8.1
The learner will be able to graph quadratic, cubic, and radical equations.
|
|
A1.8.2
The learner will be able to solve quadratic equations by factoring. .
|
|
A1.8.3
The learner will be able to solve quadratic equations in which a perfect square equals a constant.
|
|
A1.8.4
The learner will be able to complete the square to solve quadratic equations.
|
|
A1.8.5
The learner will be able to derive the quadratic formula by completing the square.
|
|
A1.8.6
The learner will be able to solve quadratic equations by using the quadratic formula.
|
|
A1.8.7
The learner will be able to use quadratic equations to solve word problems.
|
|
A1.8.8
The learner will be able to solve equations that contain radical expressions.
|
|
A1.8.9
The learner will be able to use graphing technology to find approximate solutions of quadratic and cubic equations.
|
| Problem Solving |
|
Mathematical Reasoning, Problem Solving
The learner will be able to use a variety of strategies to solve problems.
|
|
A1.9.1
The learner will be able to use a variety of problem solving strategies, such as drawing a diagram, making a chart, guess-and-check, solving a simpler problem, writing an equation, and working backwards. Example: Fran has scored 16, 23, and 30 points in her last three games. How many points must she score in the next game so that her four game average does not fall below 20 points? .
|
|
A1.9.2
The learner will be able to decide whether a solution is reasonable in the context of the original situation. Example: John says the answer to the problem in the first example is 10 points. Is his answer reasonable? Why or why not? .
|
|
A1.9.3
The learner will be able to use the properties of the real number system and the order of operations to justify the steps of simplifying functions and solving equations. Example: Given an argument (such as 3x + 7 > 5x + 1, and therefore -2x > -6, and therefore x > 3), provide a visual presentation of a step-by-step check, highlighting any errors in the argument. .
|
|
A1.9.4
The learner will be able to understand that the logic of equation solving begins with the assumption that the variable is a number that satisfies the equation, and that the steps taken when solving equations create new equations that have, in most cases, the same solution set as the original. Understand that similar logic applies to solving systems of equations simultaneously. Example: Try "solving" the equations x + 3y = 5 and 5x + 15y = 25 simultaneously, and explain what went wrong. .
|
|
A1.9.5
The learner will be able to decide whether a given algebraic statement is true always, sometimes, or never (statements involving linear or quadratic expressions, equations, or inequalities).
|
|
A1.9.6
The learner will be able to distinguish between inductive and deductive reasoning, identifying and providing examples of each. Example: What type of reasoning are you using when you look for a pattern? .
|
|
A1.9.7
The learner will be able to identify the hypothesis and conclusion in a logical deduction. Example: What is the hypothesis and conclusion in this argument: If there is a number x such that 2x + 1 = 7, then x = 3. .
|
|
A1.9.8
The learner will be able to use counterexamples to show that statements are false, recognizing that a single counterexample is sufficient to prove a general statement false. Example: Use the demonstration-graphing calculator on an overhead projector to produce an example showing that this statement is false: all quadratic equations have two different solutions. .
|