Overview
| Site: | Gladwin |
| Course: | Michigan Algebra I Sept. 2012 |
| Book: | Overview |
| Printed by: | Guest user |
| Date: | Saturday, November 23, 2024, 5:13 AM |
Description
Overview
This unit should take approximately 3-4 weeks in a traditional hourly schedule.
Students Will Be Able To
- Identify zeros of a function and their role in solutions to equations.
- Identify domain and range in context of a given situation.
- Understand linear functions have a constant rate of change and be able to identify it graphically, in a table, symbolically and verbally.
- Give the output, given an input and a function in function notation, table form or graphically.
- Identify the inverse of a function as a way to find the inputs for given multiple outputs.
- Recognize that solving equations algebraically involves substituting one equivalent form of the equation for another equation.
- Determine if a given situation can be modeled by a linear function or not, and write the function that models it, if it can be.
Prior Knowledge
A.PA.07.01 Recognize when information given in a table, graph, or formula suggests a directly proportional or linear relationship.
A.RP.07.02 Represent directly proportional and linear relationships using verbal descriptions, tables, graphs, and formulas, and translate among these representations.
A.PA.07.03 Given a directly proportional or linear situation, graph and interpret the slope and intercept(s) in terms of the original situation; evaluate y = mx + b for specific x values, e.g., weight vs. volume of water, base cost plus cost per unit.
A.PA.07.04 For directly proportional or other linear situations, solve applied problems using graphs and equations, e.g., the heights and volume of a container with uniform cross-section; height of water in a tank being filled at a constant rate; degrees Celsius and degrees Fahrenheit; distance and time under constant speed.
A.PA.07.05 Recognize and use directly proportional relationships of the form y = mx, and distinguish from linear relationships of the form y = mx + b, b non-zero; understand that in a directly proportional relationship between two quantities one quantity is a constant multiple of the other quantity.
A.PA.07.06 Calculate the slope from the graph of a linear function as the ratio of "rise/run" for a pair of points on the graph, and express the answer as a fraction and a decimal; understand that linear functions have slope that is a constant rate of change.
A.PA.07.07 Represent linear functions in the form y = x + b, y = mx, and y = mx + b, and graph, interpreting slope and y-intercept.
A.FO.07.08 Find and interpret the x and/or y intercepts of a linear equation or function. Know that the solution to a linear equation of the form ax + b = 0 corresponds to the point at which the graph of y = ax + b crosses the x axis.
A.FO.07.12 Add, subtract, and multiply simple algebraic expressions of the first degree, e.g., (92x + 8y) - 5x + y, or x(x+2), and justify using properties of real numbers.
A.FO.07.13 From applied situations, generate and solve linear equations of the form ax + b = c and ax + b = cx +d, and interpret solutions.
A.FO.08.10 Understand that to solve the equation f(x) = g(x) means to find all values of x for which the equation is true, e.g., determine whether a given value, or values from a given set, is a solution of an equation (0 is a solution of 3x2 + 2 = 4x + 2, but 1 is not a solution).
A.FO.08.11 Solve simultaneous linear equations in two variables by graphing, by substitution, and by linear combination; estimate solutions using graphs; include examples with no solutions and infinitely many solutions.
A.FO.08.12 Solve linear inequalities in one and two variables, and graph the solution sets.
A.FO.08.13 Set up and solve applied problems involving simultaneous linear equations and linear inequalities.
Mastered HSCEs
The following Michigan High School Content Expectations will be mastered in this unit:
A3.1.1 Write the symbolic forms of linear functions (standard [i.e., Ax + By = C, where B ? 0], point-slope, and slope-intercept) given appropriate information and convert between forms.
A3.1.2 Graph lines (including those of the form x = h and y = k) given appropriate information.
A3.1.3 Relate the coefficients in a linear function to the slope and x- and y-intercepts of its graph.
A3.1.4 Find an equation of the line parallel or perpendicular to given line through a given point. Understand and use the facts that non-vertical parallel lines have equal slopes and that non-vertical perpendicular lines have slopes that multiply to give -1.
Addressed HSCEs
The following Michigan High School Content Expectations will be addressed within this unit.
A1.1.1 Give a verbal description of an expression that is presented in symbolic form, write an algebraic expression from a verbal description, and evaluate expressions given values of the variables.
A1.2.1 Write and solve equations and inequalities with one or two variables to represent mathematical or applied situations.
A1.2.2 Associate a given equation with a function whose zeros are the solutions of the equation.
A1.2.3 Solve linear and quadratic equations and inequalities, including systems of up to three linear equations with three unknowns. Justify steps in the solutions, and apply the quadratic formula appropriately.
A1.2.8 Solve an equation involving several variables (with numerical or letter coefficients) for a designated variable. Justify steps in the solution.
A2.1.1 Recognize whether a relationship (given in contextual, symbolic, tabular, or graphical form) is a function and identify its domain and range.
A2.1.2 Read, interpret, and use function notation and evaluate a function at a value in its domain.
A2.1.3 Represent functions in symbols, graphs, tables, diagrams, or words and translate among representations.
A2.1.6 Identify the zeros of a function and the intervals where the values of a function are positive or negative. Describe the behavior of a function as x approaches positive or negative infinity, given the symbolic and graphical representations.
A2.1.7 Identify and interpret the key features of a function from its graph or its formula(e), (e.g., slope, intercept(s), asymptote(s), maximum and minimum value(s), symmetry, and average rate of change over an interval).
A2.2.1 Combine functions by addition, subtraction, multiplication, and division.
A2.2.2 Apply given transformations (e.g., vertical or horizontal shifts, stretching or shrinking, or reflections about the x- and y-axes) to basic functions and represent symbolically.
A2.2.3 Recognize whether a function (given in tabular or graphical form) has an inverse and recognize simple inverse pairs.
A2.3.2 Describe the tabular pattern associated with functions having a constant rate of change (linear) or variable rates of change.
A2.4.2 Adapt the general symbolic form of a function to one that fits the specifications of a given situation by using the information to replace arbitrary constants with numbers.
A2.4.3 Using the adapted general symbolic form, draw reasonable conclusions about the situation being modeled.
L1.1.2 Explain why the multiplicative inverse of a number has the same sign as the number, while the additive inverse has the opposite sign.
L1.1.3 Explain how the properties of associativity, commutativity, and distributivity, as well as identity and inverse elements, are used in arithmetic and algebraic calculations.
Source
Embracing Mathematics, Assessment & Technology in High Schools; A Michigan Mathematics & Science Partnership Grant Project