A.1 The student will solve multi-step linear equations and inequalities in one variable, solve literal equations (formulas) for a given variable and apply these skills to solve practical problems. Graphing calculators will be used to confirm algebraic solutions.
Prerequisite Knowledge:
8.4Solve practical problems involving whole numbers, integers, and rational numbers, including percents. Problems will be of varying complexities, involving real-life data.
8.5Apply the order of operations to evaluate algebraic expressions for given replacement values of specified variables.
8.16Solve multistep equations in one variable.
8.19Create and solve problems using proportions, formulas, and functions.
A.2 Represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables. Students will choose an appropriate computational technique such as mental mathematics, calculator, or paper/pencil.
A.3 Justify steps used in simplifying expressions and solving equations and inequalities using concrete objects, pictorial representations, and the properties of real numbers.
Enabling Objectives:
- Discuss orally and in writing the meaning of “solution of an equation.”
- Select an appropriate method for solving equations.
- Solve linear equations by making a table of values.
- Solve linear equations using a balance scale and manipulatives.
- Solve linear equations in one variable using the calculator.
- Find the intercept.
- Find the point of intersection on the graph.
- Find where y1 = y2 in the table.
- Solve linear equations in one variable using paper and pencil.
- Apply skills of solving linear equations to practical problems.
- Explain the meanings of the symbols for inequality.
- Explain the possible solutions for linear inequalities algebraically and graphically.
- Solve linear inequalities by hand and confirm solutions using the calculator.
- Solve literal equations (formulas) for a given variable.
A.2 The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables. Students will choose an appropriate computational technique such as mental mathematics, calculator, or paper/pencil.
Prerequisite Knowledge:
8.2Simplify numerical expressions involving exponents, using order of operations.
8.5Apply the order of operations to evaluate algebraic expressions for given replacement values of specified variables.
Enabling Objectives:
- Use the order of operations to simplify numerical expressions.
- Differentiate between the four operations by the key words in a verbal expression.
- Translate verbal expressions into algebraic expressions.
- Translate algebraic expressions into real world examples.
- Find the value of an algebraic expression using the order of operations.
- Make a table of values for a given algebraic expression.
- Make a table of values to verify if the algebraic expression makes sense.
A.3 The student will justify steps used in simplifying expressions and solving equations and inequalities using concrete objects, pictorial representations, and the properties of real numbers, equality, and inequality.
Previous Knowledge:
Prior exposure to manipulatives is helpful.
Enabling Objectives:
- Identify the properties of equality: reflexive, symmetric, transitive, substitution, addition, subtraction, multiplication, and division.
- Identify properties of real numbers: closure, commutative, associative, inverse, identity, and distributive.
- Compare multiplicative inverse to division property of equality.
- Compare additive Inverse to subtraction property of equality.
- Justify each step in simplifying expressions and solving equations using by identifying the appropriate property of equality.
- Compare and contrast both orally and in writing how the properties of equality relate to the properties of inequality.
- Justify each step of solving an inequality by identifying the appropriate property of inequality.
A.4The student will use matrices to organize and manipulate data, including matrix addition, subtraction, and scalar multiplication. Data will arise from business, industrial, and consumer situations.
Prerequisite Knowledge:
8.14Use a matrix to organize and describe data.
Enabling Objectives:
- Name matrices using the rows and columns.
- Add, subtract, and use scalar matrices with matrices.
- Solve application problems with matrices using given data.
A.5The student will create and use tabular, symbolic, graphical, verbal, and physical representations to analyze a given set of data for the existence of a pattern, determine the domain and range or relations, and identify the relations that are functions.
Prerequisite Knowledge:
8.15Investigate and describe functional relationships, including the number of sides of a regular polygon and the maximum number of possible diagonals, expressing the algebraic concept of the number of diagonals of the nth-sided polygon.
8.17Describe and represent relations using tables, graphs, and rules.
Enabling Objectives:
- Demonstrate ability to plot points (x,y) on a coordinate plane.
- Identify quadrants on a coordinate plane.
- Find the domain and range of a relation.
- Identify when a relation is a function in a table, mapping, or graph.
- Find the next three terms in a patterned sequence.
- Explain what pattern is represented in a given sequence.
- Compare the pattern formed by the domain with the pattern formed by
the range values.
- Graph the table of values to determine if the points lie in a straight line.
- Compare the pattern formed in the table to the pattern formed on the
graph (slope).
- Find the rule for the given table of values.
A.6The student will select, justify, and apply a technique to graph
linear functions and linear inequalities in two variables;
techniques will include slope-intercept, x- and y-intercepts,
graphing by transformation, and the use of the graphing
calculator.
Prerequisite Knowledge:
8.17Graph a linear equation in two variables on the coordinate plane
using a table of ordered pairs.
Enabling Objectives:
- Graph a linear function.
- Determine a table of values
- Use the x- and y-intercepts
- Use the slope and y-intercept (y = mx + b)
i. Convert the equation to slope intercept form.
ii. Identify the slope and y-intercept from the equation.
- Use a graphing calculator
i. Determine the appropriate window range.
ii. Utilize the table feature to verify points on the line.
- Recognize the unique characteristics of the equations and graphs of
vertical and horizontal lines.
c. Graph a linear inequality.
- Graph linear equations and inequalities that represent real-life situations.
A.7The student will determine the slope of a line when given an
equation of the line, the graph of the line, or two points on the
line. Slope will be described as rate of change and will be
positive, negative, zero, or undefined. The graphing calculator
will be used to investigate the effect of changes in the slope on
the graph of the line.
Prerequisite Knowledge:
8.17Graph a linear equation in two variables on the coordinate plane
using a table of ordered pairs.
A.5Analyze a given set of data for the existence of a pattern, represent
the pattern algebraically (graphically, if possible), and determine if the
relation is a function.
Enabling Objectives:
- Determine the slope given a graph, a table of values, or two points.
- Explore situations where the slope can be seen as a rate of change.
- Continue writing equations in y = mx + b form, entering these in the calculator and making predictions as to how the slope affects the slant of the line
A.8The student will write an equation of a line when given the graph
of the line, two points on the line, or the slope and a point on the
line.
Prerequisite Knowledge:
8.17Graph a linear equation in two variables on the coordinate plane
using a table of ordered pairs.
A.5Analyze a given set of data for the existence of a pattern, represent
the pattern algebraically (graphically, if possible), and determine if the
relation is a function.
A.7Determine the slope of a line when given an equation of the line,
the graph of the line, or two points on the line. Slope will be
described as rate of change and will be positive, negative, zero
or undefined. The graphing calculator will be used to investigate the
effect of changes in the slope on the graph of the line.
Recognize standard form of an equation and know how to manipulate into slope-intercept and vice versa.
Enabling Objectives:
- Write the equation of a line in slope-intercept form or standard form when given the graph of a line, two points on the line, a point and the slope, or the slope and y-intercept.
- Create and use linear models to solve problems.
A.9The student will solve systems of two linear equations in two variables, both algebraically and graphically, and apply these techniques to solve practical problems. Graphing calculators will be used both as a primary tool for solution and to confirm an algebraic solution.
Prerequisite Knowledge:
8.17Graph a linear equation in two variables on the coordinate plane
using a table of ordered pairs.
It would be helpful if students have had the opportunity to solve equations
in one variable using the table feature on the graphing calculator.
Enabling Objectives:
- Choose the appropriate method of solving linear equations.
1. Graphing
2. Substitution
- Elimination
- Graphing calculator
- Confirm algebraic results with graphing calculator.
- Use the graphing calculator to determine if a system has no solution, one solution, or an infinite number of solutions.
- Solve real-world applications involving systems of equations.
A.10The student will apply the laws of exponents to perform operations on expressions with integral exponents, using scientific notation when appropriate.
Prerequisite Knowledge:
8.2Simplify numerical expressions involving exponents, using order of operations.
Enabling Objectives:
- Evaluate powers and simplify expressions containing powers.
- Apply the laws of exponents to multiply powers, divide powers, raise a power to a power, find a power of a product and a quotient.
- Use zero and negative exponents in algebraic expressions.
- Use scientific notation to work with large and small numbers.
A.11The student will add, subtract, and multiply polynomials and divide polynomials with monomial divisors using concrete objects, pictorial area representations, and algebraic manipulations.
Prerequisite Knowledge:
Prior exposure to Algebra Tiles (or similar manipulatives) is helpful.
Enabling Objectives:
- Define polynomials, monomials, binomials, trinomials, like terms, base, exponent, coefficient.
- Add, subtract, multiply, and divide polynomials.
- Use algebra tiles to model polynomials.
- Combine like terms.
- Use patterns to multiply binomials that result in the difference of squares and perfect square trinomials.
d. Verify products and quotients on the graphing calculator.
A.12The student will factor completely first- and second-degree binomials and trinomials in one or two variables. The graphing calculator will be used as both a primary tool for factoring and for confirming an algebraic factorizations.
Prerequisite Knowledge:
Students must know how to find the area of a rectangle.
A.11Add/subtract/multiply polynomials; divide polynomials with monomial divisors using concrete objects, pictorial representations, and algebraic manipulations.
Enabling Objectives:
- Determine the degree of a polynomial.
- Factor polynomials using the greatest common factor.
- Factor polynomials by grouping.
- Identify the “special” polynomials: difference of squares, perfect square trinomials, sum/difference of two cubes.
- Factor the “special” polynomials using manipulatives.
- Factor general trinomials using manipulatives.
- Factor polynomials completely using more than one factoring technique.
- Use calculator to confirm algebraic factorization.
- Boolean logic
- Graphing
A.13 The student will estimate the square root of a whole number in
simplest radical form and approximate square roots to the
nearest tenth.
Prerequisite Knowledge:
8.6Given a whole number from 0-100, identify it as a perfect square or find the two consecutive whole numbers between which the square root lies.
Enabling Objectives:
- Identify perfect squares.
- Use algebra tiles to estimate square roots to the nearest tenth.
- Use the graphing calculator to compare decimal approximations of radicals to the estimated answers.
- Solve simple quadratic equations by finding the square root of each side.
A.14The student will solve quadratic equations in one variable both algebraically and graphically. Graphing calculators will be used both as a primarytool in solving problems and to verify algebraic solutions.
Prerequisite Knowledge:
A.12Factor completely first- and second-degree binomials and trinomials in one or two variables. The graphing calculator will be used as both a primary tool for factoring and for confirming an algebraic factorization.
Enabling Objectives:
- Write the quadratic equation in standard form.
- Solve quadratic equations using factoring, completing the square, the quadratic formula, and the graphing calculator.
- Use the graphing calculator to verify the solutions.
- table feature
- checking answers on the home screen
- Recognize the relationship between the factored form of a quadratic and its solutions.
A.15The student will, given a rule, find the values of a function for elements in its domain and locate the zeros of the function both algrbraically and with a graphing calculator. The value of f(x) will be related to the ordinate on the graph.
Prerequisite Knowledge:
8.2Simplify numerical expressions involving exponents, using order of operations.
8.5Apply the order of operations to evaluate algebraic expressions for given replacement values of specified variables.
A.2Represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables. Students will choose an appropriate computational technique such as mental mathematics, calculator, or paper/pencil.
Enabling Objectives:
- Use function notation (i.e., f(x)) to denote a function.
- Find the value of a function given an element from the domain.
- Identify a linear function, absolute value function, constant function, and quadratic function.
- Discuss both orally and in writing how the zeros of a function relate to the x-intercepts of the graph.
- Calculate the x-intercepts both algebraically and with a graphing calculator.
- Relate the value of f(x) to the ordinate on the graph.
A.16The student will, given a set of data points, write an equation for a line of best fit and use the equation to make predictions.
Prerequisite Knowledge:
8.13Use information displayed in line, bar, circle, and picture graphs and histograms to make comparisons, predictions, and inferences.
Enabling Objectives:
- Compare and contrast the graph of a line and a scatter-plot.
- Write the equation of a line of best fit using the median fit method.
- Use the equation of the line of best fit to make predictions about a given set of data.
A.17 The student will compare and constrast multiple one-variable data sets, using statistical techniques that include measures of central tendency, range, and box-and-whisker graphs.
Prerequisite Knowledge:
8.13Use information displayed in line, bar, circle, and picture graphs and histograms to make comparisons, predictions, and inferences.
Enabling Objectives:
- Find the mean, median, mode, and range for a given set of data.
- Given a set of data, determine the median, quartiles, and extremes.
- Construct stem-and-leaf plots, scatter plots, and box-and-
whisker plots for a given set of data.
- Interpret stem-and-leaf plots, scatter plots, and box-and-
whisker plots.
- Give examples of real-life situations that involve measures of central tendency.
A.18 The student will analyze a relation to determine whether a direct variation exists and represent it algebraically and graphically, if possible.
Prerequisite Knowledge:
8.1Use proportions to solve scale-model problems with fractions and decimals.
8.16 Solve multi-step equations in one variable.
Enabling Objectives:
- Solve proportions stemming from real-world situations.
- Compare and contrast direct and inverse variation.
- Recognize graphs of direct variations and inverse variations.
- Explain both orally and in writing how to determine if two variables are directly or inversely related.
- Use direct and inverse variation to solve real-world situations.