Strand

Standard

No.

Benchmark

Curriculum

Assessment

8

Number & Operation

MCA 6-8 items

Read, write, compare, classify and represent real numbers, and use them to solve problems in various contexts.

MCA 6-8 items

8.1.1.1

Classify real numbers as rational or irrational. Know that when a square root of a positive integer is not an integer, then it is irrational. Know that the sum of a rational number and an irrational number is irrational, and the product of a non-zero rational number and an irrational number is irrational.  

For example: Classify the following numbers as whole numbers, integers, rational numbers, irrational numbers, recognizing that some numbers belong in more than one category: , , , , , , . 

Glencoe McGraw-Hill

Algebra 1

Ch. 0-2

8.1.1.2

Compare real numbers; locate real numbers on a number line. Identify the square root of a positive integer as an integer, or if it is not an integer, locate it as a real number between two consecutive positive integers.

For example: Put the following numbers in order from smallest to largest:    2, , 4, 6.8, .

Another example: is an irrational number between 8 and 9.

Glencoe McGraw-Hill

Algebra 1

Ch.  0-2

8

Number & Operation

Read, write, compare, classify and represent real numbers, and use them to solve problems in various contexts.

8.1.1.3

Determine rational approximations for solutions to problems involving real numbers.

For example: A calculator can be used to determine that is approximately 2.65.

Another example: To check that is slightly bigger than, do the calculation .

Another example: Knowing that  is between 3 and 4, try squaring numbers like 3.5, 3.3, 3.1 to determine that 3.1 is a reasonable rational approximation of.

Glencoe McGraw-Hill

Algebra 1

Ch.  0-2

8.1.1.4

Know and apply the properties of positive and negative integer exponents to generate equivalent numerical expressions.

For example: . 

Glencoe McGraw-Hill

Algebra 1

Ch.  7.1

Ch.  7.2

8

Number & Operation

Read, write, compare, classify and represent real numbers, and use them to solve problems in various contexts.

8.1.1.5

Express approximations of very large and very small numbers using scientific notation; understand how calculators display numbers in scientific notation. Multiply and divide numbers expressed in scientific notation, express the answer in scientific notation, using the correct number of significant digits when physical measurements are involved.

For example: , but if these numbers represent physical measurements, the answer should be expressed as because the first factor, , only has two significant digits.

Glencoe McGraw-Hill

Algebra 1

Ch.  7.3

Algebra

MCA 24-30 items

Understand the concept of function in real-world and mathematical situations, and distinguish between linear and nonlinear functions.

MCA 4-5 items

8.2.1.1

Understand that a function is a relationship between an independent variable and a dependent variable in which the value of the independent variable determines the value of the dependent variable.  Use functional notation, such as f(x), to represent such relationships.

For example: The relationship between the area of a square and the side length can be expressed as . In this case, , which represents the fact that a square of side length 5 units has area 25 units squared.

Glencoe McGraw-Hill

Algebra 1

Ch.  1-6

Ch.  1-7

Ch.  3-4

Ch.  4-5

Ch.  4-6

Ch.  4-7

Ch.  9-1

8

Algebra

Understand the concept of function in real-world and mathematical situations, and distinguish between linear and nonlinear functions.

8.2.1.2

Use linear functions to represent relationships in which changing the input variable by some amount leads to a change in the output variable that is a constant times that amount.

For example: Uncle Jim gave Emily $50 on the day she was born and $25 on each birthday after that. The functionrepresents the amount of money Jim has given after x years. The rate of change is $25 per year.

Glencoe McGraw-Hill

Algebra 1

Ch.  3-1

Ch.  3-2

Ch.  3-3

Ch.  3-4

Ch.  3-5

Ch.  3-6

Ch.  4-1

Ch.  4-2

Ch.  4-3

Ch.  4-4

Ch.  4-5

Ch.  4-6

8.2.1.3

Understand that a function is linear if it can be expressed in the form or if its graph is a straight line.

For example: The functionis not a linear function because its graph contains the points (1,1), (-1,1) and (0,0), which are not on a straight line.

Glencoe McGraw-Hill

Algebra 1

Ch.  1-7

Ch.  3-1

Ch.  3-2

Ch.  3-3

Ch.  3-4

Ch.  3-5

Ch.  3-6

Ch.  4-1

Ch.  4-2

Ch.  4-3

Ch.  4-4

Ch.  4-5

Ch.  4-6

Ch.  9-9

8.2.1.4

Understand that an arithmetic sequence is a linear function that can be expressed in the form, where            x = 0, 1, 2, 3,….

For example: The arithmetic sequence 3, 7, 11, 15, …, can be expressed as f(x) = 4x + 3.

Glencoe McGraw-Hill

Algebra 1

Ch.  3-5

Ch.  3-6

8.2.1.5

Understand that a geometric sequence is a non-linear function that can be expressed in the form , where

x = 0, 1, 2, 3,….

For example: The geometric sequence 6, 12, 24, 48, … , can be expressed in the form f(x) = 6(2x).

Glencoe McGraw-Hill

Algebra 1

Ch.  9-6

Ch.  9-8

Ch.  9-9

Supplementary Worksheets

8

Algebra

Recognize linear functions in real-world and mathematical situations; represent linear functions and other functions with tables, verbal descriptions, symbols and graphs; solve problems involving these functions and explain results in the original context.

8.2.2.1

Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another.

Glencoe McGraw-Hill

Algebra 1

Ch.  3-1

Ch.  3-2

Ch.  3-3

Ch.  3-4

Ch.  3-5

Ch.  3-6

Ch.  4-1

Ch.  4-2

Ch.  4-3

Ch.  4-4

Ch.  4-5

Ch.  4-6

Ch.  9-9

MCA 4-6 items

8.2.2.2

Identify graphical properties of linear functions including slopes and intercepts. Know that the slope equals the rate of change, and that the y-intercept is zero when the function represents a proportional relationship.

Glencoe McGraw-Hill

Algebra 1

Ch.  3-1

Ch.  3-2

Ch.  3-3

Ch.  3-4

Ch.  3-5

Ch.  3-6

8.2.2.3

Identify how coefficient changes in the equation f (x) = mx + b affect the graphs of linear functions. Know how to use graphing technology to examine these effects.

Glencoe McGraw-Hill

Algebra 1

Ch.  3-2

Ch.  3-3

Ch.  3-4

Ch.  3-5

Ch.  3-6

Ch.  4-1

Ch.  4-4

Ch.  6-1

8.2.2.4

Represent arithmetic sequences using equations, tables, graphs and verbal descriptions, and use them to solve problems.

For example: If a girl starts with $100 in savings and adds $10 at the end of each month, she will have 100 + 10x dollars after x months.

Glencoe McGraw-Hill

Algebra 1

Ch.  3-5

Ch.  3-6

8.2.2.5

Represent geometric sequences using equations, tables, graphs and verbal descriptions, and use them to solve problems.

For example: If a girl invests $100 at 10% annual interest, she will have 100(1.1x) dollars after x years.

Glencoe McGraw-Hill

Algebra 1

Ch.  9-6

Ch.  9-7

Ch.  9-8

Ch.  9-9

8

Algebra

Generate equivalent numerical and algebraic expressions and use algebraic properties to evaluate expressions.

MCA 3-5 items

8.2.3.1

Evaluate algebraic expressions, including expressions containing radicals and absolute values, at specified values of their variables.

For example: Evaluate πr2h when r = 3 and h = 0.5, and then use an approximation of π to obtain an approximate answer.

Glencoe McGraw-Hill

Algebra 1

Ch.  1-2

Ch.  1-7

Ch.  2-5

Ch.  9-1

8.2.3.2

Justify steps in generating equivalent expressions by identifying the properties used, including the properties of algebra. Properties include the associative, commutative and distributive laws, and the order of operations, including grouping symbols.

Glencoe McGraw-Hill

Algebra 1

Ch.  1-2

Ch.  1-3

Ch.  1-4

Ch.  2-3

Ch.  2-4

8

Algebra

Represent real-world and mathematical situations using equations and inequalities involving linear expressions. Solve equations and inequalities symbolically and graphically. Interpret solutions in the original context.

MCA 10-15 items

8.2.4.1

Use linear equations to represent situations involving a constant rate of change, including proportional and non-proportional relationships.

For example: For a cylinder with fixed radius of length 5, the surface area  A = 2π(5)h + 2π(5)2 = 10πh + 50π, is a linear function of the height h, but the surface area is not proportional to the height.

Glencoe McGraw-Hill

Algebra 1

Ch.  3-1

Ch.  3-2

Ch.  3-3

Ch.  3-4

Ch.  3-5

Ch.  3-6

Ch.  4-1

Ch.  4-2

Ch.  4-3

8.2.4.2

Solve multi-step equations in one variable. Solve for one variable in a multi-variable equation in terms of the other variables. Justify the steps by identifying the properties of equalities used.

For example: The equation 10x + 17 = 3x can be changed to 7x + 17 = 0, and then to 7x = -17 by adding/subtracting the same quantities to both sides. These changes do not change the solution of the equation.

Another example: Using the formula for the perimeter of a rectangle, solve for the base in terms of the height and perimeter.

Glencoe McGraw-Hill

Algebra 1

Ch.  2-3

Ch.  2-4

Ch.  2-5

Ch.  2-6

Ch.  2-7

Ch.  2-8

Ch.  2-9

8.2.4.3

Express linear equations in slope-intercept, point-slope and standard forms, and convert between these forms. Given sufficient information, find an equation of a line.

For example: Determine an equation of the line through the points (-1,6) and (2/3, -3/4).

Glencoe McGraw-Hill

Algebra 1

Ch.  3-1

Ch.  3-2

Ch.  3-3

Ch.  3-4

Ch.  3-5

Ch.  3-6

Ch.  4-1

Ch.  4-2

Ch.  4-3

Ch.  4-4

Ch.  4-5

Ch.  4-6

8

Algebra

Represent real-world and mathematical situations using equations and inequalities involving linear expressions. Solve equations and inequalities symbolically and graphically. Interpret solutions in the original context.

8.2.4.4

Use linear inequalities to represent relationships in various contexts.

For example: A gas station charges $0.10 less per gallon of gasoline if a customer also gets a car wash. Without the car wash, gas costs $2.79 per gallon. The car wash is $8.95. What are the possible amounts (in gallons) of gasoline that you can buy if you also get a car wash and can spend at most $35?

Glencoe McGraw-Hill

Algebra 1

Ch.  5-1

Ch.  5-2

Ch.  5-3

Ch.  5-4

Ch.  5-5

Ch.  5-6

8.2.4.5

Solve linear inequalities using properties of inequalities. Graph the solutions on a number line.

For example: The inequality -3x < 6 is equivalent to x > -2, which can be represented on the number line by shading in the interval to the right of -2.

Glencoe McGraw-Hill

Algebra 1

Ch.  5-1

Ch.  5-2

Ch.  5-3

Ch.  5-4

Ch.  5-5

Ch.  5-6

8.2.4.6

Represent relationships in various contexts with equations and inequalities involving the absolute value of a linear expression. Solve such equations and inequalities and graph the solutions on a number line.

For example: A cylindrical machine part is manufactured with a radius of 2.1 cm, with a tolerance of 1/100 cm. The radius r satisfies the inequality   |r – 2.1| ≤ .01.

Glencoe McGraw-Hill

Algebra 1

Ch.  2-5

Ch.  5-5

8

Algebra

Represent real-world and mathematical situations using equations and inequalities involving linear expressions. Solve equations and inequalities symbolically and graphically. Interpret solutions in the original context.

8.2.4.7

Represent relationships in various contexts using systems of linear equations. Solve systems of linear equations in two variables symbolically, graphically and numerically.

For example: Marty's cell phone company charges $15 per month plus $0.04 per minute for each call. Jeannine's company charges $0.25 per minute. Use a system of equations to determine the advantages of each plan based on the number of minutes used.

Glencoe McGraw-Hill

Algebra 1

Ch.  6-1

Ch.  6-2

Ch.  6-3

Ch.  6-4

Ch.  6-5

Ch.  6-7

8.2.4.8

Understand that a system of linear equations may have no solution, one solution, or an infinite number of solutions. Relate the number of solutions to pairs of lines that are intersecting, parallel or identical. Check whether a pair of numbers satisfies a system of two linear equations in two unknowns by substituting the numbers into both equations.

Glencoe McGraw-Hill

Algebra 1

Ch.  6-1

Ch.  6-2

8.2.4.9

Use the relationship between square roots and squares of a number to solve problems.

For example: If πx2 = 5, then , or equivalently, or . If x is understood as the radius of a circle in this example, then the negative solution should be discarded and .

Glencoe McGraw-Hill

Algebra 1

Ch.  8-6

8

Geometry & Measurement

MCA 8-10 items

Solve problems involving right triangles using the Pythagorean Theorem and its converse.

8.3.1.1

Use the Pythagorean Theorem to solve problems involving right triangles.

For example: Determine the perimeter of a right triangle, given the lengths of two of its sides.  

Another example: Show that a triangle with side lengths 4, 5 and 6 is not a right triangle.

Glencoe McGraw-Hill

Algebra 1

Ch.  10-5

8.3.1.2

Determine the distance between two points on a horizontal or vertical line in a coordinate system. Use the Pythagorean Theorem to find the distance between any two points in a coordinate system.

Glencoe McGraw-Hill

Algebra 1

Ch.  10-6

8.3.1.3

Informally justify the Pythagorean Theorem by using measurements, diagrams and computer software.

Glencoe McGraw-Hill

Algebra 1

Ch.  10-5

Ch.  10-6

Solve problems involving parallel and perpendicular lines on a coordinate system.

MCA 3-5 items

8.3.2.1

Understand and apply the relationships between the slopes of parallel lines and between the slopes of perpendicular lines. Dynamic graphing software may be used to examine these relationships.

Glencoe McGraw-Hill

Algebra 1

Ch.  4-4

8.3.2.2

Analyze polygons on a coordinate system by determining the slopes of their sides.

For example: Given the coordinates of four points, determine whether the corresponding quadrilateral is a parallelogram.

Glencoe McGraw-Hill

Algebra 1

Ch.  4-4

8.3.2.3

Given a line on a coordinate system and the coordinates of a point not on the line, find lines through that point that are parallel and perpendicular to the given line, symbolically and graphically.

Glencoe McGraw-Hill

Algebra 1

Ch.  4-4

8

Data Analysis & Probability

MCA 6-8 items

Interpret data using scatterplots and approximate lines of best fit. Use lines of best fit to draw conclusions about data.

8.4.1.1

Collect, display and interpret data using scatterplots. Use the shape of the scatterplot to informally estimate a line of best fit and determine an equation for the line. Use appropriate titles, labels and units. Know how to use graphing technology to display scatterplots and corresponding lines of best fit.

Glencoe McGraw-Hill

Algebra 1

Ch.  4-5

Ch.  4-6

8.4.1.2

Use a line of best fit to make statements about approximate rate of change and to make predictions about values not in the original data set.

For example: Given a scatterplot relating student heights to shoe sizes, predict the shoe size of a 5'4" student, even if the data does not contain information for a student of that height.

Glencoe McGraw-Hill

Algebra 1

Ch.  4-5

Ch.  4-6

8.4.1.3

Assess the reasonableness of predictions using scatterplots by interpreting them in the original context.

For example: A set of data may show that the number of women in the U.S. Senate is growing at a certain rate each election cycle. Is it reasonable to use this trend to predict the year in which the Senate will eventually include 1000 female Senators?

Glencoe McGraw-Hill

Algebra 1

Ch.  4-5

Ch.  4-6

Ch.  9-9