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Mathematics: Pre-Algebra EightStudents in 8th Grade Pre-Algebra will study expressions and equations using one or two unknown values. Students will be introduced to the concept of a mathematical function and use functions to describe relationships between quantities. A function is a special relationship in math where a value converts to one value and only one value.The study of expressions and equations will include rational numbers and irrational (2) numbers. Additionally, students will use equations to model and solve problems.The study of figures will include the special properties of right triangles, the volume of cylinders, cones and spheres and the movement of similar shapes in coordinate geometry. Students will analyze two- and three-dimensional figures using distance, angles and congruence.After successful completion of this course, students will be eligible to take Algebra 1.Content: Identifying Rational and Irrational NumbersDuration: September (1 week)Essential Question:How is mathematics used to quantify, compare, represent and model numbers?Skills: Determine whether a number is rational or irrational. For rational numbers, show that the decimal expansion terminates or repeats (limit repeating decimals to thousandths). Convert a terminating or repeating decimal to a rational number (limit repeating decimals to thousandths).Assessment:Are the following numbers rational or irrational? 2,0,-3,34,-45,0.125,-0.5,0.3, 2,πResources:Mathematics Course 3, Prentice Hall, p. 106Standards:CC.2.1.8.E.1 Distinguish between rational and irrational numbers using their properties.Vocabulary:Irrational number – cannot be written as the quotient of two relatively prime whole numbersComments: Content: Estimating Irrational NumbersDuration: September (3 days)Essential Question:What does it mean to estimate numerical quantities?Skills: Estimate the value of irrational numbers without a calculator (limit whole number radicand to less than 144).Use rational approximations of irrational numbers to compare and order irrational numbers.Locate/identify rational and irrational numbers at their approximate locations on a number line.Assessment:5 is between 2 and 3 but closer to 2Resources:Mathematics Course 3, Prentice Hall, p. 107Standards:CC.2.1.8.E.4 Estimate irrational numbers by comparing them to rational numbers.Vocabulary:Comments: Content: One- and Two-step EquationsDuration: September (1 week)Essential Question:How can mathematical models solve real-world problems?Skills: Write and identify linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms until an equivalent equation of the form x=a, a=a, or a=b results (where a and b are different numbers).Assessment:Solve ax=b, ax+b=c.Resources:Mathematics Course 3, Prentice Hall, p. 33, 38, 261Standards:CC.2.2.8.B.3 Analyze and solve linear equations and pairs of simultaneous linear equations.Vocabulary:Comments: Content: Multi-step EquationsDuration: September (2 weeks) Essential Question:How can multi-step mathematical models represent real-world problems?Skills: Solve linear equations that have rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve an equation with variables on both sides.Assessment:Solve ax+bx=c, ax+b=c, ax=bxResources:Mathematics Course 3, p. 271Standards:CC.2.2.8.B.3 Analyze and solve linear equations and pairs of simultaneous linear equations.Vocabulary:Comments: Content: RadicalsDuration: September-October (2-3 days )Essential Question:How are irrational numbers communicated?Skills: Use square root and cube root symbols to represent solutions to equations of the form x2= p and x3= p, where p is a positive rational number.Evaluate square roots of perfect squares and cube roots of perfect cubes without a calculator. Assessment:If x2 = 25 then x=±25Resources:Mathematics Course 3, p. 107Standards:CC.2.2.8.B.1 Apply concepts of radicals and integer exponents to generate equivalent expressions.Vocabulary:Radical – the designated root of a number, such as square root or cube rootComments: Content: Integer ExponentsDuration: OctoberSeptember (3-41 weeks)Essential Question:How is mathematics used to quantify, compare, represent and model numbers?Skills: Apply one or more properties of integer exponents to generate equivalent numerical expressions without a calculator (with final answers expressed in exponential form with positive exponents).Estimate very large or very small quantities by using numbers expressed in the form of a single digit times an integer power of 10 and express how many times larger or smaller one number is than another. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Express answers in scientific notation and choose units of appropriate size for measurements of very large or very small quantities.Interpret scientific notation that has been generated by technology.Assessment:312× 3-15=3-3=133Estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109 and determine that the world population is more than 20 times larger than the United States’ population.Use millimeters per year for seafloor spreadingInterpret 4.7EE9 displayed on a calculator as 4.7×109Resources:Mathematics Course 3, Prentice Hall, p. 571Standards:CC.2.2.8.B.1 Apply concepts of radicals and integer exponents to generate equivalent expressions.Vocabulary:Comments: Content: RadicalsDuration: September (1 week)Essential Question:How are irrational numbers communicated?Skills: Use square root and cube root symbols to represent solutions to equations of the form x2= p and x3= p, where p is a positive rational number.Evaluate square roots of perfect squares and cube roots of perfect cubes without a calculator. Assessment:If x2 = 25 then x=±25Resources:Mathematics Course 3, p. 107Standards:CC.2.2.8.B.1 Apply concepts of radicals and integer exponents to generate equivalent expressions.Vocabulary:Radical – the designated root of a number, such as square root or cube rootComments: Content: VolumeDuration: October (1 week)Essential Question:How can the application of the attributes of geometric shapes support mathematical reasoning and problem solving?Skills: Apply formulas for the volumes of cones, cylinders, and spheres to solve real-world and mathematical problems.Assessment:Find the volume of cylinders, cones and spheres.Resources:Mathematics Course 3, p. 381, 389, 393Standards:CC.2.3.8.A.1 Apply the concepts of volume of cylinders, cones, and spheres to solve real-world and mathematical problems.Vocabulary:Comments: Content: TransformationsDuration: NovemberOctober (1 week) Essential Question:How are spatial relationships used to draw real situations or solve problems?Skills: Identify and apply properties of rotations, reflections, and translations.Assessment:Show that angle measures are preserved in rotations, reflections and translations.Resources:Mathematics Course 3, Prentice Hall, p. 136, 141, 146.Standards:CC.2.3.8.A.2 Understand and apply congruence, similarity, and geometric transformations using various tools.Vocabulary:Reflection – a transformation that flips a figure over a line; Rotation – a transformation that turns a figure about a fixed point; Transformation – a change in the position, shape or size of a figure; Translation – a transformation that moves each point of a figure the same distance in the same directionComments: Content: Describe TransformationsDuration: NovemberOctober (3 days)Essential Question:How are spatial relationships used to construct real situations?Skills: Given two congruent figures, describe a sequence of transformations that exhibits the congruence between them.Assessment:List the transformations that an original figure to a copy.Resources:Mathematics Course 3, Prentice Hall, p. 136, 141, 146.Standards:CC.2.3.8.A.2 Understand and apply congruence, similarity, and geometric transformations using various tools.Vocabulary:Comments: Content: Coordinate TransformationsDuration: NovemberOctober (1 week) Essential Question:How are spatial relationships used to model real situations and solve problems?Skills: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Assessment:Find the coordinates of the image of an original figure after dilation.Resources:Mathematics Course 3, Prentice Hall, p. 187Standards:CC.2.3.8.A.2 Understand and apply congruence, similarity, and geometric transformations using various tools.Vocabulary:Dilation – a transformation in which the figure and its image are similar, but the scale of the original has changedComments: Content: Compare by TransformingDuration: November (1 week) Essential Question:How are spatial relationships used to represent real situations and solve problems?Skills: Given two similar two-dimensional figures, describe a sequence of transformations that exhibits the similarity between them.Assessment:Write a rule to describe the translation of an original figure.Resources:Mathematics Course 3, Prentice Hall, p. 136, 141, 146, 187Standards:CC.2.3.8.A.2 Understand and apply congruence, similarity, and geometric transformations using various tools.Vocabulary:Comments: Content: VolumeDuration: December (1 week)Essential Question:How can the application of the attributes of geometric shapes support mathematical reasoning and problem solving?Skills: Apply formulas for the volumes of cones, cylinders, and spheres to solve real-world and mathematical problems.Assessment:Find the volume of cylinders, cones and spheres.Resources:Mathematics Course 3, p. 381, 389, 393Standards:CC.2.3.8.A.1 Apply the concepts of volume of cylinders, cones, and spheres to solve real-world and mathematical problems.Vocabulary:Comments: Content: Finding Right TrianglesDuration: DecemberNovember (3 days) Essential Question:How can the application of the attributes of geometric shapes support mathematical reasoning?Skills: Apply the converse of the Pythagorean theorem to show a triangle is a right triangle.Assessment:Is a triangle with the given sides a right triangle?Resources:Mathematics Course 3, Prentice Hall, p. 122Standards:CC.2.3.8.A.3 Understand and apply the Pythagorean Theorem to solve problems.Vocabulary:Hypotenuse – the longest side of a right triangle, opposite the right angle; Pythagorean theorem – an equation that shows the relationship between the legs and the hypotenuse of a right triangleComments: Content: Solving Right TrianglesDuration: DecemberNovember (3 days)Essential Question:How can geometric properties be used to describe situations?Skills: Apply the Pythagorean theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.Assessment:Find the missing length, either a, b, or c of a right triagle?Resources:Mathematics Course 3, Prentice Hall, p. 112Standards:CC.2.3.8.A.3 Understand and apply the Pythagorean Theorem to solve problems.Vocabulary:Comments: Content: Coordinate DistanceDuration: DecemberNovember/42-4 days Essential Question:How can geometric properties be used to model and analyze situations?Skills: Apply the Pythagorean theorem to find the distance between two points in a coordinate system.Assessment:From a graph of a right triangle, find a missing leg or the hypotenuse.Resources:Mathematics Course 3, Prentice Hall, p. 125Standards:CC.2.3.8.A.3 Understand and apply the Pythagorean Theorem to solve problems.Vocabulary:Comments: Content: Slope-interceptDuration: January (2 weeks)Essential Question:How are equations and graphs related?Skills: Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.Assessment:Given a grsaph identify the vertical intercept and the slope of the line.Differentiate between an intercept of zero and bResources:Mathematics Course 3, Prentice Hall, p. 534Standards:CC.2.2.8.B.2 Understand the connections between proportional relationships, lines, and linear equations.Vocabulary:Vertical intercept – point at which a graph crosses the y-axisComments: Content: FunctionsDuration: January (1 week)Essential Question:How is a mathematical situation represented by a function?Skills: Determine whether a relation is a function.Assessment:Mapping domain and range, does the relationship represent a function?Resources:Mathematics Course 3, p. 523Standards:CC.2.2.8.C.1 Define, evaluate, and compare functions.Vocabulary:Function – a relationship that assigns exactly one output value to each input valueComments: Content: Compare FunctionsDuration: January (1 week) Essential Question:How are functions similar?Skills: Compare properties of two functions, each represented in a different way (i.e., algebraically, graphically, numerically in tables, or by verbal descriptions).Assessment:Given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of changeResources:Mathematics Course 3, p. 527Standards:CC.2.2.8.C.1 Define, evaluate, and compare functions.Vocabulary:Comments: Content: Graph a Direct VariationDuration: FebruaryDecember (64 days)Essential Question:How are direct relationships represented mathematically?Skills: Graph proportional relationships, interpreting the unit rate as the slope of the pare two different proportional relationships represented in different ways.Assessment:Compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.Resources:Mathematics Course 3, p. 130Standards:CC.2.2.8.B.2 Understand the connections between proportional relationships,lines, and linear equations.Vocabulary:Comments: Content: Linear and Non-linear FunctionsDuration: February (1 week)Essential Question:How are mathematical situations represented by linear equations?Skills: Interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear.Assessment:Identify y=mx+b, y=x2, y=x3, and y=xResources:Algebra 1 Common Core, p. 240, 246Standards:CC.2.2.8.C.1 Define, evaluate, and compare functions.Vocabulary:Comments: Content: Model Linear FunctionsDuration: February (2 weeks) Essential Question:How are mathematical situations represented by a function?Skills: Construct a function to model a linear relationship between two quantities.Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph.Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values.Assessment:Based on a table, find the linear equation to represent it.Find the slope from two points.Resources:Algebra 1 Common Core p. 240Standards:CC.2.2.8.C.2 Use concepts of functions to model relationships between quantities.Vocabulary:Comments: Content: Sketch Graphs of FunctionsDuration: February (1 week) Essential Question:How can mathematical situation be compared graphically?Skills: Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch or determine a graph that exhibits the qualitative features of a function that has been described verbally.Assessment:Sketch graphs and interpret graphsResources:Mathematics Course 3, p. 518Standards:CC.2.2.8.C.2 Use concepts of functions to model relationships between quantities.Vocabulary:Comments: Content: Compare Similar Right Triangles By SlopeDuration: MarchDecember (3 days) Essential Question:How can similar right triangles compare slope?Skills: Use similar right triangles to show and explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane.Assessment:Given the vertical and horizontal change of two separate sets of ponts that form a right triangle, calculate the slope of each to compare.Resources:Mathematics Course 3, p. 527Standards:CC.2.2.8.B.2 Understand the connections between proportional relationships, lines, and linear equations.Vocabulary:Slope – the ratio of vertical change to horizontal change of points on a linear graphComments: Content: Scatter PlotsDuration: March (1 week) Essential Question:How can data be organized to provide insight to the relationship between quantities?Skills: Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative correlation, linear association, and nonlinear association.Assessment:Given two sets of data, determine whether the data correlate. Predict values from the display.Resources:Mathematics Course 3, p. 444Standards:CC.2.4.8.B.1 Analyze and/or interpret bivariate data displayed in multiple representations.Vocabulary:Comments: Content: Linear RegressionDuration: March (1 week) Essential Question:How does a linear relationship represent two sets of data?Skills: For scatter plots that suggest a linear association, identify a line of best fit by judging the closeness of the data points to the line.Assessment:Create a linear equation from points on a scatter plot.Resources:Mathematics Course 3, p. 444Standards:CC.2.4.8.B.1 Analyze and/or interpret bivariate data displayed in multiple representations.Vocabulary:Comments: Content: Interpret Linear RegressionDuration: March (1 week) Essential Question:How can you predict results from a scatter plot?Skills: Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.Assessment:In a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant heightResources:Mathematics Course 3, p. 444Standards:CC.2.4.8.B.1 Analyze and/or interpret bivariate data displayed in multiple representations.Vocabulary:Comments: Content: Slope-interceptDuration: December (1 week)Essential Question:How are equations and graphs related?Skills: Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.Assessment:Given a grsaph identify the vertical intercept and the slope of the line.Differentiate between an intercept of zero and bResources:Mathematics Course 3, Prentice Hall, p. 534Standards:CC.2.2.8.B.2 Understand the connections between proportional relationships, lines, and linear equations.Vocabulary:Vertical intercept – point at which a graph crosses the y-axisComments: Content: One- and Two-step EquationsDuration: December (1 week)Essential Question:How can mathematical models solve real-world problems?Skills: Write and identify linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms until an equivalent equation of the form x=a, a=a, or a=b results (where a and b are different numbers).Assessment:Solve ax=b, ax+b=c.Resources:Mathematics Course 3, Prentice Hall, p. 33, 38, 261Standards:CC.2.2.8.B.3 Analyze and solve linear equations and pairs of simultaneous linear equations.Vocabulary:Comments: Content: Multi-step EquationsDuration: December (1 week) Essential Question:How can multi-step mathematical models represent real-world problems?Skills: Solve linear equations that have rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.Assessment:Solve ax+bx=c, ax+b=c, ax=bxResources:Mathematics Course 3, p. 271Standards:CC.2.2.8.B.3 Analyze and solve linear equations and pairs of simultaneous linear equations.Vocabulary:Comments: Content: Systems of Linear EquationsDuration: March-AprilJanuary (2-3 weeks)Essential Question:How can systems of equations model mathematically situations?Skills: Solve systems of two linear equations in two variables algebraically and estimate solutions by graphing the equations.Solve simple cases by inspection.Solve real-world and mathematical problemsleading to two linear equations in two variables.Assessment:Solve x+y=ax-y=b by graphing, substitution or eliminationGiven coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.Resources:Algebra 1 Common Core, p. 364Standards:CC.2.2.8.B.3 Analyze and solve linear equations and pairs of simultaneous linear equations.Vocabulary:Comments: Content: FunctionsDuration: January (1 week)Essential Question:How is a mathematical situation represented by a function?Skills: Determine whether a relation is a function.Assessment:Mapping domain and range, does the relationship represent a function?Resources:Mathematics Course 3, p. 523Standards:CC.2.2.8.C.1 Define, evaluate, and compare functions.Vocabulary:Function – a relationship that assigns exactly one output value to each input valueComments: Content: Compare FunctionsDuration: January (1 week) Essential Question:How are functions similar?Skills: Compare properties of two functions, each represented in a different way (i.e., algebraically, graphically, numerically in tables, or by verbal descriptions).Assessment:Given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of changeResources:Mathematics Course 3, p. 527Standards:CC.2.2.8.C.1 Define, evaluate, and compare functions.Vocabulary:Comments: Content: Linear and Non-linear FunctionsDuration: February (1 week)Essential Question:How are mathematical situations represented by linear equations?Skills: Interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear.Assessment:Identify y=mx+b, y=x2, y=x3, and y=xResources:Algebra 1 Common Core, p. 240, 246Standards:CC.2.2.8.C.1 Define, evaluate, and compare functions.Vocabulary:Comments: Content: Model Linear FunctionsDuration: February (1 week) Essential Question:How are mathematical situations represented by a function?Skills: Construct a function to model a linear relationship between two quantities.Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph.Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values.Assessment:Based on a table, find the linear equation to represent it.Find the slope from two points.Resources:Algebra 1 Common Core p. 240Standards:CC.2.2.8.C.2 Use concepts of functions to model relationships between quantities.Vocabulary:Comments: Content: Sketch Graphs of FunctionsDuration: February (1 week) Essential Question:How can mathematical situation be compared graphically?Skills: Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch or determine a graph that exhibits the qualitative features of a function that has been described verbally.Assessment:Sketch graphs and interpret graphsResources:Mathematics Course 3, p. 518Standards:CC.2.2.8.C.2 Use concepts of functions to model relationships between quantities.Vocabulary:Comments: Content: Scatter PlotsDuration: March (1 week) Essential Question:How can data be organized to provide insight to the relationship between quantities?Skills: Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative correlation, linear association, and nonlinear association.Assessment:Given two sets of data, determine whether the data correlate. Predict values from the display.Resources:Mathematics Course 3, p. 444Standards:CC.2.4.8.B.1 Analyze and/or interpret bivariate data displayed in multiple representations.Vocabulary:Comments: Content: Linear RegressionDuration: March (1 week) Essential Question:How does a linear relationship represent two sets of data?Skills: For scatter plots that suggest a linear association, identify a line of best fit by judging the closeness of the data points to the line.Assessment:Create a linear equation from points on a scatter plot.Resources:Mathematics Course 3, p. 444Standards:CC.2.4.8.B.1 Analyze and/or interpret bivariate data displayed in multiple representations.Vocabulary:Comments: Content: Interpret Linear RegressionDuration: March (1 week) Essential Question:How can you predict results from a scatter plot?Skills: Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.Assessment:In a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant heightResources:Mathematics Course 3, p. 444Standards:CC.2.4.8.B.1 Analyze and/or interpret bivariate data displayed in multiple representations.Vocabulary:Comments: Content: Two-Way TablesDuration: April (1 week) Essential Question:How can mathematical decisions be made based on two collected data sets?Skills: Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible associations between the two variables.Assessment:Given data on whether students have a curfew on school nights and whether they have assigned chores at home, is there evidence that those who have a curfew also tend to have chores?Resources:Mathematics Course 3, p. 435Standards:CC.2.4.8.B.2 Understand that patterns of association can be seen in bivariate data utilizing frequencies.Vocabulary:Comments: Content: Adding PolynomialsDuration: MayApril (1 week) Essential Question:How can simplifying polynomials make calculations easier?Skills: Write variable expressions and simplify polynomialsAssessment:Write and simplify an expression for the area of multiple figures.Resources:Mathematics Course 3, p. 561Standards:A1.1.1.5.1:?Add, subtract and/or multiply polynomial expressions (express answers in simplest form – nothing larger than a binomial multiplied by a trinomial).Vocabulary:Polynomial – a multi-termed expressionComments: Content: Multiplying PolynomialsDuration: MayApril (1-2 weeks) Essential Question:How can simplifying polynomials make calculations easier?Skills: Multiplying monomials and binomialsAssessment:Simplify polynomials using multiplication.Resources:Mathematics Course 3, p. 576Standards:A1.1.1.5.1:?Add, subtract and/or multiply polynomial expressions (express answers in simplest form – nothing larger than a binomial multiplied by a trinomial).Vocabulary:Comments: ................
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