Unit

Unit Storyline

Full Objectives

Unit 1

Numbers

Timing

1-3 weeks

Objectives

PA.N.1.4

PA.N.1.5

By investigating the square roots of whole numbers from 1 to 400, students will determine  which produce rational and irrational numbers. With an understanding of the real number system students should be able to classify, compare, and order rational and irrational numbers. Also, students should be able to view operations with each and know if their results are either rational or irrational.

PA.N.1.4 Classify real numbers as rational or irrational. Explain why the rational number system is closed under addition and multiplication and why the irrational system is not. Explain why 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.

Unit 2

Measures of Central Tendency

Timing

1-3 weeks

Objectives

PA.D.1.1

PA.D.1.2

Continuing with the use of rational numbers, students will expand their understanding of calculating mean and median, and determining possible outliers. Using various displays, students  will can draw conclusions including the effects that inserting a data value or deleting a data value will have on the mean, median and range.

PA.D.1.1 Describe the impact that inserting or deleting a data point has on the mean and the median of a data set. Know how to create data displays using a spreadsheet and use a calculator to examine this impact.

Unit 3

Probabilities

Timing

1-3 weeks

Objectives

PA.D.2.1

PA.D.2.2

PA.D.2.3

While working with experimental probabilities, students will calculate both single and compound probabilities, understanding the difference between dependent and independent events and how samples are selected to best represent a larger population.

PA.D.2.1 Calculate experimental probabilities and represent them as percents, fractions and decimals between 0 and 1 inclusive. Use experimental probabilities to make predictions when actual probabilities are unknown.

PA.D.2.2 Determine how samples are chosen (random, limited, biased) to draw and support conclusions about generalizing a sample to a population.

Unit 4

Expressions

Timing

2-4 weeks

Objectives

PA.A.3.1

PA.A.3.2

Algebraic expressions from concrete models and pictorial representations are used to further develop the application of commutative, associative and distributive properties while generating equivalent expressions. Continuing with expressions and the application of the order of operations, students will substitute specified values for variables, evaluating expressions that involve all rational numbers and positive exponents.

PA.A.3.1 Use substitution to simplify and evaluate algebraic expressions.

Unit 5

Equations and Inequalities

Timing

2-4 weeks

Objectives

PA.A.4.1

PA.A.4.2

PA.A.4.3

Applying  inverse algebraic properties and equation solving skills, students will work with linear equations and inequalities that may have one solution, no solutions or infinitely many solutions. Students should be able to represent all possible solutions, graphically, and interpret how they apply in a real-world situation.

PA.A.4.1 Illustrate, write, and solve mathematical and real-world problems using linear equations with one variable with one solution, infinitely many solutions, or no solutions. Interpret solutions in the original context.

PA.A.4.2 Represent, write, solve, and graph problems leading to linear inequalities with one variable in the form px + q > r  and px + q < r , where p, q and r are rational numbers.

Unit 6

Surface Area

Timing

2-4 weeks

Objectives

PA.GM.2.1

PA.GM.2.2

PA.A.3.1*

Implementing concepts of substitution, simplifying expressions and solving equations, students will use area formulas for squares, rectangles and circles to find the surface area of 3D solids. While using nets and area of composite figures, students can relate surface area formulas of prisms and cylinders as the sum of the decomposed parts.

PA.GM.2.1 Calculate the surface area of a rectangular prism using decomposition or nets. Use appropriate measurements such as cm².

PA.GM.2.2 Calculate the surface area of a cylinder, in terms of and using approximations for π, using decomposition or nets. Use appropriate measurements such as cm².

Unit 7

Volume

Timing

2-4 weeks

Objectives

PA.GM.2.3

PA.GM.2.4

PA.A.3.1*

In Pre-Algebra, focus on 3D figures applies to rectangular prisms and cylinders. Students have used multiplication in area formulas of squares, rectangles and circles to find squared units and by stacking several figures with the same area (pad of Post-It-Notes) they establish multiplication of height and the discovery of volume, with cubed units. When applied to real-world scenarios, students will determine when values should be approximated or when they should be written in exact form, i.e. as fractions or in terms of π.

PA.GM.2.3 Develop and use the formulas V = lwh and V = Bh  to determine the volume of rectangular prisms. Justify why base area (B) and height (h) are multiplied to find the volume of a rectangular prism. Use appropriate measurements such as cm³.

PA.GM.2.4 Develop and use the formulas V = π r² h and V = Bh  to determine the volume of right cylinders, in terms of π and using approximations for π. Justify why base area (B) and height (h) are multiplied to find the volume of a right cylinder. Use appropriate measurements such as cm³.

Unit 8

Recognize Data as a Function

Timing

1-3 weeks

Objectives

PA.A.1.1

Student learning is focused on the concept of function. Students explore different types of patterns, of both linear and nonlinear functions, to understand the relationship between the independent and dependent variables. This idea of every input having exactly one output is essential. Students must understand that changing the input leads to a change in the output.

PA.A.1.1 Recognize 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.

Unit 9

Linear Functions-

Timing

2-4 Weeks

Objectives

PA.A.2.2

PA.A.2.3

PA.A.2.4

A proportional relationship is a straight line on a graph, the slope being the unit rate. Students can use various tools to translate ordered pairs from a linear graph to a data table, recognizing that the rate of change is the same in both, interpret the y-intercept as the dependent value when the independent value is zero and describe what happens to a graph when the unit rate is changed.

PA.A.2.2 Identify, describe, and analyze linear relationships between two variables.

PA.A.2.3 Identify graphical properties of linear functions including slope 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.

Unit 10

Linear Functions-

Timing

2-4 Weeks

Objectives

PA.A.1.3

PA.A.2.5

PA.A.1.2

PA.A.2.1

PA.D.1.3

PA.A.3.1*

PA.A.3.2*

Students move from their prior learning of linear relationships that are proportional to all linear functions. When students identify a situation as linear, it is essential that they are able to identify and make meaning of the slope and y-intercept, as they relate to the original context. Multiple representations of tables, graphs and equations are used to find and interpret solutions to real-world linear situations that are modeled graphically by a straight line and by a slope-intercept equation. Students should be able to calculate a table of ordered pairs that best represents data with strong linear correlation, which allows them to identify an appropriate linear model.

PA.A.1.3 Identify a function as linear if it can be expressed in the form y=mx + b or if its graph is a straight line.

PA.A.2.5 Solve problems involving linear functions and interpret results in the original context.

PA.A.1.2 Use linear functions to represent and explain real-world and mathematical situations.

PA.A.2.1 Represent linear functions with tables, verbal descriptions, symbols, and graphs; translate from one representation to another.

PA.D.1.3 Collect, display and interpret data using scatterplots. Use the shape of the scatterplot to informally estimate a line of best fit. Use appropriate titles, labels and units.

*PA.A.3.1 Use substitution to simplify and evaluate algebraic expressions.

Unit 11

Exponents

Timing

2-4 Weeks

Objectives

PA.N.1.1

PA.N.1.2

PA.N.1.3

While learning to use order of operations, students developed an understanding of exponents when evaluating terms with numerical bases. This foundation applies when performing operations on terms of variable bases with integer exponents. A combination of operations with exponential terms on numerical bases of 10, are used in expressing large and small numbers in scientific notation.

PA.N.1.1 Develop and apply the properties of integer exponents, including a⁰ = 1 (with a≠0), to generate equivalent numerical and algebraic expressions.

PA.N.1.2 Express and compare approximations of very large and very small numbers using scientific notation.

Unit 12

Pythagorean

Timing

1-3 Weeks

Objectives

PA.GM.1.1

PA.GM.1.2

PA.N.1.5*

PA.A.3.1*

PA.A.3.2*

Substitution, evaluating expressions, simplifying square roots and working with squared exponents are incorporated in the application of Pythagorean Theorem. Students will investigate the specific relationship between the three sides of a right triangle and solve for missing side lengths. When the vertices of a right triangle can be mapped onto the coordinate plane, students discover finding the longest side of the triangle, turns the Pythagorean Theorem into the distance formula for finding the distance between two points.Begin by letting students explore angles with protractors and creating polygons from triangles. Students should have mastered one step equations at this point so reinforce this process by setting up missing angle problems as an algebraic equation. Ex.  x+20+60=180. Solve for x.

PA.GM.1.1 Informally justify the Pythagorean Theorem using measurements, diagrams, or dynamic software and use the Pythagorean Theorem to solve problems in two and three dimensions involving right triangles.

PA.GM.1.2 Use the Pythagorean Theorem to find the distance between any two points in a coordinate plane.

*PA.N.1.5 Compare real numbers; locate real numbers on a number line. Identify the square root of a perfect square to 400 or, if it is not a perfect square root, locate it as an irrational number between two consecutive positive integers.

*PA.A.3.1 Use substitution to simplify and evaluate algebraic expressions.