Eighth Grade common core curriculum with D. Anzalone.doc

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Eighth Grade Curriculum

Each unit is given a range of days it will take to complete. Teachers can use this range at their discretion. I have included a 5 day buffer for assemblies and snow days. I have already counted 9 days for NYS test review and the NYS Exam, 9 days for final exam review and in-class finals, and 8 days for final exams. Depending on when the state exam is given in May, teachers may need to use the low end of the range or may be able to use the high end of the range. Number Theory (16 - 17 days)

Algebra I – Equations (16 –17 days)

Algebra II – Writing and Solving Word Problems (12 – 13 days) Algebra III – Advanced Equations and Inequalities (17 – 18 days) Graphing on the Coordinate Plane (19 – 20 days)

Geometry – Angles (14 – 15 days)

Transformational Geometry (17 – 18 days) Statistics and Probability (17 - 18 days) Geometry – Solids (11 – 12 days) Pythagorean Theorem (11 – 12 days)

Miscellaneous (31 days)

Assemblies and snow days (5 days) NYS Test Review and NYS Test (9 days) Final Exam Review and in-class finals (9 days) Final Exam Week (8 days)

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Number Theory Description

Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems. Students will understand meanings of operations and procedures, and how they relate to one another. Students will compute accurately and make reasonable estimates. Students will know that there are numbers that are not rational and approximate them by rational numbers. They will work with radicals and integer exponents. This unit will take approximately 16-18 days.

Vocabulary

Base Cube Root Radicand

Terminating Decimal Natural Number Rational Number Counting Number Non-repeating Decimal Real Number

Estimate Perfect Square Square Root

Power Standard Notation Square

Integer Subset Cube

Irrational Number Whole Number Radical

Round Unit

 Subsets of the Real Number System (1/2 day)

 Counting/Natural Numbers, Whole Numbers, Integers, Rational Numbers, and Irrational Numbers

 8.NS.1

 Rational versus Irrational Numbers (1/2 day)

 Irrational Number as a non-repeating, non-terminating decimal  8.NS.1

 Rational and Irrational Numbers on a Number Line (1 day)  8.NS.2

 Exponents-zero and negatives (2 days)  Relate to fractions and decimals  8.EE.1

 Law of Exponents (2 days)  Multiply, Divide, and Powers  With Monomials also

 8.EE.1

 Square Roots and Cube Roots (4 days)  Perfect Squares

 Non- Perfect Squares with a calculator

 Approximating square roots and irrational numbers  8.NS.2, 8.EE.2

 Scientific Notation (3 days)

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 Translate numbers from Scientific Notation into Standard Form  Compare

 Calculate  8.EE.3, 8.EE.4

 Compare Rational and Irrational Numbers (2 days)  8.NS.2

 Review and Exam (2 days)

Sample Questions from Past Assessment Exams 1. Which value of x will make the equation below true?

3x_{= 243} A. 3

B. 5 C. 8 D. 9

2. Bethany watched a movie about a spaceship that traveled 93 million miles through space. What is the number of miles the spaceship traveled in scientific notation?

F. 9.3 x 106 G. 93 x 106 H. 9.3 x 107 J. 93 x 107

3. The sum of two numbers is 6. The sum of the squares of the same two numbers is 26. What are the numbers?

F. 6 and 0 G. 3 and 3 H. 2 and 4 J. 1 and 5

4. Consider the real numbers M, N, P, and Q, and the following relationships: M< N

N < P Q> P

Which statement is always true? A. Q - P > Q - N

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5. On the line below, arrange these numbers in order from smallest to largest.

Answer

__________________________________________________________

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Algebra Description

Students will represent and analyze algebraically a wide variety of problem solving situations. Students will perform algebraic procedures accurately. Students will recognize, use, and represent algebraically patterns, relations, and functions. Students will grasp the concept of a function as a rule that assigns to each input exactly one output. They will understand that functions describe situations where one quantity determines another. They will be able to translate among representations and partial representations of functions (noting that tabular and graphical representations may be partial representations), and they will be able to describe how aspects of the function are reflected in the different representations. Students will strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use the properties of equality and the concept of logical equivalence, they maintain the solutions of the original equation. Students will solve systems of two linear equations in two variables. This unit will be broken up into three chapters lasting 12-16 days, 11-15 days and 16-20 days respectively.

Vocabulary

One-step equation Two-step equation Like terms Distributive Property Solution Greater Than

Less Than Greater than or equal to Less than or equal to

Not equal to Product Sum

Difference Quotient Variable

Coefficient Expression Check

Consecutive Integers Conversion Patterns

Sequence Arithmetic Sequence Geometric Sequence

Perimeter Set-up Let statement

Evaluate Solve

Chapter 1

 Translate verbal sentences into algebraic inequalities (.5 day)

 Write verbal expressions that match given mathematical expressions (.5 day)  Understand that numerical information can be represented in multiple ways:

arithmetically, algebraically, and graphically. (Make an equation from a table.) (2 days)

 8.F.2

 Create a graph given a description or an expression for a situation involving a linear or nonlinear relationship. Determine the rate of change and initial value of the function. (Make a graph from a table.) (2 days)

 8.F.4

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 8.F.1

 Describe a situation involving relationships that matches a given graph (Make an equation from a graph.) (2 days)

 Linear and Non-linear  8.F.5

 Compare Properties of Functions (1 day)  Qualitative Graphs (2 days)

 Review 2 step Equations (3 days)  Review and Exam (2 days)

Sample Assessment Questions

1. What is the next number in this sequence?

F 1,240 G 1,275 H 1,920 J 3,840

2. Johan is counting all the pumpkins in his family’s field just before the harvest. A diagram of the entire field divided into sections is shown below.

He counted 358 pumpkins in section A. Assuming the pumpkins are evenly distributed across the entire field, what is the best estimate for the total number of pumpkins in the entire field?

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3. A company that manufactures shoes experienced a decline in shoe sales for a 5-month period, as shown in the table below.

If sales continue to decline at the same rate, what would be the company’s sales, in thousands, for December?

F $15.0 G $13.8 H $12.6 J $11.4

4. Giovanni uses the pattern in the table below to calculate the cost of any number of shirts.

How much would 17 shirts cost? Show your work.

Answer $_________

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5. Joey needs to travel 15 miles from Smithville to Clarksville and 5 miles from Clarksville to Elmwood. The table below shows two different taxicab companies’ rates.

Joey will choose one of these four options:

1. Travel with Sunshine Cab Co. to Clarksville, then with Freedom Cab Co. to Elmwood.

2. Travel with Freedom Cab Co. to Clarksville, then with Sunshine Cab Co. to Elmwood.

3. Travel nonstop with Sunshine Cab Co. the entire way. 4. Travel nonstop with Freedom Cab Co. the entire way. Which is the least expensive way for Joey to make the trip? Show your work

Answer _____________

6. Ingrid is working with the following number pattern: 2 4 16 256 ?

Part A

According to the pattern, which number would come next? Answer __________

Part B

On the lines below, describe the pattern rule you used to find the correct answer to Part A.

Part C

Does the number pattern below obey the same rule as the first pattern of numbers? Explain why or why not on the lines below.

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7. While collecting branches from a specific kind of tree, Roland noticed a pattern where the number of leaves on the branches changed according to the length of the branch. He made the table below based on his findings.

Part A

On the lines below, describe the pattern that Roland found. Part B

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Chapter 2

 Solve special equations (negative variables, multiplying by a fraction, variables as denominators, multi-step equations) (2 days)

 8.EE.7a, 8.EE.7b

 Solve multi-step equations (combining like terms, variables on both sides of the equal sign, the distributive property) (9 days)

 8.EE.7a, 8.EE.7b  Review and Exam (2 days)

Sample Assessment Question 1. Solve the equation below for x. 20x + 5x - 20 = 21x + 4

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Chapter 3

 Writing Equations

oConsecutive Integers (2 days)

oPerimeter (2 days)

oMoney (2 days)

oGeneral (2 days)

 Solve Systems of Equations Algebraically (algebraically and word problems) (8 days)

 8.EE.8b, 8.EE.8c  Review and Exam (2 days)

1. Gwen has three jars of marbles that contain x marbles and two jars that contain y marbles. Which expression represents the total number of marbles Gwen has?

A 3x + 2y B 2x + 3y C (2 + 3) (x + y) D 3(x + y) + 2(x + y)

2. The length of a rectangle is five inches longer than the width. If the area of the rectangle is 14 square inches, which equation can be used to find the width of the rectangle?

F 2x + 2(x + 5) = 14 G x + x + 5 = 14 H x(x + 5) = 14 J x + 5 = 14

3. Jesse works in a bakery. He can make 12 pies in 96 minutes. In the space below, write an equation that can be used to find how many pies (p) Jesse would be able to make in (t) minutes.

Equation____________

If Jesse makes pies for 8 hours, how many pies could he make? Show your work.

Answer ________ pies

4. Solve the following system of equations algebraically or graphically: y = x2_{+ 2x – 4}

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Graphing Description

Students will recognize, use, and represent algebraically patterns, relations, and functions. Students will apply coordinate geometry to analyze problem solving situations. Students will use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students will recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is the slope, and the graphs are lines through the origin. They will understand that the slope (m) of a line is a constant rate of change, so that if the input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount m•A. Students will relate systems of equations to pairs of lines in the plane; these intersect, are parallel, or are the same line. Students will use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and solve problems. This unit will take approximately 18 to 22 days.

Vocabulary

Linear Equation Function Domain

Range Relation Slope

Y-intercept Table of Values Horizontal

Vertical System of Equations Common Solution

Inequality Non-linear Quadratic

Table of Values Unit

 Determine if a relation is a function (.5 day)  Vertical Line Test

 Domain and Range  8.F.1

 Distinguish between linear and nonlinear equations (.5 day)  8.F.3

 Graph a line using a table of values (including solving for y) (3 days)  8.F.4

 Determine the slope of a line from a graph and explain the meaning of slope as a unit rate. (1 day)

 8.EE.5

 Use similar triangles to explain why the slope is the same between any two distinct points on a non-vertical line in the plane. (1 day)

 8.EE.6

 Determine the intercept of a line from a graph and be able to explain the y-intercept (1 day)

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 Determine the equation of a line given the slope and the y-intercept (2 days)  8.EE.6

 Graph a line from an equation in slope-intercept form ( ) (3 days)  8.EE.6

 Solve systems of equations graphically (algebraically and word problems) (6 days)

 8.EE.8a

Sample Regents Questions

1. Solve the following system of equations algebraically or graphically: y = x2_{+ 2x – 4}

3y – 4x = 0

2. Which properties best describe the coordinate graph of two distinct parallel lines?

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Geometry - Angles Description

Students will identify and justify geometric relationships, formally and informally. Students will perform algebraic procedures accurately. Students will show that the sum of the angles in a triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines. This unit will take approximately 13-17 days.

Vocabulary

Angle Acute Angle Right Angle

Obtuse Angle Congruent Vertical Angles

Intersecting Lines Supplementary Angles Complementary

Parallel Lines Transversal Angles

Corresponding Angles Alternate Interior Angles Alternate Exterior

Straight Angle Exterior Angle Angles

Unit

 Review geometry definitions (1 day)

 Find the sum of the measures of the interior/exterior angles of triangles with algebra (2 days)

 8.G.5

 Review and classify angles and vertical, supplementary, and complementary angles (3 days)

 8.G.5

 Determine angle pair relationships when given two parallel lines cut by a transversal (3 days)

 8.G.5

 Calculate the missing angle measurements when given two parallel lines cut by a transversal (2 days)

 8.G.5

 Similarity with angle-angle criterion (2 days)  8.G.5

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1. In the diagram below, line a and line b are parallel, and line n is a transversal.

[not drawn to scale]

What is the measure of angle 1?

On the lines below, explain how you determined your answer 8

1 2

3 4

5 135° 7

a

b

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Transformations Description

Students will apply transformations and symmetry to analyze problem solving situations. Students will use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations, and ideas about

congruence and similarity to describe and analyze two-dimensional figures and to solve problems. Students will understand congruence and similarity using physical models, transparencies, or geometry software. This unit will take approximately 13-17 days.

Vocabulary

Rotate Clockwise Counter-Clockwise

Center of Dilation Direction Line Reflection Point Reflection Perpendicular Dilation

Constant of Dilation Translation x-axis

y-axis Origin Image

Corresponding Sides Corresponding Angles Similar Polygon Congruent Polygon Axis of Symmetry Point of Symmetry Translational Symmetry Rotational Symmetry One-to-one

Correspondance

Unit

 Review coordinate grid, congruence, similar triangles with proportions (2 days)

 Describe and identify transformations in the plane, using proper function notation (rotations, reflections, translations, and dilations) (1 day)

 8.G.3

 Identify the properties preserved and not preserved under a reflection, rotation, translation, and dilation (1 day)

 8.G.1a, 8.G.1b, 8.G.1c

 Draw the image of a figure under rotations (2 days)  8.G.2, 8.G.3, 8.G.4

 Draw the image of a figure under a reflection over a given line (2 days)  8.G.2, 8.G.3, 8.G.4

 Draw the image of a figure under a translation (2 days)  8.G.2, 8.G.3, 8.G.4

 Draw the image of a figure under a dilation (2 days)  8.G.3, 8.G.4

 Compound transformations to determine congruence or similarity (2 days)  8.G.4

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1. Triangle DEF is a translation of triangle ABC.

Which statement describes the translation of triangle ABC to triangle DEF? F 2 units to the right, 6 units down

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2. On the grid below, draw and label the reflection of ∆ABC in the y-axis.

List the new coordinates for ∆A’B’C’ that you just drew. Answers A’ ( , )

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3. Triangle PQR is plotted on the grid below.

Part A

On the grid above, draw the image of ∆PQR reflected in the x-axis. Label the vertices P’, Q’, and R’.

On the lines below, explain how you decided where to locate P’. Part B

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4. Figure A was transformed on a grid and renamed figure B.

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Statistics and Probability Description

Students will use a linear equation to describe the association between two quantities in bivariate data (such as arm span vs. height for students in a

classroom) informally. Students will interpret the model in the context of the data requireing students to express a relationship between the two quantities in question and to interpret components of the relationship (such as slope and y-intercept) in terms of the situation. This unit will take approximately 10-12 days. Vocabulary

Scatter plots Positive Correlation Negative Correlation Bivariate Univariate Line of best fit

Outlier Clustering Linear Approximation Quantitative Qualitative Non-linear Approximation Unit

 Construct and Interpret scatterplots for bivariate data (3 days)

 Cluster, Outlier, Positive/negative correlation, linear vs. non-linear  8.SP.1

 Create a line of best fit and analyze correlation (2 days)  8.SP.1, 8.SP.2

 Interpret real life applications of slope (2 days)  8.SP.3

 Box and Whisker and Line Plots and Analyze them (4 days)  8.SP.3

 Standard Deviation (3 days)  8.SP.3

 Use relative frequency tables to analyze bivariate data (2 days)  Two way tables

 8.SP.4

Sample Questions

1. A study showed that a decrease in the cost of carrots led to an increase in the number of carrots sold. What type of correlation does this show?

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Max Speed (mph) x

45 50 54 60 65

Max Height

(ft) y 63 80 105 118 141

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Geometry - Solids

Description: Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes. They will identify and justify geometric relationships, formally and informally. Students will solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. This unit will take approximately 5 - 8 days.

Vocabulary

Altitude Angle Area

Right Angle Base Net

Square Units Surface Circle

Three Dimensional Cubed Units Pi

Cylinder Diameter Units

Height Radius Volume

Cone Sphere

Unit

 Circle (1 day)  8.G.9

 Three-Dimensional Shapes (1 day)  Cylinders, cones, spheres with nets  8.G.9

 Surface Area of Cylinders, Cones, and Spheres (3 days)  8.G.9

 Volume of Cylinders, Cones, and Spheres (3 days)  8.G.9

 Surface Area and Volume when you change dimensions (2 days)  8.G.9

Sample Questions from Past Assessment Exams 1. Joel draws a picture of his cylinder shown below. [not drawn to scale]

7 cm

Calculate the volume of Joel’s cylinder. Round your answer to the nearest tenth. 15

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Pythagorean Theorem

Description: Students will identify and justify geometric relationships, formally and informally. Students will understand meanings of operations and procedures, and how they relate to one another. Students will compute accurately and make reasonable estimates. Students will select, apply, and translate among

mathematical representations to solve problems. They will use representations to model and interpret physical, social, and mathematical phenomena. Students will recognize and apply mathematics in contexts outside of mathematics. Students will understand the statement of the Pythagorean Theorem and its converse, and will be able to explain why the Pythagorean Theorem holds, for example, by decomposing a square in two different ways. They will be able to apply the Pythagorean Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze polygons. This unit will take approximately 11 - 14 days.

Vocabulary

Hypotenuse Pythagorean Theorem Right Triangle

Leg Right Angle Square Root

Unit

 Review Square Roots (.5 days)  8.NS.2, 8.EE.2

 Identify Right Angle, Hypotenuse, and Legs of a Right Triangle (.5 days)  8.G.5

 Develop Pythagorean Theorem and converse (1 day)  8.G.6

 Determine if a Triangle is a Right Triangle (1 Day)  8.G.6

 Find the Hypotenuse of a Right Triangle (1 day)  8.G.7

 Find the Legs of a Right Triangle (1 day)  8.G.7

 Find the distance between two points in a coordinate system (3 days)  With distance formula

 8.G.8

 Pythagorean Theorem Word Problems (2 days)  8.G.7

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Sample Questions from Past Assessment Exams

1. A window washer leans a ladder up against a wall so that the top of the ladder touches the base of the window, as shown below. The bottom of the ladder is 15 feet from the wall, and the base of the window is 26 feet from the ground.

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2. The roller coaster at an amusement park reaches a height of 170 feet before it drops down a slope to a point 10 feet above ground level, in a horizontal distance of 120 feet, as shown below.

What is the approximate length of the track, in feet, of the section of the roller coaster labeled A?

Show your work.