Grade 8 Math Connects Course 3

Chapter 1 Rational Numbers and Percent MA Standard/Strands: Time Frame:

8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. 14 days

Text (Chapter/Pages) Other Resources:

Chapter 1 (Eliminate lesson 3B, C and D)

Essential Questions Concepts, Content:

1. How can rational numbers be written as terminating or repeating decimals? The numerator divided by the denominator will give the equivalent decimal. With a mixed number it is only necessary to change the fraction part into a decimal, and to include it with the whole number part for a final answer. 2. How can decimals be changed to fractions? Decimals read correctly can be changed to the equivalent fraction with a denominator of the power of ten. Then fractions should be put into simplest form. 3. What is the standard algorithm for adding and subtracting rational numbers? When rational numbers (fractions) have the same denominator, the sum or difference is found by adding or subtracting the numerators and leaving the denominator the same. When fractions have different denominators, a common denominator and then equivalent fractions using that common denominator must be found before adding and subtracting. 4. What is the standard algorithm for multiplying rational numbers? The product of rational numbers can be found by multiplying the numerators, multiplying the denominators and then simplifying the final answer. The product can also be found by dividing out common factors between any numerator and any denominator before multiplying the numerators and denominators. 5. Why use dimensional analysis during computation? Dimensional analysis is the process of including units of measurement during computation. The units can be eliminated as common factors would be. 6. What is the standard algorithm for dividing rational numbers? To divide by a fraction, multiply by its multiplicative inverse. 7. What is the process for comparing rational numbers (in the form of fractions, decimals and percents)? (grade 6) Comparisons are easily made when the numbers being compared are in the same form. 8. How are a percent proportion and a percent equation used to solve problems? (grade 7) In a percent proportion one ratio compares the part to the whole, and the other ratio is the equivalent percent written as a fraction with a denominator of 100. In a percent equation the equivalent percent is written as a decimal and is multiplied by the whole to be equal to the part. 9. How can percents be applied to find discounts, markups, selling price and sales tax. (grade 7) If the discount is 20%, then finding 20% of the cost and subtracting that amount from the cost will give the discounted price.

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Subtracting 20% from 100% first will give the same answer by finding 80% of the original amount. The selling price with a mark-up can be found by multiplying the percent by the wholesale price and adding that amount to the wholesale price. The selling price can also be found by multiplying the mark-up percent plus 100% to the wholesale price. Sales tax is determined by adding a percent of the original cost to the original cost.

Targeted Skill(s):

Vocabulary: rational numbers, terminating decimal, repeating decimal, like fractions, unlike fractions, dimensional analysis, multiplicative inverse, reciprocal, percent proportion, percent equation 1. Students will be able to change rational numbers and mixed numbers to decimals that terminate or repeat with and without a calculator. 2. Students will be able to change decimals to equivalent fractions and simplify them. 3. Students will be able to add and subtract rational numbers with common and uncommon denominators and simplify them. 4. Students will be able to multiply rational numbers and put final answers in simplified form. 5. Students will recognize the term dimensional analysis and use it in application problems to aid in computation. 6. Students will be able to divide rational numbers by multiplying by the multiplicative inverse. 7. Students will be able order rational numbers given to them in fraction, percent and decimal form. 8. Students will be able to solve application problems using a percent proportion and/or a percent equation. 9. Students will apply percents in problems involving discounts, markups, selling prices and sales tax.

Writing:

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Grade 8 Math Connects Course 3 Chapter 2 Real Numbers and Monomials MA Standard/Strands:

8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32  3–5 = 3–3 = 1/33 = 1/27. 8.EE.2. 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 small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. 8.EE.3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate 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. 8.EE.4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. 8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. 8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., 2). For example, by truncating the decimal expansion of 2 show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

Time Frame:

19 days

Text (Chapter/Pages) Other Resources:

Chapter 2

Essential Questions Concepts, Content:

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1. What symbolism is used to write small and large numbers? Are there special names for these expressions? Small and large numbers are written as powers using a base and an exponent. Expressions to the second power are called “squared” and expressions to the third power are called “cubed”. 2. How expressions with powers evaluated? Use the order of operations to evaluate expressions. 3. How are monomials multiplied and divided? To multiply powers with the same base, add their exponents and to divide powers with the same base, subtract the exponents. 4. How are monomials raised to a power? To raise a power to a power, multiply the exponents.

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5. How is a product raised to a power? To raise a product to a power, find the exponent of each factor and multiply it by the power. 6. If exponents state the number of times a base is used as a factor, how do negative and zero exponents work? Any non-zero number to the zero power is 1and any non-zero number to a negative power n is the multiplicative inverse of the non-zero number to the nth power. 7. What is scientific notation? Scientific notation is a way to write large or small numbers as a number between 1 to 10 (including 1) times a power of 10 in exponential form. 8. How do you compute with scientific notation? Use properties of equality to associate the powers of 10 independently from the rest of the expression. Then combine both parts of the answer. 9. What are square roots and cube roots? A square root is one of two equal factors of a perfect square and a cube root is one of three equal factors of a perfect cube. By definition, if = √ . There is one cube root of any number. 10. How do you estimate the square root or cube root of a number? Finding the two nearest perfect squares (one above and one below the number), and finding their square root will tell the square root of the given number is between them. Finding cube roots can be done that way or by the “guess, check and revise” strategy. 11. How can the set of real numbers be described in a Venn diagram? Real Numbers can be divided into Rational and Irrational Numbers. Rational Numbers can be subdivided into Integers that enclose the Whole Numbers. 12. How can real numbers in different forms (decimals, fractions, radicals…) be compared? Changing real numbers into like forms, or placing them on the number line makes it possible to compare them.

Targeted Skill(s):

Vocabulary: base, cube root, exponent, power, real numbers, scientific notation, square root, monomial, perfect square, square root, radical sign, perfect cube, cube root, irrational number, real number. 1. Students will write the product of integers or the product of rational numbers using exponents. 2. Students will correctly apply the order of operations with expressions that contain exponents. 3. Students will multiply powers with the same base by adding the exponents, and divide powers with the same base by subtracting the exponents. 4. Students will raise monomials to a power by multiplying the exponent on each factor by the power. 5. Students will raise a product to a power by multiplying the exponent on each factor by the power. 6. Students can describe with properties of exponents, or patterns of powers, how negative exponents work. 7. Students will convert very large and very small numbers written in standard form, to numbers written in scientific notation and vice versa. 8. Students will multiply and divide numbers written in scientific notation by using the Associative and Commutative Properties of Equality and the properties of exponents. Students will add and subtract numbers written in scientific notation by expressing both numbers as the same power of 10 and using the Distributive Property. Final answers will be adjusted to appropriate scientific notation. 9. Students will be able to name the square roots of the perfect squares through 225, and the cube roots of perfect cubes to 125. 10. Students will be able to estimate the square roots of numbers between perfect squares to the nearest tenth by comparing square roots of the perfect squares. Students will use the “guess, check and revise” method to estimate cube roots. 11. Students will be able to categorize the Real Numbers.

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12. Students will be able to convert real numbers (fractions, decimals, percents, square roots and cube roots) to a common format and compare their size or place them on the number line.

Writing:

Assessment Practices:

Chapter 3 Equations and Inequalities MA Standard/Strands: Time Frame: Text (Chapter/Pages) Other Resources: Essential Questions Concepts, Content:

8.EE.7 Solve linear equations in one variable. 8.EE.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 17 days Chapter 3, sections 3-1 C and D first, followed by 3-1 B, 3-1 A Eliminate 3-3C, 3-4 A, 3-4 B

1. Why are equations and inequalities important in mathematics? Equations and inequalities are used to represent relationships between two quantities. 2. How is working backwards (a strategy for problem solving) similar to solving equations? Knowing the answer and the steps taken to get there allow you to reason the problem in reverse. When solving equations and inequalities, inverse operations are used to solve for the value(s) of a variable. 3. How do you make algebraic equations from verbal sentences? Reading the problem can reveal words that imply the operations used and the unknown quantity. Unknown quantities, variables should be defined. 4. How are addition and subtraction equations solved? Addition and subtraction equations can be solved with algebra tiles (as necessary) and/or inverse operations and the Subtractions and Addition Properties of Equality. 5. How are multiplication and division equations solved? Multiplication and division equations can be solved with inverse operations and the Division and Multiplication Properties of Equality. 6. What are the steps for solving a 2-step equation? The goal in solving equations is to have the unknown quantity (variable) isolated on one side of the equation. The order of operations is done in reverse order is used. 7. What are the steps for writing a 2-step equation from a verbal or pictorial situation? The unknown quantity must be determined and defined. The key words in the problem are used to determine operations and

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relationships. There is always an equal sign in an equation. 8. What is the procedure for checking a solution? Substitute the solution in place of the unknown and simplify until the identity is reached. 9. What are the steps for writing a 1 or 2-step inequality from a verbal or pictorial situation? The unknown quantity must be determined and defined. The key words in the problem are used to determine operations and relationships. There is always an inequality symbol in an inequality to represent a range of values for the solution set. 10. What is the difference between solving an equation and solving an inequality? Equations and inequalities are solved in similar ways except when dividing or multiplying an inequality by a negative value, the direction of the inequality symbol reverses. 11. What does the graph of the solution of an inequality look like on a number line? The number line solution will be a range of values with an open or closed circle, and a ray that extends in one direction. 12. What is the procedure for checking your solution to an inequality? Change the inequality to an equation and check the computation. Then select a value in the direction of the inequality and substitute it in the inequality to be sure that the statement is true. Then select a value not in the direction of the inequality and substitute it in the inequality to see a false result.

Targeted Skill(s):

Vocabulary: compound inequality, defining a variable, equation, inequality, intersection, inverse operations, 2-step equation, 2step inequality 1. Students can differentiate between equations and inequalities and know when to use each. 2. Students can work backwards to understand how a problem is solved. 3. Students will be able to read a short word problem and define a variable to use in an equation to solve the problem. 4+5 Students will be able to solve one step equations using inverse operations and/or algebra tiles. 6. Students will be able to solve two-step equations using the order of operations in reverse order. 7. Students will write two-step equations from words or pictures using variables they have defined, and then solve them. 8. Students will check their solutions in writing by substituting their answer for the variable in the original problem. 9. Students will write one and two-step inequalities from words or pictures using variables they have defined, and then solve them. 10. Students will solve equations and inequalities. 11. Students will express their solutions to inequalities as algebraic statements and on the number line. 12. Students will check their solutions to inequalities using substitution. Test values will come from both the boundary value of the solution and the region under the ray.

Writing:

Assessment Practices:

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MA Standard/Strands:

Time Frame: Text (Chapter/Pages) Other Resources: Essential Questions Concepts, Content:

7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. 7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. 8.EE.7 Solve linear equations in one variable. a. Give examples of 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). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 17 days Chapter 4 Eliminate 4-2D

1. What are the mathematical properties that can be used to simplify algebraic expressions? The properties are the: commutative, associative, additive identity, multiplicative identity, multiplicative property of zero, and the distributive property. 2. What is the Commutative Property? The Commutative Property states you can change the order of the numbers when you add or multiply and the outcome is the same. 3. What is the Associative Property? The Associative Property states you can change the grouping of numbers when you add or multiply and the outcome is the same. 4. What is the Additive Identity Property? The Additive Identity Property states that adding zero to a number does not change the value of the number. 5. What is the Multiplicative Identity Property? The Multiplicative Identity Property states that multiplying by 1 results in the original number. 6. What is the Multiplicative Property of Zero? The Multiplicative Property of zero states that multiplying by zero means the product is zero. 7. What is the Distributive Property of Equality? The Distributive Property shows that you multiply every term inside the parentheses by the number outside of the parentheses. 8. How do you write an algebraic expression in simplest form? Algebraic expressions are in simplest form when there are no parentheses and no like terms in the expression. 9. How are equations or inequalities with variables on both sides of the equal sign solved? The properties are used to simplify each side and then to isolate the variable on one side of the equal/inequality sign and the rest of the expressions on the other side. Property, simplify, counter example, distributive property, associative property, commutative property, identity, multiplicative property of zero, multiplicative identity, terms, like terms, coefficients and constants.

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Targeted Skill(s):

Students will identify each property and appropriately use them to simplify algebraic expressions. Students will solve multi-step equations.

Writing:

Assessment Practices:

MA Standard/Strands:

Time Frame: Text (Chapter/Pages) Other Resources: Essential Questions Concepts, Content:

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8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 1 8.F.3 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. 8.F.4 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. 15 days Chapter 5

1. How can a verbal description be changed to an algebraic expression? Key words (more than, times, of, quotient, etc) and defined variables to create the algebraic expression. 2. What is meant by “evaluate the algebraic expression”? Evaluate means to substitute a value for each variable in the expression and simplify it using the Order of Operations. 3. What is a coordinate plane and what parts can be labeled on it? A coordinate plane is a 2-dimensional space made up of 4 quadrants divided by the x and y-axes and is where points and functions are plotted. The origin, x-axis, y-axis and quadrants can be labeled. 4. How can the scale be created to fit a set of data onto the coordinate plane? The scale on the x-axis does not need to match the scale of the y-axis. 5. What are the domain and range of a relation? The domain is the set of first numbers or the x values in the coordinates, and the range is the set of second numbers or y values. 6. What is an arithmetic sequence and how is the common difference found? An arithmetic sequence is a list of terms that differ by a common difference. The common difference is found by subtracting the previous term from a term. 7. How can the expression for the nth term of the arithmetic sequence be determined?

Function notation is not required in grade 8.

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A data table can provide the change in the term value (y) which divided by the change in the term number (x) gives the common difference. That number as a multiplier of x, sometimes with an additional value to add or subtract gives the expression. 8. What makes a set of data or a graph linear? In a data table, the difference of two y values divided by the difference of the corresponding x values is consistent. In a graph, the slant or slope of the line is consistent. 9. How does an equation differ from an expression? There is an equal sign in an equation, but not in an expression. 10. When is a relation a function? A relation is a function when each domain value has exactly one range value, (each x has exactly one y, each input has exactly one output, each independent variable has exactly one dependent variable). 11. What is the procedure for evaluating a function rule for given domain values? Evaluate a function by substituting the domain values for the independent variable and simplify the expression according to the Order of Operations. 12. What is the interpretation of f(x) ? F(x) is function notation and means that the number in the x position replaces the x in the function rule. 13. When is a function a linear function? A function is linear when there is a constant rate of change. 14. What is the difference between continuous and discrete data? Continuous data can take on every value, so there are no gaps between points on the graph so it appears as a solid line or curve. Discrete data has space between data points and the graph has non-connected points. 15. What is the difference between a linear and a non-linear function? A non-linear function has a non-constant rate of change. 16. What is a quadratic function? Algebraically a quadratic function has an exponent of 2 on the x variable, and that is the largest exponent. Graphically a quadratic function is a parabola (u-shaped) that opens up or down. 17. How is a quadratic function graphed? A quadratic function can be graphed by generating a table of values and plotting them until the u-shape is created.

Targeted Skill(s):

Vocabulary: algebra, algebraic expression, arithmetic sequence, common difference, continuous data, coordinate plane, dependent variable, discrete data, domain, function, function table, independent variable, linear, linear function, nonlinear function, ordered pair, origin, quadrants, range, relation, sequence, term, variable , x-axis, x-coordinate, y-axis, ycoordinate 1. Students will use key words in verbal descriptions to write algebraic expressions. 2. Students will evaluate algebraic expressions. 3. Students will be able to draw, label and graph on the coordinate plane. 4. Students will draw graphs with appropriate scale so that all points will fit on the graph. 5. Students will determine the domain and range of a function.

Writing: Assessment Practices:

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MA Standard/Strands:

Time Frame:

8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways 8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; 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. 8.EE.8 Analyze and solve pairs of simultaneous linear equations. 8.EE.a and b a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. 8.F.3 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. 8.F.4 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. 20 days

Text (Chapter/Pages) Other Resources:

Chapter 6 (do it all)

Essential Questions Concepts, Content:

1. Can students identify proportional and non-proportional relationships by finding constant rate of change? Proportional relationships graph as straight lines that pass through (0,0) vs. non-proportional relationships which are curved or do not pass through (0,0). 2. How do you find the slope of a line? 3. How can direct variation be used to solve problems? 4. How can a linear equation in is slope-intercept form be graphed? 5. How can a linear equation in standard form be graphed? 6. How can a system of linear equations be solved by graphing? 7. How can a system of linear equations be solved by substitution?

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Targeted Skill(s):

1. Given a table or graph, or ordered pair, students can determine constant rate of change. 2. Given a table, students can determine a proportional relationship vs. a non-proportional relationship. 3. Given the formulas m = rise/run and m = y2-y1 / x2 – x1 or a graphed line, students can determine the slope of a line. 4. Using the formulas k = y/x and y – kx, students can determine direct variation. 5. Students can determine direct variation by viewing its graph and seeing the line go through (0,0). 6. Given the formula y = mx + b students can determine the slope and y-intercept and graph the function. 7. Given the slope and the y-intercept, students will be able to write the equation in slope-intercept form. 8. Students can rewrite an equation into slope-intercept form by isolating the variable y. 9. Students can graph a function in standard form Ax + By = C bt determining the x and y intercepts. 10. Students can identify that the intersection of a system of equations is the solution set by graphing. 11. Students can identify if a system of equations has no solution or infinitely many solutions. 12. Students can identify the solution of a set of equations by substitutions.

Writing: Assessment Practices:

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MA Standard/Strands: Time Frame: Text (Chapter/Pages) Other Resources: Essential Questions Concepts, Content:

Targeted Skill(s):

8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. 8.G.6 Explain a proof of the Pythagorean Theorem and its converse. 14 days Chapter 7 7-1 B, 7-1 C, 7-1D (sprinkled throughout), 7-2B, 7-3B, 7-3 D, 7-3E, 7-4 B

1. How are angles categorized? Angles are measured in degrees and organized by size: acute, less than 90 degrees, right, equal to 90 degrees, obtuse, greater than 90 degrees, and straight, equal to 180 degrees and by orientation in space: vertical angles and adjacent angles. 2. What are complimentary and supplementary angles? Pairs of angles whose measures add to 90 are complimentary angles and pairs of angles whose measures add to 180 degrees are supplementary angles. 3. What are the possible relationships between two lines in the same plane? Lines that never cross (intersect) are called parallel lines, and lines that meet to form right angles are called perpendicular. 4. What angles are formed when a transversal intersects parallel lines? Angles formed by a transversal are: interior, exterior, alternate interior, alternate exterior, and corresponding. 5. What is the sum of the angles in a triangle? The sum of the angles in a triangle is 180 dgrees. 6. How are triangles classified? Triangles are classified by angle measures: acute (all angles < 90 degrees), right (one 90 degree angle) and obtuse (one angle >90 degrees) and by side lengths: equilateral (3 congruent sides), isosceles (2 congruent sides) and scalene (no congruent sides). 7. What are the quadrilaterals that students should recognize and define? Trapezoids (exactly one pair of parallel sides), parallelogram (2 pair of opposite sides parallel), rhombus (parallelogram with 4 congruent sides) 8. What is a polygon? A polygon is a closed figure formed by 3 or more line segments. 9. How are polygons classified? Polygons are classified by the number of sides: 3 sides = triangle, 4 sides= quadrilateral, 5 sides = pentagon, 6 sides = hexagon, 8 sides = octagon, 9 sides = nonagon, 10 sides = decagon. Regular polygons have equal angles.

Students will be able to use appropriate vocabulary to describe angles based on their measure and orientation. Students will be able to name polygons with 3-10 sides. Students will be able to differentiate between different quadrilaterals based on their physical attributes.

Writing:

Assessment Practices:

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MA Standard/Strands:

Time Frame: Text (Chapter/Pages) Other Resources:

8.EE.2 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 small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. 8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; 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. 8.F.4 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. 8.G.1(a-c) Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. 8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. 8.G.3 Describe the effects of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. 8.G.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. 8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. 8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. 17 days Chapter 8

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Essential Questions Concepts, Content: 2.

What are similar polygons? Polygons that have the same shape that have corresponding angles that are congruent and corresponding sides are proportional. How can you find the missing measures of similar polygons? You can use the scale factor (ratio of the lengths of two corresponding sides of two similar polygons) to find missing

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3. 4. 5. 6. 7. 8.

9. 10. 11.

12. 13. 14. 15.

measures. How do parallel lines relate to similar triangles? Two parallel lines cut by two transversals can create similar triangles. What is Pythagorean Theorem? It describes the relationship between the lengths of the legs and the hypotenuse for any right triangle. How is the Pythagorean Theorem applied to right triangles? The sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse: What is the converse of the Pythagorean Theorem? If the sides of a triangle have lengths a, b and c units such that then the triangle is a right triangle. How can you find the distance between two points on the coordinate plane? You can graph the points then draw a right triangle with c as the hypotenuse and use the Pythagorean Theorem. How do you find the missing measures of a 45 -45 -90 triangle? A 30 -30 -60 triangle? In a 45 -45 -90 triangle, the length of the hypotenuse is √ times the length of a leg. In a 30 -30 -60 triangle, the length of the hypotenuse is twice the length of the shorter leg and the length of the longer leg is √ times the length of a shorter leg. How can you use a triangle to find the slope of a line? Pick two points and move vertically for the numerator and horizontally for the denominator. What are the types of transformations? Translations, reflections, rotations and dilations. What is a translation? A reflection? A translation is an image that has the same orientation as the original image but has been moved left, right up or down. A reflection is a mirror image of an image. How can you identify rotational symmetry of a figure? A figure can be turned less than a full turn about its center so that the entire figure matches the original figure. What is a rotation? A transformation in which a figure is rotated, or turned, about a fixed point. How are dilations and scale factor related? A dilation is a transformation that enlarges or reduces a figure by a scale factor. How can you draw compositions using translations, reflections and rotations? A composition of transformations is created by using several transformations.

Vocabulary Converse, Corresponding parts, Dilation, Distance formula, Hypotenuse, Legs, Pythagorean Theorem, Reflection, Rotation, Scale factor, Similar polygons, Transformation, Translation

Targeted Skill(s):

1. 2. 3.

Finding the missing measurements of similar polygons using scale factor. Using the Pythagorean Theorem to find the missing measure of a leg or hypotenuse of a right triangle. Using the Distance Formula to find the distance between two points on a coordinate plane.

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4. 5. 6. 7. 8.

Using special formulas to find the missing lengths of a 45 -45 -90 and 30 -30 -60 triangles. Graphing the translation of an image. Graphing the reflection of an image. Graphing the rotation of an image. Graphing the dilation of an image.

Writing: Assessment Practices:

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MA Standard/Strands:

8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. 8.EE.8 Analyze and solve pairs of simultaneous linear equations.

Time Frame: 5 days

Text (Chapter/Pages) Other Resources: Essential Questions Concepts, Content:

Targeted Skill(s):

Chapter 9 Units of Measure (text pages 541-560)

1. How can a literal equation be rewritten so it is equal to one of its variables? A literal equation can be made equal to any of the variables it is written with by applying the properties of equality according to the order of operations. 2. What is meant by the accuracy of a measurement? Accuracy is how close a measurement is to an accepted, true value. 3. What is meant by precision when describing a measurement? Precision is the ability of a measurement to be consistently reproduced. 4. How can accuracy and precision be differentiated? A single measurement can be accurate, but precision needs to have a group of measurements.

Vocabulary: literal equation, accuracy, precision, unit ratio, unit rate, degree, Celsius ( , Farenheit ( ), Kelvin (K) 1. Given a formula with more than 1 variable, students will solve for any variable using the Order of Operations. 2. Given a problem with measurements, students will be able to tell the difference between accuracy and precision.

Writing:

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MA Standard/Strands:

8.SP.1 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 association, linear association, and nonlinear association. 8.SP.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. 8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope

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Time Frame: Text (Chapter/Pages) Other Resources:

Essential Questions Concepts, Content:

and intercept. 8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. 12 days Chapter 10

Data Analysis and Statistics (text pages #612-631)

1. What information can be found from a scatter plot? A scatter plot show the general relationship between a data set with two variables graphed as ordered pairs on a coordinate plane. As the value of the x variable increases, the value of the y variable can increase, decrease, stay the same, or both increase and decrease without any obvious pattern. 2. How is a scatter plot made? A scatter plot is made with the independent x-variable on the horizontal x-axis and the dependent y-variable on the vertical y-axis. Scales for each axis must be made to include all the data points to be plotted, but the scales do not have to be the same for both the x- and the y-axes. 3. What is a line of best fit, and how is it created? After points are plotted in a scatter plot, the line of best fit is a line that best represents all the points. It should separate the points in half, and be located so that half the points are above it and half are below it with the distances from the points to the line about the same. Outliers can be ignored. 4. What makes a specific graphical display the best one to use? The bar graph is best used to show the number of items in specific categories, the box-and-whisker plot shows the measures of variation for a set of data, a circle graph (see page 627…)

Targeted Skill(s):

Writing:

Assessment Practices:

MA Standard/Strands:

6.G Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

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MA.1.a. Use the relationships among radius, diameter, and center of a circle to find its circumference and area. MA.1.b. Solve real-world and mathematical problems involving the measurements of circles.

Time Frame: Text (Chapter/Pages) Other Resources:

Chapter 11 Use 2D as an extension of Chapter 12 1A and 1B.

Essential Questions Concepts, Content:

1. How can geometric models be used to explore probability? The probability of landing in a region of a target is the ratio of the area of the region to the area of the target.

Targeted Skill(s):

Given a geometric figure and information about area of shaded regions within that figure, students can determine the probability of a dart hitting a certain region and express the answer as a decimal, faction or percent.

Writing: Assessment Practices:

MA Standard/Strands:

Time Frame: Text (Chapter/Pages) Other Resources: Essential Questions Concepts, Content:

8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions 8.G.9, Know the formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-world and mathematical problems. 8.EE.7 Solve linear equations in one variable. 9 Days Chapter 12

1. How is the diameter related to the radius of a circle? The diameter is twice the length of the radius. 2. Given the radius or diameter of a circle, what is the circumference of the circle?

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The circumference of the circle is equal to double the radius times pi or pi times the diameter. 3. How can one find the area of a circle? The area is equal to the radius squared times pi. 4. What constitutes a composite figure? A figure that is made up of two or more shapes. 5. Given a composite figure, how can you determine the area? Separate the figure into manageable shapes and determine the area of each shape. 6. What does the volume of a 3-dimensional figure measure? The amount of space occupied by the figure. 7. How do you calculate the volume of a prism or cylinder? The volume is equal to the area of the base times the height of the figure. 8. How is a pyramid or cone related to a prism or cylinder? A pyramid or cone is 1/3 of a prism or cylinder with the same base and height. 9. How can you calculate the volume of a sphere? The volume is equal to 4/3 times pi multiplied by the radius cubed. 10. What is the surface area of a three dimensional figure? The surface area is the area of all of the faces of the figure. 11. How can you calculate the surface area of a rectangular prism? The surface area is equal to double the length times the width plus double the length times the height plus double the width times the height. 12. How do you calculate the surface area of a cylinder? The surface area is equal to 2 times pi times the radius squared plus 2 times pi times the radius times the height. 13. If given the circumference, area, volume, or surface area of a figure, how can you find a missing measurement? Substitute the known values into the formula for the given measurement and solve for the missing measurement.

Targeted Skill(s):

Vocabulary; circle, circumference, composite figure, composite solid, diameter, radius, sphere, surface area, volume. Students will be able to: 1. Convert between diameter and radius and apply the new values appropriately. 2. Calculate the circumference of a circle given its diameter or radius. 3. Calculate the area of circle given its diameter or radius. 4. Find missing measurements given a figure’s circumference, perimeter, area, volume, or surface area. 5. Separate composite figures into manageable shapes and calculate their areas or volumes. 6. Calculate the volume of a prism, cylinder, pyramid, cone, sphere, or composite figure. 7. Calculate the surface area of a rectangular prism, cylinder, pyramid, cone, sphere, or composite figure. 8. Distinguish between a figure’s area, surface area, and volume and apply the proper measuring units.

Writing:

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Assessment Practices:

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