Linear Functions

Unit description: Students will learn to use functions to model relationships between quantities. The students will develop an understanding of congruence and similarity using physical models, transparencies, or geometry software. They will also learn to investigate patterns of association in bivariate data.

Download the complete Grade 8 Math Unit 6 framework to customize for your own planning.

Essential Outcomes of the Unit

Functions

Use functions to model relationships between quantities.

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.

Geometry

Understand congruence and similarity using physical models, transparencies, or geometry software.

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.

Statistics and Probability

Investigate patterns of association in bivariate data.

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 Understand 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 and intercept.

Essential Questions and Big Ideas

• How can I construct a function to model a linear relationship?  
  • A linear relationship is represented as y = mx+b where m represents the slope and b represents the y-intercept.  

• What kind of patterns and associations can you see from looking at a scatter plot? 
  • A scatter plot can show a linear, nonlinear, or no association.  
  • A scatter plot can show a positive or negative association.  
  • Outliers and clustering determine if a scatter plot has a weak or strong association.  

• How can I write a linear function to represent a line of best fit on a scatter plot?
  • A linear function is written in the form y = mx + b where m represents the rate of change and b represents the y-intercept.  

• How can you use a linear equation to make predictions about bivariate data?
  • The slope of a linear equation represents the rate of change in a set of data.  
  • The y-intercept of a linear equation represents the y value when the x-value is 0. 

Functions of Geometry

Unit description: In this unit the students will  learn to define, evaluate, and compare functions. They will solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. 

Download the complete Grade 8 Math Unit 5 framework to customize for your own planning.

Essential Outcomes of the Unit

Functions

Define, evaluate, and compare functions.

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.

8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Other Standards Addressed in the Unit

Functions

Define, evaluate, and compare functions.

8.F.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line. Recognize examples of functions that are linear and nonlinear.

Geometry

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

8.G.9 Given the formulas for the volume of cones, cylinders, and spheres, solve mathematical and real world problems.

Essential Questions and Big Ideas

• How do you use functions to model relationships between quantities? 
  • A function is a rule that assigns to each input exactly one output. 
  • The rule for a function determines the relationship.  

• How can algebra, graphs, tables, and verbal descriptions be used to represent and compare functions?  
  • Functions can be compared based on their rates of change. 
  • Functions can be compared based on their slopes.  
  • Functions can be compared based on their y-intercepts.  

• Are all linear equations functions? Are all functions linear? How do you know?
  • Functions written in the form y = mx+b are linear functions.  
  • Linear functions have a constant rate of change.  

• What is the relationship between volume of cones, cylinders, and spheres?
  • The volume of a cylinder can be found by 𝛑r2 h
  • The volume of a cone can be found by 𝛑r2 (h3).
  • The volume of a sphere is 43𝛑r3

• The volume of a cone is ⅓ of the volume of a cylinder.  
  • When h = 2r, the volume of a cone and a sphere together create a cylinder.  

Linear Equations

Unit description: Students will extend their understanding of equations to represent real world contexts to consider linear equations.  Students will leave this unit able to solve linear equations and recognize the number of solutions to a linear equation.  

Download the complete Linear Equations framework to customize for your own planning.

Essential Outcomes of the Unit

Expressions and Equations

Understand the connections between proportional relationships, lines, and linear equations.

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.

Analyze and solve linear equations and pairs of simultaneous linear equations.

8.EE.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms. Note: This includes equations that contain variables on both sides of the equation.

Other Standards Addressed in the Unit

Expressions and Equations

Understand the connections between proportional relationships, lines, and linear equations.

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.

Analyze and solve linear equations and pairs of simultaneous linear equations.

8.EE.7 Solve linear equations in one variable.

8.EE.7a Recognize when linear equations in one variable have one solution, infinitely many solutions, or no solutions. Give examples and show which of these possibilities is the case by successively transforming the given equation into simpler forms

8.EE.8 . Analyze and solve pairs of simultaneous linear equations.

8.EE.8a 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. Recognize when the system has one solution, no solution, or infinitely many solutions

8.EE.8b Solve systems of two linear equations in two variables with integer coefficients: graphically, numerically using a table, and algebraically. Solve simple cases by inspection.

8.EE.8c Solve real-world and mathematical problems involving systems of two linear equations in two variables with integer coefficients

Essential Questions and Big Ideas

What is a linear equation in one variable? 

• A linear equation in one variable, is one where there is only one variable and that variable does not have an exponent.  

How do I simplify terms in an equation?   

• Like terms can be combined or subtracted. 
• Like terms are terms with the same variable.  
• The distributive property can be used to expand an expression or equation.  

How can I determine the number of solutions to a linear equation?  

• If an equation ends with an untrue statement, there are no solutions. 
• If an equation ends with a true statement, there are infinite solutions.  
• If an equation ends with an x = statement, there is one solution.  

Download the complete Linear Equations framework to customize for your own planning.

Similarity

Unit description: In this unit the students will  learn to identify properties of dilations and compositions of dilations, describe the effect of dilations on two-dimensional figures in general and using coordinates. They will apply the Pythagorean Theorem to two and three dimensions using real world examples.

Download the complete Number Pairs Similarity framework to customize for your own planning.

Essential Outcomes of the Unit

Geometry

Understand congruence and similarity using physical models, transparencies, or geometry software

• 8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
• 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.

Understand and apply the Pythagorean Theorem

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.

Other Standards Addressed in the Unit

Geometry

Understand congruence and similarity using physical models, transparencies, or geometry software

• 8.G.4 Know that a two-dimensional figure is similar to another if the corresponding angles are congruent and the corresponding sides are in proportion. Equivalently, two two-dimensional figures are similar if one is the image of the other after a sequence of rotations, reflections, translations, and dilations. Given two similar two-dimensional figures, describe a sequence that maps the similarity between them on the coordinate plane.

Understand and apply the Pythagorean Theorem

• 8.G.6 Understand a proof of the Pythagorean Theorem and its converse.
• 8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Essential Questions and Big Ideas

How can you change a figure’s position or size on a 2-D coordinate plane?

• A figure can be moved using translation, rotation, reflection without changing its size or shape.
• A figure can change its size through dilation without changing its shape.

How do you recognize congruence and similarity in figures?

• Figures that are congruent if the corresponding angles and sides are the same. 
• Figures are similar if the corresponding angles are the same but the sides may be different.

How can angle relationships be used to find missing angle measures?

• Complementary angles have a sum of 90 degrees.
• Supplementary angles have a sum of 180 degrees.
• Vertical angles are equivalent.
• The sum of the interior angles of a triangle is 180 degrees.
• The sum of the exterior angles of a triangle is 360 degrees.
• Alternate interior angles are equal.
• Alternate exterior angles are equal.

What is the Pythagorean Theorem?  

• The Pythagorean Theorem is the way to solve for a missing side length in a right triangle. 
• The Pythagorean Theorem can be proved geometrically or algebraically. 

Download the complete Number Pairs Similarity framework to customize for your own planning.

Concepts of Congruence

Unit description: In this unit, students learn about translations, reflections, and rotations in the plane and how to use them to define the concept of congruence. They will learn to use and apply knowledge of rigid motions to determine similarity and congruence when solving real world problems. They will learn to identify a sequence of transformation that will map a figure onto itself. Students will learn to prove/disprove similarity/congruence using translations, reflections and rotations. Students will learn to use knowledge of angle pairs, degrees of a triangle and exterior angles to solve for missing angles. Students will learn to use the Pythagorean Theorem to find missing sides of a triangle, find distance in the coordinate plane, and solve real-world problems. 

Essential Outcomes of the Unit

Geometry- Understand congruence and similarity using physical models, transparencies, or geometry software.

• 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.

Standards

Geometry: Understand congruence and similarity using physical models, transparencies, or geometry software.

• 8.G.1. Verify experimentally the properties of rotations, reflections, and translations.
  • Verify experimentally lines are mapped to lines, and line segments to line segments of the same length.
  • Verify experimentally angles are mapped to angles of the same measure.
  • Verify experimentally parallel lines are mapped to parallel lines.
• 8.G.2. Know that a two-dimensional figure is congruent to another if the corresponding angles are congruent and the corresponding sides are congruent. Equivalently, two two-dimensional figures are congruent if one is the image of the other after a sequence of rotations, reflections, and translations. Given two congruent figures, describe a sequence that maps the congruence between them on the coordinate plane.

Geometry: Understand and apply the Pythagorean Theorem.

• 8.G.6. Understand a proof of the Pythagorean Theorem and its converse.
• 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.
• 8.G.8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Essential Questions and Big Ideas

• What are the properties of basic rigid motions?
  • Reflections, translations and rotations preserve congruence, size and sharpe, and therefore are called rigid transformations.
• How do we map the sequence between two congruent figures on a coordinate plane?
  • By testing transformations of the original figure on the coordinate plane we can determine the sequence of transformations that creates a congruent figure.
• How can we determine congruence using angle relationships?
  • Angle Sum Theorem can be used to determine congruence of angles in a triangle
  • Alternate Interior Angles Theorem can be used to determine congruence of  alternate interior angles of parallel lines cut by a transversal.
• How does the Pythagorean Theorem help solve real world problems?
  • Pythagorean theorem to find the length of a diagonal of a rectangle.
  • Pythagorean theorem to find the missing length of the side of a right triangle.

Prerequisite Standards

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

• 7.G.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
• 7.G.6 Solve real-world and mathematical problems involving area of two-dimensional objects composed of triangles and trapezoids.

Note: The inclusive definition of a trapezoid will be utilized, which defines a trapezoid as “A quadrilateral with at least one pair of parallel sides.” (This definition includes parallelograms and rectangles.)

Solve surface area problems involving right prisms and right pyramids composed of triangles and trapezoids.

Note: Right prisms include cubes.

Find the volume of right triangular prisms, and solve volume problems involving three-dimensional objects composed of right rectangular prisms.

Geometric measurement: understand concepts of angle and measure angles.

• 4.MD.5. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.
  • An angle that turns through 𝑛𝑛 one-degree angles is said to have an angle measure of 𝑛𝑛 degrees. 

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

• 4.G.1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
• 4.G.2. Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.
• 4.G.3. Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

Download the complete Concepts of Congruence framework to customize for your own planning.

Rational Numbers and Exponents

Students will deepen their understanding of rational numbers, as they investigate irrational numbers and their place in the number system.  Students will also consider exponents and how solving for a base can yield a rational or irrational number.

Essential Outcomes

The Number System

• NY-8.NS.1: Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion eventually repeats. Know that other numbers that are not rational are called irrational.
• NY-8.NS.2: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line, and estimate the value of expressions.

Expressions, Equations and Inequalities

• NY-8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. e.g.,3^2 x 3 ^ – 5 = 3 ^ – 3 = ⅓^3 = 1/27
• NY -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. Know square roots of perfect squares up to 225 and cube roots of perfect cubes up to 125. Know that the square root of a non-perfect square is irrational.  e.g., The √2 is irrational.

Other Standards Addressed in this Unit

• 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.4: Perform multiplication and division with numbers expressed in scientific notation, including problems where both standard decimal form and scientific notation are used. Use 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.

Essential Questions and Big Ideas

• What differentiates rational and irrational numbers? 
  • Rational numbers can be expressed as a fraction a/b or as a decimal that ends or repeats.   
  • Irrational numbers  cannot be represented as a fraction and as a decimal, there is no pattern.  
• How can I compare irrational numbers?  
  • Rational and irrational numbers can be compared.  
  • Irrational numbers can be placed between rational numbers based on place value.  
• What are the properties of integer exponents?  
  • When multiplying powers with the same base, the exponents are added.  
  • When dividing powers with the same base, the exponents are subtracted.  
  • When a power is raised to an exponent, the exponents are multiplied.  
  • When multiplying different bases with the same exponent, the bases can be multiplied.  
  • A negative exponent equals the power as the denominator of a unit fraction.  
• What is a square root? 
  • A square root represents a number to the exponent ½.  
  • A square root requires finding a number that multiplied by itself equals that amount.  
• What is a cube root?  
  • A square root represents a number to the exponent ⅓.  Square root of a = b where b x b = a
  • A square root requires finding a number that multiplied by itself three times equals that amount.  The cube root of a = b where b x b x b = a
• How can I use roots to solve equations?  
  • If you know an amount to the second power equals another amount, you can use the square root to find the amount.  a^2 = 36  a = square root of 36 = 6. 
• What is scientific notation and why do I use it?
  • A number is written in scientific notation when it is represented as the product of a factor and a power of 10. 
  • Scientific notation is used to make calculations with unusually large or small numbers.

Download the complete Grade 8 Math Unit 1 framework to customize for your own planning.