 Mathematics Grade 8

Strand: EXPRESSIONS AND EQUATIONS (8.EE)

Work with radical and integer exponents (Standards 8.EE.1-4). Understand the connections between proportional relationships, lines, and linear relationships (Standards 8.EE.5-6). Analyze and solve linear equations and inequalities and pairs of simultaneous linear equations (Standards 8.EE.7-8).
• "Ponzi" Pyramid Schemes
The student's task is to find the fatal catch in this sure-fire money making scheme.
• 100 People
In the 1990s researchers calculated that if there were just 100 people in the world, there would be 20 children, 25 people would not have food and shelter, 17 people would speak Chinese, and 8 would speak English. In this task, students are asked to estimate the real numbers, given that there are approximately seven billion people in the world.
• A Million Dollars
In this task, students will figure out questions such as: How much does a million Dollars in Dollar bills weigh? How many burgers can you buy for a million Dollars?
• Ant and Elephant
In this problem students are comparing a very small quantity with a very large quantity using the metric system.
• Ants versus humans
This task requires students to work with very large and small values expressed both in scientific notation and in decimal notation (standard form). In addition, students need to convert units of mass.
• Bike Ride
In this task, students interpret a distance/time graph describing a bike ride.
• Bivariate Data and Analysis: Anthropological Studies
This lesson opens with a video from an anthropologist explaining how he uses bivariate data to examine the impact that slavery had on the slave's height and weight. Students then use his data in the classroom activity which has them study the relationship between tibia and femur measurements and a person's stature. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
In this task, students will write equations to solve problems about buying bags of chips and candy bars.
• Cell Phone Plans
This task presents a real-world problem requiring the students to write linear equations to model different cell phone plans. Looking at the graphs of the lines in the context of the cell phone plans allows the students to connect the meaning of the intersection points of two lines with the simultaneous solution of two linear equations. The students are required to find the solution algebraically to complete the task.
• Chapter 1 - Mathematical Foundation (UMSMP)
This is Chapter 1 of the Utah Middle School Math: Grade 8 textbook. It provides a Mathematical Foundation for Analyzing and Solving Linear Equations.
• Chapter 1 - Student Workbook (UMSMP)
This is Chapter 1 of the Utah Middle School Math: Grade 8 student workbook. It covers analyzing and solving linear equations.
• Chapter 2 - Mathematical Foundation (UMSMP)
This is Chapter 2 of the Utah Middle School Math: Grade 8 textbook. It provides a Mathematical Foundation for Proportional and Linear Relationships.
• Chapter 2 - Student Workbook (UMSMP)
This is Chapter 2 of the Utah Middle School Math: Grade 8 student workbook. It covers Proportional and Linear Relationships.
• Chapter 3 - Mathematical Foundation (UMSMP)
This is Chapter 3 of the Utah Middle School Math: Grade 8 textbook. It provides a Mathematical Foundation for Representations of a Line.
• Chapter 3 - Student Workbook (UMSMP)
This is Chapter 3 of the Utah Middle School Math: Grade 8 student workbook. It covers Representations of a Line.
• Chapter 5 - Mathematical Foundation (UMSMP)
This is Chapter 5 of the Utah Middle School Math: Grade 8 textbook. It provides a Mathematical Foundation for Simultaneous Linear Equations.
• Chapter 7 - Mathematical Foundation (UMSMP)
This is Chapter 7 of the Utah Middle School Math: Grade 8 textbook. It provides a Mathematical Foundation for Rational and Irrational Numbers.
• Chapter 7 - Student Workbook (UMSMP)
This is Chapter 7 of the Utah Middle School Math: Grade 8 student workbook. It focuses on Rational and Irrational Numbers.
• Chapter 8 - Mathematical Foundation (UMSMP)
This is Chapter 8 of the Utah Middle School Math Grade 8 textbook. It provides a Mathematical Foundation for Integer Exponents, Scientific Notation and Volume.
• Chapter 8 - Student Workbook (UMSMP)
This is Chapter 8 of the Utah Middle School Math Grade 8 student workbook. It focuses on Integer Exponents, Scientific Notation and Volume.
• Choosing appropriate units
The purpose of this task is to use scientific notation in the context of choosing units to report quantities.
• Classifying Solutions to Systems of Equations
This lesson unit is intended to help educators assess how well students are able to classify solutions to a pair of linear equations by considering their graphical representations.
• Coffee by the Pound
Given a statement about the price of coffee, students are asked to answer a number of questions about the cost per pound and draw a graph in the coordinate plane of the relationship between the number of pounds of coffee and the total cost.
• Comparing Speeds in Graphs and Equations
This task provides the opportunity for students to reason about graphs, slopes, and rates without having a scale on the axes or an equation to represent the graphs.
• Coupon versus discount
This task involves solving equations with rational coefficients, and requires students to use the distributive law ("combine like terms"). The equation also provides opportunities for students to observe structure in the equation to find a quicker solution, as in the second solution presented.
• Different Areas?
The goal of this task is to motivate a discussion of similarity and slope via a counterintuitive geometric construction where it appears as if area is not conserved by cutting and reassembling a simple shape.
• DVD Profits, Variation 1
The first two problems in this task ask students to find the unit cost per DVD for making a million DVDs. Even though each additional DVD comes at a fixed price, the overall cost per DVD changes with the number of DVDs produced because of the startup cost of building the factory.
• Equations of Lines
This task requires students to use the fact that on the graph of the linear equation y=ax+c, the y-coordinate increases by a when x increases by one. Specific values for c and d were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question.
• Estimating Length Using Scientific Notation
This lesson unit is intended to help you assess how well students are able to estimate lengths of everyday objects, convert between decimal and scientific notation, and make comparisons of the size of numbers expressed in both decimal and scientific notation.
• Expressions and Equations
A set of short tasks for grades 7 and 8 dealing with expressions and equations.
• Expressions and Equations (8.EE) - 8th Grade Core Guide
The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 8 - Expressions and Equations
• Extending the Definitions of Exponents, Variation 1
This is an instructional task meant to generate a conversation around the meaning of negative integer exponents.
• Find the Change
This task is designed to help students solidify their understanding of linear functions and push them to be more fluent in their reasoning about slope and y-intercepts. It may also produce a reasonable starting place for discussing point-slope form of a linear equation.
• Fixing the Furnace
This task can be used to both assess student understanding of systems of linear equations or to promote discussion and student thinking that would allow for a stronger solidification of these concepts. The solution can be determined in multiple ways, including either a graphical or algebraic approach.
• Folding a Square into Thirds
The purpose of this task is to find and solve a pair of linear equations which can be used to understand a common method of folding a square piece of origami paper into thirds.
• GeoGebra: Derivation of the line
Use this file to see the derivation of the line y=mx.
• Giantburgers
Every day 7% of Americans eat at Giantburger restaurants! The student's task is to decide whether this newspaper headline can be true.
• Grade 8 Math Module 1: Integer Exponents and Scientific Notation (EngageNY)
In Grade 8 Module 1, students expand their basic knowledge of positive integer exponents and prove the Laws of Exponents for any integer exponent. Next, students work with numbers in the form of an integer multiplied by a power of 10 to express how many times as much one is than the other. This leads into an explanation of scientific notation and continued work performing operations on numbers written in this form.
• Grade 8 Math Module 4: Linear Equations (EngageNY)
In 8th grade Module 4, students extend what they already know about unit rates and proportional relationships to linear equations and their graphs. Students understand the connections between proportional relationships, lines, and linear equations in this module. Students learn to apply the skills they acquired in Grades 6 and 7, with respect to symbolic notation and properties of equality to transcribe and solve equations in one variable and then in two variables.
• Grade 8 Math Module 7: Introduction to Irrational Numbers Using Geometry (EngageNY)
Module 7 begins with work related to the Pythagorean Theorem and right triangles. Before the lessons of this module are presented to students, it is important that the lessons in Modules 2 and 3 related to the Pythagorean Theorem are taught (M2: Lessons 15 and 16, M3: Lessons 13 and 14). In Modules 2 and 3, students used the Pythagorean Theorem to determine the unknown length of a right triangle. In cases where the side length was an integer, students computed the length. When the side length was not an integer, students left the answer in the form ofx2=c, where c was not a perfect square number. Those solutions are revisited and are the motivation for learning about square roots and irrational numbers in general.
• Grade 8 Unit 2: Exponents and Equations (Georgia Standards)
In this unit student will distinguish between rational and irrational numbers and show the relationship between the subsets of the real number system; recognize that every rational number has a decimal representation that either terminates or repeats; recognize that irrational numbers must have decimal representations that neither terminate nor repeat; understand that the value of a square root can be approximated between integers and that nonperfect square roots are irrational; locate rational and irrational numbers on a number line diagram; use the properties of exponents to extend the meaning beyond counting-number exponents; recognize perfect squares and cubes, and understanding that non-perfect squares and non- perfect cubes are irrational.
• Grade 8 Unit 3: Geometric Applications of Exponents (Georgia Standards)
In this unit students will distinguish between rational and irrational numbers; find or estimate the square and cubed root of non-negative numbers, including 0; interpret square and cubed roots as both points of a line segment and lengths on a number line; use the properties of real numbers (commutative, associative, distributive, inverse, and identity) and the order of operations to simplify and evaluate numeric and algebraic expressions involving integer exponents, square and cubed roots; work with radical expressions and approximate them as rational numbers; solve problems involving the volume of a cylinder, cone, and sphere; determine the relationship between the hypotenuse and legs of a right triangle; use deductive reasoning to prove the Pythagorean Theorem and its converse; apply the Pythagorean Theorem to determine unknown side lengths in right triangles; determine if a triangle is a right triangle, Pythagorean triple; apply the Pythagorean Theorem to find the distance between two points in a coordinate system; and solve problems involving the Pythagorean Theorem.
• Grade 8 Unit 5: Linear Functions (Georgia Standards)
In this unit students will graph proportional relationships; interpret unit rate as the slope; compare two different proportional relationships represented in different ways; use similar triangles to explain why the slope is the same between any two points on a non-vertical line; derive the equation y = mx for a line through the origin; derive the equation y = mx + b for a line intercepting the vertical axis at b; and interpret equations in y = mx + b form as linear functions.
• Grade 8 Unit 7: Solving Systems of Equations (Georgia Standards)
In this unit students will understand the solution to a system of equations is the point of intersection when the equations are graphed; understand the solution to a system of equations contains the values that satisfy both equations; find the solution to a system of equations algebraically; estimate the solution for a system of equations by graphing; understand that parallel lines have will have the same slope but never intersect; therefore, have no solution; understand the two lines that are co-linear share all of the same points; therefore, they have infinitely many solutions; and apply knowledge of systems of equations to real-world situations.
• Graphing Linear Equations - Full Body Style
This Teaching Channel video and lesson plan shows students graphing a line given slope-intercept on a giant coordinate plane. (4 minutes)
• Hot Under The Collar
In this task students will compare two methods of converting temperature measurements from Celsius to Fahrenheit.
• How Many Solutions?
Given an equation students are asked to find a second linear equation to create a system of equations that has one, two, none, or an infinity of solutions.
• How old are they?
In this task, students will use equations to solve a number puzzle about three people's ages
• IXL Game: Linear Functions
Designed for eighth graders, this game will help the student understand linear functions, specifically how to graph a line from an equation. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use.
• Journey
In this task, students will read a description of a car journey and draw a distance-time graph to represent it.
• Lines and Linear Equations
This lesson unit is intended to help educators assess how well students are able to interpret speed as the slope of a linear graph and translate between the equation of a line and its graphical representation.
• Manipulating Graphs
This video demonstrates how to use the slope-intercept of a line to the graph of that line. The classroom activity has them demonstrate their understanding by finding equations for a set of lines through the origin. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Mixture Problems
Learning to think of a mixture as a kind of rate is an important step in learning to solve these types of problems. Any situation in which two or more different variables are combined to determine a third is a type of rate. Speed and time combine to give us distance. Wages and hours worked produce earnings.
• More Complicated Functions: Introduction to Linear Functions
This lesson is designed to reinforce the concept of linear functions and ask students to write functions using English, tables and algebra.
• Multiple Solutions
In this task, students will look at a number of equations and inequalities that have more than one solution.
• Orders of Magnitude
The purpose of this task is for students to develop a feel for large powers of ten, which is a critical component of working fluently with numbers in scientific notation. Note that this task develops "very large number sense"--strategies for helping students understand very small numbers are forthcoming.
• Peaches and Plums
This task allows students to reason about the relative costs per pound of the two fruits without actually knowing what the costs are. Students who find this difficult may add a scale to the graph and reason about the meanings of the ordered pairs. Comparing the two approaches in a class discussion can be a profitable way to help students make sense of slope.
• Pennies to Heaven
The goal of this task is to give students a context to investigate large numbers and measurements. Students need to fluently convert units with very large numbers in order to successfully complete this task.
• Proportional relationships, lines, and linear equations
The purpose of this task is to assess whether students understand certain aspects of the relationship between proportional relationships, lines, and linear equations. In particular, it requires students to find the slope of the line defined by the equation 4y=x and to write the equation of a line knowing its slope and y-intercept.
• Quinoa Pasta 1
This task asks students to find the amount of two ingredients in a pasta blend. The task provides all the information necessary to solve the problem by setting up two linear equations in two unknowns.
• Raising to the zero and negative powers
The goal of this task is to use the quotient rule of exponents to help explain how to define the expressions ck for c>0 and k is greater than or equal to 0. This important definition is motivated and explained by the law of exponents: adopting the definitions for the expressions c0 and c-n given in the task allows us to maintain the intuitive product and quotient rules known for all positive exponents (which this task assumes students are familiar with).
• Rate Problems
This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
• Repeating Decimals
This lesson unit is intended to help educators assess how well students are able to translate between decimal and fraction notation, particularly when the decimals are repeating, create and solve simple linear equations to find the fractional equivalent of a repeating decimal, and understand the effect of multiplying a decimal by a power of 10.
• Sammy's Chipmunk and Squirrel Observations
This task provides a context for setting up a linear equation whose solution requires some algebraic manipulation. Because the numbers involved are not too large, students can also experiment with some small values and eventually find the solution this way; a first solution with a table is provided showing this method. On the other hand, the reasoning required without using an equation is complex enough that the simplicity and elegance of the algebraic approach can be highlighted.
• Scientific Notation
Defining scientific notation and the conversion of extreme numbers into scientific notation is the focus of the animated Math Shorts video. After viewing it the students practice using scientific notation to write numbers and also create real-world problems for other students to solve. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Shelves
In this task, students must figure out how many planks and bricks are needed to build a bookcase.
• Simultaneous Linear Equations
This website offers students instruction on various methods of solving simultaneous equations and practice examples for each method.
• Slope and House Construction
A video with This Old House's carpenter demonstrates how the use of slope is critical in the building of a house. The students then practice working with slope to not only build a playground slide, but identify other real-world uses of slope. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Slopes Between Points on a Line
The purpose of this task is to help students understand why the calculated slope will be the same for any two points on a given line. This is the first step in understanding and explaining why it will work for any line (not just the line shown).
• Solving Equations
This task requires students to solve 5x+1=2x+7 in two ways: symbolically, the way you usually do with equations, and also with pictures of a balance. Show how each step you take symbolically is shown in the pictures.
• Solving Equations video
Answers the questions "what are equations?" and "how do we solve them?"
• Solving Linear Equations in One Variable
This lesson unit is intended to help educators assess how well students are able to solve linear equations in one variable with rational number coefficients, collect like terms, expand expressions using the distributive property, and categorize linear equations in one variable as having one, none, or infinitely many solutions.
• Solving Real-Life Problems: Baseball Jerseys
This lesson unit is intended to help educators assess how well students are able to interpret a situation and represent the variables mathematically, select appropriate mathematical methods to use, explore the effects of systematically varying the constraints, interpret and evaluate the data generated and identify the break-even point, checking it for confirmation, and communicate their reasoning clearly.
• Solving Systems of Equations by Graphing
This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
• Solving Systems of Linear Equations by Elimination
This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
• Solving Systems of Linear Equations by Graphing
This is a video introduction and explanation of the topic.
• Solving Systems of Linear Equations by Substitution
This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
• Sore Throats, Variation 2
The purpose of this task is to show how the ideas in the RP and EE domains in 6th and 7th grade mature in 8th grade. Parts (a)-(c) could easily be asked of 7th grade students. Part (a) asks students to do what is described in 7.RP.2.a, Part (b) asks students to do what is described in 7.RP.2.c, and Part (c) is the 7th grade extension of the work students do in 6.EE.9. On the other hand, part (d) is 8th grade work.
• Stairway Slope
Students learn how to use stairs to understand and calculate the slope of a line. The classroom activity then has students use the real-world situation of a wheelchair and a the slope of a ramp to calculate the length needed for the ramp. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Stuffing Envelopes
This task provides students with an opportunity to take the step from unit rates in a proportional relationship to the rate of change of a linear relationship. Students should already be familiar with proportional relationships from their work in prior grades.
• Summer Swimming
The purpose of this task is for students to represent relationships between quantities in a context with equations and interpret the resulting system of equations in the context. This task has a wide array of uses: it could be an introductory task to systems of equations or used in assessment.
• The Intersection of Two Lines
The purpose of this task is to introduce students to systems of equations. It takes skills and concepts that students know up to this point, such as writing the equation of a given line, and uses it to introduce the idea that the solution to a system of equations is the point where the graphs of the equations intersect (assuming they do). This task does not delve deeply into how to find the solution to a system of equations because it focuses more on the student's comparison between the graph and the system of equations.
• The Sign of Solutions
It is possible to say a lot about the solution to an equation without actually solving it, just by looking at the structure and operations that make up the equation.
• Two Lines
In this task, we are given the graph of two lines including the coordinates of the intersection point and the coordinates of the two vertical intercepts, and are asked for the corresponding equations of the lines. It is a very straightforward task that connects graphs and equations and solutions and intersection points.
• Who Has the Best Job?
Given a table students are asked to make graphs representing the relationship between the time a student worked and the money they earned.
• Writing Expressions and Equations video
How to write an equation using what we know to solve a problem we don't know. http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey  Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer  Throndsen.

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.