Several examples are presented to demonstrate the process of graphing functions, including linear, quadratic, and rational functions.

Invitations to Inquiry are short instructional activities designed to help middle and high school students work with large data sets. Teachers and students use the interactive FieldScope platform to collect, visualize, and analyze environmental data. With these new Inquiries, students can explore FieldScope's advanced mapping and graphing tools to dig deeper into data in the context of meaningful science classroom lessons.

Students use a LEGO® ball shooter to demonstrate and analyze the motion of a projectile through use of a line graph. This activity involves using a method of data organization and trend observation with respect to dynamic experimentation with a complex machine. Also, the topic of line data graphing is covered. The main objective is to introduce students graphs in terms of observing and demonstrating their usefulness in scientific and engineering inquiries. During the activity, students point out trends in the data and the overall relationship that can be deduced from plotting data derived from test trials with the ball shooter.

With the assistance of a few teacher demonstrations (online animation, using a radiometer and rubbing hands), students review the concept of heat transfer through convection, conduction and radiation. Then they apply an understanding of these ideas as they use wireless temperature probes to investigate the heating capacity of different materials sand and water under heat lamps (or outside in full sunshine). The experiment models how radiant energy drives convection within the atmosphere and oceans, thus producing winds and weather conditions, while giving students the hands-on opportunity to understand the value of remote-sensing capabilities designed by engineers. Students collect and record temperature data on how fast sand and water heat and cool. Then they create multi-line graphs to display and compare their data, and discuss the need for efficient and reliable engineer-designed tools like wireless sensors in real-world applications.

Students learn to locate the real number solution of a linear equation using tables. They also learn to locate the real number solution of a linear equation using graphical method on a Cartesian (x-y) graph.

Watch how a graph is altered when key elements of the equation change. This lesson focuses on how to manipulate the equation of a line in slope intercept form to match the graphs provided deepening the understanding of both the slope and y intercept's role in the expression.

Students learn about slope, determining slope, distance vs. time graphs through a motion-filled activity. Working in teams with calculators and CBL motion detectors, students attempt to match the provided graphs and equations with the output from the detector displayed on their calculators.

The tutorial examines graphs of functions. Students learn about the graphs of six functions through examples with detailed solutions.

Gain a basic understanding of graphing on the coordinate plane by watching this easy to understand video tutorial. Additional resources are available as part of a paid subscription service. [10:14]

This video provides an opportunity to identify a number pattern. Then organize that data in a data table and graph it. As you watch and listen to the teacher and student interact it helps clarify the thinking behind applying this concept. [4:52]

This video provides an opportunity to create a Box and Whisker Plot. A pdf worksheet is available by clicking on the hyperlink at the bottom of the page. As you watch and listen to the teacher and student interact it helps clarify the thinking behind applying this concept. [6:13]

Rate

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Solve problems involving all four operations with rational numbers.
Understand quantity as a number used with a unit of measurement.
Solve problems involving quantities such as distances, intervals of time, liquid volumes, masses of objects, and money, and with the units of measurement for these quantities.
Understand that a ratio is a comparison of two quantities.
Write ratios for problem situations.
Make and interpret tables, graphs, and diagrams.
Write and solve equations to represent problem situations.

Lesson Flow

In this unit, students will explore the concept of rate in a variety of contexts: beats per minute, unit prices, fuel efficiency of a car, population density, speed, and conversion factors. Students will write and refine their own definition for rate and then use it to recognize rates in different situations. Students will learn that every rate is paired with an inverse rate that is a measure of the same relationship. Students will figure out the logic of how units are used with rates. Then students will represent quantitative relationships involving rates, using tables, graphs, double number lines, and formulas, and they will see how to create one such representation when given another.

Gallery OverviewAllow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit's concepts or to assist students who may have fallen behind on work.Gallery DescriptionsDog and CatStudents refer to a pre-made graph showing how much dry food a dog eats and how much dry food a cat eats over the same time period. They analyze the graph and list specific information they can conclude from the graph.Faucet Rate ProblemStudents perform research on the Internet about standard water flow rates of bathroom faucets in the United States. They test faucets at home or at school and prepare responses to the questions.Shower versus BathStudents will use their knowledge of rates to figure out which uses less water, a shower or a bath.Laps, Miles, KilometersStudents use rates to convert measures given in laps, kilometers, and miles. They justify their reasons for ordering distances given in these three units.Paper ProblemStudents write and use formulas for heights of stacks of paper. They practice writing rates for h in terms of n and for n in terms of h.Three ScalesStudents use a triple number line to convert among the units laps on a track, kilometers, and miles. Given one number line, they mark two other number lines to show equal distances in the units.Water ProblemThis problem begins with a video of a cube container being filled with colored water. Students determine what information they'd need in order to figure out the volume of water in the cube at any time.

Rational Numbers

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Solve problems with positive rational numbers.
Plot positive rational numbers on a number line.
Understand the equal sign.
Use the greater than and less than symbols with positive numbers (not variables) and understand their relative positions on a number line.
Recognize the first quadrant of the coordinate plane.

Lesson Flow

The first part of this unit builds on the prerequisite skills needed to develop the concept of negative numbers, the opposites of numbers, and absolute value. The unit starts with a real-world application that uses negative numbers so that students understand the need for them. The unit then introduces the idea of the opposite of a number and its absolute value and compares the difference in the definitions. The number line and positions of numbers on the number line is at the heart of the unit, including comparing positions with less than or greater than symbols.

The second part of the unit deals with the coordinate plane and extends student knowledge to all four quadrants. Students graph geometric figures on the coordinate plane and do initial calculations of distances that are a straight line. Students conclude the unit by investigating the reflections of figures across the x- and y-axes on the coordinate plane.

Students play a game in which they try to find dinosaur bones in an archaeological dig simulator. The players guess where the bones are on the coordinate plane using hints and reasoning.Key ConceptsThe coordinate plane consists of a horizontal number line and a vertical number line that intersect at right angles. The point of intersection is the origin, or (0,0). The horizontal number line is often called the x-axis. The vertical number line is often called the y-axis.A point’s location on the coordinate plane can be described using words or numbers. Ordered pairs name locations on the coordinate plane. To find the location of the ordered pair (m,n), first locate m on the x-axis and draw a vertical line through this point. Then locate n on the y-axis and draw a horizontal line through this point. The intersection of these lines is the location of (m,n).The coordinate plane is divided into four quadrants:Quadrant I: (+,+)Quadrant II: (−,+)Quadrant III: (−,−)Quadrant IV: (+,−)Goals and Learning ObjectivesName locations on the coordinate plane.

Gallery OverviewAllow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit's concepts or to assist students who may have fallen behind on work.Gallery DescriptionDivingChen stands on top of a cliff, and a woman scuba diver dives in the ocean below. Students will determine their positions on a vertical number line that represents distance above and below sea level.Negative Numbers?Students will read about five students’ opinions about negative numbers and decide whose opinions they agree with, whose they disagree with, and why. Students will also share their own ideas about negative numbers.Temperatures in JanuaryA map shows the lowest temperatures recorded in January since 2008 for five cities. Students will locate these temperatures on a number line and compare the temperatures.Greenwich Mean TimeStudents will use positive and negative numbers and Greenwich Mean Time to find the times of different cities around the world.Numbers TimelineStudents will research the history of negative numbers and absolute value and create a timeline to show what they learned.Rational Numbers and Absolute Value VideoStudents will create a video about rational numbers and absolute value.

Students reflect a figure across one of the axes on the coordinate plane and name the vertices of the reflection. As they are working, students look for and make use of structure to identify a convention for naming the coordinates of the reflected figure.Key ConceptsWhen point (m,n) is reflected across the y-axis, the reflected point is (−m,n).When point (m,n) is reflected across the x-axis, the reflected point is (m,−n).When point (m,n) is reflected across the origin (0,0), the reflected point is (−m,−n).Goals and Learning ObjectivesReflect a figure across one of the axes on the coordinate plane.Name the vertices of the reflected figure.Discern a pattern in the coordinates of the reflected figure.

Algebraic Reasoning

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Add, subtract, multiply, and divide rational numbers.
Evaluate expressions for a value of a variable.
Use the distributive property to generate equivalent expressions including combining like terms.
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?
Write and solve equations of the form x+p=q and px=q for cases in which p, q, and x are non-negative rational numbers.
Understand and graph solutions to inequalities x<c or x>c.
Use equations, tables, and graphs to represent the relationship between two variables.
Relate fractions, decimals, and percents.
Solve percent problems included those involving percent of increase or percent of decrease.

Lesson Flow

This unit covers all of the Common Core State Standards for Expressions and Equations in Grade 7. Students extend what they learned in Grade 6 about evaluating expressions and using properties to write equivalent expressions. They write, evaluate, and simplify expressions that now contain both positive and negative rational numbers. They write algebraic expressions for problem situations and discuss how different equivalent expressions can be used to represent different ways of solving the same problem. They make connections between various forms of rational numbers. Students apply what they learned in Grade 6 about solving equations such as x+2=6 or 3x=12 to solving equations such as 3x+6=12 and 3(x−2)=12. Students solve these equations using formal algebraic methods. The numbers in these equations can now be rational numbers. They use estimation and mental math to estimate solutions. They learn how solving linear inequalities differs from solving linear equations and then they solve and graph linear inequalities such as −3x+4<12. Students use inequalities to solve real-world problems, solving the problem first by arithmetic and then by writing and solving an inequality. They see that the solution of the algebraic inequality may differ from the solution to the problem.