lesson 7 problem solving practice linear and nonlinear functions

Curriculum  /  Math  /  8th Grade  /  Unit 4: Functions  /  Lesson 7

Lesson 7 of 12

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

Lesson Notes

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Define and graph linear and nonlinear functions.

Common Core Standards

Core standards.

The core standards covered in this lesson

8.F.A.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. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

Foundational Standards

The foundational standards covered in this lesson

Expressions and Equations

7.EE.B.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.

Ratios and Proportional Relationships

7.RP.A.2 — Recognize and represent proportional relationships between quantities.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Graph a linear function using a table of values and coordinate points. 
• Graph a nonlinear function using a table of values and coordinate points.
• Understand that a linear function consists of ordered pairs that, when graphed, lie on a straight line; points on a nonlinear function do not lie on a straight line. 

Suggestions for teachers to help them teach this lesson

Lessons 7 and 8 introduce students to linear vs. nonlinear functions. In Lesson 7, students see what examples of linear and nonlinear functions look like as graphs, and in Lesson 8, students will look at examples of functions in all representations and determine if they are linear or nonlinear.

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

25-30 minutes

It costs $${$4}$$  to enter a carnival and an additional $${$2}$$  for each activity you do once inside the carnival. The equation below represents the total cost spent at the carnival, $$y$$ , as a function of the number of activities done, $$x$$ .

a.   Create a table of values that represents some inputs and their corresponding outputs and use the table of values to sketch a graph of the function.

b.   Does this function appear to be a linear function?

c.   What is the rate of change?

d.   What is the initial value?

Guiding Questions

Student response.

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The equation below represents the area of a square, $$a$$ , as a function of its side length, $$s$$ .

b.   Does this function appear to be a linear function? Explain.

A set of suggested resources or problem types that teachers can turn into a problem set

15-20 minutes

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

5-10 minutes

A chocolate factory makes 240 chocolates in 6 minutes. The total number of chocolates produced is a function of the number of minutes that the factory is running.

a.   Write an equation that describes the total number of chocolates produced ( $$y$$ ), as a function of the number of minutes ( $$x$$ ) that the factory runs.

b.   Use the equation you wrote in part a to find the number of chocolates produced when the factory runs for 3 minutes, 16 minutes, and 20 minutes.

c.   Plot the points from part b on the graph below.

d.   Is the function linear or nonlinear? Explain how you know.

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• Illustrative Mathematics Introduction to Linear Functions
• EngageNY Mathematics Grade 8 Mathematics > Module 5 > Topic A > Lesson 5 — Exercises 1-3 and Problem Set 1-3

Topic A: Defining Functions

Define and identify functions.

Use function language to describe functions. Identify function rules.

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Topic B: Representing and Interpreting Functions

Identify properties of functions represented in tables, equations, and verbal descriptions. Evaluate functions.

8.F.A.1 8.F.A.2 8.F.B.4

Represent functions with equations.

8.F.A.1 8.F.B.4

Read inputs and outputs in graphs of functions. Determine if graphs are functions.

Identify properties of functions represented in graphs.

Topic C: Comparing Functions

Determine if functions are linear or nonlinear when represented as tables, graphs, and equations.

8.F.A.1 8.F.A.3

Compare functions represented in different ways (Part 1).

Compare functions represented in different ways (Part 2).

Topic D: Describing and Drawing Graphs of Functions

Describe functions by analyzing graphs. Identify intervals of increasing, decreasing, linear, or nonlinear activity.

Sketch graphs of functions given qualitative descriptions of the relationship.

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19 Activities To Get A Grapple On Linear & Nonlinear Functions

March 23, 2023 //  by  Rachel Cruz

Mastering linear and nonlinear functions can be challenging for students, but it doesn’t have to be boring! We’ve compiled a list of 19 engaging activities that will help your students develop problem-solving skills and expand their mathematical horizons. From linear mapping models to accurate models for real-world phenomena, these activities will challenge your students to think critically and creatively. So, allow us to help you make learning fun!

1. Task Cards

These task cards are a fantastic way to engage students and help them master identifying linear functions. With various problems covering different skill levels, these cards provide a fun and interactive learning experience for students of all ages.

Learn More: Bright In The Middle

2. Card Sorting

This Card Sorting activity is an engaging and interactive way to help your students understand the difference between linear and nonlinear functions. With a set of cards to sort and classify, students will be challenged to think critically and apply their knowledge of math concepts.

Learn More: Teachers Pay Teachers

3. Tic Tac Toe

This activity combines Tic-Tac-Toe with coordinate graphing to teach students about graphing and ordered pairs. The objective is to record ordered pairs and get four in a row to win, and this activity offers various choices of mapping models for building models.

Learn More: Collected NY

4. Dino Crunch

Dino Crunch is an eighth-grade math game to help students identify linear functions. It occurs on a prehistoric battlefield and challenges learners to differentiate between linear and nonlinear graphs and equations. This fun and interactive game is a great way to reinforce algebraic concepts and improve students’ understanding of linear functions.

Learn More: Education

5. Hold The Line

Hold the Line… or Don’t is an engaging and interactive math lesson for 8th-9th graders studying Algebra 1 or Pre-Algebra. The class focuses on the concepts of linear and nonlinear functions and how they can be represented through tables, graphs, and equations.

Learn More: University of Oklahoma

6. Flippables

Flippables are interactive notebooks that can help students learn about linear and nonlinear functions. They consist of a foldable graphic organizer with information on one side and interactive activities on the other. Using them, students can engage in hands-on learning and explore critical concepts visually and interactively.

Learn More: Scaffolded Math

7. Bamboozle

Bamboozle is an online game that teaches students about functions and linear and nonlinear relationships through interactive quiz questions. Students will be challenged to identify linear and nonlinear functions, interpret graphs, and solve real-world problems involving functions.

Learn More: Baamboozle

8. Online Math Games and Simulators

The online math game and simulator series teaches students to interpret slope-intercept form as a definition of linear functions and distinguish them from nonlinear functions. The games cover exponents and varying slopes and are designed to improve engagement and academic performance.

Learn More: Legends of Learning

9. Boom Cards

Digital boom cards linear vs. nonlinear functions is an engaging and auto-graded assessment activity with 30 task cards to test your 8th-grade students’ mastery of solving multi-step, one-variable linear equations. 

Learn More: Boom Learning

10. Jeopardy

Jeopardy is an exciting game-based activity for students to learn about linear functions. It covers critical concepts such as slope-intercept form, graphing linear functions, finding slope and y-intercept, and solving linear equations. Students can play as individuals or in teams; making it a fun and interactive way to learn.

Learn More: Super Teacher Tools

11. Escape Room

This Escape room is full of 4 exciting puzzles covering topics like defining linear functions and writing equations in slope intercept form. Your students will have a blast working in teams of 2-5 to solve the challenges and escape the room. 

12. Puzzles

Put your students’ algebra skills to the test with this engaging puzzle activity! Designed to help students practice solving linear equations, this activity challenges them to match 21 questions and answers on puzzle pieces to create a given shape.

Learn More: Math Teachers Resources

13. Othello

This educational activity is designed to help students practice identifying and solving linear and nonlinear equations. Based on the classic board game Othello, this activity challenges students to place pieces on the board based on whether the corresponding equation is linear or nonlinear.

Learn More: University of Pittsburg

14. Function Machines

Function machine activities are a fun way to teach students about linear and non-linear functions. It’s an interactive activity where students have to input a number into a “machine” that performs a function on it and then outputs a new number.

Learn More: Minnesota STEM Teaching Center

15. Slope Art

This slope art linear equations project is a fun and creative way for students to practice graphing linear equations and understanding slopes. In this activity, students choose a design or image and use it to create a coordinate grid with multiple linear equations that graphically represent their chosen design. 

Learn More: Pinterest

16. Guess Who

Guess who linear functions is a game where players guess the identity of a mystery linear function by asking yes or no questions about its rate of change and initial value. The goal is to determine the function’s identity using information from tables of values, graphs, or verbal descriptions.

Learn More: I Is A Number

17. Crack The Code

Get your 8th-grade students excited about functions with this Fourth of July crack-the-code activity! Students will analyze function tables to identify if they are linear or nonlinear and then use their findings to crack a secret code and reveal a special message.

Learn More: Twinkl

Get ready to shout “BINGO” while learning about functions! Explore the world of linear and nonlinear functions with this exciting game that will have your students on the edge of their seats

19. Traveling With Linear Equations

This travel activity is a great way to make math more fun and applicable for students. Using frequent flyer miles as a basis for learning about linear equations, students can choose the airline they want to use, the dates they want to fly, and where they want to travel.

Learn More: Algebra and Beyond

Systems of Nonlinear Equations

Solving systems of nonlinear equations.

A “ system of equations ” is a collection of two or more equations that are solved simultaneously. Previously, I have gone over a few examples showing how to solve a system of linear equations using substitution and elimination methods. It is considered a linear system because all the equations in the set are lines.

What is a Nonlinear System of Equations?

On the other hand, a nonlinear system  is a collection of equations that may contain some equations of a line but not all of them. In this lesson, we will only deal with the system of nonlinear equations with two equations in two unknowns, [latex]x[/latex] and [latex]y[/latex].

There are seven (7) examples in this lesson.

Examples of How to Solve Systems of Nonlinear Equations

Example 1: Solve the system of nonlinear equations below.

This system has two equations of each kind: linear and nonlinear. Start with the first equation since it is linear. You can solve for [latex]x[/latex] or [latex]y[/latex]. For this one, let’s solve for [latex]y[/latex] in terms of [latex]x[/latex].

Substitute the value of [latex]y[/latex] into the second equation, and then solve for [latex]x[/latex]. In this problem, move everything to one side of the equation while keeping the opposite side equal to zero. After doing so, factor out the simple trinomial and then set each factor equal to zero to solve for [latex]x[/latex].

After solving the equation, we arrived at two values of [latex]x[/latex]. Substitute these numerical values to any of the two original equations. However, pick the “simpler” equation to simplify the calculation. Obviously, the linear equation [latex]x + y = 1[/latex] is the best choice!

• If [latex]x = – 3[/latex], solve for [latex]y[/latex].

Answer: (– 3, 4)

• If [latex]x=2[/latex], solve for [latex]y[/latex].

Answer: (2, –1)

Therefore, the solution set to the given system of nonlinear equations consists of two points which are (– 3, 4) and (2, –1) .

Graphically, we can think of the solution to the system as the points of intersections between the linear function [latex]\color{red}x + y = 1[/latex] and quadratic function [latex]\color{blue}y = {x^2} – 5[/latex].

Example 2: Solve the system of equations below.

The first equation is a circle with a radius of [latex]3[/latex] since the general formula of a circle is [latex]{x^2} + {y^2} = {r^2}[/latex].

What I will do is to substitute the expression of [latex]y[/latex] which is [latex]\color{blue}x+3[/latex] from the bottom equation to the [latex]y[/latex] of the top equation. Then we should be able to solve for [latex]x[/latex].

Use these values of [latex]x[/latex] to find the corresponding values of [latex]y[/latex]. I would pick the simpler equation (bottom equation) [latex]y=x+3[/latex] to solve for [latex]y[/latex].

• If [latex]x=0[/latex], solve for [latex]y[/latex].

Answer: (0, 3)

Answer: (– 3, 0)

The final answers are the points (0, 3) and (– 3, 0) . These are the points of intersections of the given line and circle centered at the origin.

Example 3: Solve the system of equations below.

This problem is very similar to problem #2. We have a line (top equation) intersecting a circle (bottom equation) at two points.

Step 1 : Solve the top equation for [latex]y[/latex].

Step 2 : Plug the value of [latex]y[/latex] into the bottom equation. You will be required to square a binomial, combine like terms and factor out a trinomial to get the values of [latex]x[/latex]. Here is the solution:

Therefore, the values of [latex]x[/latex] are

Step 3 : Back substitute these [latex]x{\rm{ – values}}[/latex] into the top equation [latex]x + y = – 1[/latex] to get the corresponding [latex]y{\rm{ – values}}[/latex].

Answer: (– 3, 2)

Answer: (2, – 3)

Step 4 : Here is the graph of the line intersecting the circle at (– 3, 2) and (2, – 3) .

Example 4: Solve the system of nonlinear equations

Substitute the expression of [latex]y[/latex] from the top equation to the [latex]y[/latex] of the bottom equation. Apply the distributive property, then move everything to the left. Factor out the trinomial, then set each factor equal to zero to solve for [latex]x[/latex].

So we have,

Since we now have the values of [latex]x[/latex], pick any of the original equations to solve for [latex]y[/latex]. The obvious choice is [latex]y=x+3[/latex] because it is much simpler than the other one.

Answer: (–1, 2)

Answer: (– 2, 1)

The graph shows the intersection of the oblique hyperbola and the line at points (–1, 2) and (– 2, 1) .

Example 5: Solve the system of nonlinear equations

Observe that the first equation is of a circle centered at [latex](-2, 2)[/latex] with a radius of [latex]1[/latex]. The second equation is a parabola in standard form with vertex at [latex](-2, 3)[/latex]. We expect that the solutions to this system of nonlinear equations are the points where the parabola (quadratic function) intersects the given circle.

We will solve this in two ways. First by the substitution method then followed by the elimination method.

I. Using the Substitution Method

It would be tempting to just substitute the value of [latex]y[/latex] from the bottom equation to the top equation. You may try it. But you should immediately realize that it makes the problem more complicated to work on. There’s a better way, though.

Isolate the term [latex]{\left( {x + 2} \right)^2}[/latex] of the second equation and plug it into the first equation.

Next, substitute this into the second equation, which gives us an equation with a single variable just in [latex]y[/latex].

Setting each factor equal to zero and solving for [latex]y[/latex], we get

Now, we want to find the corresponding values of [latex]x[/latex] when [latex]y=2[/latex] and [latex]y=3[/latex]. I will use the equation of a circle to do just that.

• If [latex]y=2[/latex], solve for [latex]x[/latex].

Answer: (–1, 2) and (– 3, 2)

• If [latex]y=3[/latex], solve for [latex]x[/latex].

Answer: (– 2, 3)

Therefore, the complete solutions are the points of intersections of a quadratic function and a circle at (–1, 2) , (– 3, 2) and (– 2, 3) .

II. Using the Elimination Method

To solve by elimination method, keep all the terms with [latex]x[/latex] and [latex]y[/latex] on the left side and move the constant to the right. Make sure that you align similar terms. In this case, only the terms with [latex]{\left( {x + 2} \right)^2}[/latex] and the constants should have similar terms.

Then subtract the top equation by the bottom equation. Don’t forget to switch the signs when you subtract, i.e., positive turns into negative, and vice versa. The term [latex]{\left( {x + 2} \right)^2}[/latex] should be eliminated after subtraction.

Since the [latex]\color{red}{\left( {x + 2} \right)^2}[/latex] term is gone, we are left with a simple quadratic equation with variable [latex]y[/latex] only then can be solved using factoring.

Start by expanding the binomial term, combine like terms, move everything to the left, factor the resulting trinomial, and set each factor equal to zero to solve for [latex]y[/latex].

Notice that we arrived at the same values of [latex]y[/latex] using the substitution method as shown above. From this point, the solution is now the same as shown above that’s why I will not show the rest of it.

The solution set consists of the points of intersections: (–1, 2) , (– 3, 2) and (– 2, 3) .

Example 6: Solve the following system

Since the [latex]y^2[/latex] terms have the same coefficient but opposite in signs, we can add the two equations together to eliminate the variable [latex]y[/latex]. This should leave us with a simple quadratic equation that can be solved easily using the square root method .

Next, divide both sides of the equation by the coefficient of the [latex]x^2[/latex] term, followed by applying the square root on both sides to get the values of [latex]x[/latex]. Don’t forget to attach the plus or minus symbol whenever you get the square root of something.

Pick any of the two original equations, and find the values of [latex]y[/latex] when [latex]\color{blue}x = \pm\, 3[/latex]. I will use the first equation because it is much simpler!

• If [latex]x=3[/latex], solve for [latex]y[/latex].

Answer: (3, 1) and (3, –1)

• If [latex]x=-3[/latex], solve for [latex]y[/latex].

Answer: (– 3, 1) and (– 3, –1)

The solutions to this system of nonlinear equations consist of the four points of intersections:

(3, 1), (3, –1), (– 3, 1) and (– 3, –1)

In fact, these are the points of intersections of the given ellipse (first equation) and hyperbola (second equation).

Graphically, it looks like the one below.

Example 7: Solve the following system

We will also solve this using the elimination method. However, multiply both of the equations first by some number so that their constants become the same but opposite in signs.

Eliminate [latex]y^2[/latex] by multiplying the first equation by [latex]2[/latex], and the second equation by [latex]3[/latex], and finally adding them together!

Now, solve for [latex]x[/latex] by dividing both sides by the coefficient of the [latex]x^2[/latex] term, and then performing the square root operation on both sides of the equation.

Back substitute the values of [latex]x[/latex] into any of the original equations to solve for [latex]y[/latex]. Let’s use the first equation.

Answer: (3, 2) and (3, – 2)

Answer: (– 3, 2) and (– 3, – 2)

The solutions to this nonlinear system are the points of intersections of the given ellipse and hyperbola.

Chapter 9, Lesson 7: Linear and Nonlinear Functions

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