How to Teach Quadratic Equations | Albert Resources (2024)

Math teachers: are you looking for a comprehensive guide for how to teach quadratic equations? Look no further. This post is for you!

Monday morning. First period. The bell rings and the students file in. You hand out the notes about quadratic equations and then hand a few out again as a few stragglers come in after the bell. Then you hear it. The age-old math question,

“When are we going to use this?”

What to say? Should you go with your speech about problem-solving skills? Perhaps you should state your analogy about exercising your brain like you exercise your muscles. What is the best way to teach quadratic equations anyway?

Math can feel like an unsolvable conundrum to students. While we may be enamored at the beauty and simplicity of an equation, our students may be calculating the number of minutes until lunch. Part of the joy in teaching math is connecting the dots for students between mathematical tasks and real life. Quadratics are a great place to talk about some adding tools into the solving equations tool belt and using equations to model real-world structures and events.

Read our article below to understand the prerequisite skills students need before taking on quadratics, tips for planning a unit on quadratic equations, and a list of real-life applications for quadratics. Let’s dive in!

Where to Begin a Quadratic Equation Unit?

Begin your quadratic equations unit with learning goals. What do you want students to be able to do when the unit is over? You should have three to five learning goals to present to students at the beginning of the unit.

These may include:

“I can solve quadratic equations using the quadratic formula”

“I can solve quadratic equations using factoring”

“I can use answer questions about real world events using quadratic equations”

“I can solve quadratic equations using multiple methods, including factoring and the quadratic formula”

Students should know what they are expected to learn and what they will be assessed on. Keeping the terminology clear and consistent throughout the unit will help students to retain information. If someone asked your students what they are learning, what would they say? Would they say “We are solving equations with x^2 with a long equation?” or “We are solving quadratic formulas?”

You decide what verbiage your students walk away with by how you present the information. Clarity and organization give students confidence about what they are doing and give students purpose behind assignments and tasks in class. Students should know the end goal of every math problem, class discussion, and homework assignment.

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What is a Quadratic Equation?

Begin by presenting quadratic equations in standard form:

y=ax^2+bx+c

In a quadratic equation, the degree is 2, so a \neq 0. Your students may not know what “degree” means, so you will need to explain that all quadratics contain an x^2 term. The quadratic may not contain x^3 or any x with an exponent above 2.

You can show your students examples and non-examples of quadratic equations. Depending on the experience of your students, you may choose to keep examples very simple and obvious or to include examples in different forms.

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Differences Between Linear and Quadratic Equations

If you are teaching quadratic equations, your students should already have experience with linear equations. Contrasting linear equations and parabolas can bridge the gap between what students already know and what they need to learn.

Linear Equation

Ex. y=3x+1

Description of Linear Graph

• Shaped like a straight line
• Constant slope (rate of change)
• One x-intercept (unless horizontal)
• Usually, no symmetry

Quadratic Equation

Ex. y=3x^2+2x+1

Description of Quadratic Graph

• Shaped like a parabola
• Changing slope (rate of change)
• Can have two x-intercepts (zero, one, or two)
• Always reflection symmetry

It is important to include visuals of the equations when beginning the study of quadratic equations. It takes time and repetition for students to connect the image of the graph with the equation of the graph.

Based on the level of your students, you can choose how in-depth you want to make your comparison of linear and quadratic equations. It is important to use terminology students will continue to use in future courses. For example, the quadratic equation is shaped like a parabola. Of course, students can easily recognize this shape is like the letter “U.” However, “parabola” is the keyword to repeat as this is what students will hear and see in future courses.

Again, using the correct verbiage will increase the likelihood that students will recall the correct information at the correct time.

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What Does it Mean to Solve a Quadratic Equation?

It is important as a teacher of mathematics to present information clearly and expose students to the various prompts they may see. Students may be asked for solutions, solution sets, roots, zeros, and x-intercepts. If students encounter all of these words without any preparation, it may feel quite intimidating! Take the time to explain to students what each word means and how students will receive prompts.

Students need to know that the roots and zeros of a quadratic equation are the x-intercepts. They should know that solving a quadratic equation means you are finding the roots or the x-intercepts. You must also explain in what format you expect students to provide answers and/or solution sets. Remember, terminology that may seem routine to us, as mathematical experts, can appear intimidating to students.

At this point, students should have experience with x-intercepts. Presenting terms students are familiar with first can make the material less intimidating. Then, explaining that the other terms, such as roots, mean the same thing will make the work feel achievable to students.

What are the Prerequisites for Solving Quadratic Equations?

Before teaching students how to solve quadratic equations, we must remember the prerequisite skills that students must master before fully understanding quadratics. It’s always useful to incorporate a review of key basic skills when presenting a new topic such as quadratic equations. This avoids taking an entire day spent in review while ensuring that students who have gaps in their knowledge are exposed to needed concepts.

Here are three prerequisite topics are key to understanding quadratic equations:

1. Principles of Graphing
2. Basic Solving Skills
3. Factoring

Principles of Graphing

Students must understand basic graphing concepts, such as coordinate pairs. This means students should understand pairs are written as (x,y) and should be able to graph coordinate pairs. Students should understand that the coordinate pairs on the graph are all true in the quadratic equation. The connection between the points on the graph and the equation may be a bit abstract for students, but a few demonstrations can strengthen the connection.

Additionally, students should understand parts of graphs. Given a graph, students should be able to identify the x-intercept(s) and y-intercepts. Students should understand when a graph is increasing, decreasing, or constant. The stronger students’ skills are in interpreting graphs, the more easily students will be able to model real-life situations using graphs.Students need to connect the graphs to the solutions of the equation, the x-intercepts.

Basic Solving Skills

In order to use the quadratic formula, students need to be able to substitute numbers for variables. Then, students need to be able to follow the order of operations. Students must be able to organize their work to solve an equation. Most students will not be used to the complexity of an expression like the quadratic equation. Some students who are used to doing work in their head will need to learn how to write their work.

Additionally, students need to understand radicals. Students must be able to take the square root of perfect squares and simplify square roots. When completing the square, students will take the square root of an expression like (x+2)^2. Students must understand that a square root has both a positive and negative solution.

Factoring

Students should have seen some factoring before encountering quadratics. Students can still be successful with quadratics and factoring even if this is their first experience with factoring, but most students will at least have experience with taking out the greatest common factor.

The stronger students’ understanding of factoring is, the easier solving quadratic equations will be, and the more quickly students can solve more complex quadratic equations. Be aware that some students may try to avoid factoring altogether if given the option to use the quadratic equation each time. Allowing students to miss out on factoring will hinder them in the long run. Factoring comes back over and over, again and again in future mathematics courses. Students will use factoring to simplify expressions in so many situations in Algebra I, Algebra II, Precalculus, and Calculus. Do not miss this opportunity to strengthen your students’ factoring skills!

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Strategies for Planning a Unit on Quadratic Equations

Always begin planning your unit with learning goals! As stated above, learning goals help your students to see purpose in everything you do, particularly if you take the time to mention which learning goal you are working with specific tasks and assignments. Creating learning goals first allows you to ensure your assessments line up with your goals. Of course, learning goals should be based on the standards.

A quadratic equations unit provides a great opportunity for both application and alternate assessments. Application problems help students to see the connection between mathematical concepts and real world experiences. Alternate assessments build confidence in students who typically struggle on written tests. You may see students thrive on alternate assessments who may surprise you with their understanding!

5 Ideas for Alternative Assessment on Quadratic Equations

Let us begin with some alternative assessments. These can be done in addition to or in place of written tests. You simply must ensure your summative assessment(s) evaluate whether or not your students accomplished all learning goals. If you created learning goals based on standards, your assessment(s) will also measure all standards.

1. Have students create a video of themselves solving a quadratic equation using one method. You can allow students to choose or you can tell them to use the quadratic equation, factoring, or completing the square. Students must ask like they are the tutor and explain each step. (Videos make great alternate assessments, but you need to set time limits or these can take forever to grade!)
2. Have students create a booklet where they solve a quadratic equation using all three methods. Allow students to explain which method they prefer and why. This can be individually or in groups. If in a group, you can use groups of three and have each person present a different method.
3. You can allow students to choose to make a video, poster, booklet, or presentation about solving a quadratic equation.
4. Take three pictures of objects or events in your life that can be modeled using quadratic equations. Students could take pictures of an arch in the school, a McDonald’s™ logo, or even a banana. Have students explain how the picture shows an example of a quadratic equation.
5. Students may use an image of a parabola and model it using a quadratic equation. You can have students use their own pictures or find one online. Students may input their picture on Desmos and create an equation that follows the path of the parabola. (Students will quickly see the usefulness of factored form!) Have students show how to solve step by step for the x-intercepts of your equation using the quadratic equation and/or using completing the square. Students can present their work or create a booklet and/or poster to submit.

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Creative In-Class Lesson Ideas

Some application ideas are better completed in class. You can create a quadratic equation to model a situation you create in class:

• Have students create a rectangle where the length is three times the width (or three times the width plus two). Students can make a variety of shapes to meet these conditions. See if students can come up with the correct equation on their own. This provides a great opportunity for discussion about the limits of mathematical models. The graphical model is only useful for considering positive and non-zero values for the dimensions and area of the rectangle.
• Take a video or picture of yourself creating the shape of a parabola using a garden hose or throwing a ball. Lead your class through the process of creating the model for your parabola. You can give the starting height (the y-intercept. This could be the height of your hand as you hold the garden hose or as you threw the ball. You can tell students how many feet away from the water or ball lands on the ground (an x-intercept). You can provide the vertex, the number of feet away from you, and the height of the highest part of the parabola. (You may need a helper as you take the video/picture for your class!). Based on this information, students should be able to construct and graph and create an equation to model the situation. Try to provide the information in logical pieces to teach students problem-solving skills. Dan Meyers’ Three Act Math Tasks provide a great model for this type of presentation.

Real-Life Applications of Quadratic Equations

Completing word problems and thinking through real-world situations can also help students to see the connection between quadratic equations and the world around them.

Quadratics and Projectile Motion

The quadratic equation h(t) = -16t^2 + vt + h models the motion of an object thrown straight up into the air where t is time measured in seconds, v is the velocity measured in feet per second, h is the initial height, measured in feet, when the object is thrown or launched, and h(t) is the height of the object measured in feet.

Using this equation, you can create multiple scenarios for students. For example, you could state that an object was thrown up into the air on the edge of the roof at your school. Let us suppose the building height at your school is 20\text{ feet}. The object was thrown at 30\text{ feet per second}. At what time will the object hit the ground?

Students must create the equation to model the situation and solve for the x-intercepts (technically, in this case, t-intercepts. First, students will substitute 20 for the initial height and 30 for the velocity.

This yields the equation: h(t) = -16t^2+30t+20

Now, students must solve for the zeros, or x-intercepts. They can do so using the quadratic formula (read our review guide on the quadratic formula).

This is a good opportunity to teach students to simplify in advance whenever possible. (You need to evaluate whether or not your students can handle an extra step. What may be helpful for some students may become a hindrance to those who are not ready). In this case, when we set y=0, we obtain the equation -16t^2+30t+20=0. We can divide all of these numbers by 2 to obtain the equation -8t^2+15t+10=0.

We can identify that a=-8, b=15, and x=10. Now we can make the substitutions into the quadratic formula.

\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

\frac{-(15) \pm \sqrt{(15)^2 - 4(-8)(10)}}{2(-8)}

\frac{-15 \pm \sqrt{225 + 320}}{-16}

\frac{-15 \pm \sqrt{225 + 320}}{-16}

\frac{-15 \pm \sqrt{545}}{-16}

\frac{15 \pm \sqrt{545}}{16}

We have two solutions, when t=\frac{15+\sqrt{545}}{16} and when x=\frac{15-\sqrt{545}}{16}. We want to know at the time the object hits the ground, so we need to use the positive solution.

The ball will land on the ground after \frac{15+\sqrt{545}}{16}\text{ seconds} or 2.4\text{ seconds} when rounded to the tenths place.

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Quadratics and Optimizing Area

Let’s suppose a farmer wants to fence in a rectangular plot of land for a certain type of crop. He only has 48\text{ feet} of fencing. What is the maximum area of land the farmer can fence in? What dimensions will his plot be?

Consider allowing students to predict what the maximum area will be. Students will certainly be able to create rectangles with a perimeter of 48\text{ feet}. When students make predictions, they have more buy-in to the solving process.

Students will also need the perimeter formula to help them develop the formula for the area of the rectangle. Students should already know these two equations about rectangles.

P=2l+2w

A=lw

Because the perimeter can be 48\text{ feet} at the most, students can substitute 48 for P to determine the relationship between length and width.

P=2l+2w

48=2l+2w

24=l+w

l=24-w

Solving for length allows students to substitute 24-w for l in the area equation, giving a quadratic equation.

A=lw

A=w(24-w)

A=24w-w^2

A=-w^2+24w

Be wary! It is tempting to solve this problem as we are accustomed to solving quadratic equations. In this case, we are not asked to the x-intercepts. We do not want to know when the area will be zero, but rather when the area will be the greatest! We actually must rewrite the equation in vertex form.

A=-w^2+24w

-A=w^2-24w

\color{blue}{\frac{-24}{2}={-12}}

\color{blue}{(-12)^2=144}

-A \color{blue}{+144}=w^2-24w \color{blue}{+144}

-A+144=(w-12)^2

-A=(w-12)^2-144

A=-(w-12)^2+144

For more information and detailed examples about changing from one quadratic form to another, check out our review article on the three forms of quadratic equations.

Now that the quadratic is in vertex form:

A=-(w-12)^2+144

…students will be able to identify the vertex is (12,144). When a parabola opens down, the vertex is the maximum point. If students have trouble grasping this concept, be sure to display a visual of the graph:

To continue: once students identify (12,144) as the highest point, they simply need to interpret the meaning of the coordinate pair to obtain the solution to the problem. The value 12 is the width. Students must remember that l=24-w to determine that if w-12, then l=12. This means the dimension of the plot of land are 12\text{ feet} by 12\text{ feet}. The value 144 is the area of the plot. The area of the farmer’s plot will be 144 \text{ square feet}.

For even more examples of real life quadratic equations, check out this additional article from MathIsFun.

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Student Practice on Quadratic Equations

For student practice questions focused on solving quadratic equations, explore Albert’s Algebra 1 practice course! All Albert questions include explanations of solutions and tips for avoiding common mistakes.

Also, check out our other detailed Algebra 1 review guides including topic reviews you can share with your students.