What is Direct Variation in Algebra? A Must-Have Guide to Teaching Direct Variation

Want to help your students answer the question: What is direct variation in Algebra? This how-to guide will help you teach your students how to tackle a direct variation problem as well as provide you with a direct variation activity you can use immediately in your classroom.

Direct variation Algebra problems can be really challenging for students. If you’re a math teacher like me, you’ve probably seen how often students miss these questions on quizzes and tests. After 20 years in the math classroom, I will give you a step by step guide on how you can successfully help your students understand direct variation.

You are going to learn how to prepare a direct variation lesson plan, how to teach different types of direct variation problems, and how to get students to solve problems on direct variation step by step.

This post is all about the best ways to help students answer the question, “What is direct variation in Algebra?”

Direct Variation Lesson Plan

Introduction to direct variation:

When you introduce a direct variation lesson, you want to engage the students immediately. So I usually ask my class:

“Let’s say that this weekend you are going to babysit for 5 hours and you will get paid $50. How much money are you getting paid per hour?”

Students will be able to answer this pretty quickly. They will say $10 the hour.

“If you plan on working next weekend babysitting and you will work 8 hours, then how much money will you earn?”

$80

“But let’s say you have a huge test and so you won’t be able to work so many hours. You’ll only work two hours instead. How much money would you earn babysitting?”

$20

“Tell me a relationship that you notice between the amount of time you babysit and the amount of money you get paid?”

Wait for students to give you a response. You are trying to get them to realize that there is a direct relationship (variation) between the amount of time they spend babysitting, and the amount of money they earn. The more hours they spend babysitting, then the more money they earn. The less hours they spend babysitting, then the less money they earn.

Then go to the board and write down the equation: m = kh and tell them that m is the amount of money they make, h is the number of hours they spend babysitting, and k is the amount of money they get paid per hour. Ask them, what is k in this situation.

k is 10

“Ok, so the equation I can make is that m = 10 h.”

“So in general, in Algebra, we call this a direction variation equation: y = kx. K is called the constant of variation, this is like your $10 the hour that you charge for babysitting. This is a constant because it doesn’t change. What changes is the y value in direct proportion to x. In other words, the more hours you babysit, the more money you make.”

Types of Direct Variation Problems

Now that you’ve introduced the lesson, the next step is to show students the types of direct variation problems they’ll need to be able to solve.

Problem #1: How to find the direct variation equation

Find the constant of variation and the direct variation equation if y = 4 when x = -2.

Ask students, what is the direct variation equation? They should say, “y = kx”. Write it on the board and tell them that you’re going to substitute the information to solve for k, the constant of variation.

y = kx

4 = k(-2)

Now how do you solve for k? They should answer: divide by -2

k = -2

Now that you’ve found k, then you can rewrite the direct variation equation to be: y = -2x

So the constant of variation is -2 and the direct variation equation is y = -2x.

Now have them try a problem on their own: Find the constant of variation and the direct variation equation if y = 28 when x = 7.

Answer: 28 = k(7), so k = 4 and the direct variation equation is y = 4x.

Problem #2: Direct variation and proportion

Y varies directly as x. If y = 7 when x = 14, find x when y = 42.

There are two ways to solve this problem. One way is to use the y and x information they give you and solve for k. Then use that k value to find the answer to the problem like this:

Since y = kx, substitute y =7 when x = 14 to get 7 = k(14). Dividing by 14 (you always divide by the number next to the variable), then k = 1/2 or 0.5.

So that means that the direct variation equation here is y = 0.5x. Now take the OTHER information they give you and substitute.

42 = 0.5x so divide by 0.5 and x = 84. And that’s the answer.

The second way you can solve this problem is by using a proportion. You can set up the problem like this:

Problem #3: Direct variation graph

Graph the direct variation equation y=(2/3)x.

Problem #4: How do you find a direct variation

For each equation below, determine whether y varies directly with x. If so, find the constant of variation.

a) y = -14x

b) y – 12x = 0

c) y = 8x – 2

d) y – 5 = 16x

So remind students that the direct variation equation is y = kx. Therefore a is an example of y varying directly as x and the constant of variation is -14.

If you solve for y in b, then y = 12x, so yes, this does show y varies directly with x and the constant of variation is 12.

Part c is not an example of direct variation since the equation shows 8x minus 2

Part d is also not an example of direct variation since if you solve for y then you get y = 16x + 5 and the adding of 5 makes this no longer a direct variation problem.

Problem #5: Direct variation word problems

Distance traveled varies directly with time. Mia is traveling to Disney World in Orlando from her house in Miami. The distance from Mia’s house to Disney World is 204 miles and it’s going to take her 3.5 hours to get there. Her friend Imani who lives in Palm Beach will have to travel 170 miles. If both Mia and Imani travel at the same speed, how long will it take Imani to get to Disney World?

Solution: This is a direct variation problem. So let’s use the equation d = r t where d is the distance traveled in miles, r = speed, and t = time in hours.

We know Mia travels 204 miles in 3.5 hours. Let’s determine her speed (a.k.a. the constant of variation).

204 = r (3.5)

Divide by 3.5

r = 58.3 mph (approximately)

So now let’s use this information to find out how long it will take Imani to get to Disney World.

d = 58.3h

170 = 58.3h

divide by 58.3

h = 2.9 hours

So it’ll take Imani about 2 hours and 54 minutes to get to Disney World.

Note: Students ALWAYS have a hard time determining minutes from hours when they get a decimal answer. Remind them that to figure out the amount of minutes, they need to take 0.9 and multiply it by 60.

Also show them that they can solve this using a proportion:

10 Examples of Direct Variation in our Daily Life

Here are some examples of direct variation. A quick search on the internet could also get you to even more examples.

1. (They’ll love this one) The more time they spend on studying for a math test, the higher their grade.
2. The more they pedal on their bicycle, the more rotations of their bicycle wheel.
3. The more exercise they do, the more calories they burn.
4. (This one is very common in math problems) The more the weight on a spring weighs, the greater the distance the spring will stretch.
5. The more money you save from your paycheck, the greater your savings will be.
6. Distance traveled and time traveled (we just saw an example of this above).
7. Hours worked and how much money you make (like in the introduction babysitting problem).
8. The more YouTube subscribers to your channel, the more revenue you make in ads.
9. Reading and the amount of vocabulary words you learn.
10. The more people you have to shop for, the more money you’ll spend.

It’s also fun to have your students come up with examples of direct variation problems.

Direct Variation Resources

If you’re looking for a fun worksheet to get your students solving direct variation problems, here’s an 18 problem color by code activity that will give them the practice they need.

Teacher Reviews:

• “Love it. My students liked it because they had the answers on the side, which helped them check their work, and they got to color as well.”
• “GREAT HOMEWORK ASSIGNMENT, SELF CHECK!”
• “My students really enjoyed this activity and for those students who like to color, they found the integration with the TpT platform a bit fun.”

You can also play a Jeopardy-like game with your students in your Algebra class or math class that includes a category on direct variation:

 

So there you have it! You now have a way to introduce a direct variation lesson, you can teach different types of direct variation problems step-by-step, and have easy, ready-to use resources that will get your students engaged in solving direct variation problems.

This blog post was all about the best way to help your students answer the question: What is Direct Variation in Algebra.