direct variation assignment answer key

Direct Variation Worksheets

How to Spot Direct Variation in Data Sets We know so may quantities that increase at a constituent rate. For example, driving your car at a consistent rate or if you save money weekly, then your savings increase at a consistent rate. If we graph these consistent values, we will get a linear equation. Here we will study the direct variation. In a direct variation, two quantities, like hours and pay, or distance and tie, decrease or increase at a constituent rate. The consistent rate that increases and decreases are known as the constant of variation. One way of spotting direct variation is by data sets. In the table given below, we will study the consistent rate of your weekly savings of $50. As this is a direct variation, we assume that you start with zero saved money. (Week| Amount) Data = 0, 0 - 1, $50 - 2, $100 - 3, $150 - 4, $200 - 5, $250. We can notice two key points about this table: 1. At the start, you have $0 at week 0. Hence, it is the first entry of the table, and it will be the first entry of any direct variation table. 2. Each week you save $50, and the amount increases by $50 at the end of every week. $50 is also known as the constant of variation as it tells us how much is being saved by the end of every week.

Basic Lesson

Demonstrates the concept of variations and the use of cross multiplication. Practice problems are provided. "x varies directly with y". If x = 5 when y = 15, find y when x = 6. The formula for direct variation is y = kx, where k is the constant of variation.

Intermediate Lesson

Explores the use of data charts that exhibit variation. Practice problems are provided. In the following chart, does one variable vary directly with the other?

Independent Practice 1

Contains 20 Direct Variation problems. Example: If 120 calories in 100 grams of a chocolate. About how many calories were in 150 grams of the chocolate that if the number of calories varies directly with the weight of the chocolate. The answers can be found below.

Independent Practice 2

Features another 20 Direct Variation problems.

Homework Worksheet

Example problems are provided and explained. Example: Write equation which represents: p varies directly with q, if p = 9 when q = 27?

10 Direct Variation problems. A math scoring matrix is included.

Homework and Quiz Answer Key

Answers for the homework and quiz.

Lesson and Practice Answer Key

Answers for both lessons and both practice sheets.

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Teaching Direct Variation

In units on linear functions , students learn about direct variation. This includes learning what the concept of direct variation in relations means, learning how to represent a direct variation using an equation, as well as graphing directly varying relations.

To make lessons on direct variation more accessible and engaging, math teachers can employ a variety of teaching strategies. In this article, we bring you a few such strategies. Use them in your class, and never worry about teaching direct variation again!

Strategies to Teach Direct Variation

What is direct variation.

You can start your lesson by providing a definition of direct variation. Explain that direct variation is a relationship between two quantities that can be represented by a function of the following form:

…where k does not equal 0. Add that k is the coefficient of x, and x and y vary directly with each other. Point out we can find the value of k as:

Write an equation on the whiteboard and ask students if it represents a direct variation. Then, demonstrate how we can find the constant of variation or k. For example, you can write the following equation:

Point out that we can rewrite the equation in the form of y = kx, which will result in:

Write another equation on the whiteboard and ask students if it’s a direct variation. Ask students to show how we can find the constant of variation k. For instance, you can write the following equation:

3x + 4y = 6

Writing Equations of Direct Variations

Explain to students how we can write the equation of a direct variation. Point out that if we are given any ordered pair (𝑥1,𝑦1) for a relationship that has a direct variation, we will find the constant of variation k and then write the equation, that is:

Provide an example of how we can write an equation of a direct variation. For instance, suppose y varies directly with x and y = 21 when x = 3. Demonstrate to students how we can write a direct variation equation relating x and y.

Graphing a Direct Variation

Point out to students that the graph of a direct variation is a straight line that passes through the origin (0,0). Present an example so that students can visualize what the graph will look like. For instance, on the image below, we can observe the graph of direct variation:

Additional Resources:

You can also enrich your lesson on direct variation with multimedia material, such as videos. For instance, use this video by Khan Academy to introduce the concept of direct variation, as well as illustrate it with examples.

This video is also a good resource to present simple guidelines on how to check whether an equation is a direct variation, as well as how to write the equation of a direct variation. Finally, use this video to show how to graph a direct variation.

Activities to Practice Direct Variation

Online jeopardy game.

This is an online game that will help students improve their skills to write a direct variation equation given x and y, and identify whether a given equation is a direct variation or find the constant of variation. Make sure that all students have a suitable device before starting.

Divide students into groups of three. Each person in the group represents a team of their own and competes against the other two players. Students should select the three-teams mode in the game when starting to play.

Players in each group take turns to answer questions related to direct variation. Each player selects the category of questions and the difficulty level. Depending on the difficulty of the question, they can score from 100 to 500 points if they answer correctly.

In the end, the student with the highest score is declared a winner. Parents who are homeschooling their children can also adjust the game to an individual one, by choosing the one-team mode when starting the game.

This game will help students hone their skills to determine if an equation is a direct variation, as well as show them how to represent a direct variation using an equation. To use this activity in your class, print out this Assignment Worksheet (Members Only).

Divide students into groups of 3 or 4 and hand out the worksheets (one worksheet per student). Provide instructions. Explain that students in each group work together to answer the question in the worksheet.

Provide some 15 minutes for students to complete the exercises. In the end, provide space for discussion and reflection. Which strategies did students use to determine if an equation is a direct variation? Were there any exercises that were particularly challenging? Why?

Before You Leave…

If you liked the strategies on teaching direct variation that we outlined in this article, we have a whole lesson that is dedicated to this topic!

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This article is based on:

Unit 5 – Linear Functions

• 5-1 Rate of Change and Slope
• 5-2 Direct Variation
• 5-3 Slope-Intercept Form
• 5-4 Point-Slope Form
• 5-5 Standard Form
• 5-6 Parallel and Perpendicular Lines
• 5-7 Scatter Plots and Trend Lines
• 5-8 Graphing Absolute Value Functions

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Direct and Inverse Variation Worksheets

Direct and inverse variation worksheets are designed for high schoolers and are divided into subtopics like identifying the type of variation by observing equations, graphs and tables, finding the constant of variation, and much more. The worksheets provide dual levels, level 1 deals with direct and inverse variations, while level 2 deals with direct, inverse, joint and combined variation. A prior knowledge of proportions will definitely be an added advantage. Kick-start your practice with our free worksheets!

Recognize Direct and Inverse Variation

The quintet multiple response pdf worksheets have exercises for learners to observe equations, graphs and recognize the type of variation as direct (linear graph) or inverse (rectangular hyperbola).

• Download the set

Direct and Inverse Variation - Equation

Equations representing the direct variation are in the form y = kx and inverse variation is in the form xy = k. Identify the type of variation in the equations featured in these printable worksheets. Also, find the constant of variation (k).

Direct and Inverse Variation - Table

State whether the table represents a direct variation or inverse variation or neither of them. Find the constant of variation as well.

Constant of Variation - Equation

These high school pdf worksheets feature exercises on finding the constant of variation (k) from the given equations, packed into two levels for easier navigation.

Constant of Variation with the given Values

Find the constant of variation (k) based on the values of the variables in the given statement. Validate your responses with our answer keys.

Write the Equation

Find the constant of variations and translate the statements into equations.

Missing Values

Navigate through the printable worksheets here to formulate an equation relating the variables to find the value of the missing entity.

Complete the Table

This set of pdf worksheets consist of exercises in tabular format. Find the constant of variation (k) and complete the table.

Direct and Inverse Variation Word Problems

Find 70+ word problems, where the student learns to apply the concept of variation to real-life scenarios.

(15 Worksheets)

Related Worksheets

» Ratio Word Problems

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» Proportions

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Direct and Inverse Variation

Direct and inverse variation answer key.

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Direct Variation Assignment #2

Mathematics.

18 questions

Introducing new   Paper mode

No student devices needed.   Know more

• 3. Multiple Choice Edit 15 minutes 1 pt In a direct variation, y = 39 when x = 3.  Find the value of y when x = 2. 26 3 58.5 2
• 5. Multiple Choice Edit 15 minutes 1 pt y varies directly with x. When y = 39,  x = 3.  Find the value of y when x = 2. 26 3 58.5 2
• 6. Multiple Choice Edit 15 minutes 1 pt Is this a direct variation? y = 2x yes no
• 9. Multiple Choice Edit 15 minutes 1 pt y varies directly with x. When y is 16, x is 32. Which equation represents this situation? y = 32x y = 16x y = 2x y = (1/2)x
• 10. Multiple Choice Edit 15 minutes 1 pt What ordered pair does a direct variation always go through when graphed? (0, 0) (1, 0) (0, 1) (1, 1)
• 12. Multiple Choice Edit 15 minutes 1 pt Eduardo counted ten seconds between seeing lightning and hearing thunder, and he knew that the lightning was about 2 miles away. What is the equation of variation? y = 10x y = 2x y = 5x y = 1/5x
• 13. Multiple Choice Edit 15 minutes 1 pt The amount of money earned on a job is directly proportional to the number of hours worked.  If $70.00 is earned in 5 hours, how much money is earned in 22 hours of work? $2 $308 $1.57 $265
• 16. Multiple Choice Edit 15 minutes 1 pt  Y varies directly with x, and y is 84 when x is 16. Which equation represents this situation? y=1344x y=100x y=5.25x y=4/21x
• 17. Multiple Choice Edit 15 minutes 1 pt Find the slope between the two points. (3, 7) and (-4, 2) 7 / 5 5 / 7 2 / 3 2
• 18. Multiple Choice Edit 15 minutes 1 pt Find the slope of the line that passes through the points (2, 4) and (6, 12) 1/2 -1/2 2 -2

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10 Direct Variation Real Life Examples

Direct variation is a mathematical concept that describes the relationship between two variables.

The value of y, therefore, changes proportionately to the change in x. This means that if x increases, y will increase, and if x decreases, y will decrease. However, the ratio between them will remain the same (constant).

1. Hourly Wages

Many workers are paid based on the number of hours they work. The more hours worked result in more money paid.

In this example, if the hourly wage was $20. $20 would be the constant (k). The total amount of money the worker will be paid is directly proportional to the number of hours they work. If they work for 5 hours, they will be paid $100 i.e. $20 x 5.

2. Distance Covered

The distance you cover when driving or walking in a specified time is directly proportional to the speed at which your vehicle is traveling.

In this example, y (the distance covered) is equal to k (the time) times x (the speed at which you are traveling).

 So, if you travel at 50 miles/hour for 2 hours, you will cover 100 miles. However, if you travel at 25 miles/hour for 2 hours, you will only cover 50 miles.

3. Amount Of Food Cooked

For example, the number of cupcakes you can prepare from a recipe will vary directly with the number of ingredients you use.

If your recipe gives measurements for 10 cupcakes and you want to make 20 cupcakes. You will have to double your ingredients in order to get more cupcakes.

4. The Amount Of Money Spent On Purchases

The amount of money spent on purchasing a specific item will vary depending on the number of items you purchase. If you purchase more of the same item, you will spend more. If you purchase fewer of the same items, you will spend more.

In this case, y is the amount of money used for the purchase and x is the cost of a single item. For example, if one gallon of milk costs $4 and you want to purchase 10 gallons, you would spend a total of $40 on milk.

5. Utility Bills

For example, the amount you are billed for electricity ( y ) is based on the amount of energy consumed ( x ) in a given period.

Electricity supply companies charge a constant rate per kilowatt-hour ( k ). Therefore, your bill will increase if you increase the amount of electric energy consumed.

6. Number Of Calories Burned in An Activity

Keeping track of the number of calories you burn during an activity is an important part of health tracking.

The number of calories you burn is not only dependent on your weight and pace/intensity, but also on the length of time or distance covered during the activity.

7. Stretching Of A Spring

A spring will stretch out in proportion to the amount of weight placed at the end of the string.

 The spring will stretch out more when a heavy object is placed at its end. It won’t stretch out as much when a light weight is placed at its end.

The spring will stretch 12cm when a 20-pound bag of rice is placed at its end.

This direct variation is applied to the design of weighing scales.

8. Consumption Of Gas

 The car will require more gas to cover a longer distance and it will require less when driving for shorter distances.

For example, if your car consumes 20 gallons of gas over 400 miles, you will consume 10 gallons of gas when driving over 200 miles.

9. Bank Balance

The more money you put into the account, the higher your bank balance will be.

10. Time It Takes To Fill A Tank Of Water Using A Pump

The volume of water in a tank is directly proportional to the length of time that the pump was switched on.

There you have it; 10 practical direct variation real life examples.

As you can see, direct variation isn’t limited to equations solved in the classroom.

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12 Examples of Linear Regression in Real Life

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Direct Variation Joke Worksheet with Answer Key

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Description

Solve 17 direct variation problems to find the answer to the joke. The first 8 problems have you finding the constant of variation. The next 4 problems give you a coordinate and have you write the direct variation equation. The last 5 problems give you two coordinates where you have to find the missing x or y.

Step-by-step answer key is included.

Great for additional practice, sub plans and remote learning.

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