Solving Distance, Speed, and Time Problems: Round Trip Adventures with Variables

Algebra 2 Grades High School 19:10 Video
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Lesson Description

This lesson explores solving distance, speed, and time problems in round trip scenarios, focusing on problems where relationships are expressed using variables (e.g., 'longer than,' 'faster than') rather than direct values.

Video Resource

Distance Speed Time Round Trip | Part 2

Kevinmathscience

Duration: 19:10
Watch on YouTube

Key Concepts

  • Distance, speed, and time relationship (d = st)
  • Translating word problems into algebraic equations
  • Solving linear equations with one variable
  • Using a table to organize information
  • Understanding relative time differences (e.g., 'longer,' 'faster')

Learning Objectives

  • Students will be able to set up equations that accurately represent distance, speed and time round trip problems where the time or speed is not explicitly given
  • Students will be able to solve algebraic equations derived from round trip distance, speed, and time problems.
  • Students will be able to organize information from word problems into a table to facilitate problem-solving.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the basic formula: distance = speed × time. Briefly discuss how this formula applies to round trip problems where the distance traveled is the same in both directions. Highlight the difference between problems where values are directly given and those where relationships are expressed using variables.
  • Video Viewing and Guided Note-Taking (15 mins)
    Play the Kevinmathscience video 'Distance Speed Time Round Trip | Part 2'. Instruct students to take notes on the problem-solving process, paying close attention to how the table is used and how equations are set up based on the given relationships (e.g., '7 hours longer').
  • Example Problem Walkthrough (15 mins)
    Work through the first example from the video step-by-step on the board. Emphasize the importance of defining variables, setting up the table correctly, and understanding what the variable represents in the context of the problem. Ask students to actively participate in each step.
  • Guided Practice (20 mins)
    Present a similar problem to the class and guide them through the process of setting up the table and writing the equation. Encourage them to work in pairs. Monitor their progress and provide assistance as needed. Have students share their approaches and solutions on the board.
  • Independent Practice (15 mins)
    Assign a set of practice problems for students to solve independently. Encourage them to use the table method and to carefully check their answers against the question's requirements. Remind students to reread the question and ensure they are answering it, as the variable they solved for might not be the answer to the problem.
  • Wrap-up and Assessment (10 mins)
    Review the key concepts of the lesson and address any remaining questions. Administer a short quiz (multiple choice and fill-in-the-blank) to assess students' understanding of the material.

Interactive Exercises

  • Interactive Table Builder
    Provide students with a partially completed table for a new round trip problem and have them fill in the missing information (distances, speeds, or times) based on the problem's description. Use a collaborative online whiteboard tool so students can work together.
  • Equation Challenge
    Present students with a round trip problem and have them race to be the first to correctly set up the equation representing the problem. Encourage peer review to check for accuracy.

Discussion Questions

  • Why is it important to define your variables clearly before setting up the equation?
  • How does the table method help in organizing the information from the word problem?
  • What are common mistakes students make when solving distance, speed, and time problems, and how can we avoid them?
  • In what real-world situations might you need to use these problem-solving skills?

Skills Developed

  • Algebraic problem-solving
  • Critical thinking
  • Translating word problems into mathematical equations
  • Organization and data representation

Multiple Choice Questions

Question 1:

A car travels to a city and back. The trip there takes 3 hours, and the trip back takes 2 hours. If the average speed on the return trip was 20 mph faster, which equation represents the relationship?

Correct Answer: 3x = 2(x + 20)

Question 2:

In a round trip problem, what is a key assumption you can make about the distances?

Correct Answer: The distance is always the same.

Question 3:

A train travels to its destination and back. The outbound trip took 5 hours. The return trip took 7 hours. The train averaged 10 mph faster on the outbound trip. If 'x' represents the return trip speed, what is the outbound trip speed?

Correct Answer: x + 10

Question 4:

What is the most important first step in solving distance, rate, time problems?

Correct Answer: Set up a table to organize information

Question 5:

A plane flies to a city and back. The trip there takes 't' hours. The return trip takes 't-1' hours. The plane averages 500 mph on the way to the city and 600 mph on the return trip. Find the time to the city:

Correct Answer: 6 hours

Question 6:

A boat travels upstream and then returns. The distance each way is 120 miles. The speed upstream is (v - 3) mph and the speed downstream is (v + 3) mph. The time to the destination upstream is 3 hours. Find the time to travel downstream on the return trip:

Correct Answer: 1.5 hours

Question 7:

A bicyclist rides to a park and back. The outbound trip took 2.5 hours and the return trip took 2 hours. On the way to the park the speed was (s + 4) mph and on the way back the speed was s mph. Find the speed of the return trip:

Correct Answer: 20 mph

Question 8:

A runner runs to a location and back covering a total distance of 20 miles. On the way there his speed was (s - 1) mph and on the way back his speed was s mph. The runner ran for 2 hours on the way there. Find the time he ran on the way back.

Correct Answer: 2.5 hours

Question 9:

A truck delivers cargo to one location and back. The location there took 6 hours. The time back took 4 hours. The speed averaged on the way there was s mph. The speed averaged on the way back was s + 10 mph. Find the speed on the way back.

Correct Answer: 40 mph

Question 10:

A train travels to its destination and back. The outbound trip took 5 hours. The return trip took 7 hours. The train averaged 10 mph faster on the outbound trip. Find the speed averaged on the way to the destination.

Correct Answer: (35 / 2) mph

Fill in the Blank Questions

Question 1:

The formula relating distance, speed, and time is distance = speed multiplied by ______.

Correct Answer: time

Question 2:

If a car travels the same distance in both directions of a round trip, the distances are said to be ______.

Correct Answer: equal

Question 3:

When setting up an equation, it's important to first define your ______.

Correct Answer: variables

Question 4:

If a trip takes 'x + 3' hours, and x = 5, the trip took ______ hours.

Correct Answer: 8

Question 5:

A table used to organize information for a round trip is important. It will contain ______ , Speed, and Time.

Correct Answer: Distance

Question 6:

If a car averages 's' mph on the way to a destination, and 's + 10' mph on the way back, then the car went ______ mph faster on the way back.

Correct Answer: 10

Question 7:

A boat went upstream for 't' hours and downstream for 't - 2' hours. So the boat's trip went ______ hours faster on the way back.

Correct Answer: 2

Question 8:

A runner covered a distance of 'd' miles on the way to the track, so he covered a distance of ______ miles on the way back.

Correct Answer: d

Question 9:

If the time in a round trip is represented as 'x' and 'x + 5', then the time that is ______ is represented by 'x + 5'.

Correct Answer: longer

Question 10:

For an airplane trip that is represented by 'x' on the outbound trip and 'x - 7' on the return, then the return trip is ______ by 7 hours.

Correct Answer: shorter

Educational Standards

A2.A.1:Represent and solve mathematical and real-world problems using nonlinear equations, systems of linear equations, and systems of linear inequalities; interpret the solutions in the original context. A2.A.2:Generate and evaluate equivalent algebraic expressions and equations using various strategies.

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