6.EE.9: Use variables to represent two quantities in a real-world...

# Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

The Millers must make a 70-mile Thanksgiving trip to visit their grandparents. Pat Miller believes in driving at a steady rate of 50 miles per hour.

(a) With Pat in the driver’s seat, how much time will the trip take?
(b) How many miles will the Millers travel in 18 minutes?
(c) Write an expression for the number of miles they will cover in $$t$$ minutes of driving.
(d) After $$t$$ minutes of driving, how many miles remain to be covered?
The Millers must make a 70-mile Thanksgiving trip to visit their grandparents. Pat Miller believes in driving at a steady rate of 50 miles per hour.

(a) With Pat in the driver’s seat, how much time will the trip take?
(b) How many miles will the Millers travel in 18 minutes?
(c) Write an expression for the number of miles they will cover in $$t$$ minutes of driving.
(d) After $$t$$ minutes of driving, how many miles remain to be covered?

﻿Temperature is measured in both Celsius and Fahrenheit degrees. These two systems are of course related: the $$Fahrenheit$$ temperature is obtained by adding 32 to 9/5 of the $$Celsius$$ temperature. In the following questions, let $$C$$ represent the Celsius temperature and $$F$$ the Fahrenheit temperature.

(a) Write an equation that expresses $$F$$ in terms of $$C$$.

(b) Use this equation to find the value of $$F$$ that corresponds to $$C=20$$.

(c) On the Celsius scale, water freezes at $$0^{\mathrm{o}}$$ and boils at $$100^{\mathrm{o}}$$ Use your formula to find the corresponding temperatures on the Fahrenheit scale. Do you recognize your answers?

(d) A quick way to get an approximate Fahrenheit temperature from a Celsius temperature is to double the Celsius temperature and add 30. Explain why this is a good approximation. Convert $$23^{\mathrm{o}}$$ Celsius the quick way. What is the difference between your answer and the correct value? For what Celsius temperature does the quick way give the correct value?
﻿Temperature is measured in both Celsius and Fahrenheit degrees. These two systems are of course related: the $$Fahrenheit$$ temperature is obtained by adding 32 to 9/5 of the $$Celsius$$ temperature. In the following questions, let $$C$$ represent the Celsius temperature and $$F$$ the Fahrenheit temperature.

(a) Write an equation that expresses $$F$$ in terms of $$C$$.

(b) Use this equation to find the value of $$F$$ that corresponds to $$C=20$$.

(c) On the Celsius scale, water freezes at $$0^{\mathrm{o}}$$ and boils at $$100^{\mathrm{o}}$$ Use your formula to find the corresponding temperatures on the Fahrenheit scale. Do you recognize your answers?

(d) A quick way to get an approximate Fahrenheit temperature from a Celsius temperature is to double the Celsius temperature and add 30. Explain why this is a good approximation. Convert $$23^{\mathrm{o}}$$ Celsius the quick way. What is the difference between your answer and the correct value? For what Celsius temperature does the quick way give the correct value?

There are three feet in a yard. Find the number of feet in 5 yards. Find the number of yards in 12 feet. Find the number of feet in $$y$$ yards. Find the number of yards in $$f$$ feet.
There are three feet in a yard. Find the number of feet in 5 yards. Find the number of yards in 12 feet. Find the number of feet in $$y$$ yards. Find the number of yards in $$f$$ feet.

You have perhaps heard the saying, “A journey of 1000 miles begins with a single step.'' How many steps would you take to finish a journey of 1000 miles? What information do you need in order to answer this question? Find a reasonable answer. What would your answer be if the journey were 1000 kilometers?
You have perhaps heard the saying, “A journey of 1000 miles begins with a single step.'' How many steps would you take to finish a journey of 1000 miles? What information do you need in order to answer this question? Find a reasonable answer. What would your answer be if the journey were 1000 kilometers?

Ms. Tan has been appointed Chief Operating Officer of Cincinnati Public Schools. Ms. Tan would like to provide each school with high speed internet access. You are the Superintendent of CPS, and you have asked Ms. Tan to collect information on the cost of various access speeds. Ms. Tan presents the table below.

Write an equation to describe the situation. Make a graph to visualize it.

Ms. Tan has been appointed Chief Operating Officer of Cincinnati Public Schools. Ms. Tan would like to provide each school with high speed internet access. You are the Superintendent of CPS, and you have asked Ms. Tan to collect information on the cost of various access speeds. Ms. Tan presents the table below.

Write an equation to describe the situation. Make a graph to visualize it.

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You are considering whether to start buying organic eggs at the grocery store. A carton of a dozen organic eggs costs 75 cents more than a regular carton. You buy about one carton per week. Write an equation to represent the additional cost of buying organic eggs over time. Make a graph to visualize.

You are considering whether to start buying organic eggs at the grocery store. A carton of a dozen organic eggs costs 75 cents more than a regular carton. You buy about one carton per week. Write an equation to represent the additional cost of buying organic eggs over time. Make a graph to visualize.

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At noon one day, the Exeter River peaked at 11 feet above flood stage. It then began to recede, its depth dropping at 4 inches per hour.

(a) At 3:30 that afternoon, how many inches above flood stage was the river?
(b) Let $$t$$ stand for the number of hours since noon, and $$h$$ stand for the corresponding number of inches that the river was above flood stage. Make a table of values, and write an equation that expresses $$h$$ in terms of $$t$$.
(c) Plot $$h$$ versus $$t$$, putting $$t$$ on the horizontal axis.
(d) For how many hours past noon was the river at least 36 inches above flood stage?
At noon one day, the Exeter River peaked at 11 feet above flood stage. It then began to recede, its depth dropping at 4 inches per hour.

(a) At 3:30 that afternoon, how many inches above flood stage was the river?
(b) Let $$t$$ stand for the number of hours since noon, and $$h$$ stand for the corresponding number of inches that the river was above flood stage. Make a table of values, and write an equation that expresses $$h$$ in terms of $$t$$.
(c) Plot $$h$$ versus $$t$$, putting $$t$$ on the horizontal axis.
(d) For how many hours past noon was the river at least 36 inches above flood stage?

The perimeter of a rectangle is 100 and its length is $$x$$. What expression represents the width of the rectangle?
The perimeter of a rectangle is 100 and its length is $$x$$. What expression represents the width of the rectangle?

Consider the sequence of numbers 2, 5, 8, 11, 14, . . ., in which each number is three more than its predecessor.

(a) Find the next three numbers in the sequence.
(b) Find the $$100^{th}$$ number in the sequence.
(c) Using the variable $$n$$ to represent the position of a number in the sequence, write an expression that allows you to calculate the $$n^{th}$$ number. The $$200^{th}$$ number in the sequence is 599. Verify that your expression works by evaluating it with $$n$$ equal to 200.
Consider the sequence of numbers 2, 5, 8, 11, 14, . . ., in which each number is three more than its predecessor.

(a) Find the next three numbers in the sequence.
(b) Find the $$100^{th}$$ number in the sequence.
(c) Using the variable $$n$$ to represent the position of a number in the sequence, write an expression that allows you to calculate the $$n^{th}$$ number. The $$200^{th}$$ number in the sequence is 599. Verify that your expression works by evaluating it with $$n$$ equal to 200.

Verify that $$(0,4)$$ is on the line $$3x+2y=8$$. Find another point on this line. Use these points to calculate the slope of the line. Is there another way to find the slope of the line?
Verify that $$(0,4)$$ is on the line $$3x+2y=8$$. Find another point on this line. Use these points to calculate the slope of the line. Is there another way to find the slope of the line?

﻿Remy walked to a friend’s house, $$m$$ miles away, at an average rate of 4 mph. The m-mile walk home was at only 3 mph, however. Express as a fraction

(a) the time Remy spent walking home;
(b) the total time Remy spent walking.
﻿Remy walked to a friend’s house, $$m$$ miles away, at an average rate of 4 mph. The m-mile walk home was at only 3 mph, however. Express as a fraction

(a) the time Remy spent walking home;
(b) the total time Remy spent walking.

How long would it take you to count to one billion, reciting the numbers one after another? First write a guess into your notebook, then come up with a thoughtful answer. One approach is to actually do it and have someone time you, but there are more manageable alternatives. What assumptions did you make in your calculations?
How long would it take you to count to one billion, reciting the numbers one after another? First write a guess into your notebook, then come up with a thoughtful answer. One approach is to actually do it and have someone time you, but there are more manageable alternatives. What assumptions did you make in your calculations?
Dependent and independent variables

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