1. Expressions & Equations
Dependent vs. Independent variables
© 2013 Meredith S. Moody

2. Objective: You will be able to…
Explain what an independent variable is in your own words
Explain what a dependent variable is in your own words
Write equations showing the relationship between the independent and dependent variables
Analyze the relationship between an independent and a dependent variable using graphs and tables and relate it to an equation
© 2013 Meredith S. Moody

3. Vocabulary
Independent variable: A variable that determines the value of other variables
Dependent variable: A variable whose value is determined by another variable
© 2013 Meredith S. Moody

4. Independent variables
Independent variables are things that don’t depend on something else
For example, time is an independent variable
No matter what happens, time occurs
Nothing influences time
© 2013 Meredith S. Moody

5. Dependent variables
Dependent variables are things that change when something else happens
For example, your age is a dependent variable
It changes with time; as time passes, your age increases
Age is a dependent variable – how old is someone? It depends on the time (time of day, week, month, year, etc.).
© 2013 Meredith S. Moody

6. Independent vs. dependent variables
How can you tell if a variable is independent or dependent?
Does it depend on something else?
If you can answer “it depends on…” then your variable is dependent
How long will it take John to drive to the store?
It depends on how fast he is driving
It depends on how far away the store is
© 2013 Meredith S. Moody

7. Example: Driving John lives in Texas; his friend lives in New York
John is going to drive to New York to visit his friend
How far will John have driven after one day of driving?
It depends on…his speed
John drives 70 miles per hour his first day of driving
How far has he gone after one day of driving?
It depends on…how long he drives
John drove for 8 hours his first day of driving
John drove 70 miles per hour for 8 hours
His first day of driving, John drove 8 x 70 = 560 miles
© 2013 Meredith S. Moody

8. Example: Graphing John’s trip
If John drove 70 miles per hour for 8 hours, how far had he gone before lunch?
Use the graph of John’s time and distance
John ate lunch after he had driven 4.5 hours
John traveled a little more than 300 miles before lunch
© 2013 Meredith S. Moody

9. Graphing rules
When graphing independent-dependent variable relationships, the independent variable ALWAYS goes on the x-axis
The dependent variable ALWAYS goes on the y-axis
How to remember: time is ALWAYS independent; time moves forward (horizontal, x-axis), not up (vertical, y-axis)
© 2013 Meredith S. Moody

10. Example: Science applications
Another subject that uses dependent and independent variables is science
In an experiment, there is an independent variable
For example, if a scientist wants to know how fertilizer affects plant growth, the fertilizer would be the independent variable
The growth would be the dependent variable
How high the plant grows would depend on how much fertilizer each plant receives
© 2013 Meredith S. Moody

11. Example: Graphing plant growth
The height of the plant depends on how much fertilizer is used
This chart shows the relationship between fertilizer and growth
How high did a plant that got 2.5 grams of fertilizer grow?
The plant grew about inches tall
© 2013 Meredith S. Moody

12. Check! Write down, in your own words, what an independent variable is
Share your explanation with a neighbor
Write down, in your own words, what a dependent variable is
Share your explanation with a different neighbor
© 2013 Meredith S. Moody

13. Guided practice
Identify which variable in the following scenarios is independent and which is dependent
Katy drives 30 miles per hour on her way to work; how far has she driven after half an hour?
Her distance depends on how long she drives and her speed (30 mph)
Independent variable: time (half an hour) & speed (30 mph)
Dependent variable: distance
A scientist wants to know how the temperature changes overnight; what is the temperature at 11pm?
The temperature depends on the time of night
Independent variable: time
Dependent variable: temperature
© 2013 Meredith S. Moody

14. Guided practice: graphing
Katy drives 30 miles per hour to work
How far has she driven after half an hour?
Independent variable: speed (30 mph)
Independent variable: time (half an hour)
Dependent variable: distance
x-axis: time/speed
y-axis: distance
She has traveled 15 miles after half an hour
© 2013 Meredith S. Moody

15. Example: Equations (John)
Relationships between independent and dependent variables can be written as equations
John drove 70 miles per hour; how far he went depended on how long he drove
John’s distance was his speed for however long he drove
His distance (d) equaled 70mph x his time (t)
d = 70t
Why multiplication?
It is repeated addition; if John drove for 2 hours, his distance would be , or 70 x 2
© 2013 Meredith S. Moody

16. Example: Equations (Katy)
Katy drove 30 miles per hour
Her distance was 30 miles per hour for however long she drove
After 1 hour, she had driven 30 miles; after 2 hours, 60 miles
Her distance (d) equaled 30mph x her time (t)
d = 30t
© 2013 Meredith S. Moody

17. Guided practice: Equations
A scientist wants to measure the effect of water on plant growth
He conducts an experiment and notices that for ever 2 oz of water he gives a plant, it grows 1 inch
The height (h) of the plant is twice the oz of water (w) he gives it
The height equals 2 times the oz of water
h = 2w
© 2013 Meredith S. Moody

18. Guided practice: Plants and water
The graph of a relationship relates to the equation
A plant grows 1 inch for every 2oz of water
The height depends on the amount of water
water: x-axis
height: y-axis
Height is 1in for every 2oz
h = 2w
© 2013 Meredith S. Moody

19. You try! Jim drives 15 miles per hour in the city
How far has he driven after 1.5 hours?
Identify the independent and dependent variables
Graph the relationship
Write an equation for the relationship
© 2013 Meredith S. Moody

20. You try! Solution Independent variables: time & speed (15 mph)
Dependent variable: distance
He has traveled 22.5 miles after 1.5 hours
d = 15t
© 2013 Meredith S. Moody