Unit 2 5th Objective Rate of Change Day 1

Essential Standard UW.9.M.F.IF.06: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

I can calculate the rate of change in a real world situation. Student Friendly I can calculate the rate of change in a real world situation.

Concepts and Skills Concepts Skills Rate of Change Table Graph Rate of Change Formula Reading Graphs Calculate Estimate Interpret

Big Ideas Average rate of change can be found from a function given symbolically, graphically or in a table. The average rate of change can be calculated using data collected from an experiment or simulation.

Essential Questions What is rate of change? How is it calculated or estimated... From a graph? From a table? What important information does rate of change tell us about the relationship between two variables? How do you explain that relationship? What does rate of change look like graphically? How do you analyze rate of change on a graph?

What is Rate of Change? A rate that describes how one quantity changes in relation to another quantity. If x is the independent variable and y is the dependent variable, then... Another name for rate of change... Slope

Independent vs Dependent A variable that stands alone and isn't changed by the other variables you are trying to measure. Dependent A variable that depends on other factors. Example Time spent studying vs test score. Time studying is independent...it does not depend on anything else. Test score depends on how much you study.

Positive or Negative? A positive rate of change: When the value of x increases and the value of y increases, the graph slants upward. A negative rate of change: When the value of x increases and the value of y decreases, the graph slants downward. A zero rate of change: When the value of x increases and the value of y remains constant, the graph is a horizontal line. An undefined rate of change: When the value of x remains constant and the value of y increases, the graph is a vertical line.

Hard to remember? Meet Slope Dude He always travels from left to right As he goes down he says, “Nice negative!” When he hikes up the hill he says, “Puff, puff, positive.” When he falls off the cliff he utters a mathematical word... “UNDEFINED!” Negative Positive undefined Zero As he hikes the straight path he says, “This is zero fun.”

Examples Zero Positive undefined Negative

Calculating Rate of Change To calculate the rate of change you need a set of data. This can come in a graph or a table. Graph Table x y or f(x) -2 2 -1 -4 -10

Calculating with a Table Use the following table to calculate the average rate of change from x = -2 to x = 0. 1. Find your values in the table 2. Calculate how y changes. x y or f(x) -2 2 -1 -4 -10 4. Divide the change in y by the change in x. y goes from 2 to -4 2 – (-4) = 6 3. Calculate how x changes. 6 ÷ -2 = -3 x goes from -2 to 0 -2 – 0 = -2 So the rate of change is -3

Calculating with a Graph The graph below shows how many cookies are being sold each day at the local bakery. What is the rate of cookie growth between day 2 and day 6? 1. Find your values on the graph 2. Calculate how y changes. 4. Divide the change in y by the change in x. y goes from 3 to 1 3 – 1 = 2 3. Calculate how x changes. 2 ÷ 4= ½ So the rate of change is ½ x goes from 6 to 2 6 – 2 = 4

Calculating from a Story Problem At 2 pm, the level of the water in the pool was 10 feet. At 6 pm, the level of water was at 2 feet. Find the rate of change of the water. Which is independent...time or water level? Time; it keeps on ticking regardless of what the water level does. So water level is dependent 2. Then plug in your numbers and calculate.

Important Note to Remember ALWAYS keep your data in the same order! Example: Using the table, find the average rate of f(x) from x = 0 to x = 4 If you do -1 – 3 , you MUST then do 4 – 0 in that order! BUT x 2 3 4 f(x) 1 -1 -1 – 3 = -4 4 – 0 = +4 = -1 3 – (-1) = +4 4 – 0 = +4 = +1

Guided Practice 1. Use the table to find the rate of change. Then graph it. Let’s pick two points from the table to find the rate of change. Change in y = 160 – 80 = 80 Change in x = 4 – 2 = 2 This also means the average rate of speed is 40 mph

Guided Practice 2. Pick two points to compare y is miles, so find the change in miles 44 – 11 = 33 x is weeks, so find the change in weeks 4 – 1 = 3 Divide change in y by change in x 33 ÷ 3 = 11 So the rate of change is 11 miles per week.

Guided Practice 3. Graph the data. Find the slope of the line. Describe what the slope means. Remember: Slope is the same as rate of change. Slope = Change in y ÷ Change in x Slope = 6 ÷ 3 = 2 What does this mean? 14 – 8 = 6 7 – 4 = 3 Each month the hair grows 2 inches.

Guided Practice 4. Which is independent? Months because time will always pass no matter what. Dependent change = $300 - $150 = $150 Independent change = 6 – 3 = 3 So the rate of change in her account is + $50 each month.

Guided Practice 5. Which is independent? Time because time will always pass no matter what. Dependent change = 72 – 88 = – 16 Independent change = 4 pm – noon = + 4 hours So the rate of change in temperature is -4°F each hour.

Guided Practice 6. Which is independent? Months because time will always pass no matter what. Dependent change = 52 – 43 = 9 Independent change = 18 – 0 = 18 So the rate of change in Jaz’s height is ½ inch each month.

Unit 2 5th Objective Rate of Change Day 2

Review What is the rate of change? A rate that describes how one quantity changes in relation to another quantity. How do we calculate the rate of change? What is another name for rate of change? Slope

Guided Practice 8-4 Week is the x value. Amount of Milk is the y value. Now we’ll just pick two points from the line to work with. Change in y = 12 – 6 = 6 Change in x = 6 – 3 = 3 So the rate of change is 2 liters of milk per week

Guided Practice 8-4 Let’s pick two points from the table to work with. Change in y = 28,500 – 21,000 = 7,500 Change in x = 4 – 1 = 3 So the rate of change is $2,500 per year.

Guided Practice 8-4 Year is the x value. Savings is the y value. Now we’ll just pick two points from the line to work with. Change in y = 60,000 – 20,000 = 40,000 why did I add extra zeros? Change in x = 5 – 15 = – 10 So the rate of change is – $4,000 per year.

Guided Practice 8-4 Let’s pick two points from the table to work with. Change in y = 44 – 22 = 22 Change in x = 4 – 2 = 2 So the rate of change is 11 employees per month.

Guided Practice 8-4 Let’s pick two points from the table to work with. Change in y = 51 – 9 = 42 Change in x = 3 – 0 = 3 So the rate of change is 14°C per minute.

Guided Practice 8-4 Weight is the x value. Price is the y value. Now we’ll just pick two points from the line to work with. Change in y = $20 – $7.50 = $12.50 Change in x = 5 – 2 = 3 So the rate of change is $4.17 per pound of food.

Independent Practice On the back of that page is your homework. This is due Thursday 9/5/13. Make sure the front is done as well!

Unit 2 5th Objective Rate of Change Day 3

Review How do you explain the relationship? When calculating rate of change, you determine how y and x are related. Example: My plant went from 2 inches to 10 inches in 4 days. What is the rate of change? The height is dependent on the time. So this means for every day the plant grew 2 inches. Y will always be a function of x (or f(x))

Guided Practice Let’s pick two points from the table to work with. Change in y = 12 – 4 = 8 Change in x = 3 – 1 = 2 So the rate of change is 4 quarts for every gallon

Guided Practice Age is the x value. Allowance is the y value. Now we’ll just pick two points from the line to work with. Change in y = 18 – 6 = 12 Change in x = 15 – 9 = 6 So the rate of change is $2 more in allowance per year older.

Guided Practice Students is the x value. Teachers is the y value. Now we’ll just pick two points from the line to work with. Change in y = 5 – 3 = 2 Change in x = 100 – 60 = 40 So the rate of change is 1 teacher for every 20 students.

Guided Practice Let’s pick two points from the table to work with. Change in y = 22 – 10 = 12 Change in x = 8 – 2 = 6 So the rate of change is 2 inches tall for every 1 foot wide.

Independent Practice

How’d You Do? Cars Washed is the x value. Profit is the y value. Now we’ll just pick two points from the line to work with. Change in y = 90 – 45 = 45 Change in x = 12 – 6 = 6 So the rate of change is $7.50 for every car washed.

How’d You Do? Let’s pick two points from the table to work with. Change in y = 12 – 8 = 4 Change in x = 4.8 – 2.4 = 2.4 So the rate of change is 1.7 yards every second.

Homework Worksheet 6-3 due Friday 9/6/13 If you need help, check your notes.

Unit 2 5th Objective Rate of Change Day 4

Shooting Targets We are going to break the class into two teams. Each team will work together to figure out the rate of change between the two targets. Once you have your answer run and hit the buzzer at the front. The team that correctly gives the rate of change earns those targets towards their score. If you’re wrong the other team gets 30 seconds to try and answer correctly, and earns and extra point if correct. You only get one guess, so work together.

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