Graph Stories 2 Teacher Notes Supplies: Task Goals: Use Graph Stories from Dan Meyer to Interpret key features of graphs Describe domain and range from a graph in context. Estimate average rate of change from a graph. Mathematical Language: function, domain, range, rate of change , intercepts , average rate of change

Practice Target Practice 4. Model with Mathematics 2-7 Solving Equations With Algebra Tiles powerpoint Practice Target Practice 4. Model with Mathematics What are some ways to represent the quantities? Where did you see one of the quantities in the task in the graph? Practice 4. Model with mathematics. 2

2-7 Solving Equations With Algebra Tiles powerpoint Learning Targets F-IFa I can understand the concept of a function and use function notation. Identify x, the input, as a quantity of the domain. Identify f(x) the output, as a quantity of the range. Apply the definition of a function to determine if an equation, a table, or graph is a function. F‐IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x). denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). 

2-7 Solving Equations With Algebra Tiles powerpoint Learning Targets F-IFb I can interpret functions that arise in applications in terms of the context. Given the graph of a linear, determine the practical domain of the function as it relates to the numerical relationship it describes. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. F‐IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F‐IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. F-­IF.6 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.

2-7 Solving Equations With Algebra Tiles powerpoint Learning Targets F-LE I can construct and compare linear, quadratic and exponential models and solve problems. 1b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Write a function that models a linear relationship. F-­LE.1  Distinguish between situations that can be modeled with linear functions and with exponential functions. 1b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

Launch How long do you think it will take to fill up the football with the pump?

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Watch the video. Graph the situation.. The picture is the video. Show the video multiple times. Twice at regular speed Once to just watch. Then to make a graph. Pause for students to label the y-axis on their papers. Again at half-speed to revise graph.

Write a question that can be answered with the graph. Teacher: Write a question that can be answered with the graph.   Possible questions: What is happening during the flat segments in the graph? What does the y-intercept mean in this graph? What does the x-intercept mean in this graph? Why does the graph start to rise at about 2.2 seconds? What will the graph look like at 20 seconds? Will the graph ever decrease?

Find the domain of the function. Find the range of the function. Make them express domain and range in sentences.   Teacher: Find the domain of the function. The time is from 2.2 to 15 seconds. Teacher: Find the range of the function. The air pressure is from 0 to 8 pounds per square inch.

Is the graph linear? Teacher: Is the graph a function? Yes because each time quantity is matched with only one air pressure quantity. Some time quantities share air pressure quantities, but the definition of a function only inspects quantities in the domain.

What can you say about the rate of change? Teacher: What can you say about the rate of change? Push students to support their responses with evidence from the graph. This graph does not have a constant rate of change. From 3 to 4 seconds the graph rate of change is zero because the man is lifting up the pump. From 10 to 11 seconds the rate of change is about 1.8 pounds per square inch each seconds. What can you say about the rate of change?

Find the average rate of change. Teacher: What can we find? What would it mean to find an “average rate of change”?   We can find an average rate of change. Focus on the word “average”. Let’s get to this understanding without symbols: Average: f(b) – f(a)   b – a Average: You need the words, because a rate compares to unit measures. 8 pounds per square inch 13.8 seconds

Watch the video. Graph the situation.. The picture is the video. Show the video multiple times. Twice at regular speed Once to just watch. Then to make a graph. Pause for students to label the y-axis on their papers. Again at half-speed to revise graph.

Answer questions with the graph. What is happening during the flat parts? Why is there a dip around 13 - 14 seconds? Which speed is faster, his climbing speed or his sliding speed? How long did he sit at the top of the slide? How high is his waist off the ground?

Find the domain of the function. Find the range of the function. Is the graph linear? What can you say about rate of change? Find the average rate of change. Make them express domain and range in sentences. Teacher: Find the domain of the function. The time is from 2.2 to 15 seconds.   Teacher: Find the range of the function. The air pressure is from 0 to 8 pounds per square inch. Teacher: Is the graph a function? Yes because each time quantity is matched with only one air pressure quantity. Some time quantities share air pressure quantities, but the definition of a function only inspects quantities in the domain. Teacher: What can you say about the rate of change? Push students to support their responses with evidence from the graph. This graph does not have a constant rate of change. From 3 to 4 seconds the graph rate of change is zero because the man is lifting up the pump. From 10 to 11 seconds the rate of change is about 1.8 pounds per square inch each seconds. Teacher: What can we find? What would it mean to find an “average rate of change”? We can find an average rate of change. Focus on the word “average”. Let’s get to this understanding without symbols: Average: You need the words, because a rate compares to unit measures.

Watch the video. Graph the situation.. The picture is the video. Show the video multiple times. Twice at regular speed Once to just watch. Then to make a graph. Pause for students to label the y-axis on their papers. Again at half-speed to revise graph.

Answer questions with the graph. How much water is in the container? When does the water fill up the container fastest? Make them express domain and range in sentences.   Teacher: Find the domain of the function. The time is from 0 to 15 seconds. Teacher: Find the range of the function. The water volume is from 0 to about 610 milliliters. Teacher: Is the graph a function? Yes because each time quantity is matched with only one water volume. Teacher: What can you say about the rate of change? Push students to support their responses with evidence from the graph. The rate of change is constant. Teacher: Find the average rate of change. Let’s get to this understanding without symbols: Average: You need the words, because a rate compares to unit measures.

How much water is in the container? When does the water fill up the container fastest? Teacher: Write a question that can be answered with the graph. Possible questions: What is happening during the flat parts? Why is there a dip around 13 - 14 seconds? Which speed is faster, his climbing speed or his sliding speed? (NOTE: speed is not connected to direction. How long did he sit at the top of the slide? How high is his waist off the ground?

Team Practice

Debrief Linear functions are a special type of functions. Compare functions to linear functions. How are both types of functions similar? How are linear functions different? This debrief should bring out this standard: F-­LE.1  Distinguish between situations that can be modeled with linear functions and with exponential functions. 1.b – Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.   Similarities: All of these graphs are functions because they meet the definition of a function (each input has exactly one output). Differences: The last one is a linear function because the rate of change is constant.

Did you hit the target? Practice 4. Model with Mathematics 2-7 Solving Equations With Algebra Tiles powerpoint Did you hit the target? Practice 4. Model with Mathematics F-IFa I can understand the concept of a function and use function notation. F-IFb I can interpret functions that arise in applications in terms of the context. F-LE I can construct and compare linear, quadratic and exponential models and solve problems.