14.1: Notice and Wonder: The Arrow (15 minutes)
The purpose of this activity is for students to practice extracting information about a context from representations that model the context. As an alternative to the prompt “What do you notice? What do you wonder?” You could simply ask students to write down everything they know about this situation based on the representations of the model. This activity is an opportunity to connect associated inputs and outputs expressed in function notation with statements about the situation modeled by the function. For example, if someone notices that the arrow is launched from a height of 18 feet, connect this to \(h(0)=18\).
Display the three representations for all to see. Ask students to think of as many things they notice as possible, and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.
An archer shoots an arrow. The arrow’s height above level ground, in feet, is modeled by the equation \(h(t)=(1+2t)(18-8t)\), and also represented by this graph and table. The time \(t\) is measured in seconds.
What do you notice? What do you wonder?
Ask students to share the things they noticed. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the representations. Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If the following ideas do not come up during the conversation, demonstrate or ask students to discuss:
- Encourage students to state the things they notice using function notation. For example, if someone notices that the arrow is launched from a height of 18 feet, connect this to the statement \(h(0)=18\).
- What does the point on the graph \((\text-\frac12,0)\) mean in the situation? Nothing, actually! If the arrow was launched at \(t=0\), only positive values of \(t\) represent something meaningful about the situation.
- For every thing that students notice in one representation, ask where they can see the information in the other representations, if possible. For example, if they use the table to know that the arrow hits the ground at 2.25 seconds, ask where they can see this information on the graph and how they can use the equation to find this information.
- What if you only knew the equation? Time permitting, demonstrate how to use graphing technology to create a graph, set an appropriate graphing window, and extract information like the coordinates of the intercepts and the maximum. Also demonstrate how to view a table of inputs and outputs.
14.2: Three Objects (15 minutes)
In this activity, students determine information about a context from a representation of that context.
To find the maximum for functions \(f\) and \(g\), you can average the zeros to get the \(x\)-coordinate of the vertex, and then evaluate the function to get the \(y\)-coordinate. (For example, for \(g\), average the zeros -0.25 and 2.5 to get 1.125, and then evaluate \(g(1.125)\).) In fact, due to the symmetry, you could average any two inputs with the same output to locate the vertex. This fact is not emphasized in the Algebra 1 lessons as students will learn to find the vertex of any quadratic function by writing the equation in vertex form in a later unit. So, if all students don’t master this idea at this time, there will be more opportunities later. For \(g\), the coordinates of the maximum of the graph can be determined using graphing technology.
Ensure that students understand that each representation pertains to a different situation. (Unlike the warm-up, where they encountered three representations of the same situation.)
Provide access to graphing technology. If students choose to create a graph of function \(g\) to help answer the questions, they are choosing appropriate tools and using them strategically (MP5).
Some different objects are launched into the air. The height of each object is modeled as a function of time in seconds.
The height, in feet, of the first object is modeled by the function \(d\) and represented by the graph.
The height, in feet, of the second object is modeled by the function \(f\) and represented by the table.
\(t\) 0 0.25 1 1.75 \(f(t)\) 14 18 18 0
- The height, in feet, of the third object is given by the equation \(g(t)=(16t+4)(2.5-t)\).
- For each object, from what height was it launched?
- For each object, how long was it in flight before it hit the ground?
- For each object, what was its maximum height and when did it reach its maximum height? If needed, give your best estimate.
Ensure that students have determined all of the correct information about the contexts from the given representations. Display each representation one at a time and invite students share their responses and explain how they determined the information. Ask questions that encourage students to relate specific features to the situation.
- “How did you figure out the launch height?” (I found the height when the time was 0. This is the vertical intercept of the graph.)
- “How did you figure out how long before the object hit the ground?” (I found the positive \(t\)-intercept which gave me the time when the height was 0.)
- “Which point on the graph tells the maximum height? What do you do if the graph is not provided?” (The vertex tells the maximum height. I can see it on the graph. In the table, I knew it was between the two largest outputs. If the graph is not provided, I can try to find halfway point between the zeros or I can use technology to graph the function.)
14.3: Comparing Two Situations with Different Representations (15 minutes)
This practice activity is similar to the last activity in the associated Algebra 1 lesson, except that students are asked to find specific information about the contexts before being asked to compare them.
To give students a greater opportunity to communicate their thinking verbally and listen to the reasoning of others, we recommend assigning each student to either work on function \(m\) or \(p\) for the first question, and then get together in groups of 2–4 to tackle the remaining questions.
Arrange students in groups of 2. In each group, one partner completes the first question for function \(m\), and the other partner for function \(p\). Then, partners should work together using the information they found in the first question to complete the remaining questions.
It’s a judgment call about whether to ask students to find the maximum of function \(m\) without a graphing calculator or provide access to graphing calculators. In the associated Algebra 1 lessons, there are times when students are asked to answer such a question without graphing.
Two objects are thrown into the air. The height of object M in meters is modeled by the function \(m(x)=(5+10x)(1.5-x)\). The height of object P in meters is modeled by the function \(p\), represented by the graph.
- For each object, determine:
- the time at which the object hit the ground
- the height from which the object was thrown
- the maximum height of the object
- the time at which the object reached its maximum height
- Which object was launched from a greater height? Explain your reasoning.
- Which object hit the ground first? Explain your reasoning.
- Which object reached a greater maximum height? Explain your reasoning.
Much of the conversation will take place within groups. Time permitting, invite students to create a visual display of their solutions and explanations. Give students an opportunity to review other groups’ work, and time to revise and refine their own work. If desired, select a few exemplary visual displays to keep posted in the classroom to support students in later lessons.