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Unit 8, Lesson 1

Finding Unknown Inputs

Algebra 1

1.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

This task reminds students that they can use a graph of a function to gain some information about the situation that the function models.

Every question can be answered, or at least estimated, by analyzing the graph. As students work, look for those who use the given equation to solve or verify the answers. For example, the first question can be answered by evaluating and the second question by evaluating . Invite them to share their strategy during Activity Synthesis.

Launch

Given their earlier work on quadratic functions, students should be familiar with projectiles. If needed, give a brief orientation on the context. Tell students that there are devices that use compressed air or other means to generate a great amount of force and launch a potato or similar-sized object.

If desired and time permits, find and show a short video clip of someone using an air-powered or a catapult-type device. Warn students that some of these devices can be dangerous and they shouldn’t try to build one without help from an adult.

Activity

None

A mechanical device is used to launch a potato vertically into the air. The potato is launched from a platform 20 feet above the ground, with an initial vertical velocity of 92 feet per second.

The function models the height of the potato over the ground, in feet, seconds after launch.

Here is the graph representing the function.

<p>Parabola facing down with x intercept at 6. Horizontal axis, time in seconds. Vertical axis, height in feet. Maximum at about 2 point 8 comma 150.</p>

For each question, be prepared to explain your reasoning.

What is the height of the potato 1 second after launch?
8 seconds after launch, will the potato still be in the air?
Will the potato reach 120 feet? If so, when will it happen?
When will the potato hit the ground?

Activity Synthesis

Focus the discussion on how students used the graph to help them answer the questions. Encourage students to use precise mathematical vocabulary in their explanation. Invite students, especially those who do not rely solely on the graph, to share their responses and reasoning.

Point out that we can gather quite a bit of information about the function from the graph, but the information may not be precise.

“Is there a way to get more exact answers rather than estimates?” (Some questions can be answered by evaluating the function. For questions that are not easy to calculate at this point, we can only estimate from the graph.)
“Which questions in the activity could be answered by calculating?” (the first two questions, about where the potato is after some specified number of seconds) “Which questions were not as easy to figure out by calculation?” (those about when the potato reaches certain heights)
“How can we verify the time when the potato is 120 feet above the ground? The exact time when it hits the ground?” (We can find the -values that make equal 120 by looking at the graph and then evaluating the function at those values of . To find the exact time the potato hits the ground, we would need to find the zeros of by solving the equation .)

Tell students that in this unit, they will investigate how answers to these questions could be calculated rather than estimated from a graph or approximated by guessing and checking.

1.2

Activity

Instructional Routines

None

Materials

To Copy (from Blackline Masters)

A Trip to the Frame Shop Handout

To Gather

Scissors

Activity Narrative

The purpose of this task is to motivate the need to write and solve a quadratic equation in order to solve a problem.

Students are asked to frame a picture by cutting up a rectangular piece of “framing material” into strips and arranging it around a picture such that they create a frame with a uniform thickness. The framing material has a different length-to-width ratio and is smaller than the picture, so students cannot simply center the framing material on the back of the picture.

Students are not expected to succeed in the task through trial and error. They are meant to struggle, just enough to want to know a way to solve the problem that is better than by guessing and checking. Some students may try writing an equation to help them figure out the right measurement for the frame, but they don’t yet have the knowledge to solve the quadratic equation. The solutions to this equation are irrational, so it is also unlikely for students to find them by chance. After 10 minutes of letting students work on the problem, move on to the Activity Synthesis.

This activity was inspired by the post “When I Got Them to Beg” by Fawn Nguyen, used with permission.

1.3

Activity

Instructional Routines

Aspects of Mathematical Modeling MLR7: Compare and Connect

Materials

None

Activity Narrative

This activity allows students to formulate a mathematical model around the framing task they saw earlier (MP4). Students are prompted to write an equation but are not expected to solve it at this point. In writing an equation and interpreting the solution in context, students practice reasoning quantitatively and abstractly (MP2).

Monitor for students who use these approaches to writing the equation:

Using the length and width of the framed picture. The length is 7 inches plus the thickness of the frame, , on either side, which gives . The height is 4 inches plus the same thickness, , for the top and bottom, which gives . The total area is , or 38, square inches, so the equation is .
Using the area of the picture plus the area of the frame. The area of the picture is 28 square inches. The area of the frame, in square inches, can be found by decomposing it into four squares (in the corners) that are by (or ) each, two rectangles that are each (top and bottom), and two rectangles that are each (left and right). The equation is , or .

As students work, monitor for different strategies students use. Invite students with contrasting approaches to share later.

This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.

Lesson Synthesis

To highlight reasons for writing and solving an equation, discuss questions such as:

“In this lesson, we saw a few examples where we knew the output of a function and were interested in finding the input that produced it. In the launched potato situation, what outputs were known?” (the potato’s heights or distances from the ground) “What inputs did we want to know?” (the times at which the potato was at certain heights)
“In the framing problem, what was the output that we cared about?” (the total area of the framed picture, which was 38 square inches) “What information was the input?” (the thickness of the frame that would produce that total area)
“Why might it be helpful to write an equation to represent a problem such as the one about framing?” (An equation helps us see the relationship between the quantities in the problem, including those whose values are unknown. We can solve for those values.)
“In the framing situation, if represents the thickness of the frame, in inches, what would a solution to the equation mean?” (the thickness of the frame that would produce a total area of 50 square inches)
"Equations such as and are called quadratic equations. Why do you think equations like these are described as quadratic?" (Each equation has a quadratic expression in it. Each one relates quantities in a quadratic function.)

Quadratic equations will be defined more formally in a later lesson. It is fine and expected that students have an informal or partial understanding of what they are at this point.)

to access Cool-down.

Student Lesson Summary

The height of a softball, in feet, seconds after someone throws it straight up, can be defined by . The input of function is time, and the output is height.

We can find the output of this function at any given input. For instance:

At the beginning of the softball's journey, when , its height is given by .
Two seconds later, when , its height is given by .

The values of and can be found using a graph or by evaluating the expression at those values of . What if we know the output of the function and want to find the inputs? For example:

When does the softball hit the ground?

Answering this question means finding the values of that make , or solving .
How long will it take the ball to reach 8 feet?

This means finding one or more values of that make , or solving the equation .

The equations and are quadratic equations. One way to solve these equations is by graphing .

To answer the first question, we can look for the horizontal intercepts of the graph, where the vertical coordinate is 0.
To answer the second question, we can look for the horizontal coordinates that correspond to a vertical coordinate of 8.

<p>Parabola facing down. X intercept at 6. Maximum at about 2 point 8 comma 150. points plotted at 0 point 1 comma 8, 1 point 9 comma 8, and 2 point 1 5 comma 0. </p>

We can see that there are two solutions to the equation and one solution to the equation .

The softball has a height of 8 feet twice, when going up and when coming down, and these instances occur when is about 0.1 and 1.9. It has a height of 0 once, when is about 2.15.

Often, when we are modeling a situation mathematically, an approximate solution is good enough. Sometimes, however, we would like to know exact solutions, and it may not be possible to find them using a graph. In this unit, we will learn more about quadratic equations and how to solve for exact answers using algebraic techniques.

Launch

Tell students that they are now to write an equation to represent the quantities in the framing problem. Consider arranging students in groups of 2 and asking them to think quietly about the questions before conferring with their partner.

If students have trouble getting started, suggest that they start by labeling the diagram with relevant lengths. Then, ask questions such as:

“How could we show that the frame is the same thickness all the way around?” (Use the same variable—for example, —to label the thickness of the strip of framing on all four sides.)
“What is the combined area of the picture and the framing material?” (38 square inches, because )
“How could we express the width of the framed picture?” () “What about the height of the framed picture?” ()
“Once the framing material is cut up and arranged around the picture, how could we express its area?” (Two rectangles with area and two rectangles with area , or )

Select work from students with different strategies, such as those described in the Activity Narrative, to share later.

Representation: Internalize Comprehension. Represent the same information through different modalities by using diagrams. Provide students with graph paper and suggest that they draw and label their diagrams with everything they know so far. Invite students to suggest types of diagrams that might be helpful to draw—such as the strips of framing material, and the picture both with and without the frame.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

Activity

None

Here is a diagram that shows the picture with a frame that is the same thickness all the way around. The picture is 7 inches by 4 inches. The frame is created from 10 square inches of framing material (in the form of a rectangle measuring 4 inches by 2.5 inches).

Write an equation to represent the relationship between the area of the framed picture and the measurements of the picture and of the frame. Be prepared to explain what each part of your equation represents.
What would a solution to this equation mean in this situation?

Student Response

to access Student Response.

Building on Student Thinking

Some students may struggle to express the overall length and width of the framed picture because of trouble combining numbers and variables. Consider drawing a segment composed of 3 pieces of length 1.5 inches, 10 inches, and 1.5 inches, and prompting students to find the length of the entire segment. Then, change each 1.5 to an and ask for an expression for the length of the entire segment ().

If students mistake the result of adding and as , consider drawing a square with side length and ask students to write expressions for the perimeter and area. Then, ask them to point out the difference between multiplying to find area and adding to find total length. If needed, draw a few more segments that are decomposed into parts, with each part labeled with a number or a variable expression, and ask students to write expressions for the total length.

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Lesson 1

Finding Unknown Inputs

Warm-up

Standards Alignment

Building On

HSF-IF.B.4

Addressing

Building Toward

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity Synthesis

Activity

Instructional Routines

Materials

To Copy (from Blackline Masters)

To Gather

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

Building Toward

HSA-CED.A.3

Standards Alignment

Building On

Addressing

HSA-CED.A.1

Building Toward