standard_description: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
problem_activity: Activity
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).
The prompt to write an equation is left relatively open to allow for different approaches. For instance, students may think in terms of:
The length and width of the framed picture. The length is 7 inches plus the thickness of the frame,
\(x\)
, on either side, which gives
\(7 + 2x\)
. The height is 4 inches plus the same thickness,
\(x\)
, for the top and bottom, which gives
\(4 + 2x\)
. The total area is
\(7 \boldcdot 4 + 4 \boldcdot (2.5)\)
or 38 square inches, so the equation is
\((7+2x)(4+2x) = 38\)
.
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
\(x\)
by
\(x\)
or
\(x^2\)
each, two rectangles that are
\(7x\)
each (top and bottom), and two rectangles that are
\(4x\)
each (left and right). The equation is
\(4x^2 + 2(7x) + 2(4x) + 28 = 38\)
or
\(4x^2 + 22x + 28= 38\)
.
As students work, monitor for different strategies students use. Invite students with contrasting approaches to share later.
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, if needed, use scaffolding questions such as:
“How could we show that the frame is the same thickness all the way around?” (Use the same variable, for example
\(x\)
, 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
\(7 \boldcdot 4 + 4 \boldcdot (2.5)=38\)
)
“How could we express the width of the framed picture?” (
\(7+2x\)
) “What about the height of the framed picture?” (
\(4+2x\)
)
“Once the framing material is cut up and arranged around the picture, how could we express its area?” (Two rectangles with area
\(x(7+2x)\)
and two rectangles with area
\(4x\)
, or
\(2x(7+2x) + 2(4x)\)
)
Reading: MLR6 Three Reads.
Use this routine to support reading comprehension, without solving, for students. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (creating a frame for a rectangular picture). If students understand the situation based on their work in the prior activity, skip to the next read. After the second read, students list any quantities that can be counted or measured, without focusing on specific values (dimensions of the picture and framing materials). During the third read, the question or prompt is revealed. Invite students to brainstorm possible strategies, referencing the relevant quantities named after the second read. This helps students connect the language in the word problem and the reasoning needed to solve the problem while keeping the cognitive demand of the task.
Design Principle: Support sense-making
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
Student Facing
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).
[IMG]
Write an equation to represent the relationship between the measurements of the picture and of the frame, and the area of the framed picture. Be prepared to explain what each part of your equation represents.
What would a solution to this equation mean in this situation?
Student Response
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Anticipated Misconceptions
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
\(x\)
and ask for an expression for the length of the entire segment (
\(10+2x\)
).
If students mistake the result of adding
\(x\)
and
\(x\)
as
\(x^2\)
, consider drawing a square with side length
\(x\)
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.
Activity Synthesis
Select students to present their equations and reasoning. If no students bring up one of the strategies shown in the Activity Narrative, bring it up. Consider asking students whether
\((7+2x)(4+2x)=38\)
and
\(4x^2 + 22x + 28 = 38\)
(or other correct equations that come up) are equivalent and how to check if they are.
Discuss what a solution to these equations represents. Make sure students understand that a solution reveals the thickness of the frame when all of the framing material (10 square inches) is used.
Solicit some ideas on how they might go about finding the solution(s). Students may mention:
graphing the equation
\(y=(7+2x)(4+2x)\)
and finding a point on the graph with a
\(y\)
-coordinate of 38
evaluating the expression on one side of the equation at different values of
\(x\)
until it has a value of 38
using a spreadsheet to evaluate the expression at different values of
\(x\)
and see what values produce an output of 38
We can see that the equations
\((7+2x)(4+2x)=38\)
and
\(4x^2 + 22x + 28 = 38\)
are each composed of a
quadratic expression
. Tell students that these are examples of
quadratic equations
. They will learn several techniques for using algebra to solve equations like this.
Formally, a quadratic equation is defined as one that can be written in the form of
\(ax^2+bx+c=0\)
where
\(a\)
is not 0. This formal definition will be introduced in the next lesson, when students work with equations of that form and make sense of them in context.


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