options: Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.
problem_activity: Activity
This activity revisits a polynomial students first encountered in an earlier lesson in this unit, relating the polynomials to the integers. The purpose of this activity is to make clear that technology allows us to identify solutions to equations that can be tedious to do by hand. The last question reinforces this point by asking students to write their own polynomial and find the
\(x\)
value that corresponds to a certain output, since they will need to estimate the solution from a graph.
The question about possible factors is meant to preview the work of the next several lessons in which students will identify factors of polynomial expressions written in standard form and eventually show that for a polynomial
\(p(x)\)
, if
\(p(a)=0\)
, then
\((x-a)\)
is a factor of
\(p(x)\)
.
Launch
Display functions
\(p\)
and
\(q\)
for all to see. Ask, “Without using a graphing calculator, what is the most points of intersection the two functions could have? The least?” Encourage students to make sketches to justify that 3 is the most and 1 is the least.
Provide access to devices that can run Desmos or other graphing technology.
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to spark students’ curiosity about a polynomial written in standard form. Display only the first sentence of this problem (“Consider the functions . . . .”) and ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving visualizing the graph, points of intersection, or possible factors. This will help students create the language of mathematical questions before feeling pressure to produce solutions.
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, to help students see how questions 1 and 2 are related, ask them to find the solutions for an equation such as
\(x^2-4=12\)
. Then, invite them to briefly reflect on the steps used. Next, ask them what they would do differently if they were told to find the
\(x\)
-values where two functions
\(f(x)= x^2-4\)
and
\(g(x)=12\)
intersect. Ask students to notice what elements were similar in each process.
Supports accessibility for: Social-emotional skills; Conceptual processing
Student Facing
Consider the functions
\(p(x) = 5x^3+6x^2+4x\)
and
\(q(x) = 5640\)
.
Use graphing technology to find a value of
\(x\)
that makes
\(p(x)=q(x)\)
true.
For the
\(x\)
-value at the point of intersection, what can you say about the value of
\(5x^3+6x^2+4x-5640\)
?
What does your answer suggest is a possible factor of
\(5x^3+6x^2+4x-5640\)
?
Write your own polynomial
\(m(x)\)
of degree 3 or higher.
Use graphing technology to estimate the values of
\(x\)
that make
\(m(x)= q(x)\)
true.
Student Response
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Anticipated Misconceptions
If students have trouble using the value of
\(5x^3+6x^2+4x-5640\)
at the
\(x\)
-value where the functions intersect to find a possible factor of
\(5x^3+6x^2+4x-5640\)
, encourage them to think about the relationship between factors of an expression and
\(x\)
-values for which the value of the expression is 0.
Activity Synthesis
The goal of this discussion is for students to see how to use technology to answer an algebraic question involving a higher degree polynomial. Invite students to share the polynomial
\(m(x)\)
that they wrote. If possible, display several versions of
\(m(x)\)
for all to see, and use the graph to estimate the
\(x\)
-values where
\(m(x)=q(x)\)
. If this is not possible, ask students to share the solutions they estimated. Students should understand that not all polynomials can be factored by hand, and that technology is a valuable tool for helping us interpret such polynomials.
Then ask students to suggest possible factors of
\(5x^3+6x^2+4x-5640\)
. After a few suggestions, ask students how they could prove something is a factor. It is important that students understand that for now, the only way to prove something like
\((x-10)\)
is a factor of
\(5x^3+6x^2+4x-5640\)
is to find the quadratic function that satisfies
\(5x^3+6x^2+4x-5640=(x-10)(ax^2+bx+c)\)
, where
\(a\)
,
\(b\)
, and
\(c\)
are real numbers. Tell students that the next lessons are about how to do this.


correct_option_index: 3

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