options: 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.

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

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.

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.

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

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.

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.

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

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

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

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
problem_activity: Activity
In this activity, students continue to work with inequalities in two variables in context. They write an inequality that represents the constraints in a situation, graph its solutions, and interpret points in the solution region.
Earlier, students saw that some points in the solution region might satisfy an inequality (mathematically) but might not work in the given situation because other considerations were at play. Here, they see another reason that some values that satisfy an inequality might be unfeasible in the situation—namely, that the pair of values must be whole numbers. When students think about the plausibility of these values in context, they engage in an aspect of mathematical modeling (MP4).
As students work, look for those who plot discrete points to represent the solution region and those who shade a part of the plane.
Launch
Arrange (or keep) students in groups of 2. Give students a few minutes of quiet time to work on the first three questions, and then time to discuss their responses before they continue with the last two questions.
Representation: Internalize Comprehension.
To support working memory, provide students with sticky notes or mini whiteboards. Some students may need extra time to explore possible combinations for multiples of 9 and 5. Encourage them to model these combinations by drawing pictures or listing multiples of each number using the whiteboard or sticky notes.
Supports accessibility for: Memory; Organization
Student Facing
A vendor at the Saturday Market makes
$
9 profit on each necklace she sells and
$
5 profit on each bracelet.
Find a combination of necklaces and bracelets that she could sell and make:
exactly $100 profit
more than $100 profit
Write an equation whose solution is the combination of necklaces and bracelets she could sell and make exactly $100 profit.
Write an inequality whose solutions are the combinations of necklaces and bracelets she could sell and make more than $100 profit.
Graph the solutions to your inequality.
[IMG]
Is
\((3,18.6)\)
a solution to the inequality? Explain your reasoning.
Student Response
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Student Facing
Are you ready for more?
Write an inequality using two variables
\(x\)
and
\(y\)
where the solution would be represented by shading the entire coordinate plane.
Write an inequality using two variables
\(x\)
and
\(y\)
where the solution would be represented by not shading any of the coordinate plane.
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Some students may have trouble graphing the line that delineates the solution region from non-solution region because they are used to solving for the variable
\(y\)
, but here the variables are
\(b\)
and
\(n\)
. Ask these students to decide which variable to solve for based on the graph that has been set up. Ask them to notice which quantity is represented by the vertical axis.
Activity Synthesis
If some students plotted discrete points and some shaded the region, choose one of each and display these for all to see. Ask students why each one might be an appropriate representation of the solutions to the inequality.
Highlight that the discrete points represent the situation more accurately, because it is impossible to sell a fraction of a bracelet or 2.75 necklaces. It is, however, tedious to plot a bunch of points to show the solution region. It is much easier to shade the entire region, but with the understanding that, in this situation, only whole-number values make sense as solutions.


correct_option_index: 2