options: Represent and analyze quantitative relationships between dependent and independent variables.

Multiply and divide within 100.

Solve systems of equations.

Represent and solve problems involving multiplication and division.

Analyze patterns and relationships.

Understand the connections between proportional relationships, lines, and linear equations.

Gain familiarity with factors and multiples.

Represent and solve problems involving addition and subtraction.

Understand properties of multiplication and the relationship between multiplication and division.

Work with equal groups of objects to gain foundations for multiplication.

Use properties of operations to generate equivalent expressions.

Create equations that describe numbers or relationships

Solve equations and inequalities in one variable.

Understand the relationship between zeros and factors of polynomials.

Perform arithmetic operations on polynomials.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Work with radicals and integer exponents.

Rewrite rational expressions.

Write expressions in equivalent forms to solve problems.

Use polynomial identities to solve problems.

Represent and solve equations and inequalities graphically.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Understand solving equations as a process of reasoning and explain the reasoning.

Understand and apply properties of operations and the relationship between addition and subtraction.

Add and subtract within 20.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Generate and analyze patterns.

Write and interpret numerical expressions.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Reason about and solve one-variable equations and inequalities.

Interpret the structure of expressions.

Work with addition and subtraction equations.

Use the four operations with whole numbers to solve problems.
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: 11

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