Grade 8 Benchmark #2

Unit Project: Say it with Symbols

The student council at Baruch is sponsoring a T-shirt sale. They plan to take orders in advance so they know how many T-shirts to have printed. They have made the following estimates of expenses and income for their sale:

·  Expense of $350 for advertising plus $195 for the using the printing press

·  Expense of $4.25 for each T-shirt

·  Income of $10 for each T-shirt

·  Income of $200 from a local business in exchange for printing the business’s logo on the back the T-shirts.

Your task is to answer the following questions:

1.  How many T-shirts must the student council sell to break even?

2.  The student council wants to run a free dance for the school for Halloween. They estimate this dance will cost them $350 to run. How many T-shirts must they sell in order to cover the costs of the dance?

3.  What would happen if you sold 40 T-shirts? Where did you find that information? You need to clearly explain your answer in detail.

4.  What price would you have to sell the T-shirts for to break even at 40 T-shirts?

Your answers to these questions must be solved in three different ways. Each method must be fully and clearly explained.

The three ways are a) Using as table

b) Using a graph

c) Solving equations

Name: ______Date: ______Class: _____

Part1: How many T-shirts must the student council sell to break even?

Make a table and graph this situation

1)

# T-shirts / Cost / Income / Profit/Loss

2) Graph the data; one line for income and one line for expenses.

Make sure you label the graph.

3) Explain how you set up your table and graph. Be sure to discuss how you chose your independent and dependent variable. Also, explain how you decided on your intervals on your table and graph.

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4) Indicate the break-even point on the table and graph. How can you tell how many t-shirts need to be sold to break even?

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5) Write an algebraic equation to find the number of t-shirts needed to break even.

Solve it and check your answer, showing all your work.

6) Explain how you set up and solved the equation.

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Part 2 How many T-shirts must they sell in order to cover the costs of the dance?

7) Use a table, graph and equation to decide how many t-shirts need to be sold to make a profit of $350.

# T-shirts / Cost / Income / Profit/Loss

Equation:

8) Which of the three methods did you find the easiest and most useful to solve these real-life situations? Explain why this is the case.

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Part 3

9) Use your answers from Parts 1 and 2 to find out what happens if you sell 40 T-shirts. Where did you find that information? You need to clearly explain your answer in detail.

10) If you want to break even at 40 T-shirts, how much does each t-shirt need to cost? Show all your work and explain your reasoning.

Skills

·  Evaluate expressions by applying the rules for order of operations

·  Write symbolic sentences that communicate their reasoning

·  Link a narrative to a corresponding graph (and link a graph to a corresponding narrative)

·  Translate a mathematical expression into a verbal expression

·  Symbolically represent a verbal expression

·  Recognize equivalent expressions

·  Solving equations with variables on both sides

Standards

·  7.A.4 Solve multi-step equations by combining like terms, using the distributive property, or moving variables to one side of the equation

·  7.A.7 Draw the graphic representation of a pattern from an equation or from a table of data

·  7.A.8 Create algebraic patterns using charts/tables, graphs, equations, and expressions

·  7.A.10 Write an equation to represent a function from a table of values

·  8.N.6 Justify the reasonableness of answers using estimation

·  8.A.1 Translate verbal sentences into algebraic inequalities

·  8.A.2 Write verbal expressions that match given mathematical expressions

·  8.A.3 Describe a situation involving relationships that matches a given graph

·  8.A.13 Solve multi-step inequalities and graph the solution set on a number line

·  8.A.14 Solve linear inequalities by combining like terms, using the distributive property, or moving variables to one side of the inequality (include multiplication or division of inequalities by a negative number)

·  8.A.15 Understand that numerical information can be represented in multiple ways: arithmetically, algebraically and graphically

Common Core Standards

·  8EE5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

·  8EE7 Solve linear equations with one variable (standards a & b)

·  8EE8 Analyze and solve pairs of simultaneous linear equations

·  8F1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

·  8F2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

·  8F4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

·  8F5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally

CCSS Standards for Mathematical Practice

·  Make sense of problems and persevere in solving them

·  Reason abstractly and quantitatively

·  Construct viable arguments and critique the reasoning of others.

·  Model with mathematics.

·  Use appropriate tools strategically

·  Attend to precision.

·  Look for and make use of structure

·  Look for and express regularity in repeated reason

Initially students will be able to draw up a table of values and attempt to solve the problem using guess and check type of approach. This will also demonstrate their ability to estimate and justify the reasonableness of their answer.

They can then translate the verbal expressions into mathematical equations and graph the equations, solving the problem graphically.

Finally they can solve the equations to solve the problem.

At each stage they can justify their methods and explain their reasoning and the mathematics involved.

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