STANDARDS FOR MATHEMATICAL PRACTICE

Question

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning

Table of Contents Page or Link
Unit Definitions
Function Notation
Page number for Interpreting Linear Functions Arising in Applications
Analyzing Linear Functions
Constructing and Comparing Linear Models
Unit Reflection Summary
Building Functions
Systems of Equations/Inequalities

Definitions:
Choose at least 10 important vocabulary words from the unit to define
Provide a model or example of each vocabulary word. (You may use symbols, graphs,
tables, or pictures.)

Systems of Equations/Inequalities:
Create a contextual situation that would illustrate the application of systems of equations
in context. Explain constraints as they apply to the context.
Create a contextual situation that would illustrate a linear inequality in two variables.
Graph the solution set to the linear inequality.

Function Notation:
Provide at least one example of a domain and range that illustrates a function and explain
why it is a function. Provide at least one example of a domain and range that is not a function and explain why.Create one real world scenario in which function notation may be used to model a linear function. Show how the function might be evaluated for inputs in the domain based on
the context of the scenario. Use the scenarios to create a recursive formula

Interpreting Linear Functions Arising in Applications:
Create a story that would generate a linear function and describe the meaning of key
features (intercepts, intervals where the function is increasing, decreasing, positive, or
negative; end behaviors) of the graph as they relate to the story.
Show the graph of your function and relate the domain to the quantitative relationship it
describes. Describe the rate of change for a linear function or the rate a change over an
interval.

Analyzing Linear Functions:
Create one linear function expressed symbolically. Graph the function using technology
(print for booklet or paste on web)
Create two different linear functions. Show one algebraically and the other using a verbal
description. Compare the two functions.

Building Functions:
Explain how to find an explicit expression, a recursive process, or steps for calculation to
complete a sequence/pattern. Write the sequence both recursively and with an explicit
formula.

Constructing and Comparing Linear Models:
Design a word problem that involves a linear model. Use a table or sequence to illustrate
the relationships described in the models.
Explain the constant rate per unit interval relative to another for the word problem that
you designed.
Construct the graphs for each model in the word problem that you designed.
Compare the linear models from your word problem. Interpret the parameters.

Reflection / Summary:
Describe your learning journey throughout the unit. Reflect on topics that you found easy
to learn and those that were most difficult.
Are there any standards that you need more help grasping? Explain. If not, which
standards do you have the best grasp? Explain.
What advice would you give to other students that will learn about linear functions in the
Future? Which task(s) did you find the most beneficial to mastering key standards?

Answer 1

Table of Contents

1. Unit Definitions
2. Function Notation
3. Interpreting Linear Functions Arising in Applications
4. Analyzing Linear Functions
5. Building Functions
6. Systems of Equations/Inequalities
7. Constructing and Comparing Linear Models
8. Unit Reflection Summary

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Unit Definitions

Vocabulary Words

1. Function: A relation where each input has exactly one output.
   Example: $f(x)=2x+1$ is a function since every x-value gives one unique y-value.

2. Domain: The set of all possible input values (x-values) for a function.
   Example: For $f(x)=\sqrt{x}$, the domain is $x\ge 0$.

3. Range: The set of all possible output values (y-values) for a function.
   Example: For $f(x)=2x+1$, the range is all real numbers.

4. Linear Function: A function that can be graphically represented as a straight line.
   Example: $f(x)=3x-5$ is linear.

5. Slope (m): The rate of change of a function, representing how much y changes for a change in x.
   Example: In $f(x)=2x+3$, the slope $m$ is 2.

6. Y-intercept (b): The point where the line crosses the y-axis, corresponding to $x=0$.
   Example: In the function $f(x)=2x+3$, the y-intercept is 3.

7. Intercept: Points where the graph intersects the axes (x-intercept, y-intercept).
   Example: For $f(x)=0.5x+2$, the y-intercept is at (0, 2) and the x-intercept at (-4, 0).

8. System of Equations: A set of equations with the same variables.
   Example:
   $$\begin{array}{rlr}y& =2x+1y& =-x+3\end{array}$$

9. Inequality: A mathematical statement that compares two expressions using $<,\le ,>,\ge$.
   Example: $3x+5>2$.

10. Piecewise Function: A function defined by multiple sub-functions, each applying to a certain interval.
    Example:
    $$f(x)=\{\begin{array}{lll}{x}^2& \text{if }x<0x+1& \text{if }x\ge 0\end{array}$$

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Systems of Equations/Inequalities

Contextual Situation for Systems of Equations

Scenario: A farmer has a plot of land where he grows two types of vegetables: tomatoes and carrots. He has a total of 100 plants with a constraint that he can only plant up to 60 tomato plants due to limited space.

Equations:
Let $t$ represent the number of tomato plants and $c$ represent the number of carrot plants.

1. $t+c=100$ (Total plants)
2. $t\le 60$ (Tomato plant constraint)

Explanation of Constraints: The farmer must follow the limitations on the number of tomato plants while maximizing the total number of plants. This creates a feasible region that can be graphed to visualize potential planting combinations.

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Contextual Situation for Linear Inequality

Scenario: A student needs to study for a test and wants to ensure she studies at least 3 hours each day, but she can only study for a maximum of 5 hours per day due to other commitments.

Inequality:
Let $h$ represent hours studied.
The inequality can be expressed as:
$$3\le h\le 5$$

Graphing the Solution Set: The graph will show the range between 3 and 5 on the h-axis, indicating the hours a student can study.

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Function Notation

Example Illustrating a Function

• Function: $f(x)=3x+2$

  • Domain: All real numbers.
  • Range: All real numbers.

This is a function because for each value of x, there is a unique value of f(x).

Example Not Illustrating a Function

• Relation: {(1, 2), (1, 3), (2, 4)}

This is not a function because the input $1$ has two different outputs.

Real-World Scenario with Function Notation

Scenario: A taxi company charges a flat fee of $3 plus $2 per mile driven.
Let $x$ be the number of miles driven.

Function notation:
$$C(x)=3+2x$$

Evaluating the Function:
If a taxi drives 5 miles, $C(5)=3+2(5)=13$.

Recursive Formula:
The cost after $n$ miles:

• $C(n)=C(n-1)+2$, starting at $C(0)=3$.

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Interpreting Linear Functions Arising in Applications

Story: A car rental agency charges $50 for a day’s rental, plus $0.20 per mile driven. The cost function can be modeled as: $$C(d,m)=50+0.20m$$ where $d$ is the number of days and $m$ is the miles driven.

Key Features:

• Y-intercept: $50 (base charge)
• Rates of Change: The cost increases by $0.20 for each additional mile.

Graph: (Copy and paste or draw the graph)
The domain reflects the number of days and miles driven. The function is increasing for all valid inputs.

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Analyzing Linear Functions

Linear Function:
$$f(x)=2x+3$$

Graph: (Use technology to graph and print)
The graph will be a straight line: slope = 2, y-intercept = 3.

Two Functions:

1. Algebraic Function: $f(x)=3x-1$
2. Verbal Description: A box with a linear price increase, starting at $10 with an added cost of $5 each time another unit is purchased.

Comparison:

• First function has a steeper slope (3 vs. 2), indicating a faster increase for every x-unit.

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Building Functions

Sequence: The sequence of numbers is defined as:

• Start with 2 and add 3 for each new term:
  $a_1=2,a_n=a_{n-1}+3$.

Recursive Formula:

1. $a_1=2$
2. $a_n=a_{n-1}+3$

Explicit Formula:
$a_n=3n-1$.

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Constructing and Comparing Linear Models

Word Problem: A company sells two types of daily subscriptions: Basic at $10 and Premium at $20. Let $b$ and $p$ be the numbers of subscriptions sold.

1. Model in Equations:
   $10b+20p$ (Total revenue from subscriptions)
2. Table/Sequence Illustration:
   | Basics Sold | Premiums Sold | Total Revenue | |-------------|----------------|----------------| | 0 | 5 | $100 | | 5 | 0 | $50 | | 3 | 2 | $70 |

Constant Rate of Change:
For each additional Premium subscription sold, revenue increases by $20, while Basic increases by $10.

Graphs: (Construct graphs for each model here)
The slope indicates the revenue increase as subscriptions increase.

Comparison:

• Basic model has a lower slope (10) than Premium (20), suggesting Premium yields more revenue per unit sold.

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Reflection / Summary

Learning Journey:
Throughout the unit, understanding linear functions became easier as I practiced their interpretation and relationships. The hardest topic for me was constructing systems of equations, particularly figuring out how to apply constraints accurately.

Standards Understanding:
I feel confident about modeling and analyzing linear functions. However, I would like to delve deeper into systems of inequalities.

Advice for Future Students:
Focus on understanding the relationships between variables. Use real-world contexts to make abstract concepts more tangible. Activities that involved graphing and comparing practical scenarios were especially beneficial in my learning.

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