F14: Nick Karavas, Dakota Doster, Brody Luke, Greg Balsam

Multiple concepts can be applied within a single word problem to assess the student's knowledge, understanding, comprehension, and communication. An important aspect of word problems is the ability to apply them across multiple subject areas. For the purpose of this section, we focused our attention on modeling word problems within Algebra. The Common Core Standards that we based our focus upon are listed below.

CCSS.MATH.CONTENT.HSA.CED.A.1
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CCSS.MATH.CONTENT.HSA.CED.A.2
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales **CCSS.MATH.CONTENT.HSA.CED.A.3**
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. **CCSS.MATH.CONTENT.HSA.CED.A.4**
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V=IR to highlight resistance R.

*Modeling Standards: Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Within this section, we will overlap standards which goes along with the modeling process.

CCSS.MATH.CONTENT.HSA.CED.A.1

Students are fine with solving equations for one variable, but when a teacher asks them to solve a word problem with one variable it is all of a sudden labeled hard. This is because students do not know how to dissect a word problem to extract the necessary information and draw correct relationships to the information. In order to help the students solve problems by creating one variable equations we need to teach students this crucial step of dissection.

First, we give them the word problem to see what conclusions they can draw. Next, we construct a picture with this information and ask questions to fill in the blanks that the students couldn’t quite get on their own. Once all the information is given, students can solve the equation.

“Suppose I have 100 feet of fencing to enclose a rectangular dog run. I will use the entire side of my house, which is 40ft long, as one of the sides of the dog run. How long should the other sides be if I want to make sure that I use all the fencing I have?”

So first you would ask the student what they notice about this problem. The will probably say things like, “there is 100 feet of fencing,” or “we are making a rectangle” maybe even, “one of the sides of the rectangle is 40 feet.” These are good observations.

Next is the picture to help illustrate the problem. So we start with a rectangle because the problem says, “rectangular dog run.” Then we ask the students if we know anything about the sides of the rectangle. Well, one of the sides is the side of the house which is 40 feet. What does that mean? If one side of the rectangle is 40 feet, then the side that is parallel with it is also 40 feet. Also, one of these sides is the house, meaning that fencing will not be used for that side. So now we have some side lengths and we know that one of the sides does not need fencing. That means the the other three sides will add up to 100 feet of fencing. We know that one side is 40 feet, but what about the other two? Hopefully, at this point the students realize that these two sides are the unknown that we need to solve. These two sides are going to be the same length as well since they are parallel lines in a rectangle. Now we can label these sides with a variable. The picture is now complete.

So we can ask the students to create and equation. It may be something like x+x+40=100 which can be turned into 2x+40=100. Now that they have the equation, it is time for the final step of solving. The students should already have these skills, but it doesn’t hurt to give them some time to work then work through it as a class.

After all of this, give them a similar word problem and see if they can work through it on their own. They should have some of the basics on dissecting a word problem. After some work time, the teacher can go through the problem with the class asking for participation. The main thing with the class participation is to see if they are dissecting the word problem correctly.

When students are introduced to equations that contain two variables most often one of the present variables is isolated on one side of the equation. An example of this would be an equation such as y=2x+3. As a result of this, students may not understand that there is a relationship between both x and y. It is important for students to understand that there is a correlation between both variables, and that understanding this correlation will help students understand different aspects of the problem with which they are presented. Furthermore, it is important for students to apply these skills not just in straightforward math problems, but also in real word applications, i.e. word problems.

An Example: (intended to be pretty simple one)

“Suppose you are in your chemistry class, and you are asked to attempt to prove that a relationship between temperature and volume exists in a particular gas. You measure the volume of the gas at 6 different temperatures and create a table with your measurements.”

“Your teacher than tasks you with two different tasks: Predict the volume of the gas when it is at 50 C and predict the temperature of the gas when it is at 72 mL.”

How would a student do this? Asking students to graph the six data points would be a good start:

By doing this, students would be able to see the type of correlation present within the data, if there is any correlation present at all. This would allow students to get a rough idea of what their prediction might look like.

Here’s a short video with a quick and decisive explanation regarding different kinds of correlation (i.e. positive, negative, no correlation): https://www.youtube.com/watch?v=CWnfwZRAuaY

Based on the information from the above video, students can now draw a conclusion regarding the relationship between the gas’s temperature and the gases volume. As one rises, the other rises. This means that there is a positive correlation between the two. Now Students must take into account the tasks asked of them. Remember, there were two:

Predict the volume of the gas when its temperature is at 50 C.

Predict the temperature of the gas when the volume is 72 mL.

How would a student go about doing this? Asking students to create a line based on the collected data would allow for accurate predictions. There are multiple ways of going about creating a line based on a data set.

Another, and more accurate, method would be to use a graphing utility, such as a graphing calculator:

Step 1, Input your data: “Stat” -> “Edit”-> Input temperature under L1 and volume under L2

Step 2, calculate a and b: “Stat” -> scroll to “Calc”-> “LinReg (ax+b)”-> “Enter”

In ax+b format, you have now calculated the slope of the line (a) and the y-intercept (b). In the case of this data set, our calculated equation is y=.25x+55.

So, how does this help predict the two values in question? To start, it is integral for students to understand that by putting the temperature values under L1 makes the Temperatures the x-values in this case, and thus the Volume is the y-value as it was put under L2.

By plugging in our new equation, y=.25x+55, into a graphing utility it can be compared to the data points.

Now that we have an equation of a line based on our data, we can calculate our two values in question: 1. Predict the volume of the gas when its temperature is at 50 C. y=.25x+55 -> y=.25(50) +55 -> y=67.5 mL

2. Predict the temperature of the gas when it is at 72 mL. y=.25x+55 -> (72) =.25x+55 -> 17=.25x -> 68 C=x *It is very important for students to understand that it is possible to find the x-value of the equation if the y-value is given. This is a point that will help students further understand the relationship between the two variables.

Overall, the point of an exercise such as this is to help students gain a better understanding of the relationships between multiple variables within the same equation. This concept is pushed even more by using it within a word problem. Students are given an opportunity to build on the understanding of correlation by graphing to find an equation allowing them to make predictions.

CCSS.MATH.CONTENT.HSA.CED.A.3

More often than not, inequalities are given a bad name by students. Now, when we add the concept of setting a constraint to a specific problem, the student has already checked out before getting through the first sentence of the word problem. One distinct way to go about presenting a problem including these said concepts to a student would be through the three acts modeling influence. The underlying meaning of this three acts modeling is the process of breaking up a given problem into three main components or acts.

Act 1: Break up the main task with as few words as possible. This could be as simple as a diagram or a table to visually show the student(s) what is given and what you want to show. Below is an example of how this can be illustrated to students.

Example:

Act 2: Define the resources that the student will need for the given problem that has been provided to them. When looking at the diagram/table and/or words given within the first act, what information will the student need to know in order to answer the question at hand.

Example:
Which inequality sign is needed for the problem? Are there any extraneous variables that can affect the result?

The student(s) at this point can effectively communicate on why they would choose the specific sign that they chose. For this given example, when you solve for w, you would save for at least that amount of weeks hence the greater than or equal to sign that would be used.

Act 3: Communication/Explanation of the problem. This is the stage that the student would apply the constraints/restrictions that they would need for the inequality problem that is presented to them. This is also the point in which the final answer is synthesized.

Example:
When solving for our given example we would say:
The student would then explain that the individual would need to save his/her allowance for at least 19 weeks, nothing less because that is a constraint, and could save up to how ever many weeks because that is the other constraint on the example at hand. Three Acts Resource for teachers
*Note: This example can also be applied to this first standard under the Common Core Standards stated above.

Interactive Technology

Another important aspect of teaching mathematics in a general sense is the ability to use interactive programs which show manipulation to problems. This is a great opportunity for teachers and students to be able to visually see what happens when specifically working with inequalities and the constraints or restrictions that are imposed with certain examples.

One of the top interactive programs that has been beneficial would be the online Desmos Calculator. The link below gives a great overview and step-by-step process with creating inequalities for students within the Desmos program. Modeling Inequalities through Desmos

To grasp a more in depth understanding of constraints/restrictions on inequalities, the link provided below gives a few examples in which teachers can base activities upon or students can use as further examples to enhance their understanding and comprehension. Modeling constraints within inequalities

CCSS.MATH.CONTENT.HSA.CED.A.4

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
Rearranging formulas is very similar to solving equations for a specific variable. A simple example of this can be illustrated using Ohm's law which has the equation V=IR. If you want to find what R equals in terms of V and I, you could divide both sides by I to get the new equation R=V/I. If you have values for V and I, you can simply plug them into this new equation and immediately find R. That is the principle behind this standard and it is applicable in multiple levels of math.

Beginner Level:
The following website allows students to practice with a simple equation such as Ohm's law to make sure they understand the principle behind this practice. If their answer to the problem is incorrect the website will tell them what the correct answer is and will give an explanation for why that is. Also provides a video explaining the methods used. Basic Ohm's Law Problems

Intermediate Level:
This standard is very useful in solving systems of equations using substitution. Here's a link to a video that describes how solve these systems using substitution. Solving linear systems with substitution

Word problems allow students to think more critically than when they are just given a system of equations. The following website provides excellent examples of how to turn a word problem into a system of equations and subsequently solve them using this standard and substitution. Solving Systems of Equations: Real World Problems

Advanced Level:
This standard has applications that go beyond a common Algebra I class. In Algebra II, this standard applies to solving systems with three equations, each with three variables. This link gives examples for systems with two variables and explains how it connects to systems with three variables. It also gives examples of how to solve systems with three variables. Solving Systems of Equations with Two and Three Variables

This standard also has connections to Calculus in problems where you want to optimize an equation with two variables, but you only know how to take the derivative of an equation with one variable. A subsequent equation will give you the necessary tools to substitute variables into the original equation that you need to optimize. Word problem examples with full explanations can be found at the following link. Optimization Problems

## Authors:

F14: Nick Karavas, Dakota Doster, Brody Luke, Greg BalsamMultiple concepts can be applied within a single word problem to assess the student's knowledge, understanding, comprehension, and communication. An important aspect of word problems is the ability to apply them across multiple subject areas. For the purpose of this section, we focused our attention on modeling word problems within Algebra. The Common Core Standards that we based our focus upon are listed below.

## Common Core Standards

High School: Algebra->Creating EquationsCCSS.MATH.CONTENT.HSA.CED.A.1Create equations and inequalities in one variable and use them to solve problems.

Include equations arising from linear and quadratic functions, and simple rational and exponential functions.CCSS.MATH.CONTENT.HSA.CED.A.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales

**CCSS.MATH.CONTENT.HSA.CED.A.3**Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.**CCSS.MATH.CONTENT.HSA.CED.A.4**Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

For example, rearrange Ohm's law V=IR to highlight resistance R.*Modeling Standards: Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Within this section, we will overlap standards which goes along with the modeling process.

Students are fine with solving equations for one variable, but when a teacher asks them to solve a word problem with one variable it is all of a sudden labeled hard. This is because students do not know how to dissect a word problem to extract the necessary information and draw correct relationships to the information. In order to help the students solve problems by creating one variable equations we need to teach students this crucial step of dissection.CCSS.MATH.CONTENT.HSA.CED.A.1First, we give them the word problem to see what conclusions they can draw. Next, we construct a picture with this information and ask questions to fill in the blanks that the students couldn’t quite get on their own. Once all the information is given, students can solve the equation.

Lets show this method with an example:

Example video that is explained below

“Suppose I have 100 feet of fencing to enclose a rectangular dog run. I will use the entire side of my house, which is 40ft long, as one of the sides of the dog run. How long should the other sides be if I want to make sure that I use all the fencing I have?”

So first you would ask the student what they notice about this problem. The will probably say things like, “there is 100 feet of fencing,” or “we are making a rectangle” maybe even, “one of the sides of the rectangle is 40 feet.” These are good observations.

Next is the picture to help illustrate the problem. So we start with a rectangle because the problem says, “rectangular dog run.” Then we ask the students if we know anything about the sides of the rectangle. Well, one of the sides is the side of the house which is 40 feet. What does that mean? If one side of the rectangle is 40 feet, then the side that is parallel with it is also 40 feet. Also, one of these sides is the house, meaning that fencing will not be used for that side. So now we have some side lengths and we know that one of the sides does not need fencing. That means the the other three sides will add up to 100 feet of fencing. We know that one side is 40 feet, but what about the other two? Hopefully, at this point the students realize that these two sides are the unknown that we need to solve. These two sides are going to be the same length as well since they are parallel lines in a rectangle. Now we can label these sides with a variable. The picture is now complete.

So we can ask the students to create and equation. It may be something like x+x+40=100 which can be turned into 2x+40=100. Now that they have the equation, it is time for the final step of solving. The students should already have these skills, but it doesn’t hurt to give them some time to work then work through it as a class.

After all of this, give them a similar word problem and see if they can work through it on their own. They should have some of the basics on dissecting a word problem. After some work time, the teacher can go through the problem with the class asking for participation. The main thing with the class participation is to see if they are dissecting the word problem correctly.

If more knowledge is needed, here are a few sources that you can consult.

Solving Equations

Solving Inequalities

CCSS.MATH.CONTENT.HSA.CED.A.2When students are introduced to equations that contain two variables most often one of the present variables is isolated on one side of the equation. An example of this would be an equation such as y=2x+3. As a result of this, students may not understand that there is a relationship between both x and y. It is important for students to understand that there is a correlation between both variables, and that understanding this correlation will help students understand different aspects of the problem with which they are presented. Furthermore, it is important for students to apply these skills not just in straightforward math problems, but also in real word applications, i.e. word problems.

An Example: (intended to be pretty simple one)

“Suppose you are in your chemistry class, and you are asked to attempt to prove that a relationship between temperature and volume exists in a particular gas. You measure the volume of the gas at 6 different temperatures and create a table with your measurements.”

“Your teacher than tasks you with two different tasks: Predict the volume of the gas when it is at 50 C and predict the temperature of the gas when it is at 72 mL.”

How would a student do this? Asking students to graph the six data points would be a good start:

By doing this, students would be able to see the type of correlation present within the data, if there is any correlation present at all. This would allow students to get a rough idea of what their prediction might look like.

Here’s a short video with a quick and decisive explanation regarding different kinds of correlation (i.e. positive, negative, no correlation):

https://www.youtube.com/watch?v=CWnfwZRAuaY

Based on the information from the above video, students can now draw a conclusion regarding the relationship between the gas’s temperature and the gases volume. As one rises, the other rises. This means that there is a positive correlation between the two.

Now Students must take into account the tasks asked of them. Remember, there were two:

- Predict the volume of the gas when its temperature is at 50 C.
- Predict the temperature of the gas when the volume is 72 mL.

How would a student go about doing this? Asking students to create a line based on the collected data would allow for accurate predictions. There are multiple ways of going about creating a line based on a data set.One method would be to use two of the data points and find a slope for the line, as explained in this quick video: https://www.youtube.com/watch?v=Zs-Fwq8HtjM

Another, and more accurate, method would be to use a graphing utility, such as a graphing calculator:

Step 1, Input your data: “Stat” -> “Edit”-> Input temperature under L1 and volume under L2

Step 2, calculate a and b: “Stat” -> scroll to “Calc”-> “LinReg (ax+b)”-> “Enter”

In ax+b format, you have now calculated the slope of the line (a) and the y-intercept (b). In the case of this data set, our calculated equation is y=.25x+55.

So, how does this help predict the two values in question? To start, it is integral for students to understand that by putting the temperature values under L1 makes the Temperatures the x-values in this case, and thus the Volume is the y-value as it was put under L2.

By plugging in our new equation, y=.25x+55, into a graphing utility it can be compared to the data points.

Now that we have an equation of a line based on our data, we can calculate our two values in question:

1. Predict the volume of the gas when its temperature is at 50 C.

y=.25x+55 -> y=.25(50) +55 -> y=67.5 mL

2. Predict the temperature of the gas when it is at 72 mL.

y=.25x+55 -> (72) =.25x+55 -> 17=.25x -> 68 C=x

*It is very important for students to understand that it is possible to find the x-value of the equation if the y-value is given. This is a point that will help students further understand the relationship between the two variables.

Overall, the point of an exercise such as this is to help students gain a better understanding of the relationships between multiple variables within the same equation. This concept is pushed even more by using it within a word problem. Students are given an opportunity to build on the understanding of correlation by graphing to find an equation allowing them to make predictions.

## CCSS.MATH.CONTENT.HSA.CED.A.3

More often than not, inequalities are given a bad name by students. Now, when we add the concept of setting a constraint to a specific problem, the student has already checked out before getting through the first sentence of the word problem. One distinct way to go about presenting a problem including these said concepts to a student would be through the three acts modeling influence. The underlying meaning of this three acts modeling is the process of breaking up a given problem into three main components or acts.Act 1: Break up the main task with as few words as possible. This could be as simple as a diagram or a table to visually show the student(s) what is given and what you want to show. Below is an example of how this can be illustrated to students.

Example:

Act 2: Define the resources that the student will need for the given problem that has been provided to them. When looking at the diagram/table and/or words given within the first act, what information will the student need to know in order to answer the question at hand.

Example:

Which inequality sign is needed for the problem? Are there any extraneous variables that can affect the result?

The student(s) at this point can effectively communicate on why they would choose the specific sign that they chose. For this given example, when you solve for

w, you would save for at least that amount of weeks hence the greater than or equal to sign that would be used.Act 3: Communication/Explanation of the problem. This is the stage that the student would apply the constraints/restrictions that they would need for the inequality problem that is presented to them. This is also the point in which the final answer is synthesized.

Example:

When solving for our given example we would say:

The student would then explain that the individual would need to save his/her allowance for at least 19 weeks, nothing less because that is a constraint, and could save up to how ever many weeks because that is the other constraint on the example at hand.

Three Acts Resource for teachers

*

Note: This example can also be applied to this first standard under the Common Core Standards stated above.## Interactive Technology

Another important aspect of teaching mathematics in a general sense is the ability to use interactive programs which show manipulation to problems. This is a great opportunity for teachers and students to be able to visually see what happens when specifically working with inequalities and the constraints or restrictions that are imposed with certain examples.One of the top interactive programs that has been beneficial would be the online Desmos Calculator. The link below gives a great overview and step-by-step process with creating inequalities for students within the Desmos program.

Modeling Inequalities through Desmos

To grasp a more in depth understanding of constraints/restrictions on inequalities, the link provided below gives a few examples in which teachers can base activities upon or students can use as further examples to enhance their understanding and comprehension.

Modeling constraints within inequalities

CCSS.MATH.CONTENT.HSA.CED.A.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Rearranging formulas is very similar to solving equations for a specific variable. A simple example of this can be illustrated using Ohm's law which has the equation V=IR. If you want to find what R equals in terms of V and I, you could divide both sides by I to get the new equation R=V/I. If you have values for V and I, you can simply plug them into this new equation and immediately find R. That is the principle behind this standard and it is applicable in multiple levels of math.

Beginner Level:The following website allows students to practice with a simple equation such as Ohm's law to make sure they understand the principle behind this practice. If their answer to the problem is incorrect the website will tell them what the correct answer is and will give an explanation for why that is. Also provides a video explaining the methods used.

Basic Ohm's Law Problems

Same website providing similar practice but with a new equation, the power equation, P=VI.

Solving Problems Involving Power

Intermediate Level:This standard is very useful in solving systems of equations using substitution. Here's a link to a video that describes how solve these systems using substitution.

Solving linear systems with substitution

Practice examples that require solving for one variable to plug that new equation into the second equation.

Practice with Solving Linear Systems Algebraically

Word problems allow students to think more critically than when they are just given a system of equations. The following website provides excellent examples of how to turn a word problem into a system of equations and subsequently solve them using this standard and substitution.

Solving Systems of Equations: Real World Problems

Advanced Level:This standard has applications that go beyond a common Algebra I class. In Algebra II, this standard applies to solving systems with three equations, each with three variables. This link gives examples for systems with two variables and explains how it connects to systems with three variables. It also gives examples of how to solve systems with three variables.

Solving Systems of Equations with Two and Three Variables

This standard also has connections to Calculus in problems where you want to optimize an equation with two variables, but you only know how to take the derivative of an equation with one variable. A subsequent equation will give you the necessary tools to substitute variables into the original equation that you need to optimize. Word problem examples with full explanations can be found at the following link.

Optimization Problems