painted cube problem using manipulatives found right here on campus! pulled from this website

Completed Tech Modeling Template by Jeff Stack, centered around the classic Painted Cube problems.

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# Better Math

## Mathematics Teachers Interested in Teaching Students to Think

#
ML Expressions & Equations

# Technology Modeling Activity Jeff Stack, who painted my cube? 6.EE

# Picture Problem Painted Cube 6.EE

# Creating Problems with Variables 7.EE.B.4, MP 4, W.7.2

**Creating Linear Problems**

### By Naomi Johnson

# Problem-based Instruction A.CED

# Theme Park Expressions 6.EE.B.6

# Let’s Head for the Hills 6.EE.C

# Frozen Yogurt Party! CCSS.MATH.CONTENT.8.EE.C.7

# Modeling Pressure: Building Functions F-BF

# High School Algebra: Systems of Equations

# Hey! Wait Up!- CCSS.MATH.CONTENT.8.EE.C.8

painted cube problem using manipulatives found right here on campus! pulled from this website

Completed Tech Modeling Template by Jeff Stack, centered around the classic Painted Cube problems.

The painted cube problem is deceptively complex, and has multiple levels of understanding applied to it. Students are shown a cube broken up into smaller sections. The cube has been painted with a particular pattern based on what faces are visible from the outside of the cube. Sections with one face exposed are painted red on that exposed side. Sections with two exposed sizes are painted pink. Sections with three exposed sides are painted green. Sections with no exposed sides are unpainted, or are clear. Student’s jobs are to count how many of each painted cube there are in a given cube, as well as to devise functions so as to quickly count out how many of a particular section there are. Their answers would be written out in table form; said tables would be given to them. Students can also be asked to find out how much surface area there is in these cubes, assuming that each section is 1 inch on each side.

CCSS.Math.Content.6.EE.A.1

Write and evaluate numerical expressions involving whole-number exponents.

CCSS.Math.Content.6.EE.A.2

Write, read, and evaluate expressions in which letters stand for numbers.

CCSS.Math.Content.6.EE.B.5

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

CCSS.Math.Content.6.EE.B.6

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

CCSS.Math.Content.8.EE.A.2

Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

2x2x2 cube | Green | Pink | Red | unpainted |

How many? | ||||

What’s the equation? |

3x3x3 cube | Green | Pink | Red | unpainted |

How many? | 12 | |||

What’s the equation? |

4x4x4 cube | Green | Pink | Red | unpainted |

How many? | ||||

What’s the equation? |

5x5x5 cube | Green | Pink | Red | unpainted |

How many? | 54 | |||

What’s the equation? |

NxNxN | Green | Pink | Red | unpainted |

How many? | 4n | 6(n-2)^2 | ||

What’s the equation? |

This lesson will focus on this picture and be used to teach standard 7.EE.B.4. This standard required students to used variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems. The students will solve the equations and figure out the answer to the last problem. We will then discuss how they came to the answers. They will then be given the chance to create their own problems with a system of equations using their own pictures to represent variables in equations. They will then show them to their peers which will show the other students the different cultures and individuality of everyone in class.

This lesson could be an integrated lesson with English Language Arts because the students could then write about their equations, what each picture represents, and why they chose to make them as they did. This lesson could also incorporate the student’s different cultures because they are allowed to use whatever picture they want and it can represent themselves however they would like. The students are given the freedom to bring themselves into the problems, and this includes their individuality and cultures. They can then share these cultures with their peers when they share their problems to the class.

**High School Algebra: Creating Equations**

This learning progression will be taught at Ellensburg High School in an Algebra 1 course. The students in the class this lesson progression is being taught in are all good students. By that I mean each student is a diligent and hard worker. Knowing this I have decided to deviate from the usual required text and use tasks from *CCSS Problem-Based Tasks for Mathematics I*. These tasks are tied directly to a CCSS and incorporate real-world applications. Knowing that my students are capable I am positive that they will transition well from their regular work from the required text book to these problem-based tasks I have found for them to do.

In between the time this learning progression starts and when the school year starter the students have learned the following:

**Reason quantitatively and use units to solve problems.**

__MATH.CONTENT.HSN.Q.A.1__

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

__MATH.CONTENT.HSN.Q.A.2__

Define appropriate quantities for the purpose of descriptive modeling.

__MATH.CONTENT.HSN.Q.A.3__

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

**Interpret the structure of expressions.**

__MATH.CONTENT.HSA.SSE.A.1__

Interpret expressions that represent a quantity in terms of its context.*

__MATH.CONTENT.HSA.SSE.A.1.A__

Interpret parts of an expression, such as terms, factors, and coefficients.

__MATH.CONTENT.HSA.SSE.A.1.B__

Interpret complicated expressions by viewing one or more of their pa rts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

The students have mastered these concepts and CCSS through activities and tasks out for their required text. Thus, the students are prepared and have the appropriate academic knowledge to begin learning concepts tied the CCSS HSA-CED.1, 3 & 4. Note the reason this learning progression does not cover HSA-CED.2 is because that has already been covered in previous lessons when teaching linear equations. As I explain the tasks for each of the CCSS 1, 3 & 4 I will explain how the students’ prior academic knowledge was necessary; as well as, how the new concepts build off of the students’ prior knowledge will help them gain a deeper understanding of the new concepts. However, before beginning the explanation of the tasks and assessments I would first like to discuss my instructional strategies.

When teaching the learning progression I will use direct instruction and the use multi-media to communicate the learning targets, the concepts, and the directions for the tasks. The initial discussion will not provide examples. I want the students to struggle and make their own inquiries about how to solve the posed problems. I will then make use of grouping to have students work together on the tasks and make inquiries as a group. Just like how these tasks are real-world applications I want the students to also learn that in real life you made need to ask someone for help and work together to solve problems. Because I know that my students are diligent workers and that they have worked well in groups before, I know that by having them tackle real-world, problem-based tasks together will yield more positive results. Lastly, this learning progression will only include formative assessment since all work will be done in groups and I want to assess discussion and inquiry not mathematical correctness. Though, before moving on there will be a section of the lessons where I will help students through effective questioning reach the correct answers almost like how the Socratic Method does it.

**Lesson 1: Problem-Based Task: Free Checking Accounts Coaching**

** **The figure to the left is of the first task of the learning progression. As shown in the figure the task is tied directly to the CCSS, A-CED.1 (HS is left off because it is understood this text is for high school mathematics 1). This text has tasks in order of what is required to learn the following concept not just in order of CCSS. Thus, this text incorporates scaffolding to effectively organize tasks and CCSS. The figure above describes a word-problem but it isn’t simply just a word-problem; it is a word-problem that describes a real-world scenario that students know they may one day be faced with creating an innate interest within them to learn how to solve the problem. On the next page of the text shown in the figure there are several questions posed related to the above word-problem:

- What is the minimum money you want to have to open your checking account?
- How much are you depositing each week?
- What inequality can be written to model the scenario?
- What is the solution to the inequality?
- What does the solution of the inequality represent in terms of the context of the problem?

When beginning this first lesson I will go over the CCSS and break it down for the students to better understand what is expected of the to learn from this task. I will then allow students to start to work through the given problems I have listed above as I walk around and use inquiry-based learning and the Socratic method to invoke deep, and critical thinking. I know that my students are capable of deriving the solution to the posed problems with minimal hints from me because I have taught them the necessary problem-solving skills and mathematical knowledge required to; however, like I said prior I did not provide any examples of procedures. Thus, students will struggle with ideas and procedures but as I walk around the room I may hint or pose effective questions to a group to help them solve a problem.

This task requires students derive an inequality and know what that inequality says about the posed problem; however, the students have not learned anything about inequalities but what they have learned is the structure of one variable equations and how to derive them from given information. The hint I will have to present to the students once I have noticed they have all gathered the information and are discussing how to express it as an equation is that an inequality is written just like an equation and would then present the appropriate syntax and notation to them. The point of this is that the students have been in deep discussion and thought about how their group can express the information after struggle to find an answer it will click for them with the information I present. Because the students know the procedure for deriving a one-variable equation from given information, I know that they are capable of deriving a one-variable inequality once I have given them that hint. But if any of the students have not derived the inequality or did not understand how their group did, I would in the last five minutes of class go over the procedure for deriving the inequality and what the correct inequality is. Their homework would then be to answer that final question (e) and be prepared to discuss it the following lesson. This would allow the students to continue to either find the solutions on their own or struggle with the concept and be confused the following lesson. Why is it good for a student to be confused the following lesson and be unable to finish his/her homework, because I know that they are on the precipice of understanding the concept and with a little nudge they will get it.

**Lesson 2: Problem-Based Tasks: Skate Constraints**

As shown in the figure the next lesson covers A-CED.3. This lesson is quite a bit more extensive and may require two days to complete. The reason the previous lesson was less extensive is because the students will be continuing to derive inequalities. The first lesson was meant to have students struggle with the idea of whether or not the procedure of deriving a one-variable equation could be applied to deriving a one-variable inequality and what an inequality represented in terms of the given information. Now the students will continue to struggle towards complete mastery of those concepts while they learn more and more about inequalities till they have all of the learning blocks necessary to have mastered inequalities. Thus, the students will be using the previous lessons knowledge in this lesson.

I will be using the same lesson outline but with a few minor changes. Instead of letting the students struggle through the posed problems immediately after going over the CCSS and directions I will go over some definitions found in the CCSS: constraints, viable and nonviable. I will need to provide an example of what a constraint is and explain why they are necessary. I would use the previous lessons inequality to do and explain how it would not make sense to have negative deposits hence there must be a constraint stating that “x” constraints must be greater than or equal to zero. I would not tie this example to the students’ posed problems in this lesson because again I want them to become confused and struggle to find the solutions. Once I had finished going over any necessary information I felt like would make the task impossible to do without I would let the students loose to begin working in their groups. The text provides these following posed problems:

- What information do you know about the amount of time needed to make the blade of a figure skate?
- What information do you know about the amount of time needed to make the blade of a hockey skate?
- How many hours each week can be spent making skate blades?
- What inequality can be use represent the amount of time it takes to make blades for both figure skates and hockey skates?
- What information do you know about the amount of time needed to make the boot of a figure skate?
- What information do you know about the amount of time needed to make the boot of a hockey skate?
- How many hours each week can be spent make skate boots?
- What inequality can be used to represent the amount of time it takes to make the boots for both figure skates and hockey skates?
- What other constraints are needed in this situation?
- What is the system of inequalities that represents this situation?
- Is it possible to construct 3 figure skates and 4 hockey skates given the constraints of this situation?

This task requires that students create two-variable inequalities which is very new because the students have not worked with two-variable equations yet so they are bound to struggle with how to express the information given as an equation; however, they have the necessary knowledge and problem-solving skills so with I am hoping that with no hints the students will derive the equations themselves. As I did in the previous lesson though, if no students have managed to derive the correct two-variable inequalities I will go over the right way to solve the first inequality maybe 30 minutes into the class so they will have to struggle with the constraints and what the inequalities represent. Then, in the following lesson I would we would go over as a class all of the problems and together answer questions (k).

**Lesson 3: Problem-Based Tasks: Bricklayers**

This is the last lesson of the learning progression. Now this lesson if important and though it does not have to do with inequalities at first glance it actually does; it has to do with all equations and expressions.

This activity is a real-world problem that uses a context that student will enjoy. They will be using variables to create and analyze a theme parks attendance. Students will be given a theme park scenario with variables for such things as the employees, visitors to the park, admissions cost and days of operation. For example, F could be the # of female employees and E for the # of male employees. The students will collect and analyze the data to determine how the park is doing financially and figure out which rides are the most efficient. They must develop math expressions to save a lot of time and energy. They will use the variables to write expressions for: 1.) The total number of people visiting the park on a given day. 2.) The total number of people in the park on a given day. 3.) The amount of money the amusement park collects from tickets on a given day if all visitors pay for a single-day pass for their respective age groups. 4.) The number of people who ride roller coasters if 3/5 of all visitors ride roller coasters on a given day. 5.) The number of visitors over age 18 who ride roller coasters during the park’s season if 1/4 of all visitors ride roller coasters on a given day.

To extend this activity students could write up a proposal about building an additional roller coaster. Students will need to use the data to select the most important variables and expressions the owners should consider to make their decision.

CCSS.MATH.CONTENT.6.EE.B.6

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

CCSS.ELA-LITERACY.W.6.2.D

Use precise language and domain-specific vocabulary to inform about or explain the topic.

CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.

CCSS.Math.Practice.MP2 Reason abstractly and quantitatively.

With hiking season right around the corner, your friends and you decide to plan a backpacking trip to a lake. During the planning process, you want to figure out how long it will take to hike the trail to your camping destination. By doing some investigative research, you are able to find a map of the trail with descriptions of the various inclines along the trail. Using your mathematical and problem solving skills, find out how long it will take you hike each part of the trail based on the incline and speed of travel to answer the ultimate question of how long it will take your friends and you to reach the camping destination at the end of the trail.

This lesson is geared for students to work in groups in determining the speed of travel along the various slopes of the trail. Students must use their mathematical reasoning and problem solving skills to calculate and solve this problem. Students can use a particular hiking trail of interest or the teacher can assign ones to the class.

This lesson is focusing on the CCSS.Math.6.EE.C.9- Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

This activity is a real-world problem that students may face in their life. It would be a great activity to introduce students to linear equations and graphing linear equations. During this activity, students can problem solve with a partner or group to figure out where they should have their frozen yogurt party based on the given information.

Attached below: A handout with directions and problems for students to solve.

This activity aligns with the the Common Core State Standards and the Math Practices:

CCSS.MATH.CONTENT.8.EE.C.7 Solve linear equations in one variable

CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.

CCSS.MATH.PRACTICE.MP4 Model with mathematics.

Handout: Frozen Yogurt Party

Pressure Sensor 400 by Vernier is a piece of technology used to calculate pressure create by chemical reactions. If were to make a lesson using this I would first speak with a science teacher and find out if you could co-create a lesson plan together using this technology. So in science the students would go over the chemical reaction and why pressure was being produced and then in math students would create a model that represents how the pressure built over time. The labs would already be set up and students would have already done it once to observe the reaction so students could reproduce the same experiment but this time focus on the model created by it. The data collected by the sensor can be sent to computer to create a model where students will interpret what type of function represents that model and derive an equation for it. This lesson help students understand how math relates to other curricula and how it can be used in the real-world. Also, this lesson will help students get used to using technology and therefor learning the set of skills required to interpret how to best use technology.

If you wanted to involve community I believe that this lesson could be done in a lab where students could set the labs up, perform the experiment, and derive functions for the given models while family and friends watch, sort of a presentation of their science and math skills. Students/lab groups could then present their findings to the audience at the end. this would allow family and friends to see what the students are learning and for students to show off what they have learned. also this would allow parents to witness your teaching first-hand. this idea however would require a lot of safety precautions since chemical reactions would be going on.

This learning progression will be applied in an Algebra I classroom where most students are in the 10th grade. This classroom has access to **CORD’s ****Algebra 1: Learning in Context** digital textbook. The common core state standards aligned with this learning progression are 8.EE.C.8, 8.EE.C.8A, 8.EE.C.8B, and HSA.REI.C.6. This learning progression is also aligned to the following mathematical practices: MP1- Make sense of problems and persevere in solving them, MP4- Model with mathematics, MP5-Use appropriate tools strategically, and MP6- Attend to precision.

**Day 1 Instructional Task 1:** In order to introduce the students to the concept of systems of linear equations, the teacher will start with by having two students volunteer to act out an example. The teacher will have them stand on opposite sides of the classroom and then walk towards each other. While they are doing that, the rest of the class will be told to watch them closely. The class will notice that at some point the students will cross paths before they continue on their respective routes. The teacher will lead a short discussion relating this walking exercise to systems of linear equations, specifically those whose graphs intersect (which the students will later learn are called consistent and independent systems). The teacher can then take this activity a step further by showing the students one way systems of linear equations can be used in the real world, in other words a way to create a system of linear equations model.

**Benchmark Assessment Day 1: **This will be in the form of an exit task. Students will be given the following question: “In your own words, explain what a common solution of a system of linear equations represents graphically and explain why it is significant algebraically.” This assessment fits in with a Standards Based Grading system and is aligned with standard 8.EE.C.8.A.

Imagine you are walking to class and you see your friend up ahead so you run to catch up with him. Can you model this situation mathematically? Of course you can! Assuming that you and your friend are moving at constant speeds, you can set up a system of linear equations. You can model this graphically by plotting distance versus time. If you were to plot the lines on the same set of axes, the point where the two lines cross would represent the physical location where you passed your friend.

It is possible to create models for situations like this in the classroom using Vernier CBR2 Motion Detectors.

In the *Hey! Wait Up!* activity, students will work in groups of four: 2 people will be the walkers, 1 person will start the motion detector and operate the stopwatch, and 1 person will mark the point where the walkers intersect. Each group will work together to collect and analyze motion data in order to determine the solution to a linear system of equations using a graphing calculator. They will then check this solution by creating a system of linear equations and solving it by hand. This activity is best done after the students have been introduced to systems of linear equations and how to solve them.

To complete the The *Hey! Wait Up!* activity each group will need 2 Vernier CBR2 Motion Detectors (with appropriate cables), a TI83 or TI84 calculator, a meter stick, and a stopwatch.

This lesson aligns with the following Common Core State Standards:

*CCSS.MATH.CONTENT.8.EE.C.8– Analyze and solve pairs of simultaneous linear equations.*

Students will be using the calculator results to find the coordinates of two points for both lines which they will then use to find the slopes of each line. Students will also use the calculator results to find the y-intercept of each line. Using this information, students will be able to create equations for the lines that they will then use to make a system of linear equations and solve.

*CCSS.MATH.CONTENT.8.EE.C.8.A– Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.*

At the end of this activity, students will be able to connect that the point where they crossed when walking is the point where the two lines intersect and that this coordinate point is the solution to the system of linear equations.

To conclude this activity, students will be given an exit slip that asks:

- In your own words, what is a system of linear equations?
- Give me an example of a real world situation (that does not involve walking/ running) that you could model using a system of linear equations.

From this you can determine and gauge your students’ conceptual understanding as well as see if they can apply the idea of using systems of linear equations to model other everyday experiences.

A benefit of using Vernier Motion Detectors when teaching the concept of systems of linear equations is that it makes the concept personal for the students. It helps bring mathematics out of the classroom and into the real world. Rather than seeing the systems of linear equations as just lines on the graph, they can see that in this situation those lines represent something, namely the students’ walks. This activity also gets the students involved in their own learning by having them get up and move around when they are doing the walking activities and collecting the data. This makes the data concrete for them rather than just some numbers, equations, and coordinates they have to work with in order to find a solution.

A Vernier CBR2 Motion Detector costs $99. To buy or to find out more about the Vernier CBR2 Motion Detector visit http://www.vernier.com/products/sensors/motion-detectors/cbr2/

To find out more about Vernier Software & Technology and explore their other products visit http://www.vernier.com/

Hey! Wait Up! worksheet: Hey! Wait Up!-Systems of Linear Equations with Vernier Motion Detectors