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 Modeling

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

# Amusement Park 7.G.B

# Inca Ancient Civilization Picture Problem 4.NBT.B.4

# COWculating – Picture Problem 8.F.B.4

# Build a playground 7.G.B.6 and MP4

# Disneyland Drive! – 8.F.B.4, 8.F.B.5

# Rocket Math: 8.F.A.3, 8.F.B.5, MP4

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# Fence Fractions – 4.NF.B.3

# How does the slope change over time for a ball thrown in the air? F.IF & F.BF

# Paper Bridges CCSS 8.EE.B.5 MP 4

**Paper Bridges**

### Standard: CCSS 8.EE.B.5 MP 4

##### By: Naomi Johnson, Rachel Van Kopp, Tracy Van Lone, Natasha Smith, and Kimberly Younger

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.

During this lesson, students will be put to the test, using their prior knowledge from the previous geometry unit on two- and three-dimensional shapes involving area and perimeter.

You and your team have just been hired by Walt Disney Parks and Resorts Worldwide to landscape a brand new amusement park with rides. WDPR has provided you with a blank map with letters E through M, representing the nine rides that you must include in your design. Your task is to map out the most effective design in order to maximize the number of visitors to each ride. Your job is to gather all of the information you can, map out each ride using GeoGebra to graph your park, and present a justification as to why your design will maximize visitors to each of your rides.

In order to complete this lesson, students will be split into groups of 3-5. The class as a whole will have a set number of rides they must include in their park, and a set area for the park itself, but each group will devise different ride sizes and configurations throughout the map in order to maximize visitors to each ride.

The image displayed above could be used in an integrated math classroom to help teach 4th grade math students about mathematical practices from ancient civilizations as well as record keeping techniques. Often times, we get caught up in the paper and pencil way of showing one’s work. By integrating a bit of Inca history in our math classroom we can elicit a different form of showing work and have some creative and artistic fun while we’re at it. A rich math task to help students model adding multi-digit whole numbers is to have students learn about Inca culture and create a quipu, which is a knot technique used to add multi-digit whole numbers.

Students would be able to read about Inca life and customs and then create a quipu using numbers of their choice. Once students are taught the fundamentals about what certain knots represent and how they are positioned, they can create their own quipu that displays an algebraic equation. An extension that can be used with this task is to have students trade quipu and determine what the equation represented is and check to make sure the answer represented is correct. Another extension would be to have students divide into stations and go around to each station and use a handout to write down the equations represented on various quipu and move about the room until they have been to each quipu station.

This task is a rich math activity because it can be done by all students of varied skill levels. Advanced students can create more complex equations, while struggling students can create much simpler equations. This task also provides multiple pathways in the sense that students can creatively represent their equations on their quipu with a variety of colors and string lengths.

The multicultural aspect of this picture activity is that it not only integrates a different form of writing math equations, but it also introduces students to Inca culture and other social studies content. Writing and literature are another integration that can be used with lesson because students read informational text about people of the Inca culture and then get an opportunity to write about how the advancements of mathematics have evolved, Inca history itself, and how students created their own quipu.

**Integrated Common Core State Standards and Mathematical Practices Addressed:**

CCSS.Math.Content.4.NBT.B.4

Fluently add and subtract multi-digit whole numbers using the standard algorithm.

CCSS.ELA-Literacy.RI.4.7

Interpret information presented visually, orally, or quantitatively (e.g., in charts, graphs, diagrams, time lines, animations, or interactive elements on Web pages) and explain how the information contributes to an understanding of the text in which it appears.

CCSS.ELA-Literacy.W.4.2.d

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

CCSS.Math.Practice.MP4 Model with mathematics.

CCSS.Math.Practice.MP5 Use appropriate tools strategically

CCSS.Math.Practice.MP6 Attend to precision.

This picture – taken from Freedom Works UK – becomes the center of a fun and highly interactive mathematics lesson where students get to think what it might be like to be a farmer! Students will be assigned a set number of cows that eat a set square footage of grass per day, and must determine how much land is needed in order to sustain the cows. Furthermore, students must consider the amount of time needed in order for grass to grow back, but they must have enough already grown for the cattle to eat in the meantime.

This lesson has students focus working on the Common Core State Standards:

CCSS.MATH.CONTENT.8.F.B.4

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.

CCSS.MATH.CONTENT.8.F.B.5

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.MATH.PRACTICE.MP2

**Reason abstractly and quantitatively.**

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to *decontextualize*—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to *contextualize*, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

This lesson correlates to the culture of the Ellensburg area, for a large portion of this area is populated by those in the agricultural profession. While most of them are in the hay production/distribution realm, students could connect to this rural lesson nonetheless.

**By: Kimberly Younger, Rachel Van Kopp, Lizzie Englehart and Naomi Johnson**

This lesson is focused on a 7^{th} grade standard CCSS.Math.Content.7.G.B.6 but could be used for 6^{th} through 8^{th} grade depending on the application. This lesson focuses on the use of formulas to find area and problem solving of a real-world problem with the use of technology.

The prompt is “The school district is building a new playground for the new elementary school down the road. They have hired Playgrounds R’ Us to build it, but the supervisor wants to know what students would want on a playground. Create a playground with the following requirements.”

The students are building on their knowledge of area and perimeter formulas and applying it to a problem. The students are given a square footage for the playground, they must use three or more different shapes to represent their equipment, and the total square footage of the equipment must cover 30% of the playground’s area or more.

The students are given a packet which includes direction, a rough draft grid paper, final draft grid paper (submitted for approval), a screen shot of their Geogebra playground and a write up about their playground.

Below is an example of the packet students received. (link to the packet)

Below is a student’s sample playground

**Rough Draft ****Blueprint Final Draft Blueprint Geogebra Blueprint**

**Table for Blueprint**

Extension for “Build a Playground”:

As an extension to this lesson, students will later be able to work with 3 dimensional figures and nets to build the playground they have constructed in our lesson. This lesson emphasized finding and working with area of various geometrical figures and special reasoning. Using the knowledge, they have gained through our lesson, the students will be able to create the net that would best fit the equipment shape that they have presented to us on their “blue prints”.

In order to create the appropriate net, students will need to understand that the 3-dimensional shapes base will be the shape they have placed on their map in the lesson “Build a Playground”. This extension will cover CCSS.math.content.7.g.b.6 which states “Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.” This extension will help students make connections to the relationships between 2-dimensional figures and 3-dimensional figures, and connect the concepts of surface area and volume to real-world scenarios.

Students are put to the test in this fun and highly interpretive summative review of the previous unit: linear functions!

You and your family are going on a trip to Disneyland, but you must drive there! Your parents have decided that they don’t want the wear-and-tear of such a trip on their own vehicle, so they have narrowed it down to two (2) rental vehicles. However, they are having a hard time figuring out which one would be the most efficient/economical choice. Your job is to gather all the information you can, and construct a minimum of five (5) equations, graph them on GeoGebra, and lastly must decide which car your parents should choose for your vacation!

In order to complete this lesson, students will be split into groups of 3-5 members. The class as a whole will have a set gas price, but each group will have different vehicles as well as a monetary cap that they must remain under.

**Rocket Math**

By: Natasha Smith, Mariana Rosas, Paloma Vergara, & Melisa Sanchez-Leyva

This modeling lesson is for an 8th-grade classroom and is focused on the standards CCSS.MATH.CONTENT.8.F.A.3 and CCSS.MATH.CONTENT.8.F.B.5. This lesson introduces students to the concept of nonlinear functions. In the lesson, students will be able to explore the concept of a nonlinear function and expand their knowledge of what a function can look like.

This lesson follows a similar format to Dan Meyers’ 3 Acts. Students will start by watching a video of a model rocket launch. Individually, they will quickly draw a graph of what they think the relationship between the height of the rocket and time is. Next, they will work in groups to plot the points estimating the relationship of the height of the rocket at each second. Lastly, the teacher will take them outside and launch a model rocket to prove or disprove students’ theories. The model rocket will have a Pocketlab attached to it which will provide an exact graph of the height of the rocket at each time point. Students will compare their graphs to the Pocketlab graph. We decided to launch the rocket again instead of providing students with a graph from the original launch in the final act as it adds an element of excitement to the activity and the students will enjoy going outside to watch the rocket launch.

This lesson incorporates multiple types of technology. For the video used in the lesson, teachers would achieve best results by filming their own rocket launch as they will want to use the exact same type of model rocket in both the video and the in-class launch. The video representation should be similar to this video. Students will also be using the website Desmos to graph points. Lastly, the teacher will be using a Pocketlab. The Pocketlab is a wireless sensor that can be attached to different objects and will record different types of data and output graphs. For this lesson, the Pocketlab can be attached to the model rocket and will record the height (altitude) of the rocket as time passes.

Students are engaged in real world application of mathematical modeling when creating a fence using fractions that have unlike common denominators to add to make a whole. Tommy is building a fence and has one side length finished (1 whole) and needs help finding the other sides to his fence. The students will be given different fractions to analyze on geogebra–an online tool for students to use to help conceptualize fractions with unlike denominators. This lesson is designed for a 4th grade class modeling addition of fractions. Students will be working individually to help Tommy create his fence.

This will be taught in a calculus class. Using the software and tools as shown in the following links: https://www.vernier.com/ https://www.vernier.com/products/software/lp/, the students will use the equipment to throw a ball in the air and track its position as time goes on. The students will use the data to draw a graph of position vs time for the ball and create an equation which give the position as it is related to the time. The students will then take the slope at different points and use the data to draw a slope vs time graph. They will also create a function for this graph. This will be an introduction activity to derivatives. The students will create a functions of the position vs time as well as create a function for the slope vs time. We will then go into covering the material for derivatives. The students will understand the procedure of how to create a function from a graph or data. They will also understand the concept of what a derivative is and how it relates to the real world problem I have provided. Below you will find the handout I will give the students to guide them through the activity.

The CCSS for this lesson are as follows:

- CCSS.MATH.HSF-IF.B.4: Interpret functions that arise in applications in terms of context
- CCSS.MATH.HSF-CED.A.2: Create equations that describe numbers or relationships.
- CCSS.MATH.HSF-BF.A.1: Build a function that models a relationship between two quantities.

Ball Throwing Activity

Name:

Date:

- Set up your iPad such that it is far enough away to get a video of the ball as you throw it straight up in the air. You should make sure the ball stays in the view of the iPad when it is thrown for you should not move the iPad during the experiment.
- Open the Vernier Probes and Software program. Take a video of the ball as you lightly toss it straight up in the air.
- Sketch a graph of a position vs time for the ball in the space below. Create a function for this graph.

- Take the slope of the graph at t=0, at a point before the vertex, at the vertex, at a point after the vertex before the end, and finally at the point just before the ball hits the graph. What are the units of these slopes? What does this tell you about what the number means? Sketch a graph of the slopes as a time vs slope graph. Is it linear? What does that mean? Write an equation for the graph.

**Problem**

Paper Bridges is looking at the correlation between the number of pennies that a number of paper bridges could hold. To find this, we folded pieces of paper that were all the same size in the same way. We then put one paper, folded as a bridge, on two equal size books, an inch of paper was on each book. We then placed pennies on the bridge until it caved in 1/8 of an inch or collapsed, whichever came first. We then counted the pennies and recorded the number.

We repeated these steps with 2, 3, 4, and 5 paper bridges laying on top of each other to see how more paper effected the number of pennies that were able to be help.

**Analysis**

Table 1: Data collected from our experiment.

To analyze our data, we inputted the data from Table 1 into our calculators. We performed a linear regression on the data as well as an exponential regression. We were given the following equations:

Linear Equation: y = 12x – 2.2

Exponential Equation: y = 5.9259 * 1.6537^x

We plotted all our data and graphed our equations. Figure 1 shows what this looked like.

Figure 1: Our data plotted with our two equations. The red line represents the linear equation and the blue line represents the exponential equation.

After looking over Figure 1, we had a hard time determining which equation did a better job of representing our data. Neither line was incredibly close or far away from the points. At this point in time, we believed both equations could possibly be good representations of the data.

We decided to find the absolute value of the difference between each data point and their expected linear and exponential value. We then averaged the differences. Table 2 shows what we found.

Table 2: The absolute value of the difference between each data point and their expected linear and exponential value. In the bottom line is the average of the differences.

We were than able to see that the linear equation seemed to better match our data. All the linear differences compared to the exponential differences were smaller expect for with the first data point. The average exponential difference was more than two times the average linear difference. It appeared that the exponential equation could be growing too fast.

While we may have found reason to believe the linear equation was the better fit, we still felt like we needed more data. When doing this activity in a classroom, we would recommend extending the domain and creating more data points. We would also recommend having students do multiple trials of the experiment or having students share their data with the class.

**Extension **

A lot of variables come into play when a structure collapses. For instance, our paper bridges collapse as more and more pennies are placed in the middle. Interestingly, even though all groups of students involved in the investigation in our class used the same paper, folded the same way, and collapsed their bridges with pennies weighing exactly the same amount, no two teams’ bridges collapsed under the same number of pennies. Some teams gently “placed” their coins on their bridges, enabling their bridge to hold more weight than the teams that “dropped” their coins on the bridges. Some teams spread their coins a bit wider along the center of their bridge than others, enabling more coins to stack up on their bridge prior to its collapse. Other teams folded their bridge papers more sharply and precisely, causing the structure to be more rigid. Many variables add up to change the dynamics from one bridge to another despite seemingly identical bridge designs and project execution instructions.

A real-life structural collapse occurred on 1987 when UW built an addition to Husky Stadium. Nine temporary guy wires were holding the stadium roof up while additional structure was built underneath. The stadium was well designed and should not have collapsed. But a collection of variables added up to change the dynamics from the stadium addition as designed to the stadium addition as executed. Construction specifications were not very specific about when the appropriate time would be to remove the temporary guy wires and how many should be removed at a time. Ask students to read these articles and identify the variables they see impacting the collapse. Would the structure have stood if only 5 guy wires had been cut? What if only 4 guy wires were cut and the hollow steel tubular beams had been filled with concrete as specified by the designer? Would the stadium have stood? What about the cat?