NP - Geometry - Indirect Measurement with TRIG
When teaching geometry and doing the chapter on trig, students would encounter problems with a triangle, one side and one angle given, and they had to find another side, often the height of a building or tree. What I found is that they could do it in the book but when I took them outside, the "question in the book" became a real "problem" real with real challenges and required them to think more deeply into what they had to do.

After doing the problems in the book, give them something outside to find the height of. Don't give them step-by-step directions. Let them problem solve.

Divide the students into teams of 3-4 students, give each group a 100' tape measure, clinometer, calculator, paper, and pencil, they set out to find the height of a building, the height of a lamp post in the parking lot, or other tall objects available. Going outside and doing "real life" problems seem to be motivating for most students.

Tip: Show them how to use a clinometer and make sure that they look at their measuring tapes and find the "zero" and can read the measures. I have found that even high school students need to go over this - they don't do it much anymore at home! Take the time to do this.

After doing the problems in the book, give them something outside to find the height of. Don't give them step by step directions. Let them problem solve. (See CCSS Math Practice #1)

Indirect Measurement with Similar Triangles
Want to do some indirect measuring with similar triangles using shadows or mirrors - click this link for more - INDIRECT Measurement

1. PJ - Calculus - This lesson is an introduction to finding volumes of revolution. By the end of the lesson students should be able to

Visualize and determine the best method in order to find the volume of the revolved solid

Compute both Riemann Sums of discs as well as the Integral method for volume

Explain the relationship between Limits with Riemann Summations to find volume and Integration to find volume.

Overall students should show the following skills during the lesson

Make sense of problems and persevere in solving them

Reason abstractly and quantitatively

Attend to precision

1) The best way to really teach this topic is to jump in feet first and start with a problem. In order to make sure that all the students feel comfortable, the first example is going to be simple: starting off with discussion on the characteristics of the function and finding the area under this curve on a specific interval. There are two main strategies that could be used to find the area, either a Riemann sum or taking the Integral. So what happens if this area is rotated it around the x-axis, creating a solid? What does this look like? How could we find the volume of this?

2) Taking the Riemann Summation Approximation method for the original function, and then rotating this “bar representation” of the area under the curve, we get a solid made up of small discs, which we can find the volume of. Using the physical model, students need to get up and interact with not only me, but with their classmates; taking the model apart, they will work in groups to find the volume of their respective cylinders. Some groups will get one cylinder, others two, but they need to measure the radius and then find the volume. As a class, we will then add all the volumes up to get our first approximation for the specific volume of revolution pertaining to the given function rotated around the x-axis from 0 to 3.

3) Since this is at best a rough estimate of the actual volume, what should we do in order to get a more accurate answer? Creating more and more slices would make our answer closer to the actual value, so this would be equivalent to taking the limit of the summation as the number of pieces (n) we are adding goes towards infinity. This is the jump that students should have made earlier in the semester when covering limits and integration, because as n gets larger and larger, the summation becomes closer and closer to the actual area under the curve, so specifically for revolutions it would be the integral of the function over the specific interval. This is an extremely powerful and insightful equation, and students need to understand how we got to this point and how it can be used for all types of different problems.

4) In order to hammer down the relationship between the different representations for revolved volumes, the next thing that they will do is play a sort of matching game on the SMART Board. Students need to match the graph of the original function with its corresponding solid of revolution, as well as to the corresponding integral formula for volume.

Closure: What connections do solids of revolutions have with what we have learned previously in Calculus? How are Riemann sums and Integrals related when it comes to finding the volume of these solids? Where does the idea of a Limit come into play? How might we apply this to the real world? What are some applications? What are some examples of things that we could find the volume of? For a preview into what is coming next, what would happen if we rotated around a different line or axis? How would finding the volume change? How would we go about finding it?

3. (ES)-Geometry: Reflections-In Geometry class, students learn about transformations. These transformations include reflections, translations, rotations, dilations, and tessellations. When learning about reflections, it is difficult for students to understand the definition of and properties of a reflection if they do not have a hands-on example. They have difficulty understanding that the segment connecting the preimage to the reflection line has to be the same length as the segment connecting the image to the reflection line. When they see the definition, students just memorize the properties instead of understanding why the properties are true. In order to help students understand reflections I created a lesson which includes the investigation of reflections with MIRA.

Materials: MIRA (one for every student), Reflection Worksheet

Introduce the topic of reflections with a Human Mirror Video by Improv Everywhere. The video has people mirroring each other in the New York subway. Allow students the opportunity to mirror each other. I found that some of my more interactive and energetic classes really enjoyed a couple of minutes of attempting to mirror each other.

Divide the students into groups of 3-4. Pass out the Reflections Worksheet. The worksheet asks students to use MIRA to reflect preimages and write down any observations. As students work on the worksheet, walk around and ask them guided questions. Have the students discuss their observations and write their observations on the board in the classroom to use as a discussion topic at the end of class.

Once students complete the worksheet, have a class discussion about their observations. Then introduce the definition and properties of a reflection.

As closure, return to the Human Mirror Video and ask students to identify the preimage, image, reflection line, and properties seen in the video.

Tips:

Some classes may not want to attempt to be Human Mirrors like the video.

Provide some examples of observations before passing out the worksheet because some students may not understand what to write as an observation.

Common Core Math Standards:

G-CO-4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G-CO-5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g. graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

CCSS Math Practices:

(2) Reason abstractly and quantitatively.

(3) Construct viable arguments and critique the reasons of other.

(7) Look for and make use of structure.

4. (KR) – Algebra II: Scatter Plots of Lines of Regression – When learning about scatter plots and lines of regression students encounter problems with understanding the importance of learning how to gather data, finding the correlation between the two variables, and predicting a value using the prediction equation. They also have difficulty using the graphing calculator when it comes to inputting data and becoming comfortable with using their new technology. To help the students understand scatter plots and also become more comfortable using their graphing calculators I created a lesson which consists of students gather data and creating their own scatter plots and, using their calculators, finding the prediction equation and prediction value.

Materials: SmartBoard (attached), Ti-83/Ti-84/Ti-84 Silver Plus Graphing Calculator, Ti-nspire ViewScreen, Tape Measures, Paper and pencil

Explain how the students are going to connect the importance of scatter plots to the real world by finding and collecting data like researchers do every day.

Break students into groups of 4-5 members and have them create a data table of their heights and arm spans in inches.

They then are to plug this data into their graphing calculators and create a scatter plot, find a regression equation, plot the regression line, determine whether the equation fits the data well, and predict what Shaquille O’Neill’s arm span would be in regards to their data when he is 7’1” tall.

As the students work in their groups, walk around and answer any questions in regards to using the graphing calculator and interpreting their results.

Each student is to show their calculator with the window in which they have the scatter plot with the regression line and TRACE used to find Squaquille’s arm span. This will emphasize their participation.

Reveal that Shaquille O’Neill’s arm span is 87 inches and determine whose predictions were close and whose weren't.

3. Tips:

Example: A group of four girls with the same arm spans and heights found Shaq to have an arm span of 56 inches. When looking at their prediction equation they noticed that their slope was zero because their points plotted together formed a horizontal line. Therefore, even though their data collection was correct, their predictions were off. Examples like these can be used to explain why researchers gather information from large quantities.

To emphasize the importance of gathering a lot of data for research, have the students come in the next day with the same questions answered using the data from the class as a whole. Note the prediction becomes more accurate.

4. CCSS Math Practices:

Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

5. (GL)- Algebra II: Systems of Equations- When learning systems of equations, it is very important to introduce a real-world application so that students can fully see the purpose behind why they exist. Though for some it may be easy to see the purpose behind why we use systems of equations or the process for which they are solved, it is very understandable how one could also get lost in the work. To enhance their learning for this section, I have them role-playing a real-life scenario for which they can clearly see how systems of equations are applicable.

Materials: A few bags of candy, a bucket, fake (or fabricated) money, and the following worksheet:

Introducing the lesson will involve looking at different real-life scenarios for which systems of equations are relevant. It is crucial to mention that all systems of equations involve finding missing values. It just so happens that often it ends up being that we need to use what we know to find what we don't.

The students will be separated in half. One half will be the vendors and the other half will be the buyers. The vendors will reach into a bucket full of candy (for which there are two types) and take a few handfuls. They will record how much total candy they have, but not the individual amounts.

That will be their first equation: x + y = total amount of candy.

Then the vendors will determine a dollar value for which they will sell each candy; and then sell the candy to the buyers, They will then record how much money they earn to create their second equation.

(Price of Candy A)x + (Price of Candy B)y = Total Profit

Once they create both their equations, they can use those to determine how much of each candy they originally had.

This will be followed by a class discussion on how they came to the numbers they did. It will lead them to discuss what they did correct and incorrect and give them the opportunity to correct their own personal mistakes. This would make for a good closure to the lesson.

Tips:

DO NOT let the students eat the candy, as that could compromise the data. Use the candy as an incentive to complete the lab in a timely manner. That way, they can have it when they have complete the lesson

Let them have the freedom to teach themselves about this. Help them when they need it, but give them the opportunity to learn on their own.

CCSS:

Understanding Systems of Equations

Providing Real-World Applications

Making use of Structure

6. (NP) Getting ready to introduce modeling or exponential functions? Try this skittles experiment to model population growth/decay. It's a great one for getting students to realize that the parts of the equation had some real meaning that could be explained. Download the PowerPoint (or SMART Board notebook or .pdf), grab some Skittles, paper plates, and your calculator and enjoy a sweet treat in your math classroom!

Indirect Measurement with TRIGWhen teaching geometry and doing the chapter on

trig, students would encounter problems with a triangle, one side and one angle given, and they had to find another side, often the height of a building or tree. What I found is that they could do it in the book but when I took them outside, the "question in the book" became a real "problem" real with real challenges and required them to think more deeply into what they had to do.Don't give them step-by-step directions.Let them problem solve.100' tape measure, clinometer, calculator, paper, and pencil, they set out to find the height of a building, the height of a lamp post in the parking lot, or other tall objects available. Going outside and doing "real life" problems seem to be motivating for most students.Indirect Measurement with Similar TrianglesWant to do some indirect measuring with s

imilar trianglesusing shadows or mirrors - click this link for more - INDIRECT Measurement1. PJ - Calculus - This lesson is an introduction to finding volumes of revolution. By the end of the lesson students should be able to

2) Taking the Riemann Summation Approximation method for the original function, and then rotating this “bar representation” of the area under the curve, we get a solid made up of small discs, which we can find the volume of. Using the physical model, students need to get up and interact with not only me, but with their classmates; taking the model apart, they will work in groups to find the volume of their respective cylinders. Some groups will get one cylinder, others two, but they need to measure the radius and then find the volume. As a class, we will then add all the volumes up to get our first approximation for the specific volume of revolution pertaining to the given function rotated around the x-axis from 0 to 3.

3) Since this is at best a rough estimate of the actual volume, what should we do in order to get a more accurate answer? Creating more and more slices would make our answer closer to the actual value, so this would be equivalent to taking the limit of the summation as the number of pieces (n) we are adding goes towards infinity. This is the jump that students should have made earlier in the semester when covering limits and integration, because as

ngets larger and larger, the summation becomes closer and closer to the actual area under the curve, so specifically for revolutions it would be the integral of the function over the specific interval. This is an extremely powerful and insightful equation, and students need to understand how we got to this point and how it can be used for all types of different problems.4) In order to hammer down the relationship between the different representations for revolved volumes, the next thing that they will do is play a sort of matching game on the SMART Board. Students need to match the graph of the original function with its corresponding solid of revolution, as well as to the corresponding integral formula for volume.

3. (ES)-

Geometry: Reflections-In Geometry class, students learn about transformations. These transformations include reflections, translations, rotations, dilations, and tessellations. When learning about reflections, it is difficult for students to understand the definition of and properties of a reflection if they do not have a hands-on example. They have difficulty understanding that the segment connecting the preimage to the reflection line has to be the same length as the segment connecting the image to the reflection line. When they see the definition, students just memorize the properties instead of understanding why the properties are true. In order to help students understand reflections I created a lesson which includes the investigation of reflections with MIRA.Materials: MIRA (one for every student), Reflection WorksheetLesson Plan:Tips:Common Core Math Standards:CCSS Math Practices:4. (KR) –

Algebra II: Scatter Plots of Lines of Regression– When learning about scatter plots and lines of regression students encounter problems with understanding the importance of learning how to gather data, finding the correlation between the two variables, and predicting a value using the prediction equation. They also have difficulty using the graphing calculator when it comes to inputting data and becoming comfortable with using their new technology. To help the students understand scatter plots and also become more comfortable using their graphing calculators I created a lesson which consists of students gather data and creating their own scatter plots and, using their calculators, finding the prediction equation and prediction value.Materials:SmartBoard (attached), Ti-83/Ti-84/Ti-84 Silver Plus Graphing Calculator, Ti-nspire ViewScreen, Tape Measures, Paper and pencilLesson Plan:- Explain how the students are going to connect the importance of scatter plots to the real world by finding and collecting data like researchers do every day.
- Break students into groups of 4-5 members and have them create a data table of their heights and arm spans in inches.
- They then are to plug this data into their graphing calculators and create a scatter plot, find a regression equation, plot the regression line, determine whether the equation fits the data well, and predict what Shaquille O’Neill’s arm span would be in regards to their data when he is 7’1” tall.
- As the students work in their groups, walk around and answer any questions in regards to using the graphing calculator and interpreting their results.
- Each student is to show their calculator with the window in which they have the scatter plot with the regression line and TRACE used to find Squaquille’s arm span. This will emphasize their participation.
- Reveal that Shaquille O’Neill’s arm span is 87 inches and determine whose predictions were close and whose weren't.

3.Tips:- Example: A group of four girls with the same arm spans and heights found Shaq to have an arm span of 56 inches. When looking at their prediction equation they noticed that their slope was zero because their points plotted together formed a horizontal line. Therefore, even though their data collection was correct, their predictions were off. Examples like these can be used to explain why researchers gather information from large quantities.
- To emphasize the importance of gathering a lot of data for research, have the students come in the next day with the same questions answered using the data from the class as a whole. Note the prediction becomes more accurate.

4.CCSS Math Practices:5. (GL)-

Algebra II: Systems of Equations- When learning systems of equations, it is very important to introduce a real-world application so that students can fully see the purpose behind why they exist. Though for some it may be easy to see the purpose behind why we use systems of equations or the process for which they are solved, it is very understandable how one could also get lost in the work. To enhance their learning for this section, I have them role-playing a real-life scenario for which they can clearly see how systems of equations are applicable.SystemsLab.docx

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- Introducing the lesson will involve looking at different real-life scenarios for which systems of equations are relevant. It is crucial to mention that all systems of equations involve finding missing values. It just so happens that often it ends up being that we need to use what we know to find what we don't.
- The students will be separated in half. One half will be the vendors and the other half will be the buyers. The vendors will reach into a bucket full of candy (for which there are two types) and take a few handfuls. They will record how much total candy they have, but not the individual amounts.
- That will be their first equation: x + y = total amount of candy.

- Then the vendors will determine a dollar value for which they will sell each candy; and then sell the candy to the buyers, They will then record how much money they earn to create their second equation.
- (Price of Candy A)x + (Price of Candy B)y = Total Profit

- Once they create both their equations, they can use those to determine how much of each candy they originally had.
- This will be followed by a class discussion on how they came to the numbers they did. It will lead them to discuss what they did correct and incorrect and give them the opportunity to correct their own personal mistakes. This would make for a good closure to the lesson.

- DO NOT let the students eat the candy, as that could compromise the data. Use the candy as an incentive to complete the lab in a timely manner. That way, they can have it when they have complete the lesson
- Let them have the freedom to teach themselves about this. Help them when they need it, but give them the opportunity to learn on their own.

- Understanding Systems of Equations
- Providing Real-World Applications
- Making use of Structure

6. (NP) Getting ready to introduce modeling or exponential functions? Try this skittles experiment to model population growth/decay. It's a great one for getting students to realize that the parts of the equation had some real meaning that could be explained. Download the PowerPoint (or SMART Board notebook or .pdf), grab some Skittles, paper plates, and your calculator and enjoy a sweet treat in your math classroom!Materials: A few bags of candy, a bucket, fake (or fabricated) money, and the following worksheet:Lesson Plan:Tips:CCSS:If you need calculator directions for scatter plots, visit the Graphing Calculator Help page.