Author: cospers

7.G.B.4,6 and 8.G.C.9: Volume and Surface Area

cospers Assessment of CCSS-Math, CCSS-Math Learning Progressions, Common Core State Standards Mathematics, Geometry Curriculum Aligned with CCSS-Math

This learning progression is designed for a high school Applied Geometry course in which the students have access to computers. The Common Core State Standards that align with this learning progression are: 7.G.B.4, 7.G.B.6, and 8.G.C.9. The reason that we combined 7th and 8th grade standards is because the students need a chance to review and solidify their understanding of the basics of area and volume but also be challenged by having to find real-world applications. The students will also be assessed on the following standards of mathematical practice: MP4; model with mathematics, MP6; attend to precision, and MP7; look for and make use of structure.

pizza

The unit will begin with a discovery based lesson about pi. The students will be required to partner up and measure the circumference and diameter of two different objects. Each partner will be assigned different circular objects. There will be a master list at the front of the room that the students can record their measurements on and then write what the circumference/diameter of their objects was. The purpose of this lesson is to show the students that, no matter what the size of the circle, if they take the circumference/diameter of their object they will get approximately 3.14, or pi. By realizing this pattern, the students are looking for and making use of structure (MP7). The students will also be required to graph the data, with the circumference of the objects being represented on the y-axis and the diameter on the x-axis. This will create a relatively linear line. The students will then be asked to identify the slope of this line, which will be 3.14. Finally, the students will be required to answer some questions in order to elicit further evidence of their understanding. Some of the questions that will be asked are:

1. What is the average ratio of circumference to diameter?

2. What observations did you make?

3.  Can you derive the formula for circumference of a circle by knowing Pi is the ratio of diameter to circumference?

After the students understand the origin of pi, they will be required to use the concept of pi to find the area of circles. They will also learn how to find the surface area and area of rectangles and triangles and their respective 3-D prisms. Finally, the students will learn what the difference is between surface area and area. They will be asked what real-world applications that are related to area and surface area. In using the equations to correctly calculate volume and surface area, the students are attending to precision (MP6). The students will be formally assessed via a worksheet. Problems on this worksheet include

1. Find the surface area of the origami figure you made in class

2.  circle      radius:                     Equation:                      Area:

3. What do you notice about the shapes of a cylinder when calculating surface area?

 

Once the students understand how to find the area and surface area of these basic objects, they will move on to finding both surface area and volume of more advanced objects such as pyramids, cones, spheres, triangular prisms, cubes, and cylinders. The students will be provided with a list of the necessary equations to know. The purpose of this lesson is for the students to be able to apply the given formulas and to identify how volume and surface area apply to the real-world. In order to allow the students to practice the equations for finding the volume and surface area of these objects, they will use Google Earth. The students will download the Volume of Solids kmz file from realworldmath.org. They will then follow the directions to virtually travel the world and find the volume and surface area of different buildings in different countries. The program itself will give the necessary dimensions for the equations. By having the students find real-world applications of volume and surface area, they are modeling with mathematics (MP4). The students will record their work and answers on a worksheet that they will be given. One of the questions on the worksheet looks like this:

Leaning Tower of Pisa

pisa     Surface Area:

 

Volume:

 

In order to assess the students ability to connect this lesson to the real world, they will also be asked on their worksheet “In what career would you use volume and surface area? Why do you think that?”. This is an important question since the main purpose of this lesson is to get the students to realize the real-world applications of volume and area.

 

Learning Progression: LearningProgression

Discovering Pi lesson plan: discovering pi

Area and Surface Area lesson plan: LP Area

Google Earth Volume and Surface Area lesson plan: GeoMath lesson plan

REI.B.3 & 4: Quadratic Formula, Completing the Square, and Factoring. Oh my!

cospers Assessment of CCSS-Math, CCSS-Math Learning Progressions, Common Core State Standards Mathematics, HS Algebra, Uncategorized

wizard of oz

Just like Dorothy, the Tin Man, and the Scarecrow were scared of lions, and tigers, and bears in the Wizard of Oz, 10th grade students will most likely be scared of solving quadratic equations by completing the square, the quadratic formula, and factoring. However, through proper instruction, enough practice, and effective formative assessments, solving linear equations and inequalities and quadratic equations will be nothing to worry about.

x equals 2

This learning progression would be taught in a beginning 10th grade algebra classroom. The Common Core State Standards that will be addressed are HSA.REI.B.3, HSA.REI.B.4.A, and HSA.REI.B.4.B. Additionally, MP.4, MP.6, and MP.7 are the mathematical practice standards that align with this learning progression.

Since this is such a dense Common Core cluster, there will be four lessons that make up this learning progression. Students have previously learned what equations and expressions are and how they are different, how to categorize polynomials, steps to solving simple linear equations, and how to graph linear equations. The progression will begin by teaching the students how to solve more advanced linear equations and inequalities. This lesson may take a couple of days depending on how advanced the students are. The students will learn how to graph inequalities and how to attend to precision in writing their answers and labeling their graphs. For the second lesson, students will learn what completing the square means and how it is useful in finding roots of quadratic equations. Each time they complete the square, they will be required to draw a picture to show how they came up with the number that they are adding to both sides. An example of a problem on their worksheet, which is used as their formative assessment for this lesson, is provided below. Once students understand how to complete the square, they will move on to the third lesson and will learn what the quadratic formula is, how it is used, and how it can be derived from completing the square. Finally, the fourth lesson entails teaching students how to factor a quadratic equation when the leading coefficient is 1. By using frequent formative assessment techniques such as worksheets, class activities, and exit slips, the students and the teacher will both be able to track the students’ progress towards the learning targets.

completing the square

 

 

The equations will be written on the board and the students will copy them from the board onto their equation box. They will then complete the picture and solution. They will have 4 pages scaffolded like the worksheet shown on the left. Picture taken from http://mathequalslove.blogspot.com.au/2015/05/algebra-2-solving-quadratics-inb-pages.html

 

 

 

 

Learning Progression: Learning Progression

S-ID Texting Champion

cospers Assessment of CCSS-Math, HS Statistics & Probability, Uncategorized

Task

A student in Ms. Smith’s class claimed that girls text faster than boys. Naturally, the boys disagreed. Therefore, Ms. Smith’s statistics class did an activity where each student calculated the average words per minute they could text. The data is recorded below.

 

boysgirls

 

1.    Sketch two side by side box plots to compare the average words per minute of girls and boys.   Make sure to include the minimum, maximum, first and third quartiles, and median. Additionally, calculate the mean for each data set.

 

2.   Why is the mean less than the median in both boys and girls?

 

3.      Compare and contrast the two box plots. What does this mean when we are comparing words per minute between boys and girls?

 

4.      If I was trying to describe the center of these distributions, would the mean or median be more appropriate?

 

5.    Based upon the data presented above, which gender texts faster? Support your answer with statistical data.

 

 

An assessment commentary and solution is on the attached document: IM problem

7.G.B.4: Delicious Math

cospers HS Geometry, HS Modeling, ML Modeling, Picture Problems

Even when writing essays, humor is frequently used as an attention grabber to engage readers and to put them in a positive mindset for reading the rest of the essay. This idea also applies to engaging students in the math classroom. The pictures featured in this blog includes many different possible math applications.pizza mathimportant math

 

 

 

 

One such problem is using different shaped and sized pizzas to determine surface area and perimeter, and to improve comparison and mathematical reasoning skills. A common math problem requires the students to compare the size and price of a circular pizza to the size and price of a square and/or rectangular pizza. This will have the students calculate the surface area of each pizza and compare them to see which one is larger. Next, the students will have to compare the prices of each pizza to see which pizza is a better deal. Additionally, you can have the students discover pi by having them compare the circumference and surface area of small, medium, and large pizzas. A couple questions that you can ask the students are:

  • What is the surface area of each pizza?
  • Which pizza is the better deal?
  • What is the circumference of a circular pizza with an area of ____ square inches?
  • If the crust is 1 inch wide, what is the surface area of the pizza excluding the crust?
  • If the pizza has an average of 4 pieces of pepperoni per square inch, how many pieces of pepperoni are on each of the pizzas?
  • What is the ratio of circumference to surface area of the circular pizzas? Why is that number familiar?
  • To have the students show examples of their reasoning skills, you can ask them to explain how they got their answer for any of the problems just listed.

To further engage the students, you can actually order the pizza and have the students perform their calculations on actual pizzas and then let them eat the pizzas.

There are many common core state standards that you could teach through a pizza lesson, but the following standard is the one that fits the activity described above.

CCSS.MATH.CONTENT.7.G.B.4
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

HSN.Q.A.1 and HSA.CED.A.1 Flying Function

cospers Algebra Teaching Issues, Uncategorized

archery-2Archery is considered to be one of the oldest sports in the world. Bows and arrows have been being used for hunting and for sport for over 25,000 years. So when trying to think of a way to incorporate the real world into a mathematics lesson, I thought what better way than to use archery to help students hit their learning target (pun intended). The unique flight of each arrow, not to mention the archer’s paradox, provides a math teacher many options when it comes to applying math to archery. Not only does archery have many connections to math, but it is something that a lot of the students are interested in. As the archery director at a summer camp, I had campers begging me to spend more time at the archery range. Even in my field experience classroom, at least 50% of my students are hunters and are constantly telling each other hunting stories.

Recent advances in technology makes analyzing arrow flight simple and fun. The Logger Pro software by Vernier allows students to take a small clip of them shooting an arrow and analyze it frame by frame and to perform statistical analysis of the data. The students will be able to find the velocity of the arrow as it leaves the bow and compare it with the velocity of the arrow as it hits the target. This will hopefully spark a conversation about air resistance and other real-world factors that will affect mathematical predictions. Students will also be able to track the flight of the arrow and find the function that best fits that flight. No matter what type of function you choose, Logger Pro will come up with the constant multipliers for that function that best fits the flight of the arrow. This can open a discussion about margin of error.

Activity

Materials needed:

  • Recurve or Longbow
  • Flu Flu arrows (preferably with yellow fletching)
  • Computers with Logger Pro software
  • Targets
  • Phone with video capabilities or camera
  • Worksheet

First, I will have the students take a guess at how quick they think that the arrow is flying. I will discuss with them that, in archery, speed (velocity) is commonly described by feet per second. We will have a short review about how when something is moving only forward, speed and velocity are the same thing. I will then have the students convert their feet per second guess into miles per hour and yards per hour. I will also have the students guess which type of function would best model the flight of an arrow. I will not require them to come up with an exact function, that is, they will not need to guess constants, but I am looking for an answer of linear, quadratic, exponential, cubic, quartic, quantic, etc. After the students have made their predications, we will go out to the football field and set up the targets. This is when I will teach them about safety and proper shooting form and they will be able to take a couple practice shots. By teaching them about safety, I am preparing students to be responsible citizens and archers. It’s up to the teacher how far away they want the students to shoot from. The farther away, the more curve you will see in the arrow. However, the farther away you are the less likely you are to see the arrow fly. I suggest shooting from about 20-30 yards. The students will take turns shooting and for safety reasons, I will be the one to take the video. All other students should be behind the shooting line. The videos will look similar to the one below.

IMG_1822

When the students analyze the speed on Logger Pro, they will realize that the initial speed of the arrow is going to be faster than the speed of the arrow as it hits the target. In my example, the arrow was initially going 43.848 yards/sec and then as it hit the target it was going 20.823 yards/sec. The students answers will initially be in yards/sec form and then I want them to convert it to feet/sec and then to miles/hour. I will then ask the students why they think the arrow slows down. Next, the students will analyze the video by graphing points on the actual video using Logger Pro. The students will then be able to compare the function they predicted to the function that Logger Pro says best fits their arrow flight. Note: you will probably have to teach the students how to use Logger Pro prior to this activity.

Capture Capture2

Barriers to Implementation/Adaptations

Since archery can be considered a dangerous sport, a teacher must be certified in order for students to be allowed to shoot archery on school grounds. There is an organization called National Archery in the Schools Program (NASP) which certifies teachers to be able to do just that. Typically, PE teachers are the ones that are certified; however, anyone is able to be certified and certification does not expire. I personally am certified so with support from a principle I would be able to implement this activity using real bows and arrows. If you’re not certified and have no desire to get certified, you can see if the PE teacher at your school is certified and if they are interested in supervising. If there is no way to have anyone certified present, you could also do this activity with a cheap nerf gun or have the students create a small homemade bow out of pencils and rubber bands. You could even shoot one video yourself on your own time and then have all the students analyze the same video instead of shooting their own. This activity could even be done with the flight of any other items, basketballs, baseballs, shoes, javelins, etc. So you do not necessarily have to do this activity with an arrow.

Parental and Community Involvement

My hope is that the students will be so excited about having shot arrows during math class that they will go home and tell their families all about it, thus, involving their families. Part of my reason for having the students convert their guesses and the actual speed of the arrow into miles per hour is so that the general community would understand the units with which they were talking about. Aside from archers, I’m not sure who commonly refers to distance in only yards and refers to speed in feet/sec. I want the students to not only be able to make the conversion but also be able to discuss the activity with others. Also, I will definitely be alerting the parents prior to this activity that the students will be shooting arrows during class. Even though flu flu arrows are not very dangerous at all in comparison to regular arrows, I still want the parents to be aware and involved. Hopefully, by letting the parents know that this activity is going to happen, they will be eager to ask their child about the activity and the student will be able to share with them.

The hunting and archer community is very unique in that, if you are a fellow archer or hunter, you’re almost automatically friends. People in this community love talking and telling each other stories, sometimes for hours upon end. By having the students shoot a couple arrows during class, I am supplying the students with stories and an experience that they will be able to contribute to the conversations if they are ever around other archers.

arrow

This activity aligns with the following common core state standards.

CCSS.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.

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.

You can download the free demo version of Logger Pro at http://www.vernier.com/downloads/logger-pro-demo/

 

Worksheet:Flying Function worksheet

8.G.C.9 GeoMath: Finding Volume and Surface Area

cospers Geometry Curriculum Aligned with CCSS-Math, ML Modeling

“Today’s students are used to and expect continuous connection to information and people. Forcing them to put their personal technology away during class contradicts their lives and gives students one more reason to expect what they learn in school will have little relationship to reality.” – Robert Beichnergoogle earth

It’s that exact expectation from students, that what they are learning in math class has little to do with the real world, that is going to drive me to the brink of insanity. I know that my geometry students are in desperate need of an answer to their two daily questions “when will I ever use this?” and “why do I have to learn this?” Well, I finally took the advice of Robert Beichner and decided that, for this lesson, I won’t ask the students to put away their “personal technology”. Instead, I am encouraging them to use their computers to learn about the importance and relevance of finding surface area and volume of different geometric objects by applying it to the real world.

How? Well, it’s quite simple really. If we want students to understand real world connections, what better way than to look at the ‘real’ world? Okay, Google Earth isn’t technically the real world. However, it’s a pretty good visual representation. Teachers and student can now go to the website www.realworldmath.org to find a wide variety of ways in which you can use Google Earth to teach math, science, and history. One of the lessons involves “flying” around the world to different famous buildings to find the volume and surface area of each. Google Earth even provides a brief history lesson along the way. Students will be able to see the applications of different formulas for volume and surface area unfolding before their eyes. Google Earth and the link mentioned above are great resources to get the students involved in real world applications of math all while using their beloved technology. I’ve used Google Earth in the classroom before and it’s quite amazing how willing students are to stay on task because navigation the app is fun, visual, and mathematical relevance is obvious.  You never know, you may get one day’s break from the “when will I ever use this?” and “why do I have to learn this?” questions. Trust me, the opportunities are endless. Literally…Google Earth even includes outer space.

blockhead

googleearthpyramid

 

 

 

 

 

Lesson Plan:  GeoMath lesson plan

Worksheet: GeoMath- google earth worksheet