Circles and Squares: 7.G.B.4

By: Nick Spencer, Elizabeth Englehart, Sam Marcoe, and Grayson Windle

Introduction:

In Dan Meyer’s “Circles and Squares” video found at http://www.101qs.com/2859, we find a square and a circle on a number line ranging from 0 to 40.  We also notice that the sizes of the two geometrical figures change depending on their position on the number line, indicating some sort of relationship between the two shapes and their respective positions.

 


Group Findings:

The video raises several questions, including the obvious “What is this relationship?”.  Our group first began by trying to answer another question, “When are the areas of the two shapes equal?”  In the above screenshot of the video, we see that the circle and the square seem nearly equal in size in this position.  After some investigation, we found that the areas of the two shapes are equal when the distance from where the shapes touch and the end of the number line is 21.2 to the left and 18.8 to the right.  We were able to test this by visually seeing they are nearly the same in size at 20, and then by measuring the side length of the square and the radius of the circle with some guess-and-check methods.

To find the precise answer, however, we found that you could use the quadratic formula, as well as a couple of other ways through investigating.

Classroom Adaptation:

This problem can be used in a variety of grade levels, particularly in a middle-school classroom.  This problem utilizes algebraic, geometric, conceptual and problem-solving skills to solve.  Teachers can aim for their students to approach this problem in a variety of ways, depending on whether they want this to be an algebra-heavy problem or more of a conceptual experience.  This is a great example of a problem for students to practice the quadratic formula, but even better a problem for students to be able to find connections in mathematics, and use those connections to derive explanations for their findings.

CCSS:

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.

Shape Sorter Activity CCSS.Math.Content.6.SP.B.5.b

By: Nick Spencer, Sam Marcoe, Grayson Windle, Elizabeth Englehart

With the Shape Sorter activity, which can be located at this website, we enable our students to work with several different concepts such as venn diagrams and geometric patterns.  The students work with a venn diagram in which they can set up two “rules” for each side of the diagram, and then are tasked with organizing the shapes accordingly to the rules.  If a shape meets both standards set by the rules, the student can place the shape in the center of the venn diagram, and if the shape doesn’t meet the criteria for either rule the student can leave it outside of the venn diagram.

In the figure above, my group of teacher candidates created our own venn diagram with these two rules:

  1. The figure has at least one line of symmetry (left side)
  2. The figure has rotational symmetry (right side)

In the beginning of this activity, we have several different shapes and begin organizing them in the matching part of the venn diagram based on their geometric characteristics.  The figures on the left side of the diagram have at least one line of symmetry while the figures on the right have rotational symmetry.  The figures in the center have both characteristics set by our rules, while the figures on the outside have neither lines of symmetry nor rotational symmetry.

This online educational activity allows the students to further develop conceptual understanding, and practice procedural fluency for geometric shapes and laws, as well as develop understanding for venn diagrams.  By allowing students to combine geometry with venn diagrams, students can open their perspective that venn diagrams can be utilized in various and unique situations.

 

Coke vs. Sprite CCSS.Math.Content.6.RP.A.1

By: Nick Spencer, Sam Marcoe, Elizabeth Englehart, Grayson Windle

This activity is based off of Dan Meyer’s “Coke vs. Sprite” activity.  In this activity, Dan has a glass of coke, and a glass of sprite, containing equal amounts.  Dan uses a dropper to take some of the sprite and drop it into the coke, creating a coke/sprite mixture.  He then uses the dropper to take some of the coke/sprite mixture, and drops it back into the original sprite.

Dan then asks us this: Which glass contains more of its original soda?

Step 1: Our group began our investigation with a visual representation via drawings in order to solve this problem.  In the figure below, the top two circles represent our original glasses of coke and sprite, each containing 100mL of themselves.

Step 2: Next, we have the dropper extract 10mL of the sprite, and drop it into the coke.  Now we are left with a glass containing 100mL of coke and 10mL of sprite, and a glass containing 90mL of sprite.

Step 3: For the next step, our dropper takes 10mL of the coke/sprite mixture, which we will say contains 9.1mL of coke and 0.9mL of sprite, and drops this into the original sprite glass.

Step 4: Here we do some math.  After adding the coke/sprite mixture into the sprite glass, we find that the original coke glass now has 90.9mL of coke with 9.1mL of sprite, and our sprite glass has 90.9mL of sprite with 9.1mL of coke.

Conclusion: So, which glass contained more of the original soda?
Coke Glass: 90.9mL Coke & 9.1mL Sprite
Sprite Glass: 90.9mL Sprite & 9.1mL Coke

We find that the glasses actually end up containing equal amounts of their original sodas.  We also discovered that this outcome would be the same regardless of the amount of extraction from the glasses.  Had we began with taking say 20mL from the Sprite (leaving 80mL) and putting it into the Coke (which now has 100mL of Coke and 20mL of Sprite), and then taken 20mL of the Coke/Sprite mixture (lets say the dropper has 15mL of Coke and 5mL of Sprite) and then dropped this into the Sprite glass, we would simply find that each glass now contains 85mL of the original soda, and 15mL from the other soda.

CCSS: CCSS.MATH.CONTENT.6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

Art from a Different Angle 7.G.B.5


This math problem is one in which students are using their knowledge of different kinds of angles (supplementary, complimentary, vertical, adjacent) to find as many angles as they can within some sort of abstract art.

The main idea is to locate as many angles of each time as they can within the art-piece, as well as give reasoning in written form of how they know they are correct, based on the different laws of angles and not just a protractor.  Students can also be expected to identify patterns during discussions as well as in written forms, and can present their arguments to the class.  Students can be given various samples of art to work with, and be separated into groups, so that at the end of the lesson the groups can present their art and which angles they found and where.

This lesson is an integration of math into art, and a great way for students to explore how math really is a part of more than most would believe.  This lesson could also lead into the idea that there are different kinds of math within all art, and students could begin trying to identify the various mathematics involved in art and widen their understanding of mathematics in the real world.

CCSS.MATH.CONTENT.7.G.B.5
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

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

CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.

CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.

CCSS.MATH.PRACTICE.MP6 Attend to precision.

CCSS.ELA-LITERACY.W.7.1.B
Support claim(s) with logical reasoning and relevant evidence, using accurate, credible sources and demonstrating an understanding of the topic or text.

Laser-Tag-Mania! CCSS.MATH.CONTENT.HSA.CED.A.1 andCCSS.MATH.CONTENT.HSA.CED.A.2

The story problem is about a game of laser tag where you have a certain amount of life points and you lose an amount of life points every time you get shot. The students will work in groups and as a group they will decide what they want their life point to be and how much will be lost per shot. Once they have decided that they will come up with an equation to model the situation. After they have their equation they go onto Geogebra and use it to graph their equation. Then each group puts their equation and graph on the white board and as a class we have a discussion about which equation students think would make the best game of laser tag and why.

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

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

CCSS.MATH.CONTENT.HSA.CED.A.2

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

LaserTagLessonPlan and Worksheet