Engaging students: Dilations

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Candace Clary. Her topic, from Geometry: identifying dilations.

How could you as a teacher create an activity or project that involves your topic?

Dilations are types of transformation. One activity that I could create for my students is a matching game. I can create cards with index cards, or sheets of paper that have been cut up, that have pictures on them. Each one will be labeled and the students must classify them as dilations, why they are considered dilations, and how they were dilated. As a follow up to this activity, I could assign a topic to create their own city, or small town. They would be required to draw out their town, as well as model it using common crafts. After they do this, they will need to be able to dilate the buildings, and other such things, to make a life size city. They will not have to make the city with a model, but instead, they will need to make a blue print using their model in mind. On this blue print, they will need to inform me of the size of the dilations.

How can this topic be used in your students’ future courses in mathematics or science?

Dilations can be used in many different subjects. Dilations can be used to find sides of a triangle when learning about the triangle congruence theorems. These can be useful in algebra when finding side lengths of figures. This may not happen very often, but it is displayed in algebra. Another place that it will help, although it may not be math, it will help in math classes for architectural students, as well as help people in construction. Many science classes require science projects that work and simulate something real. Dilations can be used when making these projects because you can’t make a real river, but you can structure something that is a smaller figure to the real thing, same thing as a volcano. With architecture, dilations can help with making blue prints and can help in building these blue prints with dilations in mind. With construction, those are blue prints too. I’m not saying in order to build something you must know how to dilate something, but it will help tremendously.

How has this topic appeared in pop culture?

To get the students engaged in the topic, I could bring up the Disney channel movie ‘Honey I Shrunk The Kids.’ This will bring up a discussion with the kids when I ask them what the dad did with his shrink ray. Some ideas that may come up will be that he made them smaller, and then at the end of the movie he made then bigger, back to normal. But in the people were still the same people, they didn’t change, only the size did. At least I hope that is what happens in the discussion. I could then instruct the students into pretending that they had a shrink ray and ‘shrink’ some shapes, as well as other students. This activity, and their answers will be recorded on a chart that they will turn in at the end of class. They, themselves, can decide what size they want to shrink to, but they have to remember to bring the student back to normal at the end of class. I think this activity will be fun for the kids because they will never forget what a dilation is, since they have been ‘dilated’.

Engaging students: Finding the area of a right triangle

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Andrew Wignall. His topic, from Geometry: finding the area of a right triangle.

What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

To introduce the topic of the area of a right triangle early in a lesson, we can first examine the area of a rectangle, which students should already know how to do.

Say you have a large rectangular garden, 60 feet wide and 10 feet long. Home Depot sells sod (which is a pre-grown grass on a net that can be spread on the ground) at a rate of $3/square foot. What is the area of the garden, in square feet? How much sod should you order? How much would it cost to cover the entire garden with sod?

Instead of having the entire garden covered with sod, suppose you wanted to cover part of the garden with sod and leave the rest as soil for planting flowers. To make it more visually interesting, you decide to set the sod as a triangle? The sod triangle will have a base of 60 feet and a height of 10 feet. What is the area of this triangle in relation to the area of the entire garden? What is the area of this triangle? How much sod should you order? How much would it cost to cover the triangular area with sod?

Through this activity, we can investigate a relationship between right triangles and rectangles, and also the relation of the area of a triangle compared to the angle of a rectangle.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website

One tool to show the area of a right triangle quickly and easily is the Area Tool on Illuminations (http://illuminations.nctm.org/Activity.aspx?id=3567). With trapezoids, parallelograms, and triangles available, you can click and drag the three vertices of a triangle and instantly see how the area is affected. You can create a quick table and keep a running tally of the base, height, and area, so you can recalculate in front of the class.

Illuminations has a sample lesson plan available online for discovering the area of triangles, and integrates this tool into the plan. If not using this tool as part of a similar plan, we must understand that this tool will not be great for introducing the lesson, as there is no button to lock onto a right triangle. However, there is a button to lock the height, so when you move the vertex opposite the base, you can see how the area does not change, see how the height can be outside the triangle, and extend the formula for the area of a right triangle to the area of any triangle. This tool can then be used in further lessons when discussing the area of parallelograms and trapezoids.

How can this topic be used in your students’ future courses in mathematics or science?

Since triangles are one of the most basic shapes, the area of triangles comes up time and time again. Triangles will also be used to find the area of more complex polygons, such as hexagons and irregular polygons, by breaking down complex shapes into simple triangles and quadrelaterals. Trigonometry uses right (and non-right) triangles extensively; in Precalculus, we will revisit the area of triangles, and learn how to find the area of triangles without explicitly being given the base and height.

Outside the classroom, the area of a triangle is used extensively in architecture, as triangles are strong, and triangular trusses and frames are used in many steel structures. As the inside empty area of the triangle increases, then the stress on the triangle increases, and architects must take this into consideration.

Triangles are also used in 3d computer graphics, as the 3d shapes they design actually consist of lots of little triangles, and they have to fit textures of a certain size (say 512 pixels x 512 pixels) onto a few triangles, so it is important that they know how and where for these textures to lie.

References

Math is Fun, “Activity: Garden Area”. http://www.mathsisfun.com/activity/garden-area.html

Illuminations: Resources for Teaching Math, “Discovering the Area Formula for Triangles”. http://illuminations.nctm.org/Lesson.aspx?id=1874

Illuminations: Resources for Teaching Math, “Area Tool”. http://illuminations.nctm.org/Activity.aspx?id=3567

Home Depot, http://www.homedepot.com/p/StarPro-Greens-Centipede-Southwest-Synthetic-Lawn-Grass-Turf-Sold-by-15-ft-W-rolls-x-Your-L-2-97-Sq-Ft-Equivalent-RGB7/202025213

Math is Fun, “Heron’s Formula”. http://www.mathsisfun.com/geometry/herons-formula.html

Maths in the City, “Most stable shape – triangle”. http://www.mathsinthecity.com/sites/most-stable-shape-triangle

Andre LaMothe, “Texture Mapping Mania”. http://archive.gamedev.net/archive/reference/articles/article852.html

Engaging students: Defining the terms perpendicular and parallel

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Allison Metlzer. Her topic, from Geometry: defining the terms perpendicular and parallel.

B1. How can this topic be used in your students’ future courses in mathematics or science?

The concepts of perpendicular and parallel will be implemented in many of my students’ future mathematics courses not only in high school, but also in college. In algebra, the students are asked to find the slope or the rate of change. In looking at the slope, students are asked to find if it’s parallel or perpendicular to another function’s slope.

In geometry, many shapes have properties that define them as having parallel or perpendicular sides (i.e. squares, rectangles, parallelograms, etc.). Also, in order to decide if triangles are similar, their corresponding sides must be parallel. In order to use the Pythagorean Theorem, the triangle must be right angled or have the two legs perpendicular to one another.

In calculus, students are asked to find orthogonal vectors which are also defined as perpendicular vectors. Also, calculus incorporates concepts from algebra and geometry which in turn, include parallel and perpendicular lines.

Therefore, many, if not all of my students’ future math courses will use the topics parallel and perpendicular. Thus, it would be important for me to teach them the two concepts correctly now so that there wouldn’t be any misconceptions in the future.

C3. How has this topic appeared in the news?

One big thing the news talks about every two years is the Olympics. Using the concept of parallel and perpendicular, the constructions are made for all of the different events. Apparent examples of events incorporating parallel lines are track, speed skating, and swimming. The one I will focus on is swimming, namely because it is a very popular Olympic event and one of my favorites. Pictured below is an Olympic swimming pool of 8 lanes. Do the lanes appear to be parallel? Two things that are parallel are defined as never intersecting while also being continuously equidistant apart. One can clearly see the lanes of the pool never intersect. If they did, then the contestants could interfere with one another. Also, because the Olympics is a fair competition, the lanes are equidistant in order to give each contestant a fair and equal amount of room.

Because the Olympics is a well-known event featured in newspapers, articles, and on TV, the students will be able to understand this real world application of parallel and perpendicular.

E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Before I would play the video, https://www.youtube.com/watch?v=vnnwfcDcNlY, I would first ask the students to think of as many examples they can of parallel and perpendicular in the real world. After about a couple of minutes, I would tell them to keep those in mind and see if the video included any they didn’t think of. I would play the video from 1:25 to 3:05 which is the portion that displays all of the examples. It has clear pictures of recognizable objects which incorporate parallel or perpendicular lines. Also, the video has labels on the pictures to even more clearly describe where the components of parallel and perpendicular lines are. I believe that the initial brainstorm along with this video would get the students thinking about the importance of parallel and perpendicular lines. Also, I would make the connection that those examples would not be considered parallel or perpendicular unless they met the following definitions. Then I could explicitly define both parallel and perpendicular.

Thinking of real world examples, and seeing pictures of them will help the students understand what parallel and perpendicular lines should look like. After they have this initial understanding, they then could get a better grasp of the definitions. Also, they would recognize the importance of following the definitions to correctly construct objects involving parallel and perpendicular lines.

References:

Detwiler, dir. Intro to Parallel and Perpendicular Line. YouTube, 2010. Web. <https://www.youtube.com/watch?v=vnnwfcDcNlY >.

Medicine’s Uncomfortable Relationship With Math: Calculating Positive Predictive Value

Taken from: http://archinte.jamanetwork.com/article.aspx?articleid=1861033&utm_source=silverchair+information+systems&utm_medium=email&utm_campaign=archivesofinternalmedicine%3aonlinefirst04%2f21%2f2014

In 1978, Casscells et al¹ published a small but important study showing that the majority of physicians, house officers, and students overestimated the positive predictive value (PPV) of a laboratory test result using prevalence and false positive rate. Today, interpretation of diagnostic tests is even more critical with the increasing use of medical technology in health care. Accordingly, we replicated the study by Casscells et al¹ by asking a convenience sample of physicians, house officers, and students the same question: “If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5%, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about the person’s symptoms or signs?”

Approximately three-quarters of respondents answered the question incorrectly (95% CI, 65% to 87%). In our study, 14 of 61 respondents (23%) gave a correct response, not significantly different from the 11 of 60 correct responses (18%) in the Casscells study (difference, 5%; 95% CI, −11% to 21%). In both studies the most common answer was “95%,” given by 27 of 61 respondents (44%) in our study and 27 of 60 (45%) in the study by Casscells et al¹ (Figure).

Encouraging Students to Tinker

A recent blog post from Math Ed Matters had the following pedagogical insight:

How do we encourage students to tinker with mathematics? As a culture, it seems we are afraid of making mistakes. This seems especially bad when it comes to how most students approach mathematics. But making and then reflecting on mistakes is a huge part of learning. Just think about learning to walk or riding a bike. Babies are brave enough to take a first step even though they have no idea what will happen. My kids fell down a lot while learning to walk. But they kept trying.

I want my students to approach mathematics in the same way. Try stuff, see what happens, and if necessary, try again. But many of them resist tinkering. Too many students have been programmed to think that all problems are solvable, that there is exactly one way to approach each problem, and that if they can’t solve a problem in five minutes or less, they must be doing something wrong. But these are myths, and we must find ways to remove the misconceptions. The first step is to encourage risk taking.

A few months ago, Stan Yoshinobu addressed this topic over on The IBL Blog in a post titled “Destigmatizing Mistakes.” I encourage you to read his whole post, but here is a highlight:

Productive mistakes and experimentation are necessary ingredients of curiosity and creativity. A person cannot develop dispositions to seek new ideas and create new ways of thinking without being willing to make mistakes and experiment. Instructors can provide frequent, engaging in-class activities that dispel negative connotations of mistakes, and simultaneously elevate them to their rightful place as a necessary component in the process of learning.

Here are a few related questions I have:

How do we encourage students to tinker with mathematics?

How do we destigmatize mistakes in the mathematics classroom?

How do we encourage and/or reward risk taking?

What are the obstacles to addressing the items above and how do we remove these obstacles?

Source: http://maamathedmatters.blogspot.com/2014/04/encouraging-students-to-tinker.html

Prepare to Have Your Mind Blown by a Balloon and a Minivan

For more information, see http://io9.com/prepare-to-have-your-mind-blown-by-a-balloon-and-a-mini-1565303363?utm_campaign=socialflow_io9_facebook&utm_source=io9_facebook&utm_medium=socialflow

See also Smarter Every Day’s channel: http://www.youtube.com/user/destinws2

Percentage points

An unorthodox way of solving quadratic equations

This post concerns an unorthodox but logically correct technique for solving a quadratic equation via factoring. I showed this to some senior math majors as well as graduate students in mathematics; none of them had ever seen this before. Suppose that we want to solve

$6x^2 - 13x - 5 = 0$

without using the quadratic formula. Trying to solve this by factoring looks like a pain in the neck, as there are several possibilities:

$(x + \underline{\quad})(6x - \underline{\quad}) = 0$ ,

$(x - \underline{\quad})(6x + \underline{\quad}) = 0$ ,

$(2x + \underline{\quad})(3x - \underline{\quad}) = 0$ ,

$(2x - \underline{\quad})(3x + \underline{\quad}) = 0$ .

So instead, let’s replace the original equation with a new equation. I’ll get rid of the leading coefficient and multiply the constant term by the leading coefficient:

$t^2 - 13t - (5)(6) = 0$ , or

$t^2 - 13t - 30 =0$ .

This is a lot easier to factor:

$(t - 15)(t+ 2) = 0$

$t = 15 \quad \hbox{or} \quad t = -2$

So, to solve for $x$ , divide by the original leading coefficient, which was $6$ :

$x = 15/6 = 5/2 \quad \hbox{or} \quad x = -2/6 = -1/3$ .

As you can check, those are indeed the roots of the original equation.

This technique always works if the quadratic polynomial has rational roots. But why does it work? I’ll give the answer after the thought bubble.

The original quadratic equation was

$6x^2 - 13x - 5 = 0$

Let’s make the substitution $x = t/6$ :

$6 \displaystyle \left( \frac{t}{6} \right)^2 - 13 \left( \frac{t}{6} \right) - 5 = 0$

$\displaystyle \frac{t^2}{6} - \frac{13t}{6} - 5 = 0$

Multiply both sides by $6$ , and we get the transformed equation:

$t^2 - 13t - 30 = 0$

Although I personally love this technique, I have mixed feelings about the pedagogical usefulness of this trick… mostly because, to students, it probably feels like exactly that: a trick to follow without any conceptual understanding. Perhaps this trick is best reserved for talented students who could use an enrichment activity in Algebra II.

Neil Armstrong on Being a Nerd

Opting Out of High-Stakes Assessments

In response to the growing movement of parents who have opted out of high-stakes testing, Michelle Rhee wrote a defense of the (commercial) enterprise in the Washington Post. This op-ed piece was brilliantly deconstructed, point by point, at http://curmudgucation.blogspot.com/2014/04/wapo-wastes-space-on-that-woman.html. I encourage you to read the whole thing. A few excerpts:

[Michelle Rhee]: No, tests are not fun — but they’re necessary. Stepping on the bathroom scale can be nerve-racking, but it tells us if that exercise routine is working. Going to the dentist for a checkup every six months might be unpleasant, but it lets us know if there are cavities to address. In education, tests provide an objective measurement of how students are progressing — information that’s critical to improving public schools.

Except that the current crop of Standardized Tests are not like stepping on a scale or going to the dentist. They are like trying to find out a child’s weight by waterboarding him. They are like having your teeth checked by a blind blacksmith. Because, in education, tests NEVER provide an objective measure of anything, because tests are made by people. Yes, tests are useful– but only good tests. And do you know what good tests are useful for? They are useful for providing information critical to helping further the education of students.

I am not a Systems True Devotee. STDs believe that we just have to create a well-oiled precision machine and it will spit out Smarterer Student Products like toasters off an assembly line. I would stop to further develop the point, but we’re only one paragraph in. These woods are dark and deep, but we have miles to go.

From this diving board, That Woman proceeds to register her stunned amazement that in various places, there’s a movement that is convincing parents to pull kids out of these tests! Really!!! These marvelous tests that will tell us how schools are doing!! What in the name of God are they thinking!?!?!!?

[Michelle Rhee:] This makes no sense. All parents want to know how their children are progressing and how good the teachers are in the classroom. Good educators also want an assessment of how well they are serving students, because they want kids to have the skills and knowledge to succeed.

Allow to help you comprehend this, O She. You are correct that parents and educators do want to know these things. Your mistake is in believing that they can only know this by looking at standardized test results.

Yes, the Great and Powerful Woman Who No Longer Has a Curtain To Hide Behind imagines a world where parents sit at home after eight months of school, wringing their hands and saying, “Oh, jehosephat, I wish we knew how Janey was doing in school. But we have no idea.” Meanwhile, at school, teachers sit and the lounge and say, “Yeah, I’ve been with this kid for eight months but I just don’t know how he’s doing. Thank God we’re going to be giving a high stakes high pressure badly written unproven standardized test soon so that I’ll know how it’s going.”

In That Woman’s universe, parents and teachers (sorry– public school parents and teachers) are dumber than dirt. In fact, the list of People Standing in the Way of Educational Excellence gets longer and longer. Parents, teachers, democratically elected school boards– reformy fans have an enemies list that keeps lengthening.

[Michelle Rhee:] We don’t need to opt out of standardized tests; we need better and more rigorous standardized tests in public schools.

Yes!! When you’re doing something stupid and bad and non-productive, do it More Harder!!

[Michelle Rhee:] We also shouldn’t accept the false argument that testing restricts educators too much, stifles innovation in the classroom or takes the joy out of teaching. That line of thought assumes that the test is the be-all and end-all — and if that’s the perspective, the joy is already long gone.

Here’s a multiple choice test for you, dear, exhausted reader. Select which statement best reflects the meaning of the above excerpt:

1) Do not assume that the test is the be-all and end-all. It will just be-all the way we decide to end-all teaching careers, school existence, and student futures.

2) You cannot claim that this year’s testing is sucking up all the joy of teaching, because we actually drained that lake long ago and killed the fish flopping in the mud with fire and big pointy sticks.