How will you teach students to use the Pythagorean Theorem to solve problems in alignment with the CCSS?
Coherence is a critical aspect of the Common Core. It’s defined as thinking across grades and linking to major topics within grades. How will you teach students to use the Pythagorean Theorem to solve problems in a way that develops coherence across Algebra and Geometry topics?
I like to make an Algebra-Geometry connection with Pythagorean Theorem (PT) by using the converse of the PT and ultimately the concept of Pythagorean Triples. The PT states that if a right triangle has sides a, b and and c, then a2 + b2 = c2. The converse states that if a2 + b2 = c2 for a given triangle with sides a, b and c, then that triangle is a right triangle. In both of cases, we assume that c is the longest side (and thus the hypotenuse). Also, any set of integers, a, b and c that satisfy the equation are called Pythagorean Triples. This brings in some number theory to complement the Geometry.
First, I begin teaching the Pythagorean Theorem in Geometry by going to its origins. To the Greeks, this theorem showed that the area of the square constructed on the hypotenuse of a right triangle is equal to the sum of the areas of the two squares constructed on the legs of said right triangle. The Greeks did not use Algebra – they thought of all problems in geometric terms.
So when we say in Algebra “a-squared plus b-squared equals c-squared,” we think of squaring as the process of multiplying a number by itself. The word “squared”, though, comes from the idea that we are making a square using a given length. So “5-squared” means the construction of a square of side length 5. The area of that square is 25 which is calculated as 5 times 5 (squaring). So the algebraic term “squaring” comes from the geometric concept of the area of a square with a given side length.
Now to the business of Pythagorean Triples. There are several ways in which a Pythagorean Triple can be generated using formulas. Here is one such method:
If m and n are any two positive integers, with m < n, then a = n2 - m2, b = 2nm, c = n2 + m2 will produce a Pythagorean Triple of the form a, b, c.
For example, if m=1 and n=2, we get:
a = 22 - 12 = 4 – 1 = 3, b = 2(2)(1) = 4 and c = 22 + 12 = 4 + 1 = 5. So we get the triple 3, 4, 5. Thus a triangle with sides of length 3, 4 and 5 would be a right triangle.
It is a great Algebraic exercise for students to verify why these formulas work by showing that the formulas for a, b and c satisfy the requirement that a2 + b2 = c2. By substituting the formulas we get (n2 - m2)2 + (2mn)2 = (n2 + m2)2. Students can perform the expansions of both sides and verify that they are equivalent. I find this to be nice way to visit the topic of the Pythagorean Theorem in a way that is not typically covered in textbooks.
The first teaching of the Pythagorean Theorem in Algebra is always a gratifying unit. For some reason the formula sticks with students. Perhaps, it is because of its elegance or that it sings when you say it: “a_squared plus b_squared equals c_squared.” Teaching the Pythagorean Theorem is also fun because there are so many great visual “proofs”. You can fold patty paper. You can make a square by putting four right triangles end to end and investigate the areas. Or, you can use square cut outs with sides a,b and c that correspond to sides of right triangles and allow the students to physically cut the a and b squares to fit into the c square. Not only can you show that it works under certain conditions, equally exciting is showing that it doesn’t work by repeating the same exercises with acute or obtuse triangles. Basically, you can approach introducing the Pythagorean Theorem from just about any learning style.
The Pythagorean Theorem has built-in coherence. What is the very next thing we typically teach? In most curriculums it is the distance formula. I tell my students this is just the Pythagorean Theorem in a tuxedo, dressed slightly differently but certainly the same. One of the standards in the common core is recognizing structure. The more we can point out common structures in the mathematics we teach, the more coherence is built. Building coherence isn’t just about scaffolding their learning, it is also critical for them to apply their knowledge to new problems. Coherence is modeling how to take what they know to the next level and the Pythagorean Theorem follows them into many of the next levels.
As students move from Algebra to Geometry and beyond, the Pythagorean Theorem will be the foundation for several topics. These topics include: special triangles, right triangle trigonometry, and the equation for circles in Geometry, trigonometric identities and the study of conics in Advanced Algebra, related rates or using the disc method in Calculus, the computation of standard deviation and the least squares regression equation in Statistics to name a few. So, how do I teach students to solve problems using the Pythagorean Theorem to develop coherence?
The answer is threefold: through reflection, demonstration and re-telling the story. Each time a new concept is introduced, reflection questions can be asked. Questions like: “What does this remind you of?” or “How does this relate to former learning (in this case the Pythagorean Theorem)?”, or “Do you notice a pattern or structure in this problem that we have used in the past?” The second method I deploy is that of demonstration. Showing students how this seemingly new concept is an extension or an application of what we have done before. The use of modeling with technology like Java applets, Geometer’s Sketchpad or Fathom can create great visualization for students. Finally, I believe in the power of the story. The Pythagorean Theorem has a rich history and the Pythagoreans provide enough intrigue and scandal to rival modern entertainment. Telling the story of how those real-life, often eccentric, characters built the mathematics we use today engages students and builds coherence, not just across mathematical subjects but across the entire curriculum.
The fifth standard for mathematical practice calls for students to “use appropriate tools strategically.” The Pythagorean Theorem is as ubiquitous as any tool in the secondary math classroom, giving us the opportunity to teach the idea that a theorem or an algebraic rule is indeed a mathematical tool, and to teach students to look for and discover opportunities to use this tool in successively more complex, unexpected, and novel ways. In an early algebra class, this means showing students that while they do have the option of memorizing the distance formula, the Pythagorean Theorem might be a more streamlined way to think about it. Students are so often bogged down in trying to memorize the distance formula (“Which number is x1? Which is y2?”) that they can get stuck on a problem. These students are so excited to see that they didn’t have to memorize anything after all: they just had to consider the right triangle whose legs are the vertical and horizontal displacements between two points. This application lays groundwork for a later exploration of the relationship between the tangent ratio and slope.
Aside from its applications in coordinate geometry, the Pythagorean Theorem welcomes exploration from the perspective of proof in the geometry classroom and extension in trigonometry. As students learn the characteristics of vigorous proof, they can examine some of the famous proofs (and false ones: http://.cut-the-knot.org/pythagoras/FalseProofs.shtml) of the theorem. They can then be encouraged to use the theorem in new proofs, which is an application of a theorem as a tool.
In my experience, the Pythagorean Theorem is one of the handful of mathematical ideas that nearly any student can recite from memory, even if they don’t know exactly how to use it. “A squared plus b squared equals c squared.” It is our job to continue to leverage this as a piece of prior knowledge. Doesn’t the formula for a circle look like something you’ve seen before? Yes - and wow, why is that? The magnitude of a complex number? It's pretty much just the distance formula again. Wouldn’t it be cool if you could use the Pythagorean Theorem in a triangle without a right angle? Yes, let’s tackle that, and in doing so we’ll develop the Law of Cosines, which will of course require us to flex our proof muscles, so it’s a good thing we did this in Geometry. And what’s up with this whole thing about squaring complex numbers and getting two-thirds of a Pythagorean Triple?
The more we ask these types of questions, the more coherent our curricula will be. Teachers should use the Pythagorean Theorem as case study in what this looks like, and work to find more topics that offer the same opportunities.
The Pythagorean Theorem is one of the most critical concepts that students learn in school mathematics. It begins in middle school with a basic introduction to the concept, and from there builds in complexity as students progress through the grades. As such, having a coherent vertical approach to teaching the theorem is very important.
Students often have a natural intuition with measuring length, but not so with area. For example, if you show a student a segment with length 2, a segment with length 3, and then a segment with length 5, you can pretty easily convince them the two smaller segments add up to the larger segment. However, this is not necessarily the case with areas. In fact, often our visual perception tricks our mind. Do two squares of areas 9 and 16, really appear to be the same size as a single square of area 25? From our perspective as teachers, the answer is obvious, yes it does! But, for students this question is not nearly as obvious as adding two smaller lengths to get the larger. As such, I firmly believe that it is critical that teachers in middle school emphasize that the Pythagorean Theorem is as much a statement about areas, as it is lengths of a right triangle.
A great activity to use when first introducing the theorem is having them stack crackers (I like to use Cheese-Its) on the edge of a given right triangle. Working with manipulatives gives the students a concrete “proof” of the theorem, enough to convince them that it is true. Then, as students move into Geometry, formal proofs of the theorem should be explored more in depth. As Steven Strogatz discusses in his wonderful book The Joy of x, the meaning of an “elegant” proof can naturally be discussed with the Pythagorean theorem, because of the multitude of accessible proofs available to students.
Once students reach Algebra 2, it is time to transfer the knowledge of the Pythagorean Theorem to develop notions of the conic sections, and in particular using it to develop the distance formula and the equation of a circle. Once students are introduced to the concepts of trigonometric ratios, this notion can be extended even further to the Pythagorean identities, and the law of cosines. When I teach each of these concepts, I constantly refer back to the Pythagorean Theorem during my mini-lecture in order to connect to prior learning.
For students to demonstrate their mastery of the Pythagorean Theorem not only within the current grade level, but also across grade levels, students must be able to articulate the connections. Each year, in the review unit of my AP Calculus class, we spend a day on the Pythagorean Theorem and how the theorem connects to so many different concepts in mathematics.
To start, I go through a process which helps activate their memory. First, I give the students 2 minutes (set a timer) to write down everything they know about the Pythagorean Theorem. I then reset the timer for another 2 minutes, where they add to the list of what they know by discussing and comparing with a partner. Most often, students don’t get beyond the basic geometric interpretation. I probe the student further by giving an example (usually the distance formula, or definition of circle). I give them another 2 minutes to work with their partner to add to their list.
Once their list is fairly complete, their task is to develop a concept map illustrating the connections formed by the Pythagorean Theorem. For each “node” on the concept map, the students have to create a challenging problem to represent the use of the Pythagorean Theorem in each setting. I collect the concept maps (with the problems) and give them feedback.
This process allows me to assess the students understanding and knowledge of how the Pythagorean Theorem is connected across grades/classes. It also allows the students to gain a better understanding of these important connections, so that they are more prepared for the challenges of higher level mathematics, including the calculus. Students within a program that follows a coherent approach to teaching the Pythagorean Theorem should be able to complete this assessment.
When students make connections between current learning and their prior knowledge, they are more likely to retain that new learning, develop a deeper understanding of the mathematical ideas foundational to the learning, and be able to apply their learning to new and unfamiliar contexts. Students often have heard about the Pythagorean Theorem (PT) in casual references by another math teacher or from older siblings or friends prior to their first formal lesson on PT. But the PT offers fruitful connections and applications around which content in several different courses can be built.
Prior to the PT, many students are taught the distance formula in a course like Algebra 1 where the purpose is to drill order of operations, exponents, and square root operations. The following year, Geometry students learn about the PT and even “prove” the theorem by one of the more common proofs with combinatorics, but these proofs lack relevance for students in the absence of an organic understanding of the PT as a generalization from more basic, intuitive cases. Geometry and Algebra 2 / Trig students solve triangles, including non-right triangles, but usually do not view the PT as a special case of the Law of Cosines. Trig students learn all about Pythagorean Identities but usually cannot explain where they come from despite their suggestive name. Conic sections, related rates, complex numbers, and other topics would also benefit from greater coherence in students’ mathematical experiences, particularly around the PT. We must do better to elucidate the common thread of the PT in all of these topics; the use of multiple representations, and utilizing the process of generalizing from more simple cases to the variable form, provide a powerful way to bring coherence across topics and grades.
In Algebra 1, I start with a numeric approach using special cases, measuring and estimating distances, and then lead into the generalization of the distance formula and it’s identical twin the PT. Using a real or generic map, I ask students how far it would be to walk from their house, for now located at the origin, to their school (or some other location of interest) at the point (3,4). Staying on the streets, this is easy – just count the boxes/blocks on each street. But what if we could walk across the land directly to the school? Using various estimation and measurement techniques with string, rulers, etc. students can approximate the actual distance as being close to 5 units. But exactly 5 units? How do we know? Repeating this process with other Pythagorean triples, including multiples of the fundamental triples (6-8-10, 9-12-15, and so on are all multiples of the 3-4-5 triple), students can recognize patterns and relationships through repeated reasoning. Then, I move their home around to different points but keep the school at (3,4), such that the distance that can be modeled by another Pythagorean triple (say 5-12-13), but now students must subtract coordinates in the x- and y- directions to find the length of each side of the right triangle before estimating and then trying to compute the length of the hypotenuse. Now I begin to use points that are not modeled by Pythagorean triples – but now the key is to carry out the calculations without simplifying the coordinate values along the way. Through repeated reasoning again, I can seamlessly replace one or a few x- and y- coordinates with variables and have students follow the same algebraic procedures as they did with constant values – this process facilitates the move toward generalization. From this generalization of the distance formula, squaring both sides gives the PT. Connections back to the map that students used and the process of estimating from special cases, before generalizing with variables, is critical to developing students’ reasoning habits and mathematical thinking skills.
I must quickly but briefly note the importance of using memorable problems, including but not limited to real-world contexts, that can be modeled with the PT. Conditioning students to ask themselves “Have I seen a problem similar to this one before?” is a productive way for students to reflect back on their prior knowledge, determine a productive entry point into a problem, and illuminate connections to the problem they are trying to solve.
Throughout the study of right triangle trig, Pythagorean Identities are pervasive. The connections to the unit circle with reference angles forming right triangles with the x-axis should be a focal point. Given the unit circle of radius 1, the opposite leg from the reference angle is sin(theta) / 1 , the adjacent leg is cos(theta) / 1 , and the PT gives the well-known identity sin^2(theta) + cos^2(theta) = 1. From this identity, the other Pythagorean Identities can be derived through dividing every term by sin^2(theta) or cos^2(theta). When students can reduce the amount of memorizing they need to do, and instead can re-derive a formula when they need to use it, that is a victory for conceptual understanding.
The PT should also be viewed as a special case of the Law of Cosines when solving non-right triangles is taught. Noting that the PT works only for right triangles, an “adjustment” to the PT will accommodate non-right triangles: c^2 = a^2 + b^2 – 2ab*cos(C); we derive this equation in class without me telling students first. When the angle C = 90 degrees, cos(90)=0, the last term multiplies out to 0, and the PT remains!
Throughout these courses, the Pythagorean Theorem is one of those “big ideas” around which a comprehensive curriculum should be framed. Identifying the “big ideas” of a course reduces the barriers that teachers face when trying to draw connections among math topics and in the frequent remediation that is required with students who have not been presented with mathematically significant opportunities to problem-solve with the PT. When students struggle with new learning related to the PT, this framework permits the teacher to return to the numeric and graphical representations to scaffold and rebuild the student’s understanding that is requisite for mastering the new learning, and students will be more responsive to this intervention as a result of the meaningful problem-solving opportunities they have engaged in along the way. This framework also enables students to see the structure behind the distance formula and PT, then realize that the two really are one-and-the-same.
Pythagorean theorem is used to find the lengths of sides of right triangles, can link to area, can also link to the distance formula as well as to the triangle inequality theorem where they can figure out the range of values that the missing side can be and evaluate their answers based on this. Additionally the relationship between can be used to look at tangents and triangles when creating right triangles with the diameter / radius; chords within a circle...
In teaching our district math curriculum I frequently use project-based learning and inquiry learning. This gives my lessons a natural coherence as students develop new approaches for each challenge by building on what they already understand and feel confident doing. I compare it to helping someone learn to play a musical instrument. There are some basic skills necessary to get appropriate sounds, but learning to read sheet music happens because the student wants to play new or more complicated pieces, not because it is the next chapter or standard. As the student becomes more accomplished she/he may try a new instrument or may try creating an original work, building on what they've already mastered.
The Pythagorean theorem is interesting to teach to older students because they have already had some exposure to it, know the words and may even remember a2+b2=c2. Unfortunately these students rarely understand where this theorem comes from or how it can be applied as a problem-solving tool. I begin by identifying what their level of understanding is and what misconceptions they hold, by asking each student to briefly describe how they would use the Pythagorean Theorem to find the diagonal of a flat-screen TV, given the lengths of its sides. Their responses give me a good idea what additional instruction is needed. I anticipate that most seniors can plug numbers in to a calculator successfully, but many will hesitate unless given specific information about which lengths should be a, b and c. In addition, some will forget or not understand the process of taking the square root of the result and others will struggle with when and how to square the lengths of the sides.
After reviewing my students’ responses, I follow through on the initial challenge, explaining that I am planning to buy a new TV and entertainment center online, but need to be sure that the TV will fit properly in the entertainment center. I add that the dimensions of the TV are given in terms of the diagonal, but the entertainment center uses the lengths of the sides for the TV space. As we brainstorm how to use the Pythagorean Theorem to help solve this problem, I expect someone will suggest I just draw it out and measure it or call the company and ask them if the TV will fit. I acknowledge these suggestions and describe circumstances when I’ve needed to figure something out and couldn’t measure or call because of time or material constraints.
We return to the Pythagorean Theorem and I ask the students how we can identify which sides are a, b and c. We focus on determining which side should be c since it’s all by itself. It’s easy to say “go with the longest side”, but since these are seniors who should already know this, that method hasn’t worked so far. Instead I use a hands-on activity that is fairly quick and easy. I distribute pre-cut pieces of pipe cleaner, heavy paper, or whatever else I have handy. These items are cut to lengths of Pythagorean triples and are in color-coded sets. Each student in a given team receives a different set so they can compare results. I ask the students put their pieces together into a right triangle. Once everyone has accomplished this I ask them to mark two of the pieces so they can remember which ones made the 90° angle. I check to make sure they’ve all gotten this done and then suggest that they try to line up their pieces in any way to make two pieces equal in length to the third. When everyone has finished this task, I instruct them to share what they’ve discovered with their teammates and then journal briefly about how they can remember that the longest side is not a side making up the right angle and that it is equal to the sum of the lengths of the other two sides.
I refer back to my original TV problem and ask if they can now tell which side should be c. I ask students who are still struggling to think about how my TV relates to the triangles they just worked with. Sometimes they need a diagram or picture to recognize that the diagonal makes a right triangle across my TV.
Now we can move on to identifying which remaining side should be a and which should be b. To help students understand that it really doesn’t matter, I use a simpler example of addition like 2+3=5. Then I ask anyone who knows another way to write that equation to put it on the board. They will post all of the alternatives, and because this is math they fell comfortable with, they will generally peer edit any incorrect postings. These variations provide an opportunity to write a2+b2=c2 under my initial equation and put matching equations under each of the posted variations. There will be a few students who argue that it’s not the same because my equation uses squares, in which case we repeat the process with known squares like 22 and 32. I instruct students to make a brief journal note about what they’ve learned about identifying the sides of a right triangle.
Coming back again to my original TV problem, I ask students to work independently to find out whether or not the TV I’ve selected will fit the entertainment center, using their understanding of the Pythagorean Theorem to see how the TV diagonal compares to the diagonal of the space in the entertainment center. When they’ve finished I ask them to share their results with their team and then ask the teams to reach a consensus to share with the class.
What we’ve done so far has been to build on prior knowledge of the Pythagorean Theorem and reinforce basic computational skills. I assume that earlier instruction included a demonstration or activity that involved making actual squares with sides the length of each side of the right triangle and then cutting squares a and b to fit square c. If that hasn’t ever happened I might use it to help students connect to squaring the sides. In addition, I would now move on to other applications using the Pythagorean, both geometrically with shapes and algebraically with the distance formula and graphing. By using an assortment of real-world problems, both teacher-generated and student-generated, I will help my students become confident and capable in their application of the Pythagorean Theorem across the disciplines of mathematics. This also prepares them for later applications such as complex numbers, trig identities, and the law of cosines.
The Pythagorean Theorem is a very important concept in mathematics. It can be related to many topics in mathematics. As such, students engage the Pythagorean Theorem throughout their mathematical career, starting in middle school. Initially, students learn that the Pythagorean is an algebraic representation of the relationship of the sides of a right triangle. Students are then asked to prove that relationship using visual aids or writing indirect proofs or two-column deductive proofs. Students learn that the Pythagorean Theorem
has a relationship with slope as well as distance on the coordinate plane, using the distance formula. Students use the Pythagorean Theorem to model simple real life scenarios to solve for distance or height and/or angles of elevation and/or depression.
As a teacher of trigonometry, I utilize the Pythagorean Theorem in several ways to extend students’ depth of understanding. Of course, students use it to determine the value of sine, cosine and tangent in special right triangles. We also use it to derive the Law of Cosines and the Pythagorean Identities.
The Law of Cosines is a generalization of the Pythagorean Theorem. Hence, the Pythagorean Theorem, using geometry, algebra and right triangle trigonometry, can be used to prove the validity of the Law of Cosines. Students are provided an oblique triangle and are expected to construct a proof. In a similar way we can apply the Pythagorean Theorem to a right triangle in the unit circle to derive the Pythagorean Identities.
Using Pythagorean Theorem to solve problems:
I generally come back to Pythagorean Theorem in a multiplicity of topics as I teach and tutor students throughout the various levels of mathematics: As we move from basic geometry into the Cartesian plane and analyze change in distance across two dimensions. As a tool when superimposing right triangles onto real world scenarios from overview of a plan to change in elevation, right up to looking at vector analysis in both two and three dimensional space.
I teach Pythagorean Theorem as a tool in simple machine analysis, particularly inclined planes and wedges, but also in static force analysis when looking at truss design and structural analysis. This theorem moves throughout so many areas of applied mathematics that we never really stop talking about it in one way or another. Whether it is finding vector magnitude values, looking at circle sectors and polar graphs, converting to rectangular coordinates, right up to trig substitution in advanced calculus integration techniques, we never stop noticing it as a collective body of mathematicians. My educational philosophy is there is just a little you must memorize in mathematics to understand and recognize all the things you can deduce or calculate. Pythagorean Theorem is one of those golden nuggets we treasure, the more you see its familiar presence...the more math anxiety quiets and the mind can relax and function optimally.
The Pythagorean Theorem is first introduced in middle school, yet it is explored in more depth in high school geometry. It also has several applications within geometry, and it is also relevant to topics in Algebra II and Precalculus (namely trigonometry).
When the Pythagorean Theorem is first introduced, it is important for students to see that the c in a2 + b2= c2 represents the length of the hypotenuse in a right triangle. Students are taught that this side is opposite from the right angle and the longest side. Several word problems can be solved, and students should be able to discern from the word problem whether they are solving for a, b, or c. A classic type of word problem is a ladder leaning against a wall. The length of the base of ladder to the wall, the height of the ladder on the wall, and the length of the ladder form a right triangle. A question could be posed with any two of the three measurements given. Students should write their own problems of this type.
In geometry, the exploration of the proofs of the Pythagorean Theorem becomes more relevant than the basic calculations. We consider the areas a2, b2, and c2 in a proof. For instance, the square of side c can be shown inscribed in a larger square of side (a+b), forming four right triangles with base a and height b. A congruent square can then be created using a square of side b, a square of side a, and the same four right triangles rearranged. By discarding the four right triangles from each diagram we have the resulting squares that yield the Pythagorean Theorem. In coordinate geometry, the Pythagorean Theorem comes up again with the distance formula in the plane. And in three dimensional space, it can be shown that the distance formula for opposite vertices of a rectangular prism is really just an application of the Pythagorean Theorem twice: once using a right triangle with the length and width as bases, and a second time using the resulting hypotenuse and the height as two new bases. Another use of the Pythagorean Theorem within geometry is with the special right triangles – the relationships between the sides in a 45-45-90 triangle and in a 30-60-90 triangle. With the 45-45-90 triangle, a variable can be assigned to the base (the bases are congruent), and the Pythagorean Theorem can then be used to derive the hypotenuse (the product of square root two and the base). With the 30-60-90 triangle, a variable can be assigned to the shorter base, the hypotenuse is twice that variable, then the Pythagorean Theorem is applied to derive the longer base (the product of square root three and the shorter base).
This basic understanding of the hypotenuse becomes important again in geometry and trigonometry. On the unit circle, we can draw a right triangle from the origin, to (x, 0), and to (cos(x), sin(x)). This can be connected to the classic ladder problem where we now have a ladder of length one, distance from wall cos(x), and height sin(x). Now we can visualize the basic identity sin2(x) + cos2(x) = 1.
In Algebra II when complex numbers are studied, the absolute value of a complex number is introduced as its distance from the origin in the complex plane. This is yet another application of the Pythagorean Theorem. Suppose the complex number is in the first quadrant, we can then draw a right triangle which brings us back to the classic ladder visual: The base of the ladder to the wall is the real part, the height of the ladder on the wall is the real part, and the length of the ladder is the absolute value.
My response concerns two specific standards that I address in teaching 12th Grade Math.
http://.corestandards.org/Math/Content/HSG/GPE/A/3">CCSS.Math.Content.HSG-GPE.A.3 Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
Firstly, I address the misconception that “a” and “b” must always represent the legs of the right triangle. Students who memorize a2 + b2 = c2 have a hard time understanding how the theorem applies to an ellipse, when “a”, half of the major axis, is larger than “c”, the distance from the center to a focus. (In this context, b2 + c2 = a2.) This is a helpful lesson because students need to understand that variables like a, b, c, x, and y, due to the frequency with which they are used, are often interchangeable in different circumstances (like when the x-axis and y-axis go by different names).
http://.corestandards.org/Math/Content/HSF/BF/A/1">CCSS.Math.Content.HSF-BF.A.1 Write a function that describes a relationship between two quantities.
Another way in which I teach problem solving via the Pythagorean Theorem is by alluding to it as a “hidden piece of background info” useful in some related-rate problems. Students who can easily work implicit differentiation exercises in Calculus often struggle with the basic Algebraic and Geometric processes in related-rate word problems, simply because no one explicitly says, “Hey, use the Pythagorean Theorem here to help write your equation!” I encourage them to include such powerful theorems in their problem-solving “bag of tricks”.
The Pythagorean theorem discusses how the sides of a rightt triangle are related to each other and thus enables us to identify any unknown side in any right triangle, given that the other two sides are known. This topic is usually integrated in an Algebra course by replacing the unknown side with a variable, say x, and thus the Pythagorean theorem becomes a solvable quadratic equation.
Geometry comes in, as it provides visuals, to put some context to the Pythagorean theorem. Before the sides of the right triangle, say a for altitude, b for base and c for hypotenuse) are squared to fit into that Pythagorean theorem, we could remind the students that squaring the side could be visualized such that the side a is actually a side of another figure - in this case a square which is adjacent to the right triangle. The geometric interpretation of a squared is then the area of this square with side a, and the same process could be replicated for the two other sides (b and c) of the right triangle.
In terms of how this could look like in the classroom, we could provide the students cut outs to fill in the square of side a as well as the square of side b and then see that these pieces combined could completely fill in the square of side c. The geometrical representation aims to verify that the combined area of the two smaller squares is exactly identical to the area of the biggest square, or in essence the Pythagorean theorem.
As an extension, two or more sides of the right triangle could be expressed in a single variable and once substituted into the pythagorean theorem, results into a more complex quadratic equation of a single variable. depending on the complexity and ability of the students, this could even be extended further, for instance by using two variables, with the task of expressing one in terms of the other.
How this comes out geometrically is equally interesting and challenging in that by having the sides be expressed by a single variable and then plugging in to the Pythagorean theorem, we are able to determine conceptually some characteristics of the problem, such as how much wider the square sides are compared to each other or as a starting conversation to the idea of pythagorean triples.
Another extension is by verifying that the pythagorean theorem also paves way to another geometric discovery that the areas of any other figures formed having a, b and c as sides, for as long as all the three figures formed are all similar to each other, the area of the figure formed with side a, added to the area of the figure formed with side b will be exactly equal to the area of the figure formed with side c.
The Pythagorean Theorem shows up numerous times from its introduction in Grade 8 through the end of the high school curriculum, so it becomes easy to constantly refer back and apply the core concept of the theorem and continually build upon the foundation that was laid during its introduction. By continually referring back to this theorem over many topics during many years, you can succeed at showing students that this is a tool that is continually used over many different and seemingly unrelated topics. That’s really what coherence is about. The topics that we cover from grade to grade are related and intertwined in ways that aren’t always necessarily obvious when they’re introduced
For example, most of the problems in 8th grade deal with knowing two sides of a right triangle and using the theorem to find the third. This may also extend to using the formula to prove that a given triangle is a right triangle (the converse) and related application problems like finding the diagonal of a rectangle. These are direct uses where we’re presented with a triangle and its sides and have to apply the theorem.
In Geometry, the theorem shows up repeatedly in much less obvious ways because it is a tool instead of the actual subject of the problem. Here’s where coherence shows up! It is used in problems related to sine, cosine, and tangent. It is the entire basis of the distance formula, as you can show that (x2 – x1)^2 is really just a^2 in the Pythagorean Theorem and likewise for the other parts. You will use the Pythagorean Theorem to find the area of a regular polygon in certain circumstances. (For example, given a radius and a side length, the apothem is needed.) It can be used in many parallelogram problems to find altitudes and diagonals for measurement related problems and proofs that use these parts. It can be used to find missing information in cones and pyramids.
Continuing into Algebra, you can relate the Pythagorean Theorem to the Law of Cosines by showing that it is really just a generalized version of the formula. (The Law of Cosines is c^2 = a^2 + b^2 – 2ab cos C and in a right triangle, Cos 90 = 0 and the last term disappears, leaving you with the Pythagorean Theorem!) The Pythagorean Theorem can be used to derive the basic equation of a circle which is used heavily in Algebra and Geometry.
For this particular topic, coherence is not a difficult task because of the frequency in which it shows up. You will have many very natural and unforced opportunities to use the Pythagorean Theorem. The responsibility of the teacher will be to naturally weave this into their lesson during topics where it is less obvious (for example, Law of Cosines or the distance formula.)
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