Chapter 6

Spirolateral design of the word “mathematics”, made with segments of horizontal and vertical lines, both of which can be described as polynomials in two variables.

 

Linear Polynomials In Two Variables

 

 

While looking for a best fit line in the last chapter, the vertical distances between data points and points on the line, were referred to as errors, and were measured by their squares.  The distance between any two points, in any direction, can be thought of as the error between the two points, and can be measured by the squared error.

image of distance between any two points, in any direction, can be thought of as the error between the two points, and can be measured by the squared error.

The problem is, how do we find the area of a tilted squared?

For now, we’ll just look at the squared error between points, with integer coordinates. This way, the vertices of the tilted square will have integer coordinates as well.

  • Points with integer coordinates, are referred to as lattice points.
  • Shapes with vertices which are lattice points, are referred to as lattice polygons.

 

image depicting lattice points and lattice polygons

Below is a snapshot of an applet, which will challenge you to discover how to find the area of any lattice polygon.

snapshot of an applet, which will challenge you to discover how to find the area of any lattice polygon.

This computer shaped icon indicates the directions which follow, are part of this 8th grade course. Learners are expected to fulfill the requirements of those directions, and to sustain a determined effort over a period of time when necessary. Run the applet  “lattice.html”  until you find a pattern between the area of the polygon, and the number of inside points and boundary points.

The pencil shaped icon indicates the following directions involve recording and defending work and discoveries on paper, in order to fulfill the requirements of the previous directions (templates for doing this will be provided by the teacher). Use a copy of the pdf  appletnotes2g  to record examples and your discoveries.

The smiley face icon is a friendly stop sign, to remind the learner to defend their work, participate in mathematical discourse with at least one other student, keep the applet/program open, and then check with their math teacher before continuing After finding a formula for the area, based on the number of inside and boundary points, go on to the next page.

As we discovered, the number of inside points, plus half of the boundary points, minus one, gives us the area of any simple lattice polygon (one without holes or crossing edges) .  This is called Pick’s Theorem, named after Georg Pick.

Pick’s Theorem:  The area of a simple lattice polygon is given by the formula  i + b/2 – 1, where i represents the number of inside points, and b represents the number of boundary points.

snapshots of an applet, which uses Pick’s Theorem to determine the area of the square, which is then used to determine the length of the blue line segment.To the right, are snapshots of an applet, which uses Pick’s Theorem to determine the area of the square, which is then used to determine the length of the blue line segment.

computer shaped icon Run the applet  “distance.html”, choose a line segment, and determine its length using Pick’s Theorem.

smiley face icon After making the word “Correct” appear, for an example different from the one above, go on to the next page.

 

Pick’s Theorem only works for lattice polygons, or polygons which have vertices with integer coordinates.  This means Pick’s Theorem will only help us find the distance between two points, if the two points have integer coordinates as well.

snapshot of an applet, which will allow you to discover another way of determining the squared distance between any two points.So we still need to discover a way of determining the area of the squared distance between two points with noninteger coordinates.

Since all points have both x and y coordinates, we can talk about the horizontal and vertical errors between any two points.

To the right, is a snapshot of an applet, which will allow you to discover another way of determining the squared distance between any two points.

computer shaped icon Run the applet  “shortcut.html”  , until you discover another way of finding the squared distance between two points.

pencil shaped icon Use a copy of the pdf appletnotes2g  to record an example and your discoveries.

smiley face icon After finding a way of determining the squared distance, without using Pick’s Theorem, go on to the next page.

image depicting Pythagorean Theorem  A2 + B2 = C2As we discovered, between two points, the horizontal distance squared, plus the vertical distance squared, is the same as the actual distance squared.

In other words, all we have to do is use the Pythagorean Theorem  A2 + B2 = Cto determine the actual distance squared, where “A” is the horizontal distance, “B” is the vertical distance, and “C” is the actual distance.

 

Below is an example of a “proof without words”, for the Pythagorean Theorem.

example of a “proof without words”, for the Pythagorean Theorem.

computer shaped icon pencil shaped iconUse a copy of the pdf  booknotes2g, to fill in the words for the above proof of the Pythagorean Theorem.

smiley face icon After defending how one of the above pictures, or how both of the above pictures together, proves the Pythagorean Theorem  A2 + B2 = C, go on to the next page.

Since finding the distance between two points is just an application of the Pythagorean Theorem, all we need are the coordinates for the two points to determine the horizontal and vertical distances between them.

Below are snapshots of an applet, which will let you discover how to determine the distance between two points, without using lattice points.  To truncate to the 100ths place, means to shorten the decimal to the 100ths place, without rounding.

snapshots of an applet, which will let you discover how to determine the distance between two points, without using lattice points.

computer shaped icon Run the applet  “distanceformula.html”  , until you can find the distance between two points, without using the lattice.

pencil shaped icon Use a copy of the pdf  appletnotes to record an example and your discoveries.

smiley face icon After making the word “Correct” appear, and recording the picture and work for that example, go on to the next page.

As we discovered, the horizontal and vertical distances between two points, are the positive differences between the x coordinates of each point, and the y coordinates of each point.

image showing that he horizontal and vertical distances between two points, are the positive differences between the x coordinates of each point, and the y coordinates of each point.

Since the horizontal and vertical distances end up being squared, finding the negative difference between the x coordinates and/or the negative difference between the y coordinates, will also work (even though technically, distances can’t be negative). In other words, finding the change in y and the change in x, as we would for determining slope, is good enough.

Since the Pythagorean Theorem gives the square of the actual distance between two points, we eventually have to take a square root.  This is why we will usually see the distance formula written as follows.

distance formula written

Since all we need are the coordinates of two points to determine the distance between them, the distance formula will still work, even if the coordinates don’t have integer values. Below are snapshots of a program, which will randomly generate 5 line segments. The challenge is to determine their lengths (truncated, or shortened, to the 10ths place).

 snapshots of a program, which will randomly generate 5 line segments.

computer shaped icon Run  program 601  , until you correctly determine the lengths of at least 3 of the 5 line segments.

pencil shaped icon Use a copy of the pdf  programnotes  to record the examples and your work.

smiley face icon After correctly finding the lengths of at least 3 of the line segments, and defending your work, go on to the next page.

Using the Pythagorean Theorem will show, the point ( 6 , 8 ) is 10 units away from the origin.  The point ( 6 , 8 ) isn’t the only point which is 10 units away from the origin however.

Below are snapshots of an applet, which will allow you to find other points which are 10 units away.

snapshots of an applet, which will allow you to find other points which are 10 units away.

computer shaped icon Run the applet  “tracer.html”  , until you discover something about all of the points 10 units away from the origin.

pencil shaped icon Use a copy of the pdf  appletnotes2g to record your discoveries.

smiley face icon After defending and recording your discoveries, go on to the next page.

 

The Pythagorean Theorem can be used to determine the distance of a point from the origin. As we discovered, if we consider all of the points which are the same distance away from the origin, the Pythagorean Theorem relates the horizontal and vertical distances of the points on a circle with the radius. In other words, circles are graphs of quadratic polynomials in two variables.

circles are graphs of quadratic polynomials in two variables.

The graphs of quadratic polynomials in one variable are always parabolas (smilelike or frownlike graphs). As we just discovered, a quadratic polynomial in two variables, can have a circle for a graph.

Below is a snapshot of an applet, which will allow you to explore the graphs of quadratic polynomials in two variables.

snapshot of an applet, which will allow you to explore the graphs of quadratic polynomials in two variables.

computer shaped icon Run the applet  “quadsin2vars.html”  , and record your discoveries about graphs of quadratics in two variables.

pencil shaped icon Use a copy of the pdf  appletnotes2g  to record sketches and polynomials, and your discoveries.

smiley face icon After recording and defending your discoveries about quadratic polynomials in two variables, go on to the next page.

 

As we discovered, quadratic polynomials in two variables, can have different types of graphs. They are also referred to as conics or conic sections, due to the shapes of cross sections of hollow right circular cones (the shapes created where the cones and the planes intersect each other).

examples of conics or conic sections

image 1: horizontal cross sections create circles
image 2: steeper cross sections create ellipses

examples of conics or conic sections part two

image 3: cross sections even with the slant create parabolas
image 4: steeper to vertical cross sections create hyperbolas

A book shaped icon that indicates the following directions are opportunities for further information, exploration, or learning. These opportunities are optional, unless the learner finds them helpful, interesting, and/or necessary for algebra credit. To further explore the above examples, by rotating and/or modifying them, open the Grapher file  “conics.gcx”  .

book shaped icon To see the above shapes as the cross sections of cone shaped shadows, open the pdf conic shadows.

While we were exploring quadratic polynomials in two variables, if we had chosen a coefficient of zero for each of the three degree 2 terms, x2, y2, and xy, we would have seen a line. This is because, with no degree 2 terms, we’re left with a linear polynomial in two variables, of the form Ax + By = C.

Below are snapshots of an applet, which will allow you to explore the graphs of linear polynomials in two variables.

snapshots of an applet, which will allow you to explore the graphs of linear polynomials in two variables.

computer shaped icon Run the applet  “linearin2vars.html”  , until you see a connection between the polynomial and the intercepts.

pencil shaped icon Use a copy of the pdf  appletnotes2g  to record examples and your discoveries.

smiley face icon After recording and defending the connection between the polynomial and the intercepts, go on to the next page.

As we discovered, whether in one variable or in two variables, the graphs of linear polynomials are always lines (as the name linear suggests).  Instead of using the y-intercept and the slope to uniquely describe a line, linear polynomials in two variables use both the y-intercept and the x-intercept to uniquely describe a line.  A linear polynomial in two variables of the form  Ax + By = C, is referred to as the standard form of a line (where “A” is nonnegative).

Below are snapshots of a program, which will randomly generate 10 linear polynomials in standard form.  The challenge is to determine the x- and y-intercepts.

snapshots of a program, which will randomly generate 10 linear polynomials in standard form.

computer shaped icon Run  program 602  until you correctly determine the x- and y-intercepts, for at least 8 of the challenges.

pencil shaped icon Use a copy of the pdf  programnotes to record the polynomials, the intercepts, and any discoveries.

smiley face icon After correctly determining the x- and y-intercepts for at least 8 of the 10 challenges, go on to the next page.

As we discovered, to find the x-intercept, the y-value has to be zero (otherwise, the point wouldn’t be on the x-axis), and to find the y-intercept, the x-value has to be zero (otherwise, the point wouldn’t be on the y-axis).

Example of a linear polynomial in standard form

In other words, we just figure out, 2 times what equals -18, and -3 times what equals -18. We could also divide by -18 to find the reciprocals of the intercepts instead (if -3 • 6 = -18, then -3 is 1/6 of -18). This would turn the linear polynomial into intercept form, or  x/a + y/b = 1 form, where “a” is the x-intercept, and “b” is the y-intercept (which we know, is just an equivalent equation after a reflection about the x-axis and scaling vertically).

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Below are snapshots of a program, which will randomly generate 7 lines for you. The challenge, is to describe the line in standard, or  Ax + By = C  form.

snapshots of a program, which will randomly generate 7 lines for you.

computer shaped icon Run  program 603  , until you correctly describe at least 5 of the lines, in standard form.

pencil shaped icon Use a copy of the pdf  programnotes2g to record your work and any discoveries.

smiley face icon After correctly describing at least 5 of the 7 lines, in standard form, go on to the next page.

 

Depending on the situation, there are advantages and disadvantages to describing lines as polynomials in one variable versus two variables.  For instance, when knowing the slope is important, describing the line as a polynomial in one variable, or in slope-intercept form, is more helpful.  On the other hand, vertical lines can only be described by linear polynomials in two variables, or in standard form (the slope would be undefined otherwise).  In slope-intercept form, it’s easy to plot as many points as we need when graphing.  In standard form though, we don’t have to work with fractions or decimals.  As we will soon discover, It’s helpful to be able to switch from one form to the other when needed.

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Below are the steps for converting standard form, or Ax + By = C form, into slope-intercept form, or y = mx + b form.

here are the steps for converting standard form, or Ax + By = C form, into slope-intercept form, or y = mx + b form.

Below are the steps for converting slope-intercept form, or y = mx + B form, into standard form, or Ax + By = C form.

here are steps for converting slope-intercept form, or y = mx + B form, into standard form, or Ax + By = C form.

Having to go through all of the geometric steps on the previous two pages to convert forms of a line, isn’t practical. Memorizing the resulting formulas at the bottom of the previous two pages, isn’t practical or helpful either. Since all of the geometric steps are vertical or horizontal transformations, we can simply convert directly, algebraically.

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Below are snapshots of a program, which will randomly generate 12 linear polynomials for you to convert.

snapshots of a program, which will randomly generate 12 linear polynomials for you to convert.

computer shaped icon Run  program 604  , until you correctly convert at least 10 of the 12 linear polynomials.

pencil shaped icon Use a copy of the pdf  programnotes to record your work and any discoveries.

smiley face icon After correctly converting at least 10 of the linear polynomials, go on to the next page.

 

We now know three ways to write linear polynomials, one way in one variable, and two ways in two variables. Below are snapshots of an applet, which will allow you to see the similarities and differences between all three ways.

snapshots of an applet, which will allow you to see the similarities and differences between all three ways.

computer shaped icon Run the applet  “stringart.html”  , until you are familiar with the similarities and the differences between the 3 forms.

pencil shaped icon Use a copy of the pdf  appletnotes  to summarize your observations.

In case you were wondering, the envelope formed by the lines (the shape tangent to the lines), is a parabola. Any two line segments with the same number of equally spaced tick marks, will produce a “string art” parabola.

book shaped icon To explore “string art” parabolas, run  program 605  .

smiley face icon After summarizing the similarities and differences between the above three forms of a line, go on to the next page.

If we picture the lines in the “string art” creations, as creases, it’s not too surprising to find out a parabola can be created by folding paper. Below is a snapshot of an applet, which will explain the steps for creating a paper folded parabola.

snapshot of an applet, which will explain the steps for creating a paper folded parabola. computer shaped icon Run the applet  “paperparabola.html”  , and make a paper folded parabola.

book shaped icon If you thought the picture at the beginning of the chapter was interesting, run  program 608  .

program 608

Below is a snapshot of the chapter 6 programs menu program.

snapshot of the chapter 6 programs menu program.

book shaped icon Run the program  “6menu.bas”   , or its alias to access every program used in chapter 6.

 

 

 

 

 

 

 

 

“I often admired the mystical way of Pythagoras, and the secret magic of numbers” — Sir Thomas Browne —

“Mathematics rightly viewed possesses not only truth, but supreme beauty” — Bertrand Russell —

“We cannot hope that many children will learn mathematics unless we find a way to share our enjoyment and show them its beauty as well as its utility.” — Mary Beth Ruskai —

“Might is geometry; joined with art, resistless” — Euripides —

“It is here [in mathematics] that the artist has the fullest scope of his imagination” — Havelock Ellis —

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