High School Mathematics-Physics SMILE Meeting
1997-2006 Academic Years
09 December 1997: Bill Lilly [Kenwood HS]
He showed a new calculator-based-ranger [TI-CBR], which is used with TI-82 calculators to detect motion and measure distance. It can be incorporated into programs on the calculator for automatic data-taking. The device represents an improvement on the older CBL devices. He talked about an application of measuring a motion of a pendulum. The minimum distance for detection is about 0.5 meter. We ran out of time during Bill's presentation, and we hope he can return to show it next year.
07 April 1998 Bill Lilly [Kenwood High School; Math Dept]
He demonstrated the capabilities of the TI-CBR Calculator-Based Ranger, [http://www.calculatorsinc.com/ti-cbr.aspx] which is a sonic motion detector that provides information in digital and easily convertible form. The device is made by Texas Instruments for use with their calculators TI-82, 83, 85, 86, or 92. The price is $90, and seems to be dropping with time. The "innards" of the device are the same as a Polaroid auto-focus system. The apparatus was demonstrated by hanging a bottle on a swing and swinging it back and forth as a pendulum. He was able to obtain [interpolated] numerical values of x, v, and a at time intervals of about 0.1 seconds, so that motions could be studied quantitatively, and graphs easily generated.
02 March 1999: Fred Schaal [Lane Tech HS Math]
He investigated determinants and the magic squares. Using the TI graphing calculator's internal ability to solve determinants he put in a 5x5 array and showed some to the abilities of the TI graphing calculator. Fred tried the numbers forming a "golden square" as well as a Fibonacci sequence, and in neither case did the determinant vanish.
Porter commented that the result was not what Fred expected because the elements did not have the proper relationship where an element or the ray wasn't a factor of the other numbers. Determinants can be thought of in geometric terms, where the points can be identified as vectors, and the vectors are added. there can be a solution when they reside in the same plane. You can write two equations in two unknowns in the form
A1 · X = B1
A2 · X = B2
The object is to determine the two dimensional vector X when its scalar products with the vectors A1 and A2 are given. There is no solution when A1 and A2 are parallel to one another, or when the determinant of coefficients vanishes. So it goes...
20 April 1999: FJ Schaal [Hi Tech Low Tech Lane Tech High School]
He first placed a grid on the board by using the chart that has the holes to allow chalk dust to produce a grid. He placed points (2,13) and (16,3) on the grid and used the formula
He explained the steps and keys to press on the calculator: ( ( 2 -16 ) ( 2 + ( 13 -3 ) ( 2 ) ( = distance. Now, adding an additional point of (-12,-6), using the above formula we get distances:
Hero [HERON] formula for the area of a triangle of sides a, b, c:
This gives 224 [exactly!!], whereas by graphing and counting the number of fully enclosed squares plus 1/2 the number of partially enclosed squares we get 219.
04 May 1999: Porter Johnson [IIT]
Porter took off on the use of the calculator and commented about not rounding off! By using integers and keeping the calculation to as many places as possible we may get an answer that will result in integers. If you round off the answer will be an decimal, and hide the fact that the answer from a problem starting with integers will be integers. You may not want to round the working numbers in a calculators to a few significant places and thus hide some of significance of the answer. The example consisted in calculating the area of a triangle with vertices at integer-valued x- and y-coordinates. The area of the triangle, which will come out as either an integer or half-integer, can be computed geometrically by enclosing a rectangle around the triangle, and calculating the area of the rectangle and the other right triangular pieces. Simple, non??
28 September 1999: Fred Schaal (Lane Tech)
Fred drew our attention to inequalities. For example, for a straight line y = m x + b, we might have y = (1/2) x + 2. On a graph of y vs x, below the line y < (1/2) x + 2, and above the line y > (1/2) x + 2. Our hand-held calculators seem not to enable us to display a band of (x, y) pairs that lie between a minimum and maximum set of values. Is there any way to do such a thing on any hand-held calculator? Interesting question!
12 October 1999: Walter Macdonald [CPS Substitute; Medical
He demonstrated the versatility of the HP48GX Programmable Calculator [http://www.johann-sandra.com/surveying/hp-48gx-49-33s-calculator.htm], with 128K of internal RAM [random access memory] connection with the software package SPICE48, which allows one to program circuit diagrams [although somewhat tediously, it seems] and to specify the voltage source as DC, sinusoidal AC, sawtooth, or "whatever". The calculator permits display of output currents and such in graphical form, in order to gain insight into the effects of changing resistances, capacitances, and inductances, or switching in diodes. The device is better suited to providing its gentle user with graphical insight, rather than mere numbers.
Walter also commented that the new x-ray machine in his hospital has a digital interface, so that familiarity with PCs/Macs is required in routine usage. Porter observed that the world is "going digital", giving these instances:
09 November 1999: Bill Colson (Morgan Park HS - Math)
drew 2 lines that intersect to form an angle. In his geometry class Bill uses a book titled Line Design. Some students have difficulty understanding what an angle is, and by getting them involved in such constructions using straight lines, they develop a feeling for the idea of angle. Example: start with two straight lines intersecting at 90o, and mark off equal divisions beginning at the intersection. Draw two additional lines connecting the divisions (1 & 2) and (2 & 1). The two lines intersect at one point. Now repeat this on a new drawing, making 4 lines connecting (4 & 1), (3 & 2), (2 & 3) and (1 & 4). These 4 lines intersect at 3 points. By repeating this process next for divisions 6, 5, 4, 3, 2,1 - it becomes quite clear that the intersecting straight lines come to more and more closely define a curved line. And the idea of approaching something as the limit of a process repeated an "infinite" number of times becomes an experience that is useful in more advanced math. Bill passed around examples of beautiful art work his students made using these ideas. Very nice!
29 February 2000: Fred Schaal (Lane Tech HS)
showed us classroom use of plotting calculators, and he used one to project to the screen so we could see what was going on. Betty Roombos followed Fred's directions to do this, and it worked well. Fred had her (and others with calculators he had passed out) plot the equations of straight lines and find their intercepts with the x and y axes. He used number pairs to determine a set of lines:
|10||1||y = -(10/1)x+10|
|9||2||y = -(9/2)x+9|
|8||3||y = -(8/3)x+8|
|7||4||y = -(7/4)x+7|
|6||5||y = -(6/5)x+6|
|5||6||y = -(5/6)x+5|
|4||7||y = -(4/7)x+4|
|3||8||y = -(3/8)x+3|
|2||9||y = -(2/9)x+2|
|1||10||y = -(1/10)x+1|
The intersections of successive pairs of lines were determined by using the plotting computers, and one could even zoom up close to the intersections to see more detail and better determine the actual number pair locating each intersection. He then used the intersection points as data, and did a linear, quadratic, and cubic fit to those points on a calculator. He compared the results with the "tangent curve" described below by Porter Johnson. A fine phenomenological math lesson!
29 February 2000:
Comments by Porter Johnson (IIT)
The formula for line segment lying in first quadrant and intercepting the x-axis at x = a and intercepting the y-axis at y = b - a, where 0 < a < b, is
Let us regard the parameter b as being fixed, while a varies continuously between the values 0 and b. A solid region in the first quadrant is filled by these lines. To determine the boundary of that region, we solve the above relation for y, to obtain
In this relation, keep x fixed, and vary a. The maximum value of y under such variation can be calculated by setting the derivative dy/da to zero:
We substitute this value of a into the expression for y(a) to obtain
To show that this is indeed the maximum value of y(a), calculate the quantity ymax - y(a):
Clearly, the right side is non-negative, and it is zero when a is chosen as above. The formula for ymax gives the greatest value that y can have for a given value of x, and thus lies at the top of the region traced out by the straight lines described above. Thus, the top of the region is given by
This curve, which represents the envelope of all the straight lines, can also be written as
The curve is a parabola with the line of symmetry [axis] lying along the line x=y. It is symmetric under interchange of the variables x and y. The point closest to the origin has coordinates
This is the "symmetry point of the parabola", and is the "tangent point" for the curve with a = b/2, namely
Every other point on this curve is a "tangent point" for one and only one of the straight lines described above, which have intercepts a and b-a, respectively.
Here is an Excel-generated image of the lines and the asymptotic curve:
Incidentally, these "envelope curves" occur frequently in geometrical optics, in which light rays move in straight lines in a uniform medium. Clearly, the "bundle" formed by all light rays can have a nontrivial structure. The boundary of that "bundle" is called a caustic in geometrical optics. As an example, see http://www.math.harvard.edu/archive/21a_spring_06/exhibits/coffeecup/index.html The Coffeecup Caustic. Here is a brief description of the effect, taken from that reference.
You are drinking form a cylindrical cup in the sunshine. Sometimes, when the sun shines into the cup, you can see a crescent of light as the sunshine reflects from the inside of the cup onto the surface of the drink. A picture of a real cup is shown, and you can do your own on-line computer simulations of the effect. Check it out!
14 March 2000: Porter Johnson (IIT Physics)
explained Coffee Cup Caustics to us with some mathematical detail, but nicely presented.
Coffee Cup Caustics
The Coffee Cup Caustic is shown on the website http://www.sciencedaily.com/releases/2009/04/090414160801.htm, on which the reflected image from a real coffee cup is shown, as well as a "Monte Carlo" simulation of the event.
Let us suppose that rays parallel to the y-axis strike an upper semi-circle of radius R from the inside, and are reflected. If we let the angle between the reflection point be q, the coordinates of the point of reflection are x = R cos q and y = R sin q. The angle of incidence of the ray, relative to the normal, is p/2 - q, and the angle of reflection has that same value, as well.
The reflected ray travels at an angle p/2+2
q relative to the positive x-axis, and
strikes the circle again at a point an angle
3 q from the horizontal, at the point
(x = R cos3 q,
y = R sin 3 q). The equation for
the reflected ray is
A template of a 360o Protractor was handed out, and participants made their own caustic by drawing lines from the positive x-axis [right on the middle] from angles q to 3 q. Here is a set of angles relative to the horizontal to use:
The right side of the caustic will arise, as if per magic!
The full curve, which was produced using the software package EXCEL, is given below. An excellent reference on using EXCEL in graphics is the book EXCEL for Scientists and Engineers by William J Orvis, Second Edition [ISBN 0-7821-1761-9].
01 February 2000: Fred Schaal (Lane Tech HS)
invited a number of us up front, gave us each a piece of polar graph paper (paper with a set of concentric circles on it, and radii drawn in every 10o). He had each of us cut out a segment of angle, one of us 10o, next 20o, then 30o, etc. We then taped the edges together to form a set of cones. Fred proceeded to lay out a set of equations for the lateral area (LA) of a cone, and ended up with a result that the altitude of a constructed cone lying on the table is:
We used sticks of spaghetti to poke down through our cones' tops to the table below, marked the stick, and got an experimental value of h. And then we calculated h from the above expression. Both observed and calculated values matched surprisingly well! A beautiful phenomenological math lesson! Thanks, Fred!
28 March 2000: Fred Schaal (Lane Tech HS)
Fred brought us back to something he had done with us earlier: The truncated cone. Start with a circle of radius R cut from paper. Draw two radii on the circle from its center, and so define an angle of 360o - ko between them. Cut out the sector between them, as shown:
s = 2 p r = p [k/180] R
Tape the radii together, to form a truncated cone surface with the paper. Let r be the radius of base of the cone, and h its altitude. Then
h = (R2 - r2)1/2 = R * [ 1 - (ko/360o)2]1/2.
= p R3/3 * [ko/360o]2 * [ 1 - (ko/360o)2 ]1/2
"What value should the angle ko have in order for volume V to be a maximum?" was the question Fred asked us. But lacking for any volunteers, Fred left us with the unanswered question. Maybe next time somebody will have an a solution?
02 May 2000: Porter Johnson
If you take a sheet of paper with a circle of radius R, and cut out a sector of opening angle ko, the volume of the cone formed by that sheet is
The formula may be simplifying by defining the ratio x = ko/360o, so that for radius R = 1
One may show by elementary calculus that the volume is maximized by the choice
corresponding to angle k = 293.9388o. The maximum volume is
Here is a set of steps that will result in the data table and the graph: First, type these descriptive words in lines #2 and #3
At this point you should save the EXCEL table [after all, you've worked hard to create it!].
The next objective is to make a graph of the volume versus angle using EXCEL. To do so, complete the following steps: EXCEL
02 May 2000: Fred Schaal (Lane Tech HS)
did the Law of Sines with us as an exploration by hand-held calculator. He drew a triangle on the board and identified its three angles as A, B, C, and the lengths of the sides opposite as a, b, c, as shown:
Choosing values for A, and keeping c and B fixed but letting the side a increase with A, he obtained the following table of values, which he wrote on the board:
|A||a / sin A|
Triangles were actually drawn for the various values of A, and a was then measured, and the above ratios were calculated based on these experimental measurements. Fred noted that the ratio does not change much, and wondered at this.
Fred set up the calculation of the curve for x(T), y(T), where
He displayed the result as a set of points with the aid of a TI graphing calculator, a transparency projector and LCD display. The graph looked like a roll of carpeting viewed end-on, a kind of spiral around the origin. Most interesting! A good example of how to connect the abstract to the concrete so as to make math "real" for students!
02 May 2000: Bill Lilly (Kenwood HS)
gave us a handout titled "Pendulum," which was a lab exercise. Using a TI-CBR Sonic Ranger, high frequency sound from the Ranger was reflected from a swinging balloon [held on the end of string by Pearline Scott (Franklin School)]. The output from Ranger (for 10 s) was stored in the calculator and shown on the projector screen as a graph with the aid of Fred Schaal's LCD display. It looked like a sine wave, and Bill described how students could analyze the data for frequency, effect of shortening string, lengthening string, amplitude effects, velocity minima and maxima with position, etc. Another beautiful example of concrete/abstract connections!
26 September 2000 Fred Schaal (Lane Tech HS)
presented "92s to the Rescue." He gave one hand-held TI 92 to Betty Roombos (Lane Tech HS) to enter commands and data. The LCD display of the TI 92 was projected onto the screen for all to see. Fred placed on the board: a/sinA = ? - referring to the Law of Sines. He instructed Betty how to draw a triangle and label its vertices with angles A, B, C, and the sides opposite with a, b, c. Betty was a whiz, and with Fred's further direction she soon had the angle A labeled with its value of 41.99o, and the side a = 2.28 cm, taken as a "measured" value. The ratio R = a/sinA was calculated as 3.41 cm. Then, as Betty dragged the top vertex, C, around the screen of the TI 92 (as one does with a mouse on a PC), the triangle changed its linear and angular dimensions, and the changing value of R was seen to range from 3.10 to 4.77 and higher. When the sizes of angles were measured to be in radians instead of degrees, the range of values of R continued to be the same. Beautiful! What a great way for students to develop an intuition for the Law of Sines and what it tells us about the geometry of plane triangles. Thanks, Fred, for phenomenological math!
26 September 2000 Fred Farnell (Lane Tech HS)
walked carefully across the front of the room in front of us and asked,
Next, Fred repeated the experiment, but this time ran away
the CBR. The D vs t graph of his motion appeared
of constant acceleration motion. When he found the best fit to
the parabolic equation,
10 October 2000 Fred Farnell (Lane Tech HS)
explained why the fit of a straight line equation to experimental constant velocity points (and its subsequent v vs t graph) at the last meeting was so poor. It turned out that the constant (non zero) velocity occurred over an 8 second period, but the data taking ran for 15 seconds, and the velocity was zero for the last 7 seconds! When Fred ran new data and kept only the portion for non-zero velocity, the fit turned out great! He did this "live". Same story for motion of constant acceleration. Thanks for restoring our faith, Fred!
24 October 2000 Fred Schaal (Lane Tech HS)
set up the projector to display the output of a TI 92 calculator so we all could see it, and gave the TI 92 to Betty Roombos to manipulate. Fred asked Betty to draw a triangle and label its vertices with A, B, C and we saw it as it happened. Then he advised Betty what to do to construct the bisectors of the angles at the vertices A and B. The angle bisectors intersected at a point which was labeled F. Next, he had Betty construct a ray connecting vertex C with the point of intersection, F. The angles ACF and BCF were "measured" by manipulation of the TI 92 and labeled with their values, which both happened to be 37.05 o! This showed that the constructed ray was a bisector of the angle at vertex C, and was convincing evidence that the bisectors of all three angles intersect in a common point! Wow!
Then Fred asked Betty to "grab" the vertex point C and move it around. As this was done, we could see the shape of the triangle ABC change, and the values of the two angles also changed, but remained equal. Very nice! Similar exercises may be done to show that medians meet at a point; perpendicular bisectors of sides meet at a point. Porter Johnson asked: What about the "Nine Point Circle Theorem? Anyone know? Check the website http://en.wikipedia.org/wiki/Nine-point_circ.
A perfect complement to what you did last time, Fred! (...the actual construction on the white board, which did not display it well.)
07 November 2000 Fred Schaal (Lane Tech HS)
sketched out on the board a kind of geometry problem he will do next time using the TI 92 calculator. He drew two straight lines (representing mirrors), intersecting at some angle. Then he drew a triangle with vertex points A, B, C and sketched its reflection across the nearest of the mirrors. He pointed out that the order of the points, A, B, C was ccw (counter clockwise) for the triangle, but cw (clockwise) for the vertex points - A' , B' , C' - of its reflection. Sketching the reflection of the those points across the second mirror (straight line) located vertex points A", B", C" - which followed a ccw order. It could be interesting to see how Fred will do this with the TI 92. But this provoked discussion about reflections in mirrors, and brought out that it takes two mirrors to see your reflection as others see you!
21 November 2000 Fred Schaal [Lane Tech HS]
displayed the output of his TI 92 Calculator on the screen. He chose the "geometry" option and drew a triangle. Then he added a line, and reflected the triangle about the line. He noted that the second triangle could not be superposed on the first one, because it was "left-handed". Next, he added another [non-parallel] line, and made reflection of the second triangle. He showed that the third triangle could be superposed by rotation onto the first one. Thus, two reflections correspond to a spatial rotation.
10 December 2002: Fred Schaal [Lane Tech HS,
Mathematics] TI-92 Graphics Calculator Attacks
Fred used the TI-92 calculator to illustrate basic geometrical ideas; specifically, general properties of triangles. First he hooked a calculator to a translucent imaging screen placed on the overhead projector, so that the entire class (we) could see the calculator display. Then he gave the calculator in question to one of the participants, who used the menu to bring up "geometry", and then "insert triangle". The participant then clicked on three vertex points in succession, and the triangle appeared. Fred then showed her how to shift one of the vertices of the triangle by using click and drag. Then she clicked on an icon to construct the altitudes of the triangle, which were seen to intersect at a point, the ortho-center of the triangle. When she moved a vertex, the altitude lines and ortho-center changed continually. Then she clicked on another icon to obtain perpendicular bisectors of each side, which intersect at a point, the circum-center of the triangle. Again, as a vertex was moved, the circum-center also changed. Interestingly, neither the ortho-center nor the circum-center must always lie inside the triangle. These calculators make geometry fun, Fred!
25 February 2003: Walter McDonald [CPS Substitute -- VA Hospital
Approximation of Functions
Walter showed us how to do Taylor Series polynomial expansions of trigonometric functions on his HP 48GX Programmable Calculator. The idea is to truncate the expansion of, say, sin x in powers of x after a few terms, and to plot the results on the calculator. The expansion:
11 March 2003: Fred Schaal [Lane Tech HS,
TI Interactive Software
Fred had intended to transfer images from his laptop computer to the screen in order to display TI Interactive Software, but discovered that, without a projector, there was no hope of doing so. Such transfers of images from one type of system to another are always problematical, and the safest step is to use your own system for the entire process. Thanks for trying, Fred!
25 March 2003: Fred J Schaal [Lane Tech HS,
Capers: TI Interactive
Fred showed the TI Interactive Software, which emulates a TI-83 Calculator on a Windows-based PC. With the assistance of Monica Seelman, Fred went into the graphics mode, and first displayed a graph of the function y = x2 - 8, using the ranges (-10 £ x , y £ +10). A very nice upward parabolic curve appeared, which left the region of graphing at x » ± 4.2. Next Fred plotted the graph of y = 8 - x2, which corresponded to the same parabolic curve, but inverted. The two curves intersected at the line y = 0, for x = ± Ö 8 » ± 2.8. Finally, Fred asked for the area between the two curves, which can be formally written as the integral
Very visual and user-friendly, Fred!
08 April 2003: Fred Schaal [Lane Tech HS,
Green Line '92-ing!
Fred had been playing around with his calculator while riding the el, and began studying the following problem: For a given triangle with vertices (A, B, C), draw the following lines:
Fred, you have constructed an unusual problem! Very Good!
07 October 2003: Fred Schaal [Lane Tech HS,
Lubbock*** [TI-83] Helper
Fred used his calculator and plotter to extend his lesson of 29 February 2000 concerning tangent lines and asymptotic curves. This lesson had, in turn, been an extension of a lesson by Bill Colson on 11 November 1999 on producing intricate geometrical patterns. These comments by Porter Johnson are excerpted from the first lesson:
The formula for line segment lying in first quadrant and intercepting the x-axis at x = a and intercepting the y-axis at y = b - a, where 0 < a < b, is
Fred's trusty programmer, Bill Colson, developed and displayed the graphs for xp, where the real, non-integer variable p increases from 1 to 2. The curves all passed through x = 1, but became steeper below that point for x < 1 as p was increased. For x > 1, it was the other way around. Fred showed the graphs for negative x as well. Surprisingly, for p = 1.1, 1.3, 1.5, 1.7, and 1.9 the curves were negative, whereas for p = 1.2, 1.4, 1.6, and 1.8 they were positive. How come?
Comments by Porter Johnson: For negative x, there is a real, positive value of p whenever p is a rational fraction of the form a / b, where the integer b is even. Furthermore, there is a real, negative value of xp when p=a / b is rational, when b is odd. The gremlins and lubbockians*** of calculator programming were definitely on your side for this one, Fred.
Q: How many Texas Tech football supporters does it take to change a light bulb?Bigger plots are definitely better! Good, Fred!
A: Don't be silly; there's no electricity in Lubbock*** [home of Texas Instruments™]!
21 October 2003: Babatunde Taiwo [Dunbar Vocational HS,
Graphics Calculator and Motion Sensor
Babatunde showed that his TI-83 graphing calculator, CBL Ranger motion sensor, and Vernier Logger Pro [http://www.vernier.com/soft/lp.html] graphical interface could be used to display graphs of idealized motion, as well as record the motion of moving objects. First he displayed a distance-time graph that looked like the one shown here, with distance marks in meters and time marks in seconds.
He set up the motion sensor and software so that it would record the position of an person walking across the room. The objective was to reproduce the graph given above with one's own motion --- by being at the right place at the right time. After three trials, Babatunde was able to reproduce the curve pretty well. Walter MacDonald then tried it with this curve, as well as the curve given below. He was able to reproduce the curves on his first attempts.
Babatunde then showed the curves of velocity and acceleration versus time, as determined numerically from the positions. He then pointed the motion sensor downwards, dropped a ball from a height of about a meter, and used the motion sensor to record its vertical position versus time. He was able to follow the ball through several bounces. Then he showed the graphs of velocity and acceleration versus time. The acceleration was negative, and of modest relative size, when the ball was in flight, whereas when the ball struck the floor it became quite large and positive. [Because of the intrinsic time resolution of the apparatus, we were seeing some time averaged acceleration of the ball, rather than its instantaneous acceleration.] These are very visual exercises in the kinematics of motion, which allow us to experience the direct connection of motion with distance-time graphs. Thank you for showing this to us, Babatunde!
For additional information see the websites: Activity Investigating Motion: http://www.math.mtu.edu/gk-12/investigatingmotion.html and Utilizing the Graphing Calculator in the Secondary Mathematics and Science Classroom: http://www.esc4.net/math/.
Babatunde also showed a spinning top toy that he had recently obtained at American Science & Surplus [ http://www.sciplus.com/category.cfm?category=13]:
Here, Kitty Kitty ... A top that’s fun! A top that doesn’t need string or practice or coordination. Twist the winder, push the button, and the colorful plastic top bounces around on its spring bottom, lights up brightly (batteries included), and spins for a very long time. A nice distraction for a jaded child or a bored tabby cat
18 November 2003: Fred J Schaal [Lane Tech HS,
STO FRM Not That Bad!
Fred asked if anybody knows the standard "slope-intercept" equation for a straight line. Somebody suggested ...
Bill Shanks mentioned that an elliptical curve with semi-major axes (a,b) can be written in a similar form:
Thanks for showing us the neat Algebra, Fred!
04 May 2004: Fred Schaal [Lane Tech HS,
Fred reminded us of a universal feature of programming algorithms on digital computers, in that they perform arithmetic exactly with integers, whereas with real numbers the calculations may be inexact because of round-off error. Consequently, arithmetic with fractions can be done exactly -- provided the numbers that arise are not too large. On his trusty TI-81 programmable calculator, Fred went into fraction mode, and was able to verify the relation: