Preface: I went on a Desmos kick this afternoon and my brain was flooding with ideas of ways to use it in both little bits (lesson intros, aids, etc) and big bits (whole-class period exploration and inquiry activities).  So, even though I don't teach this course anymore, I am going through my Math Analysis (PreCal) concept list and writing down ways Desmos can be used as a tool to help students discover and explore math concepts and/or aid in providing a visual or model for the math concept.  There are a lot of concepts below  that are introduced or initially taught in Algebra 1, Geometry, or Algebra 2, so this list in not just for Math Analysis / Pre-Cal.  At the very end there is an “other” section for topics not related to those in my concept list.

Please check it out and feel free to add your own ideas!  This is a work in progress - let’s start collaborating!

You can use Ctrl-F (PC) or Command-F (Mac) to search for specific topics or concepts within the document

You will find three types of things below, color coded accordingly:

1. Ideas for use, but nothing developed (if you have something, let me know!)
2. Pre-Created Desmos graphs (by myself and others) that help to explore or model the concept.  Some of these include basic instructions for use with graph.
3. Guided Discovery or Inquiry activities that help students explore the concepts.  These are ready to use as lessons in class.  Most of these do not come with pre-created Desmos graphs as the students type in and manipulate the equations themselves.  (However, for scaffolding and differentiation purposes, pre-created graphs could be made)

This document can be found at bit.ly/kirchdesmos

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MATH ANALYSIS HONORS list of concepts from 2013-2014

Note: There are a lot of concepts below  that are introduced or initially taught in Algebra 1, Geometry, or Algebra 2.

Textbook sections are from the Yellow Precalculus with Limits: A Graphing Approach Textbook

A Introduction to Functions  1.1-1.2

1.Finding the slope of a line given two points & 2.Writing the equation of a line given slope and a point or two points

3.Writing the equations of parallel and perpendicular lines given a line and a point

• Plot original line, make table with given point. Make equation y=mx+b, use sliders to find value for 'm' that is visually perpendicular (check algebraically).  Then, play with 'b' to find the value that will allow the line to also cross through given point.  Check and solve algebraically

4.Identifying functions while looking at ordered pairs or a graph (vertical line test)

• Students can plot sets of ordered pairs using a table to visually see if they represent a function

5.Evaluating functions with numbers or variable expressions

6.Writing linear models and evaluating for word problems

• Use table feature to create table of data in problem

7.Writing and solving Cost, Profit, and Revenue word problems

8.Evaluating piecewise functions (2,3,4 pieces)

•  graph all functions, evaluate different pieces so students can see each x-value relates to a specific part of the function

9.Finding the domain and range of a function (rational, even/odd radical, polynomials) in interval notation

•  look at graph to find visually, relate to algebraic way to find it

10.Evaluating the difference quotient (linear and quadratic)

B  Characteristics of Functions 1.3-1.4

1. Finding relative minimum and relative maximum values of a polynomial with a graphing calculator & 2.Identifying intervals of increase and decrease of a polynomial; identifying concavity

• Plug the equations into Desmos.  Desmos automatically puts dots at all key parts of the graph that you can click on to get the ordered pair.  Student still has to identify and analyze, but not spend all the time learning how to push buttons on a calculator.  You can much easier see the graph and manipulate the window to be able to do intervals of increase and decrease
• Sample graph to find extrema and intervals

3.Graphing piecewise (2 and 3 pieces) functions

• Plug the equations in with restrictions.  The only part that Desmos doesn't do is the open/closed circles on the fences.  Great way to help students visualize what piecewise functions actually are.
• 2piece piecewise graph visual and manipulation pre-created Desmos graph here (source unknown)
• 4piece piecewise graph manipulation with restrictions pre-created Desmos graph here

4.Understanding and applying vertical and horizontal shifts, vertical stretches and shrinks, and reflections within function notation & 5.Sketching parabolas, absolute value, & cubic graphs with shifts

C Operations with Functions 1.5-1.6

1.Adding and subtracting functions & 2.Multiplying functions & 3.Dividing functions & 4.Composing functions (including variables and values)

• Plug in f(x) and g(x) into Desmos.  Then, plug in the operation (i.e. f(x)+g(x) or f(g(x)). ) See resulting graph.  Plug in resulting graph to Desmos to see if it lines up.
• For composing with values, you can identify the value on the resulting graph.
• See pre-created Desmos activity here (source unknown)

5.Finding inverse functions algebraically

• We talk about what inverse functions are algebraically, numerically, and graphically. Students can see graphically by inputting both functions and the line y=x and seeing that they do reflect each other.  They can create tables of values for both functions and see that the x's and y's are switched.
• Pre-created Desmos graph of inverse functions graphically here, here, here and here (source unknown)

6.Verifying inverse functions using composition

• Once you have verified algebraically, plug both equations in with y=x to verify graphically.  Use a table of values (one Desmos-created, the second one self-created by choosing the y-values from previous table) to verify numerically.

7.Restricting the domain of functions to make it one-to-one

• Sketch the function to see if it passes the Horizontal Line Test.  If not, play with the restrictions to find ways for it to work.  Graph both the original and the inverse (non-restricted and restricted) to see it visually.
• See pre-created desmos graph here

D Factoring 2.1

1.Completing the square (a=1, a=not 1, fractions NOT included)

2.Completing the square(a=1, a=not 1, fractions required)

4.Factoring sum and difference of cubes

• Visual of formula for difference of cubes pre-created Desmos graph here (source unknown)

5.Factoring difference of squares

7.Factoring by grouping

• For all: plug original and factored equations into Desmos to see if they make the same graph.  Factoring is just a different way of writing the same function!  See pre-created Desmos graph here

E Basics of Polynomials2.1-2.2

1.Identifying x-intercepts, y-intercepts, vertex (max/min), axis of quadratics and graphing them. Quadratics in standard form.

• Verifying and visualizing all answers, including typing in standard form equation + vertex form conversion to make sure they are equivalent.  See Desmos pre-made graph here.

2.Finding maximum and minimum values of quadratic applications using calculator, interpretation of solutions. [path of football]

• Graph on Desmos to identify key points.  So much easier to manipulate window!

3.Finding maximum and minimum values of quadratic applications using calculator, interpretation of solutions. [maximizing area]

4.Using the leading coefficient test for polynomials to write end behavior (limit notation)

• Exploration activity with different degrees of polynomials for students to discover the patterns with end behavior and ask questions regarding the “middle” behavior (why so many humps? why so few humps? etc)

5.Finding zeroes and multiplicities of polynomials using factoring

• Graph equation to verify answer and observe what happens on the graph  (part exploration activity trying to find pattern of when the graph goes through, bounces, or curves through?)

6.Writing equations of polynomials given zeroes and multiplicities

• Graph original and factored equations, as well as table of values for zeroes, and verify they are equivalent.

7.Graphing polynomials, including: x-int, y-int, zeroes (with multiplicities), end behavior. All polynomials will be factorable.

• Verify that graphical and algebraic answers match up in terms of all key parts.

F Finding zeroes of polynomials 2.3-2.5

1.Polynomial long division

2.Polynomial synthetic division

3.Using remainder theorem- take one known zero of cubic polynomial to find remaining zeroes

4.Finding all possible real zeroes (p's and q's)

5.Finding possible + and – real zeroes (Descartes)

6.Given polynomial of 4th or 5th degree, find all zeroes (all real) [utilize Rational Roots Thm, Descartes Rule of Signs]

• Graph polynomial and confirm answers visually

7.Adding and subtracting complex numbers  8.Multiplying and dividing complex numbers  9.Plotting complex numbers

• Complex number graphing & addition pre-created Desmos graph here (source unknown)

10.Given polynomial of 4th or 5th degree, find all zeroes (real and complex zeroes)

G Rational Functions 2.6-2.7

1.Find vertical asymptotes (equation and limit notation) of rational functions

2.Find horizontal asymptotes (equation and limit notation) of rational functions

3.Find slant asymptotes (equation) of rational functions

4.Find holes (ordered pairs) of rational functions

5.Find domain (interval notation) of rational functions

6.Find x-intercepts and y-intercepts (ordered pair) of rational functions

7.Graph rational functions (1V1H, 2V1H, 1V1S, 2V1S, OV1H) with all parts

• All above: analyze visually with Desmos.  Plug in asymptotes as equations to check and see how graph interacts with asymptotes.  Plot holes as ordered pairs.

8.Minimizing area problem

H Introduction to exponential and logarithmic functions 3.1-3.2

• Visual of Logs/Exponential Graphs reflecting across the identity line.  Pre-created Desmos graph here. (Source unknown)

1.Converting from logarithmic to exponential form and vice versa

2.Evaluating logs on a calculator (base 10 and e)

3.Using inverse properties of logs

4.Change of base formula

5.Expanding logs

6.Condensing logs

7.Finding logs given approximations

8.Solving exponential equations

• Graph both sides of the equation as two separate functions and see where they meet

9.Solving logarithmic equations

• Graph both sides of the equation as two separate functions and see where they meet

I Graphs & apps of exp. & log functions 3.3-3.4

1.Graphing exponential functions and identifying x-intercept, y-intercept, asymptote, domain, range (3 points on graph minimum)

2.Graphing logarithmic functions and identifying x-intercepts, y-intercepts, asymptote, domain, range (3 points on graph minimum)

3.Compound interest

4.Continuously compounding interest

5.Investment application problem (Pert)

J Systems of Equations7.1-7.4

1.Solving systems of 2-variable linear equation; understanding solutions

2.Solving systems of 2-variable nonlinear equations; understanding solutions

• Graph system on Desmos to verify solution graphically

3.Solving three-variable systems with Gauss-Jordan elimination/matrices/row-echelon form/back-substitution

4.Solving non-square systems

5.Partial Fraction decomposition with distinct factors

6.Partial Fraction Decomposition with repeated factors

K Sequences and Series8.1-8.3

1.Evaluating terms of a given sequence

2.Writing summation notation and finding partial sums of sequences/series

3.Finding general terms of sequences that are neither arithmetic nor geometric

4.Understanding Fibonacci and other recursive sequences

5.Finding the nth term of an arithmetic sequence given just about anything

6.Finding partial sums of arithmetic sequences

7.Finding the nth term of a geometric sequence given just about anything

8.Finding partial sums of geometric sequences

9.Finding sums of infinite geometric sequences

10.Writing a repeating decimal as a rational number using geometric series

11.Evaluating word problems involving arithmetic and geometric sequences and series

L Binomial Expansions, Counting Principles, and Probability 8.5-8.7

optional Mathematical Induction

1.Expanding binomials using binomial theorem

2.Finding a specific term or specific coefficient in a binomial expansion

3.Expanding with the difference quotient

4.Calculating Possibilities with FCP

5.Calculating Possibilities with Combinations (one event)

• nCr calculation on Desmos here

6.Calculating Possibilities with Combinations (two events)

• nCr calculation with two events on Desmos here

7.Calculating Possibilities with Permutations

• nPr calculation on Desmos here

8.Calculating Possibilities with Distinguishable Permutations

9.Basic Probability (one event)

10.Probability of Independent “AND” Events (two events, with replacement)

11.Probability of Independent “AND” Events (two events, WITHOUT replacement)

12.Probability of “OR” Events (mutually exclusive)

13.Probability of “OR” Events (NON-mutually exclusive)

14.Probability of events that require Combinations to determine sample space

M Conic Sections9.1-9.3

1.Completing the square with TWO variables

2.Classifying Conics given equations

3.Graphing circles given equation (must complete the square first) and identifying all parts (center, radius)

• Pre-created Desmos graphs here  (source unknown)

4.Graphing parabolas given equation (must complete the square first) and identifying all parts (vertex, focus, directrix, axis)

• Pre-created Desmos graphs here and here (source unknown)

5.Graphing ellipses given equation (must complete the square first) and identifying all parts (center, focus, major axis, minor axis, vertices, covertices, eccentricity)

6.Graphing hyperbolas given equation (must complete the square first) and identifying all parts (center, focus, conjugate axis, transverse axis, vertices, covertices, asymptotes, eccentricity)

N Angles and the Unit Circle4.1,4.2,4.4

3.drawing angles in standard position (positive and negative angles)

4.finding supplementary and complementary angles

5.finding coterminal angles (positive and negative)

6.identifying reference angles

7.knowing all degrees and radians around the unit circle, knowing all the ordered pairs around the unit circle, understanding and applying ASTC to the Unit Circle

8.finding exact values of all 6 trig functions when given angle in degrees or radians (using Unit Circle)

9.finding angles when given exact value of any of the 6 trig functions (using Unit Circle)

O Right Triangle Trigonometry4.3,4.8

1.evaluate angles in degrees or radians on a calculator

2.finding angles on a calculator using inverse trig functions

3.solving basic trig equations with non-exact answers

4.finding all six trig ratios given point on terminal side

5.finding all six trig ratios when given one trig ratio (must draw right triangle)

6.using SOHCAHTOA to find missing pieces of right triangles (given 2 sides, given angle and side)

7.using the 30-60-90 triangle

8.using the 45-45-90 triangle

9.solving basic right triangle word problems

10.solving angle of elevation and depression word problems

P Non-Right Triangle Trigonometry6.1-6.2

1.Law of Sines AAS or ASA

2.Law of Sines SSA (one, none, two solutions)

3.Law of Cosines SSS or SAS

4.Area of an oblique triangle

5.Heron's Area Formula

6.Applications with Law of Sines

7.Applications with Law of Cosines

Q Trigonometric Identities Part 15.1-5.3

1.using fundamental identities to simplify and verify expressions (simple, one or two step identities)

2.find all trig functions when given one trig function and quadrant (using identities)

3.trigonometric substitution

4.solving trig equations that have more than one function in it at first (basic substitution)

5.using various simplification methods to verify, such as GCF, substitution of identity, multiplying by conjugate, combining fractions w/ binomial denominator, separating fractions with monomial denominators, and factoring. More complex.

R Trigonometric Identities Part 25.4

1.Exact value of sums or differences (might have to find what it adds or subtracts to)

2.Using sum and difference formulas when given values (right triangles)

3.Trig of an inverse trig function (use sum and difference to prove)

4.Simplifying sum and difference identities

5.Solving trig equations with sum and difference

S Trigonometric Identities Part 3 5.5

1.Writing products as sums

2.Writing sums as products

3.Using power-reducing formulas

4.using half-angle formulas

5.Finding function values with double angles (right triangles)

6.Solving multiple angle equations (using multiple-angle identities)

7.solving equations with half-angle formulas

T Graphing Trig Functions4.5-4.7

1.graphing sine and cosine

2.graphing secant and cosecant

3.graphing tangent and cotangent

U Limits 11.1,11.2,11.4

1.Continuous functions and types of discontinuities (point, jump, infinite)

2.Limits numerically

3.Limits graphically

4.Writing one-sided limits for jump and infinite discontinuities

5.Direct substitution

6.Dividing out

7.Rationalizing

8.Limits at infinity

V Derivatives and the Area Problem 11.3-11.5

1.Finding the derivative function (the limit of the difference quotient)

2.Finding the slope of the tangent line at a specific x-value

• Tangent line to a curve visual pre-created Desmos graph here  and here (source unknown)
• Another tangent line to a curve visual pre-created Desmos graph here  and here (source unknown)

3.Finding equation of tangent line at specific point

4.Finding when tangent line is horizontal

5.Approximating area under a curve

Other

• Linear Regressions
• even and Odd functions pre-created desmos graph here (source unknown)
• Parametric Vertical Motion pre-created Desmos graph here (source unknown)
• Just for fun - the love formula here