Whether teaching remedial, mainstream, or honors classes, in a segregated or integrated program, these 224 dynamic animations bring to life algebraic topics from pre-algebra through pre-calculus. Packaged on a CD-ROM (with a basic license for 4 on-site computers to use at a variety of levels), Algebra In Motion animations perform equally well on either the Windows or Macintosh platform. They must be run by The Geometer’s Sketchpad v4 or v5 (no prior versions), owned and sold by McGraw-Hill Education, on either Windows or Macintosh platforms. Although a detailed instruction manual is included on the CD-ROM (PDF format), most of the animations can be run successfully using only the on-screen information.

THE BASICS
INTRODUCING THE COORDINATE PLANE
MULTIPLICATION & FACTORING
GEOMETRY - see video on home page
CONNECTING SOLUTIONS TO GRAPHS
ADVANCED GRAPHING
CONICS & THEIR APPLICATIONS
TRIGONOMETRY
THEOREMS
MORE APPLICATIONS & VERTICAL TEAMING

   
  



















 

 

 

  

 
 

 

  THE BASICS
Visually explore or review fractions (meanings, comparisons, improper, LCD, adding, decimals, percents), signed number operations, absolute value, and introduce the concept of an equation as a balance of values and use that balance to solve preset equations or create your own.
 
  absolute value
adding signed integers
adding signed numbers
subtracting signed numbers
         
  fractions of 1 whole unit comparing fractions(1) comparing fractions(2) fractions of a group
         
  improper fractions finding LCDs adding fractions decimal fractions
         
    percent fractions common uses of fractions  
         
  equation balance
(+ 3 different preset examples,
1 adapts to your own examples)
equation balance – “backwards”
(+ 2 different preset examples,
1 adapts to your own examples
)
     
  the Real Number System - define
the Real Number System - practice
     
  geometric transformations - reflect*, rotate*, translate*, dilate*, summary, and an example using reflection

 

 

INTRODUCING THE COORDINATE PLANE & GRAPHING
Introduce students to the basic vocabulary and characteristics of the coordinate plane (+ history). Dynamically display the definition of slope. Practice graphing lines from y = mx+b, Ax+By = C, and y-y1 = m(x-x1) forms. Explore the relationship of parallel and perpendicular lines to slope. Develop the formulas for midpoint and distance. Test relations using an animated vertical line test
Present 4 different graphing grids on the same screen.
 


axes~quadrants~coordinates
vertical line test, domain/range
exploring domain/range
with any function

       
  visualizing slope
graph y = mx + b
graph Ax + By = C
       
  graph y-y1 = m(x-x1)
parallel and perpendicular
developing the midpoint formula
       
  developing the distance formula
4 grids on 1 screen
evolution of a polynomial
       
  parabola evolution
parabola 3 graphing forms
  
       
  piecewise explanation
piecewise absolute value
piecewise - create your own
       
  transform f(x) to f(x) + a
transform f(x) to af(x)
transform f(x) to f(x-a)
       
 
transform f(x) to f(ax)
transformation practice
(line)
transformation practice
(absolute value)
       
  transformation example
(parabola)
transformation practice
(sine)
transformation practice
(exponential)

 

 

MULTIPLICATION & FACTORING
The distributive property is geometrically demonstrated for products of all combinations of monomials, binomials, and trinomials. The factored form of a2 – b2 is developed and proved. Special emphasis is given to (x+h)2 ≠ x2+h2 and (x+h)3 ≠ x3+h3.  "Completing the square" is modeled physically.

  FOIL (a+a)(x+b)

 (a+b)(c+d) = ac+ad+bc+bd
 (a+b+c)(d+e+f)
       
   a2 - b2 = (a+b)(a-b)
(x+h)2 = x2 + 2xh + h2
(x+h)3 = x3 + 3x2h + 3xh2 + h3
   completing the square
 
   
       
 

GEOMETRY - added in 2016

ANGLES
definitions of all types of single angles and angle pairs, including all transversal angles

TRIANGLES
all parts and definitions; the 180 degree sum; the Exterior Angle Theorem; SSS, ASA, AAS, and SAS; the Isosceles Triangle Theorem;
visualizing overlapping triangles; altitudes, medians, and angle bisector segments along with the centers they determine and the
characteristics of the loci of those centers including the Euler Line and the 9-point circle; the Triangle Inequality Theorem;
the Geometric Mean; 30-60-90 degree triangles; and 45-45-90 degree triangles

CIRCLES

all parts and definitions; the circle's equation in the coordinate plane; major vs. minor arcs & semicircles; central angles; inscribed angles;
angles whose vertices are in the interior or exterior of the circle; quadrilaterals inscribed in circles; chord, secant, tangent properties and theorems;
rolling out pi; arc lengths and sector areas; plus a few extra explorations for students who need a bit of a challenge

QUADRILATERALS, POLYGONS, SOLIDS
definitions and properties of each in the "family" of quadrilaterals (trapezoid, parallelogram, rhombus, rectangle, square) plus a summary;
sums of interior and exterior angles in polygons; regular n-gons; all area formulas;
definitions, properties, volumes and surface areas of prisms, cylinders, pyramids, cones, and spheres

For a quick view of these 40 new "geometry-specific" animations added to the Algebra In Motion collection in 2016,
see them in action in the GEOMETRY video accessible from the home page (or the online demo page).





CONNECTING SOLUTIONS TO GRAPHS
Reinforce meaningful understanding of solutions to sentences with absolute value, systems of 2 linear equalities or inequalities, and finding the roots of a quadratic equation (including complex roots) using its related parabola.
  |ax+b| ≥ c
system of linear equalities
system of linear inequalities
       
  complex roots of quadratic equations
polynomial root dragging
(set of 7 animations)
 
       
  ADVANCED GRAPHING
Dynamically graph points in 3D space or on the complex number plane (history included). Control coefficients of a polynomial to “morph” it from a constant function up through a 5th degree polynomial. Similarly “morph” graphs of logarithmic & exponential functions, parametric & polar graphs, greatest integer functions, and inverses. Explore how composites of functions are created, and create linear programming examples.
 
  points in xyz-space
points in complex plane
"morphing polynomials"
       
  "morph" exponential functions
"morph" logarithmic functions
parametric graphs
       
  polar graphs
your choice
greatest integer functions
       
 
more greatest integer

inverses
creating composites
(adapts to any example)
       
  composite ex. 1
composite ex. 2
composite ex. 3
       
  linear programming
(create your own example)
linear programming
(preset example 1)
linear programming
(preset example 2)
 
       
  CONICS & THEIR APPLICATIONS
Dynamically create each conic section from its definition. Graph and “morph” all features of each. Alter coefficients of the general equation or eccentricity to “morph” one conic into another. Explore applications to satellite dishes, elliptical pool tables, whisper chambers, & falling objects.
 
  overview
construct circle by def. & graph
construct parabola by def.
       
 
construct ellipse by definition
construct hyperbola by definition

graphing parabolas
       
 
graphing ellipses

graphing hyperbolas
"morphing" from general equation
       
  parabaloid, ellipsoid, hyperboloid

family of hyperbolas & ellipses

mutually orthogonal
       
  reflections & collections
falling projectile
altering eccentricity
       
  TRIGONOMETRY
In an environment where rotation is real, not merely imagined, thoroughly investigate the unit circle’s angles, coordinates, and ratios. Literally unwrap the unit circle to form sine and cosine waves. Dilate and translate trigonometric graphs to explore amplitude, period, and shift. Explore and prove Pythagorean identities, the Law of Sines, and the Law of Cosines. Convincingly demonstrate that sin (a+b) can’t be (sin a + sin b).
 
 
unit circle angles
aim62.gif (81293 bytes)

sine, cosine, tangent, definitions
sin, cos, tan, sec, cec, cot on the unit circle
       
  special angles of the unit circle
aim22.gif (81293 bytes)
unwrapping the unit circle
"morphing" trig graphs
       
  Pythagorean Identities
Law of Sines
Law of Sines- ambiguous case
       
  Law of Coines
sin (a+b), Q1
sin (a+b), Q2 both acute
       
 
sin (a+b), Q2 acute+obtuse

cos (a+b), Q1
 sin (a+b), cos (a+b)
geometric approach

 

 

THEOREMS
Dynamically explore conventional theorems such as the Pythagorean Theorem (+ history) along with 7 different visual proofs of it. In addition, discover a large selection of unusual and unexpected theorems concerning tangents to parabolas, cubics, and quartics that will amaze and fascinate your students while laying an excellent foundation for more advanced mathematical study.

 
Pythagorean Theorem

More Pythagorean Theorem
any 3 tangents to any parabola

3 tangent proof (part 1)
         
  3 tangent proof (part 2)
parallel tangent at midpoint
inflection point at midpoint
use 2 roots to find 3rd (+ extend to imag. rts)
         
  ratio of areas in cubic
more ratios of areas in cubic
ratios of areas in quartic
PT – shearing proof
         
  PT – Chinese proof PT – Pythagoras proof PT – Bhaskara proof PT – DaVinci proof 
         
 
PT – Garfield proof

PT – Generalized
any 2 tangents to any parabola

2 tangent proof (part 1)
         
 
2 tangent proof (part 2)
any parabola with any 3 tangents
2 parabolas & common tangents
 
         
  MORE APPLICATIONS & VERTICAL TEAMING
Build basic graphing sense using intriguing questions about real-world situations that animate at the click of a button. Thoroughly explore classic problems such as the “open box” (vary the size of squares removed from the corners of the original rectangle), the sliding ladder, flying kite, etc. As the level of the class increases, more and more features can be explored at the click of a button. Finally, even calculus students will benefit by experiencing the varying rates of change in familiar favorites – perfect for vertical teaming!
 
 
Optimization - semicircle

Optimization- can

Optimization - cylinder

Optimization - any f
         
  pendulums - period, LCM
pendulums - waves
pendulums - graphs, rates
 
         
 
truck with square wheels
path of pebble
caught in wheel

mapping a Ferris wheel

escalators
         
  4 bugs
logarithmic spirals
dog & marching band
 
         
  distance = (rate)(time)
distance = (rate)(time)
midpoints
more midpoints
         
 
Aristotle’s Wheel paradox

linear vs. angular velocity
linear vs. angular velocity in gears
 
         
 
vertical team-open box
vertical team - clock
(convert time)*
vertical team - clock
(geometry)
vertical team - clock
(trig. & adv. graphing)
         
  vertical team – ladder
basic PT application
vertical team – ladder
trig. application
vertical team – ladder
rag & graph application
vertical team – ladder
calculus application
         
  vertical team – kite
basic PT application
vertical team – kite
trig. application
vertical team – kite
function graphs application
vertical team – kite
calculus application
         
  vertical team – balloon
basic formulas (approx. pi)
vertical team – balloon
basic formulas (pi exact)
vertical team – balloon
dimensional analysis
vertical team – balloon
angle elevation/depression
         
  vertical team – balloon
trig. application
vertical team – balloon
function graphs application
vertical team – balloon
calculus application