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# Sine of the Times

The Math Education Blog

## Sitemaps

- [XML Sitemap](https://www.sineofthetimes.org/sitemap.xml): Contains all public & indexable URLs for this website.

## Posts

- [A New Fractal Construction](https://www.sineofthetimes.org/a-new-fractal-construction/)
- [A Transformations Approach to Complex Numbers](https://www.sineofthetimes.org/a-transformations-approach-to-complex-numbers/) - This post, inspired by the work of Al Cuoco, uses Web Sketchpad to explore a transformations approach to complex numbers.
- [A Double Spiral from David Henderson](https://www.sineofthetimes.org/a-double-spiral-from-david-henderson/)
- [The Dog Leash Investigation](https://www.sineofthetimes.org/the-dog-leash-investigation/) - Marta Venturini describes how a Sketchpad model can help students visualize the solution to a problem involving a dog whose movements are limited by a leash.
- [Collaborating on an Extension to a Little-Known Theorem](https://www.sineofthetimes.org/collaborating-on-an-extension-to-a-little-known-theorem/) - Stephen Fuchs explains how encouraging mathematics student inquiry, questioning, and self-initiated investigations can fuel collaboration in solving complex algebra and geometry problems.
- [Cartesian and Polar Graphs](https://www.sineofthetimes.org/cartesian-and-polar-graphs/) - This Sketchpad activity relates to a May 2013 Mathematics Teacher article on Graphing Polar Curves.
- [Introducing Dynagraphs](https://www.sineofthetimes.org/introducing-dynagraphs/)
- [Algebra Cross Number Puzzles](https://www.sineofthetimes.org/algebra-cross-number-puzzles/)
- [Cross Number Puzzles](https://www.sineofthetimes.org/cross-number-puzzles/) - This post explores how cross number puzzles can be used to introduce young learners to addition, subtraction, the commutative property, and place value.
- [Exploring Multiples on a Dynamic Number Grid](https://www.sineofthetimes.org/exploring-multiples-on-a-dynamic-number-grid/) - Dynamic number grids with adjustable dimensions allow students to investigate patterns relating to multiples and least common multiples.
- [Adding and Subtracting on a Dynamic Number Chart](https://www.sineofthetimes.org/adding-and-subtracting-on-a-dynamic-number-chart/) - Students use a dynamic number grid to model addition and subtraction strategies.
- [Learn to Multiply Like No Bunny's Business](https://www.sineofthetimes.org/learn-to-multiply-like-no-bunnys-business/) - Daniel Scher describes a new game for desktop Sketchpad and Sketchpad Explorer for the iPad that develops children's understanding of multiplication.
- [A Quartet of Ellipse Constructions](https://www.sineofthetimes.org/a-quartet-of-ellipse-constructions/) - Daniel Scher presents four Web Sketchpad models for drawing ellipses. Using a mix of algebra and geometry, students can prove each method yields an ellipse.
- [Innovative Approaches to Computer-Based Assessment, Part Three](https://www.sineofthetimes.org/innovative-approaches-to-computer-based-assessment-part-three/) - In part three of an ongoing series, Daniel Scher examines what meaningful assessment might look like with Web Sketchpad.
- [A Double Dissection from The New York Times](https://www.sineofthetimes.org/a-double-dissection-from-the-new-york-times/) - Daniel Scher presents an interactive Web Sketchpad model for visualizing a dissection puzzle from the New York Times Numberplay column.
- [Dilation Games: Assessment That's Fun](https://www.sineofthetimes.org/dilation-games-assessment-thats-fun/) - Scott Steketee presents a Web Sketchpad game for sharpening students' understanding of dilation. The game treats dilation as a geometric example of functions.
- [Revisiting the Hundred Chart](https://www.sineofthetimes.org/revisiting-the-hundred-chart/) - With a few adjustments, we can make the Hundred Chart more intuitive and more useful for students. This post explains why the improvements are needed and describes how students can build a physical model that more accurately corresponds to the number system.
- [The Folded Circle Construction](https://www.sineofthetimes.org/the-folded-circle-construction/)
- [A Plethora of Hyperbola Constructions](https://www.sineofthetimes.org/a-plethora-of-hyperbola-constructions/)
- [Injecting Surprise Into the Triangle Midline Theorem](https://www.sineofthetimes.org/injecting-surprise-into-the-triangle-midline-theorem/) - This post examines the role of surprise in mathematics, with a particular focus on the triangle midline theorem.
- [A Triple Number Line Model for Visualizing Solutions to Equations](https://www.sineofthetimes.org/a-triple-number-line-model-for-visualizing-solutions-to-equations/) - A triple number line model helps students to visualize whether there is 0, 1, or an infinite number of solutions to algebraic equations.
- [Virtual Cuisenaire Rods](https://www.sineofthetimes.org/virtual-cuisenaire-rods/) - Guest contributor Stavroula Patsiomitou shares ideas about using virtual Cuisenaire rods built with Web Sketchpad.
- [Dynagraphs of Linear Functions](https://www.sineofthetimes.org/dynagraphs-of-linear-functions/)
- [Some Triangle Shearing Investigations](https://www.sineofthetimes.org/some-triangle-shearing-investigations/)
- [Constructing Rectangles with Constant Perimeter and Constant Area](https://www.sineofthetimes.org/constructing-rectangles-with-constant-perimeter-and-constant-area/)
- [Exploring a Perimeter-Area Relationship](https://www.sineofthetimes.org/exploring-a-perimeter-area-relationship/)
- [Catching the Vibe](https://www.sineofthetimes.org/catching-the-vibe/)
- [Protect the Sheep](https://www.sineofthetimes.org/protect-the-sheep/) - A game of enclosing sheep and wolves in fences helps children to develop their conceptual understanding of polygons.
- [A Hidden Polygons Puzzle](https://www.sineofthetimes.org/a-hidden-polygons-puzzle/) - Daniel Scher offers web-based Sketchpad puzzles in which students are challenged to find a hidden square and equilateral triangle from a collection of points.
- [A Compendium of Elementary Web Sketchpad Activities](https://www.sineofthetimes.org/a-compendium-of-elementary-web-sketchpad-activities-2/) - Daniel Scher presents a compilation of over 25 Web Sketchpad activities designed for the elementary grades.
- [Arranging Addends Puzzles Revisited](https://www.sineofthetimes.org/arranging-addends-puzzles-revisited/) - Daniel Scher revisits a puzzle that combines logic and addition practice, offering a twist that breathes new life into the puzzle.
- [The Return of the Odometer](https://www.sineofthetimes.org/the-return-of-the-odometer/) - Daniel Scher presents the second of two interactive Web Sketchpad odometers that provide students with engaging place-value puzzles to ponder and solve.
- [Exploring Triangle Area with Custom-Built Tools](https://www.sineofthetimes.org/exploring-triangle-area-with-custom-built-tools/) - Daniel Scher presents two interactive Web Sketchpad models that help students to understand that every triangle has three bases and three corresponding heights.
- [Constructing an Ellipse with Web Sketchpad Tools](https://www.sineofthetimes.org/creating-an-ellipse-with-web-sketchpad-tools/) - Daniel Scher provides a Web Sketchpad model that allows you to build an ellipse starting from a blank sketch.
- [Putting the Power of a Point Theorem to Work](https://www.sineofthetimes.org/putting-the-power-of-a-point-theorem-to-work/) - Daniel Scher illustrates two Web Sketchpad applications of the power of a point theorem.
- [Pythagoras Plugged In](https://www.sineofthetimes.org/pythagoras-plugged-in/) - Teachers in a professional development workshop use Web Sketchpad to explore a dynamic proof of the Pythagorean Theorem.
- [Make Your Own Fractions](https://www.sineofthetimes.org/make-your-own-fractions/) - Build interactive area models of fractions picking any numerator or denominator you like and "animate" the numerical values to see how they affect the fraction.
- [The Varied Paths to Constructing a Square](https://www.sineofthetimes.org/the-varied-paths-to-constructing-a-square/) - This post profiles Web Sketchpad's innovative tool interface in the context of constructing a square.
- [Digging Deep Into Varignon's Theorem](https://www.sineofthetimes.org/digging-deep-into-varignons-theorem/) - Varignon's theorem can benefit from dynamic geometry software in ways that rarely get explored. This post explores some of those missed opportunities.
- [Deducing the "Mystery" Fraction](https://www.sineofthetimes.org/deducing-the-mystery-fraction/) - Students deduce the fractional location of a point on a number line by estimating its location using successive estimates to refine their guess.
- [A Geometry Gem from Twitter](https://www.sineofthetimes.org/a-geometry-gem-from-twitter/) - A geometry problem from Twitter lends itself to a dynamic approach with several nice solutions as well as a connection to the Law of Cosines.
- [Interactive Array Explorations](https://www.sineofthetimes.org/interactive-array-explorations/)
- [The Scaled Maps Problem](https://www.sineofthetimes.org/the-scaled-maps-problem/)
- [A Dynamic Approach to Finding Pirate Treasure](https://www.sineofthetimes.org/a-dynamic-approach-to-finding-pirate-treasure/)
- [Circle Tracer Challenges](https://www.sineofthetimes.org/circle-tracer-challenges/)
- [Dividing and Subdividing](https://www.sineofthetimes.org/dividing-and-subdividing/)
- [The Expanding Circle Construction](https://www.sineofthetimes.org/the-expanding-circle-construction/)
- [Tweaking the Expanding Circle Construction](https://www.sineofthetimes.org/tweaking-the-expanding-circle-construction/)
- [Constructing the Pi-Petal Rose](https://www.sineofthetimes.org/constructing-the-pi-petal-rose/)
- [Visual Models for Adding and Subtracting Fractions](https://www.sineofthetimes.org/adding-and-subtracting-fractions-on-a-number-line/)
- [Pirate Treasure Awaits](https://www.sineofthetimes.org/pirate-treasure-awaits/) - The geometric properties of hyperbolas are key to uncovering the location of a pirate's treasure.
- [Race to the Burning Tent](https://www.sineofthetimes.org/race-to-the-burning-tent/) - An ellipse holds the key to this unusual solution to the classic Burning Tent problem.
- [The Cowgirl Problem](https://www.sineofthetimes.org/the-cowgirl-problem/) - A problem traditionally solved with geometric reflection yields to a beautiful two-ellipse solution as well.
- [Symmetry Challenges](https://www.sineofthetimes.org/symmetry-challenges/) - In this blog post, we leverage symmetry properties to construct triangles and quadrilaterals using curated collections of geometric construction tools.
- [A Paper Folding Investigation from Connected Geometry](https://www.sineofthetimes.org/a-paper-folding-investigation-from-connected-geometry/) - This post shares a paper-folding activity from the Connected Geometry curriculum and describes how it can be modeled and explored with Web Sketchpad.
- [Euclid Walks the Plank](https://www.sineofthetimes.org/euclid-walks-the-plank/) - Using Web Sketchpad, students construct a boardwalk path of equal-length planks to explore the key concepts behind Euclid's Proposition 1.
- [Special Quadrilaterals and Their Diagonals](https://www.sineofthetimes.org/special-quadrilaterals-and-their-diagonals/) - What types of quadrilaterals can you build starting from two segments and their midpoints and using the segments as the diagonals of a quadrilateral?
- [Slope of the Sine Function, Part 1](https://www.sineofthetimes.org/slope-of-the-sine-function-part-1/)
- [Slope of the Sine Function, Part 2](https://www.sineofthetimes.org/slope-of-the-sine-function-part-2/) - This blog post, a continuation of the prior post, provides tools to hone our estimate for the slope of the sine function at any given point on the graph.
- [Teaching with Web Sketchpad](https://www.sineofthetimes.org/teaching-with-web-sketchpad/)
- [Classifying Quadrilaterals](https://www.sineofthetimes.org/quadrilateral-pretenders/)
- [Tracing the Sine and Cosine Functions](https://www.sineofthetimes.org/tracing-the-sine-and-cosine-functions/)
- [Rotation Designs](https://www.sineofthetimes.org/rotation-designs/)
- [Visualizing a Dynamic Triangle](https://www.sineofthetimes.org/visualizing-a-dynamic-triangle/)
- [Modeling with Dynamic Algebra Tiles](https://www.sineofthetimes.org/modeling-with-dynamic-algebra-tiles/)
- [Drawing Conic Sections Point by Point](https://www.sineofthetimes.org/drawing-conic-sections-point-by-point/)
- [Building with Pattern Blocks](https://www.sineofthetimes.org/building-with-pattern-blocks/)
- [A Beautiful Application of Shearing](https://www.sineofthetimes.org/a-beautiful-application-of-shearing/)
- [Exploring Conic Sections with Sketchpad](https://www.sineofthetimes.org/exploring-conic-sections-with-sketchpad/) - Daniel Scher gives an overview of his Sketchpad curriculum module, Exploring Conic Sections with The Geometer's Sketchpad.
- [A Trio of Parabola Constructions](https://www.sineofthetimes.org/a-trio-of-parabola-constructions/)
- [Experiments with a Color Calculator](https://www.sineofthetimes.org/experiments-with-a-color-calculator/) - Daniel Scher provides an interactive color calculator for exploring decimal representations of fractions based on an model from Nathalie Sinclair.
- [Drawing an Ellipse with Pins and String](https://www.sineofthetimes.org/drawing-an-ellipse-with-pins-and-string/) - Daniel Scher presents an interactive Web Sketchpad model for modeling the well-known pins-and-string construction for drawing ellipses.
- [Drawing Ellipses with a Congruent Triangle Construction](https://www.sineofthetimes.org/drawing-ellipses-with-a-congruent-triangle-construction/) - Daniel Scher presents an interactive Web Sketchpad model for drawing ellipses based on a congruent triangle linkage from the mathematician Frans van Schooten.
- [Pattern Block Puzzles](https://www.sineofthetimes.org/pattern-block-puzzles/)
- [Adding and Subtracting on an Open Number Line](https://www.sineofthetimes.org/adding-and-subtracting-on-an-open-number-line/)
- [Transformation Dances](https://www.sineofthetimes.org/transformation-dances/) - As a penguin moves, you, as a frog, must match its behavior with the challenge of dancing as a reflection, rotation, or dilation of the penguin's path.
- [The Mysteries of Polygon Flats](https://www.sineofthetimes.org/the-mysteries-of-polygon-flats/)
- [Introducing the Elipso](https://www.sineofthetimes.org/introducing-the-elipso/)
- [A Ping-Pong Puzzle From a Sketchpad Editor](https://www.sineofthetimes.org/a-ping-pong-puzzle-from-a-sketchpad-editor-and-friend/) - Daniel Scher presents a Sketchpad model of a puzzle created by Dan Bennett for the New York Times Numberplay column.
- [What's the Appropriate Role for the Teacher when using Computer Games?](https://www.sineofthetimes.org/whats-the-appropriate-role-for-the-teacher-when-using-computer-games/) - Rick Gaston writes about the important roles that teachers serve when using computer math games in the classroom to support the learning of all students.
- [Discovery by Dragging](https://www.sineofthetimes.org/discovery-by-dragging/) - Use a prepared sketch to distinguish parallelograms from parallelogram pretenders. How can this activity best address the Standards for Mathematical Practice?
- [A Pythagorean Tree Grows in Your Web Browser](https://www.sineofthetimes.org/a-pythagorean-tree-grows-in-your-web-browser/) - Daniel Scher presents an interactive Pythagorean Tree built with The Geometer's Sketchpad.
- [Understand the Sine Function by Dancing It](https://www.sineofthetimes.org/understand-the-sine-function-by-dancing-it/) - Scott Steketee shows how students deepen their understanding of the sine function by "dancing" it, moving y to keep up with x and experience the rate of change.
- [Multiplication Arrays](https://www.sineofthetimes.org/multiplication-arrays/) - This post features several interactive Web Sketchpad models for exploring multiplication using an array model.
- [Binomial Multiplication and Factoring Games](https://www.sineofthetimes.org/binomial-multiplication-and-factoring-games/) - This sequence of Web Sketchpad activities provides a collection of games that develop an area model approach to binomial multiplication and factoring.
- [Adding and Subtracting Integers](https://www.sineofthetimes.org/adding-and-subtracting-integers/)
- [The Interior Angle Sum: An Embodied Investigation](https://www.sineofthetimes.org/triangle-angle-sum-embodied/) - Sarah Stephens describes a lesson she created that leverages embodied cognition to help students develop abstract mathematical concepts.
- [Introducing Web Sketchpad at the 2021 NCTM Annual Meeting](https://www.sineofthetimes.org/nctm-2021-wsp-workshop/) - This blog post provides an overview of three Web Sketchpad-themed sessions at this year's virtual NCTM Annual Meeting.
- [Exploring Scaled Polygons](https://www.sineofthetimes.org/scaledpolygons/) - This post describes an approach for drawing a polygon and a scaled copy of the polygon simultaneously.
- [Generalizing the Pythagorean Theorem](https://www.sineofthetimes.org/generalizing-the-pythagorean-theorem/) - This post describes how students can explore an extension to the Pythagorean Theorem by using The Geometer's Sketchpad.
- [Hats Off to This Aperiodic Tiling](https://www.sineofthetimes.org/hats-off-to-this-aperiodic-tiling/) - This post examines the role of social media in promoting the discovery of an aperiodic monotile.
- [Adventures of the Scooting Tick Mark](https://www.sineofthetimes.org/adventures-of-the-scooting-tick-mark/)
- [Zooming in on Place Value](https://www.sineofthetimes.org/zooming-in-on-place-value/) - Daniel Scher presents an interactive Web Sketchpad of a "zooming" number line that introduces students to place value and decimal concepts.
- [The Origami-Math Connection](https://www.sineofthetimes.org/the-origami-math-connection/)
- [Construct a Building: Modeling Multiplication with Arrays](https://www.sineofthetimes.org/construct-a-building-modeling-multiplication-with-arrays/)
- [The Art of Parametric Equations](https://www.sineofthetimes.org/the-art-of-parametric-equations-2/) - Daniel Scher provides an interactive Web Sketchpad model of a parametric curve that displays beautiful visual patterns when its coefficients are varied.
- [The Swimming Pool Problem](https://www.sineofthetimes.org/the-swimming-pool-problem/) - An optimization problem lends itself beautifully to a geometric solution involving ellipses, introducing notions from calculus in an intuitive manner.
- [A Sliding Rulers Approach to Adding and Subtracting](https://www.sineofthetimes.org/a-sliding-rulers-approach-to-adding-and-subtracting/)
- [Pi Day 2015: Pieces of Pi](https://www.sineofthetimes.org/pi-day-2015-pieces-of-pi/) - Scott Steketee provides piecewise function tools and a tool to fill the region between two graphs, and challenges you to use them to draw the character π.
- [Tribute to Zalman Usiskin](https://www.sineofthetimes.org/tribute-to-zalman-usiskin/) - Scott Steketee describes his presentation at the Symposium Honoring Zalman Usiskin and presents both a movie and an interactive sketch.
- [Transformation Games](https://www.sineofthetimes.org/transformation-games/) - This post presents an abundance of games that find their inspiration in three geometric transformations: reflection, rotation, and dilation.
- [Introducing Algebra Mazes](https://www.sineofthetimes.org/introducing-algebra-mazes/)
- [Simultaneous Equations in Elementary School? You Bet!](https://www.sineofthetimes.org/simultaneous-equations-in-elementary-school-you-bet/) - Daniel Scher offers an interactive Web Sketchpad game for exploring simultaneous equations in elementary school.
- [Polar Graphing](https://www.sineofthetimes.org/polar-graphing/) - This post describes how to use Sketchpad to graph a polar function and create a family of polar functions.
- [Soccer Challenges: Angling for a Shot on Goal](https://www.sineofthetimes.org/soccer-challenges-angling-for-a-shot-on-goal/) - Just in time for the 2014 World Cup, Scott Steketee uses Web Sketchpad to present two geometric soccer challenges related to a striker's shooting angle.
- [Exploring Place Value With an Interactive Odometer](https://www.sineofthetimes.org/exploring-place-value-with-an-interactive-odometer/) - Daniel Scher presents an interactive Web Sketchpad model of an odometer that offers varied practice in developing place-value concepts.
- [Iteration in the Complex Plane](https://www.sineofthetimes.org/iteration-in-the-complex-plane/) - Scott Steketee uses Sketchpad to make it easy for you to conduct experiments and create artistic designs described in a recent article from Mathematics Teacher.
- [How do you make ... a pentaflake?](https://www.sineofthetimes.org/how-do-you-make-a-pentaflake/) - Scott Steketee provides a bit of help to a twelve-year-old Sketchpad user, and learns about a beautiful fractal in the process.
- [Pentaflake Chaos](https://www.sineofthetimes.org/pentaflake-chaos/) - Scott Steketee responds to a comment, and shows a couple of other ways to create pentaflakes in Sketchpad.
- [Create Parametric Curves Graphically and Kinesthetically](https://www.sineofthetimes.org/create-parametric-curves-graphically-and-kinesthetically/) - Nate Burchell describes how to use his tactile function approach to make parametric functions come alive to his students.
- [Discovering the Angle Sum and Difference Identities](https://www.sineofthetimes.org/discovering-the-angle-sum-and-difference-identities/) - Scott Steketee shares an inquiry task, created in collaboration with one of his Math Methods students, leading to an elegant discovery of the angle sum formulas.
- [Multiplication Is Not Repeated Addition](https://www.sineofthetimes.org/multiplication-is-not-repeated-addition/)
- [Innovative Approaches to Computer-Based Assessment, Part Four](https://www.sineofthetimes.org/innovative-approaches-to-computer-based-assessment-part-four/) - Daniel Scher examines how the diversity of approaches to a computer-based assessment question can be gleamed from something as simple as the placement of three points on the graph of a parabola.
- [Innovative Approaches to Computer-Based Assessment, Part Two](https://www.sineofthetimes.org/innovative-approaches-to-computer-based-assessment-part-two/) - In part two of an ongoing series, Daniel Scher examines what meaningful assessment might look like with Web Sketchpad.
- [Refutation in a Dynamic Geometry Context](https://www.sineofthetimes.org/refutation-in-a-dynamic-geometry-context/) - Guest blogger Michael de Villiers describes a false conjecture that appears to be true until examined under the microscope of Dynamic Geometry software.
- [Creating Mosaics Inspired by a Pattern from Sultan Ahmed Mosque](https://www.sineofthetimes.org/building-mosaics-inspired-by-the-sultan-ahmed-mosque/) - Mirek Majewski explores the geometry of the mosaic in the entrance of the Sultan Ahmed Mosque in Istanbul using Web Sketchpad.
- [What Is All the Fuss About Lines?](https://www.sineofthetimes.org/what-is-all-the-fuss-about-lines/) - Daniel Scher considers why students have a hard time learning to form the equations of lines and offers an interactive model as a fresh way to think about them.
- [Beam of Light](https://www.sineofthetimes.org/beam-of-light/)
- [Revisiting a Childhood Addition Code with Sketchpad](https://www.sineofthetimes.org/revisiting-a-childhood-addition-code-with-sketchpad/) - Daniel Scher describes a code from his childhood math class and then provides a Sketchpad version of the code that delves deeper into the mathematics.
- [Find the Secret Number](https://www.sineofthetimes.org/find-the-secret-number/)
- [Isosceles Triangle Puzzles](https://www.sineofthetimes.org/isosceles-triangle-puzzles/) - Daniel Scher presents interactive Web Sketchpad puzzles that challenge students to develop strategies for navigating an isosceles triangle through a grid.
- [A Follow-Up to the Interior Angle Sum](https://www.sineofthetimes.org/the-interior-angle-sum-follow-up/) - This follow-up to Sarah Stephens' post on using embodied cognition to make sense of the triangle interior angle sum theorem introduces Web Sketchpad support.
- [Revisiting the Isosceles Triangle Challenge](https://www.sineofthetimes.org/revisiting-the-isosceles-triangle-challenge-2/) - Dynamic geometry and Web Sketchpad provide a perfect environment for exploring an intriguing Japanese geometry problem.
- [Around and Around: Investigating Multiples](https://www.sineofthetimes.org/around-and-around-investigating-multiples/) - Daniel Scher presents interactive Web Sketchpad models of locks that motivate investigations of multiples and least common multiples.
- [Unlocking the Patterns of Multiples and Factors](https://www.sineofthetimes.org/unlocking-the-patterns-of-multiples-and-factors/) - Daniel Scher shares an interactive Web Sketchpad puzzle that challenges students to open a safe using their knowledge of multiples and factors.
- [Writing Mathematics with Sketchpad](https://www.sineofthetimes.org/writing-mathematics-with-sketchpad/) - Sketchpad is to mathematics as composition is to English. Students can use Sketchpad like a visual programming language to compose their mathematical creations.
- [A Mathematical Mystery Story with Web Sketchpad](https://www.sineofthetimes.org/a-mathematical-mystery-story-with-web-sketchpad/) - Students use Web Sketchpad to create a story involving mystery patterns. The challenge is to solve the mystery by gathering clues and explaining the patterns.
- [Stars, Polygons, and Multiples](https://www.sineofthetimes.org/stars-polygons-and-multiples/) - Students create star and polygons with Web Sketchpad and explore the underlying visual and numerical patterns and relationships.
- [More Fun with Number Bases](https://www.sineofthetimes.org/more-fun-with-number-bases/)
- [The Pulse of Contemporary Turkish](https://www.sineofthetimes.org/the-pulse-of-contemporary-turkish/)
- [When Factoring Gets Personal](https://www.sineofthetimes.org/when-factoring-gets-personal/) - Daniel Scher describes how a Sketchpad factoring game stimulates students' creativity and problem-solving skills by allowing them to create and share puzzles.
- [Factor Patterns at Your Fingertips](https://www.sineofthetimes.org/factor-patterns-at-your-fingertips/) - Daniel Scher offers an interactive factor array for exploring factor patterns in your web browser.
- [Eigenvectors of 2 x 2 Matrices: A Geometric Exploration](https://www.sineofthetimes.org/eigenvectors-of-2-x-2-matrices-a-geometric-exploration/) - Shiva Gol Tabaghi presents an interactive Web Sketchpad model for exploring eigenvectors from a geometric perspective.
- [Surprise, Surprise: Tangent Circles Produce an Ellipse](https://www.sineofthetimes.org/surprise-surprise-tangent-circles-produce-an-ellipse/) - Daniel Scher presents an interactive Web Sketchpad model that draws ellipses using tangent circles as part of the underlying construction.
- [Arthur Ganson and the Excitement of Construction](https://www.sineofthetimes.org/arthur-ganson-and-the-excitement-of-construction/) - Daniel Scher offers an interactive Web Sketchpad model inspired by the kinetic sculptures of Arthur Ganson.
- [Questioning Some Dynamic Geometry Angle Wisdom](https://www.sineofthetimes.org/questioning-some-dynamic-geometry-angle-wisdom/) - Daniel Scher examines the role of measurement with Sketchpad and concludes that for elementary-age students, it's worth challenging some well-worn assumptions.
- [Logic Puzzles Made Visual](https://www.sineofthetimes.org/logic-puzzles-made-visual/) - Daniel Scher presents interactive Web Sketchpad logic puzzles that build students' deductive reasoning abilities.
- [The Brouwer Fixed Point Theorem](https://www.sineofthetimes.org/the-brouwer-fixed-point-theorem/) - Daniel Scher describes and illustrates with Web Sketchpad an accessible and beautiful geometry problem that relates to the Brouwer Fixed Point Theorem.
- [A Balance Model Approach to Algebraic Equations](https://www.sineofthetimes.org/a-balance-model-approach-to-algebraic-equations/) - Daniel Scher presents a Web Sketchpad pan balance model for introducing young learners to algebraic equations in an intuitive, hands-on way.
- [The Dynamic Ebbinghaus Illusion](https://www.sineofthetimes.org/the-dynamic-ebbinghaus-illusion/) - Daniel Scher presents an interactive Web Sketchpad model of the award-winning Dynamic Ebbinghaus illusion.
- [Reasoning with Multiples to Find the Mystery Number](https://www.sineofthetimes.org/reasoning-with-multiples-to-find-the-mystery-number/) - Daniel Scher describes an interactive Web Sketchpad model that challenges elementary students to find a "mystery" number based on clues about multiples.
- [Exploring Factor Rainbows](https://www.sineofthetimes.org/exploring-factor-rainbows/) - Daniel Scher presents a Web Sketchpad model for representing factors and multiplication visually in the context of a bunny jumping along a number line.
- [Can Computer-Based Assessment Model Worthwhile Mathematics?](https://www.sineofthetimes.org/can-computer-based-assessment-model-worthwhile-mathematics/) - Spurred by a post by Dan Meyer, Daniel Scher examines what meaningful computer-based assessment might look like with Web Sketchpad.
- [A Coordinate Plane Logic Puzzle](https://www.sineofthetimes.org/a-coordinate-plane-logic-puzzle/) - Daniel Scher presents a Web Sketchpad-based coordinate plane logic puzzle from the Everyday Mathematics curriculum.
- [Understanding Fraction Multiplication](https://www.sineofthetimes.org/understanding-fraction-multiplication-2/) - Daniel Scher presents an interactive Web Sketchpad area model for understanding and performing fraction multiplication.
- [Comparing and Identifying Fractions Visually](https://www.sineofthetimes.org/comparing-and-identifying-fractions-visually/) - Daniel Scher presents a Web Sketchpad activity in which students compare the size of two fractions, each represented as a part of a circle.
- [An Equivalent Fractions Game](https://www.sineofthetimes.org/an-equivalent-fractions-game/) - Daniel Scher presents a Web Sketchpad game for exploring equivalent fractions.
- [An Interactive Approach to Finding nth Roots](https://www.sineofthetimes.org/an-interactive-approach-to-finding-nth-roots/)
- [Raz's Magic Multiplying Machine](https://www.sineofthetimes.org/razs-magic-multiplying-machine/) - Students explore the behavior of multiplication using a "magic" number line in which the product of two values, a and b, moves in response to dragging them.
- [Splitting Arrays](https://www.sineofthetimes.org/splitting-arrays/)
- [Tanton's Two-Pan Balance Puzzle](https://www.sineofthetimes.org/tantons-two-pan-balance-puzzle/)
- [A New Twist on Arranging Addends](https://www.sineofthetimes.org/a-new-twist-on-arranging-addends/)
- [The Polar-Cartesian Connection](https://www.sineofthetimes.org/the-polar-cartesian-connection/) - This model connects polar and Cartesian representations of functions by animating the independent variable and tracing out the same function simultaneously.
- [A Geometric Interpretation of Euler's Identity](https://www.sineofthetimes.org/a-geometric-interpretation-of-eulers-identity/)
- [Slope Games Aplenty](https://www.sineofthetimes.org/slope-games-aplenty/)
- [Maximizing Triangle Area](https://www.sineofthetimes.org/maximizing-triangle-area/)
- [Wait for a Date](https://www.sineofthetimes.org/wait-for-a-date/) - Daniel Scher provides a web-based Sketchpad model for exploring a classic probability problem through geometric modeling and simulation.
- [What's New with Web Sketchpad in 2019](https://www.sineofthetimes.org/whats-new-with-web-sketchpad-in-2019/)
- [Bunny Times Brings Conceptual Understanding to Multiplication](https://www.sineofthetimes.org/bunny-times-brings-conceptual-understanding-to-multiplication/) - Bunny Times presents a visual and interactive approach for children to learn, with understanding, the multiplication facts 2 × 2 through 12 × 12.
- [Arranging Addends Puzzles](https://www.sineofthetimes.org/arranging-addends-puzzles/) - Daniel Scher offers an interactive web browser-based game that gives students practice in mental arithmetic while engaging them in thought-provoking puzzles.
- [Constructing Daisy Designs](https://www.sineofthetimes.org/constructing-daisy-designs/)
- [Danny's Ellipse](https://www.sineofthetimes.org/dannys-ellipse/) - Daniel Scher shares an ellipse construction technique discovered by high-school student Danny Vizcaino and challenges readers to prove that the method works.
- [Constructing Equal-Area Triangles](https://www.sineofthetimes.org/constructing-equal-area-triangles/) - Daniel Scher presents an interactive Web Sketchpad problem focusing on triangle area drawn from the 1990s-era textbook, Connected Geometry.
- [Dissecting Rectangles Into Squares](https://www.sineofthetimes.org/dissecting-rectangles-into-squares/) - How can an arbitrary rectangle be dissected into a square? This blogpost explores several methods and situates the solutions within a larger dissection problem.
- [Zooming Integers: Magnifying the Number Line](https://www.sineofthetimes.org/zooming-integers-magnifying-the-number-line/) - Daniel Scher presents an interactive model for estimating the location of integers by repeatedly magnifying a number line.
- [A Geometry Challenge from Japan](https://www.sineofthetimes.org/a-geometry-challenge-from-japan/) - Dynamic geometry and Web Sketchpad provide a perfect environment for exploring an intriguing Japanese geometry problem.
- [Creating Origami Whirlpool Designs with Sketchpad](https://www.sineofthetimes.org/creating-origami-whirlpool-designs-with-sketchpad/) - Daniel Scher describes how to use The Geometer's Sketchpad to create origami whirlpool patterns based on the work of Tomoko Fuse.
- [Estimating Angle Measurement](https://www.sineofthetimes.org/estimating-angle-measurement/) - Students estimate the measurements of random angles using Web Sketchpad. Tick marks at every 10 degrees serve as an aid, but are gradually stripped away.
- [Creating Animated Factorization Diagrams](https://www.sineofthetimes.org/creating-animated-factorization-diagrams/) - Coaches worked with the company Metamorphosis to study and pilot an activity on animated factorization diagrams with Web Sketchpad.
- [The Perimeter Problem](https://www.sineofthetimes.org/the-perimeter-problem/)
- [π Day 2014](https://www.sineofthetimes.org/π-day-2014/) - Born on π Day eve, Scott Steketee shares his favorite dynamic ways to relate the area of a circle to its radius and circumference. Check out the explosion!
- [Playing with Triangular Decompositions](https://www.sineofthetimes.org/playing-with-triangular-decompositions/) - Guest blog author Juan Camilo Acevedo presents a collection of interactive Web Sketchpad challenges that illustrate the educational role of play.
- [Dilation Challenges](https://www.sineofthetimes.org/dilation-challenges/) - Scott Steketee describes new Sketchpad activities connecting geometric transformations and linear functions and shows a web-based intro that sets the scene.
- [Covering Your Bases: An Interactive Dials Model](https://www.sineofthetimes.org/covering-your-bases-an-interactive-dials-model/) - Daniel Scher revives the exploration of bases other than 10 as a valuable educational topic through the use of two interactive Web Sketchpad models.
- [A Pythagorean Dissection](https://www.sineofthetimes.org/a-pythagorean-dissection/) - Guest blogger Nathan Dummitt describes a dissection problem that foreshadows the Pythagorean Theorem and is ideal for kickstarting the first day of class.
- [Dancing Unknowns: You Haven't Seen Simultaneous Equations Like These!](https://www.sineofthetimes.org/dancing-unknowns-you-havent-seen-simultaneous-equations-like-these/) - Daniel Scher offers an interactive Web Sketchpad activity in which elementary-age students solve for unknowns as the unknowns "dance" across the screen.
- [Moving Beyond Formulas When Investigating Triangle Area](https://www.sineofthetimes.org/moving-beyond-formulas-when-investigating-triangle-area/) - Daniel Scher presents two interactive Web Sketchpad models to demonstrate Everyday Mathematics' approach to developing students' understanding of triangle area.
- [WCYDWT: A Ball, a Trash Can, and Web Sketchpad](https://www.sineofthetimes.org/wcydwt-a-ball-a-trash-can-and-web-sketchpad/) - Scott Steketee borrows one of Dan Meyer's WCYDWT pictures and creates some Web Sketchpad tools for students to use in their investigation.
- [Constructing Morley Triangles](https://www.sineofthetimes.org/constructing-morley-triangles/) - Guest blogger Adrienne Barrett has used Sketchpad to explore the Morley Triangles. In this post she helps the reader construct the first three Morley Triangles.
- [Hello Spring!](https://www.sineofthetimes.org/hello-spring/) - Ninth grade student Anna Nguyen contributes this week's sketch to welcome the start of spring using colors, shapes, animation, and music.
- [Solving Simultaneous Equations with Common Sense](https://www.sineofthetimes.org/solving-simultaneous-equations-with-common-sense/) - Daniel Scher describes a Web Sketchpad-based activity that allows elementary-age children to solve simultaneous equations by leveraging their common sense.
- [Reflecting on the Annual NCTM Meeting](https://www.sineofthetimes.org/reflecting-on-the-annual-nctm-meeting/) - Daniel Scher offers two interactive Web Sketchpad models relating to reflection.
- [The Broken Stick Puzzle](https://www.sineofthetimes.org/the-broken-stick-puzzle/) - Daniel Scher uses Web Sketchpad to explore the probability that a stick broken randomly at two locations can be assembled to form a triangle.
- [International Congress for Mathematics Education Part 2](https://www.sineofthetimes.org/international-congress-for-mathematics-education-part-2/) - ICME is the biggest international math conference, held every year. In this part I report on my own presentation, and I include one of the websketches.
- [Function Dances at NCTM](https://www.sineofthetimes.org/function-dances-at-nctm/) - In this networking lounge presentation, we enacted geometric transformations in the form of dances, with our bodies as the variables.
- [Connected Geometry](https://www.sineofthetimes.org/connected-geometry/) - Connected Geometry, a high-school geometry curriculum developed at EDC in the 1990s, is now available for free from the Internet Archive.
- [The Wavy Wavy Bridge](https://www.sineofthetimes.org/the-wavy-wavy-bridge/) - Learn how to use Sketchpad to construct the Wavy Wavy Bridge special effect. This special effect adds a wave effect to any picture or other object.
- [Binomial Multiplication and Concreteness Fading](https://www.sineofthetimes.org/binomial-multiplication-and-concreteness-fading-2/) - A progression from two-digit multiplication to binomial multiplication illustrates the pedagogical concept of "concreteness fading," where concrete materials gradually and explicitly fade toward more abstract ones.
- [Introducing the Dynamic Number Project](https://www.sineofthetimes.org/introducing-the-dynamic-number-project/) - Daniel Scher provides an overview of the all-new Dynamic Number website that features 70 free Sketchpad activities for grades K through 11.
- [International Congress for Mathematics Education Part 1](https://www.sineofthetimes.org/international-congress-for-mathematics-education-part-1/) - ICME is the biggest international math conference, held every year. Here are a few highlights in Part 1. In Part 2 I'll report on my own presentation.
- [Connecting Functions in Geometry and Algebra](https://www.sineofthetimes.org/connecting-functions-in-geometry-and-algebra/) - The Feb. 2014 issue of Mathematics Teacher features an article connecting geometry and algebra through functions and uses interactive illustrations throughout.
- [Exploring Tessellations with Web Sketchpad](https://www.sineofthetimes.org/exploring-tessellations-with-web-sketchpad/) - Daniel Scher offers three Web Sketchpad models for creating tessellations of regular polygons and arbitrary triangles and quadrilaterals.
- [Branding isn’t about math. Are you sure about that?](https://www.sineofthetimes.org/branding-isnt-about-math-are-you-sure-about-that/) - Karen Coe discusses the math and use of Sketchpad behind Key Curriculum's new logo, which is a visual representation of pi.
- [The Mathematics of Logos](https://www.sineofthetimes.org/the-mathematics-of-logos/) - Karen Greenhaus explores company logos created using mathematical constructs and suggests using Sketchpad to recreate and discover the mathematics behind logos.
- [Early Childhood Math Routines](https://www.sineofthetimes.org/early-childhood-math-routines/)
- [Sketchpad Reflection Puzzles](https://www.sineofthetimes.org/sketchpad-reflection-puzzles/) - Daniel Scher describes how Sketchpad 5's transformation capabilities can be used to create letter recognition puzzles by cropping letters of the alphabet and reflecting what remains.
- [Oh, the Fractions You'll See!](https://www.sineofthetimes.org/oh-the-fractions-youll-see/) - Daniel Scher describes his work in the Dynamic Number project as students use Sketchpad to create and explore fractions they would not typically encounter.
- [Catching Up with New Web Sketchpad Functionality](https://www.sineofthetimes.org/catching-up-with-new-web-sketchpad-functionality/)
- [Picturing Functions and Functions of Pictures](https://www.sineofthetimes.org/picturing-functions-and-functions-of-pictures-at-the-baltimore-nctm-regional/) - This movie provides a short introduction for the presentation I will make on Functions and Pictures at the NCTM 2013 Regional in Baltimore.
- [Pass the Candy! Sweetening the Study of Probability](https://www.sineofthetimes.org/pass-the-candy-sweetening-the-study-of-probability/) - Kathy Shafer uses TinkerPlots to simulate uniform and nonuniform distributions to help teachers make deep connections among probability concepts.
- [Exploring the Law of Large Numbers with TinkerPlots](https://www.sineofthetimes.org/exploring-the-law-of-large-numbers-with-tinkerplots/) - Kathy Shafer describes how she used TinkerPlots in a lesson for teachers about the Law of Large Numbers, and demonstrates how technology can be used to explore "what-if" questions.
- [Sketchpad Activities, Cognitive Demand, and Differentiation](https://www.sineofthetimes.org/sketchpad-activities-cognitive-demand-and-differentiation/) - How can a Sketchpad activity best promote deep mathematical thinking? Four versions of the same activity try to address this problem. What do you think?
- [A Funhouse Mirror of Sketchpad Transformations](https://www.sineofthetimes.org/a-funhouse-mirror-of-sketchpad-transformations/) - Daniel Scher borrows an idea from the book Math Bytes by Tim Chartier to transform pictures into funhouse mirror images using polar coordinates and Sketchpad.
- [Aligning Objects in Sketchpad](https://www.sineofthetimes.org/aligning-objects-in-sketchpad/) - You can align text and other objects in Sketchpad. This video shows how to align text on a page and how to make objects line up from one page to the next.
- [What Does Rowing Have to Do with Teaching Mathematics?](https://www.sineofthetimes.org/what-does-rowing-have-to-do-with-teaching-mathematics/) - How to make a connection between the perseverance and effort required demanded by sport and the Common Core Standards for Mathematical Practice.
- [Exponential Harmony with Sketchpad](https://www.sineofthetimes.org/exponential-harmony-with-sketchpad/) - My Math Methods class helped brainstorm this activity using exponential functions to explore frequencies of harmonious notes and possible piano tunings.
- [A Swan Song for Sweet Karen Coe](https://www.sineofthetimes.org/a-swan-song-for-sweet-karen-coe/) - Andres Marti describes how Karen Coe brought Ignite talks to the mathematics education community, and features a video from Key's number one fan, Jennifer North Morris.
- [ICME: The Nature of Students’ Mathematical Thinking](https://www.sineofthetimes.org/icme-the-nature-of-students-mathematical-thinking/) - Scott Steketee reflects on research presented at ICME that mathematical concepts in students' minds are connected to their real-world sensory-motor experiences.
- [The Standards Are Not The Curriculum](https://www.sineofthetimes.org/the-standards-are-not-the-curriculum/) - Jamila Riser urges educators to focus on teaching mathematics and not individual standards. Mathematics educators should focus on the goal of creating mathematically competent and literate citizens.
- [ICME: A Sensory-Motor Experience of Korea](https://www.sineofthetimes.org/icme-a-sensory-motor-experience-of-korea/) - Scott Steketee reflects on his experiences at the ICME (International Congress on Mathematical Education) conference in Seoul, South Korea.
- [Preparing Tomorrow's Data Scientists Through Data Games](https://www.sineofthetimes.org/preparing-tomorrows-data-scientists-through-data-games/) - Rick Gaston discusses the field test of one of the new online Data Games that KCP Technologies is developing, which help students develop data analysis and modeling skills.
- [Parents, Children, and Functions in Sketchpad](https://www.sineofthetimes.org/parents-children-and-functions-in-sketchpad/) - Scott Steketee describes parent/child relationships in Sketchpad as functions. Students enhance function understanding by making sense of these relationships, .
- ["Data Games"—Getting the Math Right in Math Games](https://www.sineofthetimes.org/data-games-getting-the-math-right-in-math-games/) - Rick Gaston describes Data Games—a KCP Technologies project to create high-quality, engaging games that help students learn data analysis and math skills.
- [What's up with Key? A Response to Dan Meyer](https://www.sineofthetimes.org/whats-up-with-key-a-response-to-dan-meyer/) - Andres Marti responds to Dan Meyer's tweet by explaining how Key has shifted its focus from textbook publishing to educational technology software development.

## Pages

- [About](https://www.sineofthetimes.org/about/) - Sine of the Times is a K-12 mathematics education blog that explores the learning opportunities made possible by Web Sketchpad technology.
- [Privacy Policy](https://www.sineofthetimes.org/privacy-policy/) - A privacy policy for the blog Sine of the Times, covering information recorded, use of information, and disclaimers about off-site linking.

## Categories

- [Uncategorized](https://www.sineofthetimes.org/category/common-core-general/)
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- [Conferences](https://www.sineofthetimes.org/category/conferences/)
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- [Real-World Math](https://www.sineofthetimes.org/category/real-world-math/)
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- [Geometric Functions](https://www.sineofthetimes.org/category/geometric-functions/)
- [Algebra](https://www.sineofthetimes.org/category/algebra/)

## Tags

- [Fathom](https://www.sineofthetimes.org/tag/fathom/)
- [Dynamic Data](https://www.sineofthetimes.org/tag/dynamic-data-software/)
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- [artificial intellligence](https://www.sineofthetimes.org/tag/artificial-intellligence/)
- [A.I.](https://www.sineofthetimes.org/tag/a-i/)
- [fractal](https://www.sineofthetimes.org/tag/fractal/)