The Dynamic Number Project
The Dynamic Number project is writing curriculum to support its software development efforts. All of the Dynamic Number activities below use the Geometer’s Sketchpad, Version 5. You can download a preview version of Sketchpad to try the activities. We welcome your input on these curriculum materials, especially if you have the opportunity to use them with students. Contact us with your feedback.
The mathematical themes described below served as the basis for our curriculum development. For each theme, we explain the underlying pedagogy and mathematical models that guide our work.
Our work with elementary number properties focuses predominantly on multiple representations for learning about multiples and factors. We’ve written extensively about our factor activities on Key Curriculum’s blog, Sine of the Times. In particular, read Exploring Factor Patterns in an Interactive Array, When Factors Put on Their Dancing Shoes, and When Factoring Gets Personal.
This collection of activities also includes several models that relate to addition, including ten frames and addition on an open number line.
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To develop facility with and understanding of fractions, students need to construct and manipulate a variety of fraction representations. But creating accurate visual representations of fractions is not so easy: Drawing an area model of 2/3 is challenging for most students. Drawing an area model of 2/17 is near impossible for students and teachers alike.
Fractions can be represented as area models (parts of a circle or parts of a rectangle) or as locations on number lines. Different curricula place emphasis on different representations, and it’s important for students to work with both, so we provide activities that encompass both area and number line models of fractions.
Our Sketchpad activities provide tools that allow students to construct accurate models of fractions, including ones they normally might not encounter. These include fractions like 1/99, 2/17, 38/7, and 13/57. All can be created simply by clicking a numerator and a denominator from a list of numerical values. The ease of construction gives students (and teachers) a handy toolkit that allows them to think about “what-if” fraction questions and explore them with our Sketchpad tools.
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Real numbers, by their nature, measure continuous quantities. But number as an elementary school construct is only occasionally continuous; it is most often discrete. In particular, the familiar number-line model employed in elementary curricula is tied to a discrete view of number. This representation has consequences: When asked to name a number between 3 and 4 on a number line, students might say 3.5, but would be unlikely to propose 3.782. Why do we not give attention to such “messy” decimals: those that cannot be represented in exact tenths or with familiar names like 0.25 or 0.75? Reasons include the challenges of working with place value and the difficulty in identifying a location like 7.5362 on a traditional number line.
In our collection of decimal and place-value activities, we take advantage of the visualizations afforded by Sketchpad to create number lines where it is possible to explore decimals at a micro level—where pinpointing the location of 7.5362 is no more difficult than finding 1.5. These kinds of experiences are very useful in broadening and deepening students’ conception of our number system.
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We have pursued several approaches to visualizing multiplication, but have spent the most effort promoting a dynamic array model that focuses on partial products: on decomposing the numbers being multiplied and recomposing the partial products thus generated.
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Early Algebra and Logical Thinking
The work of mathematician and educator W. W. Sawyer serves as a basis for many of our early algebra activities. In his Vision in Elementary Mathematics (Pelican Books, 1970), Sawyer writes:
It is quite possible to use simultaneous equations as an introduction to algebra. Within a single lesson, pupils who did not know what x meant can come not merely to see what simultaneous equations are, but to have some competence in solving them. No rules need to be learnt; the work proceeds on a basis of common sense. The problems the pupils solve in such a first lesson will not be of any practical value. They will be in the nature of puzzles.
In keeping with Sawyer’s recommendation, we’ve fashioned most of our early algebra activities in the form of puzzles whose goal is to determine the “secret” numerical values of letters, or of blanks to be filled in with numbers. The puzzles can be solved without formal algebra and rely entirely on the logical thinking available to most third graders. There is high replay value in these puzzles. Sketchpad can randomly generate an abundance of challenges for students to solve, or students can create puzzles for each other on the “Make Your Own” pages of the models.
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Students have well-known difficulty generalizing the concepts of variable and function. Technology-supported student explorations of geometric functions (functions defined by geometric transformations that take a point to a point) provide valuable opportunities for students to directly construct, manipulate, and observe variables and functions. Students can drag the independent variable, observe and record covariation as a picture on the screen, restrict the domain, identify families, use function notation, and compose functions as they move toward a mature generalized conception of functions as objects.
In our collection of Geometric Function activities, we focus on eight themes through a careful progression of sequenced interactions with Sketchpad. These themes are: Varying Variables, Observing Covariation and Rate of Change, Exploring Function Domain and Range, Distinguishing Function Families, Using Function Notation, Composing Functions, Transitioning to an Object Conception of Function, and Connecting Geometric and Algebraic Functions.
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The Dynamic Number project focuses on grades 2–8 mathematics, but we have included several algebra games in our collection that grew as on offshoot of the array models we developed for multiplication. These games focus on factoring trinomials using dynamic algebra tiles. The activities are notable not only for their visual representation of partial products, but for their representation of products like (2x – 4)(3x – 6) using the concept of tiles with negative area.
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