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The Dynamic Number activities range from ten-frame models suitable for kindergarten to algebra models for grades 7 and higher. Below are descriptions of the eight mathematical themes that define the Dynamic Number project.


The Dynamic Number activities range from ten-frame models suitable for kindergarten to algebra models for grades 7 and higher. Below are descriptions of the eight mathematical themes that define the Dynamic Number project.

Addition & Subtraction
Grade Level K–4

The addition and subtraction activities focus on conceptual understanding and fluency in mental mathematics. They use three dynamic representations (ten frames, the 99 chart, and the number line) and two puzzles (cross number puzzles and paired sums).

The ten-frame activities concentrate on composing and decomposing numbers using tens. The 99 chart activity develops students’ mental math fluency as they trace colored paths through the chart to add and subtract numbers. The number line activities bring dynamic visualization of addition and subtraction strategies to number-line navigation. The puzzles encourage strategies for doing mental mathematics and using place value.

Grade Level 2–5

The multiplication activities develop students’ conceptual understanding of multiplication through a variety of array models.

In the construct-a-building activity students use a rectangular array model of multiplication to learn, with understanding, the multiplication facts 2 × 2 through 11 × 10. The bug multiplication activity introduces multiplication as a scaling operation. In the two-digit multiplication activities, students decompose two-digit multiplication problems into tens and ones, manipulate an array to change the number of tens and ones in each dimension, find the four partial products, and add the partial products to find the total area (that is, the product).

Early Algebraic & Logical Thinking
Grade Level 2–8

The early algebra and logical thinking puzzles introduce students to the concepts of equations and unknowns through engaging puzzles that embed algebraic reasoning in contexts where it arises naturally.

Two-pan balance activities contextualize the steps needed to isolate and solve for an unknown by having students move and remove weights on a scale, actions that students undertake without an introduction to formal procedures. “Mystery” sum activities and a “sneaky” sum activity focus on solving for multiple unknowns and are inspired by the work of the mathematician W. W. Sawyer.

Multiples & Factors
Grade Level 3–6


The multiples and factors activities develop students’ conceptual understanding by employing dynamic number grids and arrays. Students develop fluency through strategies such as adding on, skip counting, and breaking problems down into “friendly” facts.


The dynamic number grid activities display multiples of any two numbers simultaneously, with the ability to change the dimensions of the grid to reveal patterns of common multiples. An interactive array activity displays all of the numbers from 1 to n in color-coded form: Those numbers that are factors of n are shaded blue; the remaining numbers are shaded orange. Through manipulating the array, students uncover mathematical relationships in the pattern of factors.

Grade Level 3–6

The fraction activities use dynamic, student-controlled visual models that promote both conceptual understanding and computational fluency through area representations (both rectangular and circular) and number-line models.

The activities specifically address estimating and comparing fractions, creating fractions greater than one, creating fractions that add up to one, and adding and subtracting fractions. In each activity, students use special tools to easily and accurately construct fractions of their own choice. Many of the activities are designed as games or puzzles, providing additional motivation and interest.

Decimals & Place Value
Grade Level 3–7

The decimals and place value activities reveal the structure of our number system by providing ever-finer magnified views of number lines and by making it easy to explore the patterns contained in the decimal representations of fractions.

The number line activities allow students to repeatedly “zoom in” on intervals along a number line to refine their initial estimate of a point’s location. For example, a student may refine an estimate of 3.5 to 3.52, and then to 3.517. Through this process, students discover that divisions of tenths, hundredths, thousandths, and beyond are all structurally similar. They see that no matter how fine the magnification, there are always more precise values that can be viewed by zooming in yet again.

Grade Level 8–9

The algebra activities focus on multiplying binomials and factoring trinomials using dynamic algebra tiles. The models are notable for their adjustably-sized x tiles, for their representation of partial products, and for their support of negative areas.

Using an area model, students drag sliders to control the unit length and x-length of the tiles and drag points to control the coefficients of the terms. The binomial multiplication activities build from problems like (+ 2) (+ 3) to problems like
(3– 4) (2x – 3). The trinomial factoring games develop the reverse process of factoring trinomials by progressing from problems like x2 + 5+ 6 to problems like 6x2 – 7x – 5.


Geometric Functions
Grade Level 7–11

Geometric Functions provide a compelling entrée to function concepts by developing geometric transformations as functions, thus emphasizing sensory-motor experiences of functions while strengthening understanding of geometric transformations.

Students create points as their independent and dependent variables and label these variables using function notation. They then drag the independent variable to observe the variables’ relative rates of change and restrict the independent variable to a path to investigate domain and range. Students also compose functions by turning the output of one function into the input of another and use composition to create and understand inverse functions.


1. Breadcrumbs

Navigation format shows you where you are in the website

2. Video Player

Video that demonstrates how to implement the activity

3. Notes & Worksheets

PDF files with teacher notes and student worksheets

4. Sketchpad Files

Sketchpad documents that accompany each activity

5. Objectives

Expected student outcomes for the activity

6. Common Core Connections

Common Core mathematical practices and content standards for the activity

7. Related Activities

Activities with similar content and learning objectives

8. Learn More

Additional information about the topic

Do I need to buy the Dynamic Number activities?

No, all of the Dynamic Number activities and supporting materials are free and can be downloaded from this website.

Do I need to buy The Geometer's Sketchpad to use these activities?

Yes, you do need to purchase Sketchpad to use the activities with desktop or laptop computers. You can download a free evaluation copy here, and get pricing information here.

Can I use the Dynamic Number activities with iPads?

Yes, many of the activities are suitable for the iPad. Sketchpad Explorer is a free iPad app that students can use to do any of the Dynamic Number activities marked as iPad compatible. You can find out how to make the activities available through the iPad here.

Do I need to invest time in learning how to use Sketchpad before my class can start exploring the Dynamic Number activities?

No, you can dive right into any Dynamic Number activity without prior Sketchpad experience. The instructions for using each Sketchpad document are contained in the activity notes and demonstrated in the video that accompanies the activity.

Do I need a full set of lab or classroom computers to use the Dynamic Number activities?

While our activity notes describe students working in pairs at computers, this is by no means the only option. A single classroom computer connected to a projector can be an effective way for your entire class to engage in a Dynamic Number activity.

Do I need to devote an entire class period to each Dynamic Number activity?

No, the activities can be adapted for a variety of uses. For example, you could use a Dynamic Number model of fractions as a whole class, five-minute introduction to a lesson from your textbook.

My classroom time is limited. Are these Dynamic Number activities designed as enrichment or can they replace some of my existing lessons?

The Dynamic Number activities focus squarely on Common Core mathematics content and practices. You can use many of the activities as replacements for your standard lessons. The activities are also ideal for students who are struggling with a particular concept or who would benefit from added challenges.

Are there other Sketchpad activities suitable for elementary and middle-school students?

You'll find more Sketchpad resources and activities here.

This material is based upon work supported by the National Science Foundation under KCP Technologies Award ID 0918733, with grant period September 1, 2009 through August 31, 2013. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.