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.

Ten Frames

In this pair of activities, students begin by working with a single ten frame that contains anywhere from 0 to 10 counters. They practice to see how quickly they can determine the number of additional counters needed to fill the frame with ten counters in all. In the second activity, students move and arrange counters with a collection of ten frames in ways that make it easier to identify the total number of counters without needing to count them one by one.

You can download the two ten-frame activities to use with students. Download: Activities.zip

You can also watch a movie demonstrating how to use the ten frame activities. Download: TenFrames.mov

Addition on the Open Number Line

Students add two-digit and three-digit numbers on an open number line. They devise different strategies for decomposing and composing numbers to make it easier to find the sum. These strategies include using landmark numbers, jumping by multiples of 10, and splitting addends into smaller “friendlier” numbers.

You can download the Open Number Line activity to use with students. Download: Activity.zip

You can also watch a movie demonstrating how to use the Open Number Line activity. Download: Open Number Line.mov

Dynamic Number Grids

An interactive number grid serves as a unifying model for this pair of activities. The grid displays multiples of any two numbers simultaneously, showing multiples of one number in orange, the other number in blue, and common multiples in an orange-blue combination. By experimenting with a variety of number pairs and different grid dimensions, students will be able to identify least common multiples and analyze visual and numerical patterns associated with common multiples. The emphasis on visual patterns helps students who are better at remembering spatial patterns than sequences of numbers.

You can download the two dynamic number grids activities to use with students. Download: Activities.zip

You can also watch a movie demonstrating how to use the dynamic number grids. Download: Number Grids.mov

Dancing Factors

Students pick a number from 1-36 and see all the factors of that number “dance” around it in a circle. The ease of changing the numbers and watching new dance partners join the circle and old partners bow out allows students to delve into the properties of factors.

You can download the Dancing Factors activity to use with students. Download: Activity.zip

You can also watch a movie demonstrating how to use the Dancing Factors activity. Download: Fractions.mov

Factor Arrays

Students find all the factors of any given number by manipulating a visual model that displays the number as circles in a rectangular array. Dragging the corner of the array changes the number of circles per row. Students examine which dimensions of the array yield the same number of circles in every row.

You can download the Factor Arrays activity to use with students. Download: Activity.zip

You can also watch a movie demonstrating how to use the Factor Arrays activity. Download: Arrays.mov

Area Models of Fractions

Area models of fractions typically use circles or rectangles to represent the whole. While it is easy to draw reasonably accurate pictures of 1/2 or 1/3 by hand, it is much harder to draw fractions like 5/7 or 11/30. This set of fraction activities gives students simple Sketchpad tools that allow them to create accurate pictures of any fraction whatsoever, whether it be 11/30, 29/100, or fractions larger than one. With the world of fractions at their fingertips, students can focus on core mathematical concepts like equivalent fractions, improper fractions, and the ordering of fractions, and experiment with fractions they normally would not encounter in their textbooks.

You can download seven activities that represent fractions as parts of one or more circles. Download: Activities.zip

You can also download the same set of seven activities with the fractions represented as parts of one or more rectangles. Download: Activities.zip

Before you try either set of activities, watch a movie demonstrating how to use them. Download: Fractions.mov

Measuring with Fractions

In this activity, students name the locations of unmarked points along a 0–1 number line using unit fractions (fractions with a numerator of 1) as measuring sticks.

You can download the Measuring Fractions activity to use with students. Download: Activity.zip

You can also watch a movie demonstrating how to use the Measuring Fractions activity. Download: Fractions.mov

Fractions on a Number Line

Fractions are just numbers, and as such, it makes sense to think about them using a number-line approach. This collection of fraction activities allows students to create segments whose lengths are represented as fractions. These segments can be attached to number lines and added and subtracted by placing them end to end. A student might, for example, create the sum 1/4 + 2/5 on one number line, and then on the number line below it, build the same sum as 5/20 + 8/20. This set of activities provides a virtual workspace for adding and subtracting fractions and makes it easy to build accurate, visually rich representations with just a few clicks of the mouse.

You can download five Fractions on a Number Line activities to use with students. Download: Activities.zip 

You can also watch a movie demonstrating how to use the fraction activities. Download: Fractions.mov

Dividing and Subdividing

These activities develop students’ understanding of fractions not just as locations on a number line, but as the result of dividing and subdividing a unit segment into equal parts. One way, for example, to construct 1/6 is to split a unit segment into six equal parts. Another way is to first split the segment in half and then split each half into thirds. Alternatively, the segment may first be split into thirds and then those thirds split in half. These activities give students the tools to explore such approaches to constructing fractions.

You can download two Dividing and Subdividing activities to use with students. Download: Activities.zip

You can also watch a movie demonstrating how to use the Dividing and Subdividing activities. Download: Fractions.mov

Mystery Fractions

In this pair of activities, students develop and refine their estimation skills to identify “mystery” fractions represented by unlabeled tick marks on a 0-1 number line.

You can download two Mystery Fractions activities to use with students. Download: Activities.zip

You can also watch a movie demonstrating how to use the Mystery Fractions activities. Download: Fractions.mov

Zooming Decimals

In this pair of activities (inspired by an idea from the curriculum ThinkMath!), students explore tenths, hundredths, thousandths, and ten-thousandths by repeatedly “zooming in” on a number line to obtain increasingly accurate estimates of a point’s location.

You can download two Zooming Decimals activities to use with students. Download: Activities.zip

You can also watch a movie demonstrating how to use the Zooming Decimals activities. Download: Zooming.mov

Reach the Target

In this activity, students use their estimation skills and knowledge of place value to identify the locations of unlabeled tick marks, exact to units, tenths, hundredths, thousandths, ten thousandths, or hundred thousandths on a collection of customizable number lines.

You can download the Reach the Target activity to use with students. Download: Activity.zip

You can also watch a movie demonstrating how to use the Reach the Target activity. Download: Target.mov

Fill a Box with Chocolate

Students subdivide a rectangular array of chocolate into strips of size 1, 2, and 5. They then skip count to determine the total number of squares in the array. In other variants of the model, students are given the number of rows or columns in the array and the total number of squares and must determine the missing dimension.

You can download the chocolate box activity to use with students. Download: Activity.zip

You can also watch a movie demonstrating how to use the chocolate box activity. Download: Chocolate.mov

Construct a Building

In this this activity, students use a rectangular array model of multiplication to learn, with understanding, the multiplication facts 2 x 2 through 11 x 10. Students play games in which they are asked to find the total number of rooms in a building when given the number of rooms per floor and the number of floors. Students construct a visual model of the building—a rectangular array—to help them solve each problem. As they progress, students develop strategies for finding products and hone their multiplication skills. The activity ends with a multiplication “marathon” to assess students’ progress.

You can download the Construct a Building activity to use with students. Download: Activity.zip

Bug Multiplication

In this activity, students explore multiplication as a scaling operation to learn and practice multiplication facts up to 10 x 10.

You can download the Bug Multiplication activity to use with students. Download: Activity.zip

Array Models of Multiplication

In this pair of activities, students develop strategies for solving one- and two-digit multiplication problems using interactive models of arrays. The models give students the flexibility to decompose multiplication problems into simpler problems with numbers whose products they already know.

You can download the two array activities to use with students. Download: Activities.zip

You can also watch a movie demonstrating how to use the array activities. Download: Arrays.mov

Two-Digit Multiplication with Base-Ten Blocks

In this pair of activities, students use virtual base-ten blocks to model two-digit multiplication.

You can download the two multiplication activities to use with students. Download: Activities.zip

You can also watch a movie demonstrating how to use the multiplication activities. Download: Multiplication.mov

Binomial Multiplication

In this set of four activities, students use virtual algebra tiles to multiply binomials as well as factor trinomials.

You can download the four binomial multiplication activities to use with students. Download: Activities.zip

You can also watch a movie demonstrating how to use the binomial multiplication activities. Download: Binomial.mov

Find the Secret Number

In this game students are detectives who gather clues to find a secret number on a number line. For each clue, they drag a pointer to their guess and then ask Sketchpad to check that guess. The clue appears, telling them whether the secret number is less than the guess, equal to it. or greater than it. Students keep records of their guesses and the clues they've gotten in order to eliminate more and more possible suspects until they are certain of the secret number's identity. As students play, they develop, discuss, and analyze search strategies, and use their strategies to engage in a team competition. This activity helps younger students become familiar with the > and < symbols and it gives all students practice in reasoning, strategizing, and collaborating with classmates. The Make Your Own page provides students the opportunity to create their own game, and is also an excellent way for students to investigate ordering on number lines that include negative numbers. The Two Players page allows students to cooperate in trying to guess both secret numbers with as few clues as possible, and then to play as a team against another team.

You can download the Find the Secret Number activity to use with students. The sketch includes a page of activity objectives and two pages of teacher notes. The worksheet gives students a way of recording what they've learned using paper and pencil as well as onscreen. Download: Find the Secret Number.zip

You can also watch a movie that gives an overview of the activity and of ways of using the sketch. Download: Find the Secret Number.mov 

Balance Scale

This collection of six balance scale activities introduces students, in an intuitive way, to equations, variables, and solving for unknowns.

You can download six balance scale activities to use with students. Download: Activities.zip

You can also watch a movie demonstrating how to use the balance scale activities. Download: Balance Scale.mov

Mystery Sums

This collection of three Mystery Sums activities combines logical and early algebraic reasoning as students develop methods for uncovering the secret numerical values of letters. Using random challenges generated by Sketchpad as well as puzzles created by their classmates, students develop strategies for solving for unknowns that introduce them to techniques from algebra.

You can download three Mystery Sums activities to use with students. Download: Activities.zip

You can also watch a movie demonstrating how to use the Mystery Sums activities. Download: Mystery Sums.mov

Sneaky Sums

Sneaky Sums is a game of logic that develops students’ early algebraic reasoning skills as they use logic to deduce the numerical values assigned to four geometric shapes. A 4 x 4 grid is randomly populated with circles, triangles, squares, hexagons, and blank spaces. Each shape has been assigned a secret numerical value. To determine these values, students ask for clues: the sum of the shapes in any row or column. By piecing together the clues, students determine the numerical values of the shapes. Each new random assignment of shapes to the grid creates a fresh challenge to solve.

You can download the Sneaky Sums to use with students. Download: Activity.zip

You can also watch a movie demonstrating how to use the Sneaky Sums activity. Download: Sneaky Sums.mov

Geometric Functions

Middle school and introductory algebra students can gain a deeper view of functions and avoid common misconceptions by exploring functions that are geometric in nature. Though students do not generally think of transformations and loci as functions, both turn geometric input into geometric output. By using Sketchpad, students can vary an input point continuously, observe the variation in the output, develop a working definition of function, explore families of functions, see how a function maps domain to range, produce exciting graphic effects by applying functions to pictures, and directly experience a profound connection between geometry and algebra.

More information about geometric functions, including Sketchpad activities, teacher notes, worksheets, and movies, can be found on the Geometric Functions page.

Watch a one-hour recorded webinar to learn all about geometric functions.

Transforming Functions

In this trio of activities, students transform a point, stretching or translating it by performing arithmetic on its coordinates. They use this work to define a custom transformation that can stretch or translate an entire picture — applying the transformation not just to a single point, but to a collection of points. Students then apply the transformation to the graph of a function, stretching and translating the graph. Finally, they use their mathematical definition of the point’s transformation to solve for the transformed y-value in terms of the transformed x-value, deriving the familiar formula y' = a f((x'-h)/b) + k. Because the activities set the stage for this final derivation, students are able to do the derivation themselves rather than memorizing a formula.

You can download a collection of three function transformation activities to use with students. Download: Activities.zip.