Active, Engaging, Effective Learning

FogStone Isle engages late elementary through middle school students in creatively crafting their own island, with math intrinsic to all activities.

The program constantly adapts to each students’ level, adjusting the difficulty of the problems and the concept being learned as they progress through the game.

Our team of teachers, neuroscientists, and game designers build on research-supported approaches, working to teach current curriculum while also preparing kids for future learning, illustrating the conceptual progressions that tie mathematical topics together.

Below, learn more about a few of our activities, with examples of the underlying conceptual learning supported.

Fraction Equivalence

An understanding of the concept underlying equivalent fractions is at the root of all operations with fractions, for example fraction comparison, simplification, and addition by finding common denominators. Without that conceptual understanding, students tend to just memorize procedures.

In learning fraction equivalence and simplification, students build chimneys to keep Gruffins (the local animals) warm on cold winter nights. Building a chimney involves finding matched pairs of tiles in order to build a symmetric structure. The tiles are of fractional sizes, so pairs of tiles that are equivalent fractions must be found.

Understanding the concept of fraction equivalence begins with a visual approach. Tools are available to aid understanding, based on the commonly used “big one” approach, with a clear visual impact of moving between equivalent fractions based on the “big one”.

Figures 1 and 2 show the comparison of two round shapes – and make it clear that despite different subdivisions, the same amount of each round is shaded. The tool enables the player to physically change the subdivisions. Over time, students relate visual fractions to symbolic fractions, as seen in Figures 3 and 4. Ultimately, players gain practice comparing pairs of symbolic fractions. (Figs 5 and 6). In all cases, the same tool is available – helping transfer the visual knowledge to symbolic knowledge.

FIGURE 1
FIGURE 2
FIGURE 3
FIGURE 4
FIGURE 5
FIGURE 6

Fraction Addition

When crafting buildings, students are immersed in fraction addition and subtraction. Using fractional-sized bricks, students build houses for Gruffins, the friendly creatures that inhabit their island. Fraction addition and subtraction are taught conceptually using fraction bars and number lines. Want your building to stand up straight? You’ll need to use the appropriately sized bricks. Scaffolding tools (called the “sketchpad”) allow students to use visuals and virtual manipulatives to help them make meaning of the math. In particular, the visual model used for equivalent fractions is extended with fraction bars and number lines, allowing students to explore ways to use equivalent fractions to find like denominators. That relationship to the earlier tool from the fraction equivalence game helps students transfer their earlier learning of equivalent fractions to the search for like denominators. Over time, students will sequence through a series of levels in which they solve more complex combinations of unlike denominators.

Figures 7 – 11 show the use of the sketchpad to find a like denominator as students add rows of bricks to their house.

FIGURE 7
The player would like to complete the second row from the bottom. First, the fractions must be added, then the sum must be subtracted from 2 (the size of the row) to determine which brick to use to fill the row.
FIGURE 8
Now the player brings up the sketchpad tool. The equivalent fraction tool (on the bottom line) is now used to operate on the fraction bars representing the bricks on the number line, helping the student envision fractions as magnitudes.
FIGURE 9
Here, the player has found a way to use equivalent fractions to get to like denominators.
FIGURE 10
Like denominators made the addition easy. Notice that to find the brick that can complete the row, the player must again use their fluency with equivalent fractions to find a brick that fits (in this case, the 7/6 brick).
FIGURE 11
Here is the completed row.

Fraction Multiplication

Research projects have reinforced what teachers already know about teaching fraction multiplication — the best way to illustrate the concept behind fraction multiplication is with an area model.

In our activity, the area model has become a farm. The students engage in an exciting turn-based strategy game to take over their farm, while battling weeds. With each turn, students estimate whether they have enough water to irrigate the section of the farm they would most like to irrigate, and make strategic considerations, comparing multiple possible moves (see Figure 12).

This process brings into play many different areas of the brain, as they simultaneously deal with the area model of the farm, numerical fractions representing their choices, the resulting symbolic equation that must be solved, estimation of water needed for their move, and comparison to the available water in their tank, represented in decimal. On the average, kids make two moves per minute – thinking through this entire process repeatedly, as problems get harder (due to different subdivisions of the farm field).

Students learn about cross-simplification and experience how that can make fraction multiplication easier. The visual tool to support cross-simplification builds on the tools they used earlier for fraction equivalence and fraction addition – thereby showing the conceptual relationship to earlier learning. (see Figure 13). This new tool helps extend the concept of equivalent fractions to the concept that they can divide out shared factors between any of the numerators and any of the denominators. In the later levels of the fraction multiplication activity, students learn to move between mixed numbers and improper fractions.

FIGURE 12
FIGURE 13

Fraction Division

Teachers are well aware of the difficulty of teaching the underlying concepts behind the invert-and-multiply approach to fraction division. Multiple master teachers, researchers, and game designers collaborated on this design, which carefully steps through the underlying concepts – first showing why the denominator of the divisor becomes a numerator (working with unit fraction divisors), and then showing why numerators become a denominator (illustrating with non-unit numerators). Finally, everything is put together, as the student gradually increases their understanding and succeeds in solving harder problems with fractional answers. Since release, many teachers have praised our approach as an important breakthrough.

All of this is encapsulated in the action of a fun and engaging game: the Gruffins raise Rollerdillos, a feisty creature that tends to run amok if not kept corralled. In this exciting game, students use fraction division to create a pen around the rolling Rollerdillos before they escape. Each time a piece of fence is built, the student is solving a fraction division problem. Fraction division is taught conceptually using the panels of the fence as the divisor and the fence length as the dividend. The activity goes through a series of fraction division problems to illustrate why you invert and multiply when solving a fraction division problem.

Ratio and Proportion

To build useful products for the Gruffins, students can create factories within existing buildings. Students will solve different types of ratio/proportion problems with various unknowns. For example, they may need to calculate the amount of ingredients that are needed given the ingredient ratio and the number of products to be manufactured, or maximize the number of items manufactured given various constraints on availability of ingredients. The variations of problem types encourage the students to think about ratios/proportions from various viewpoints.

In the example shown in this video, the student has discovered how to use equivalent ratios to simplify manufacturing – creating products requiring a final parts ratio of 6:4 by running through two phases of stamping out parts in a 3:2 ratio – thereby cutting in half the number of machines needed to manufacture their product.

Adding and Subtracting Decimals

This game gives students practice adding and subtracting decimals to the thousandths. By choosing from a variety of wallpaper colors and styles, students decorate the inside walls of the houses they have built on FogStone Isle. Students are given tools to add wallpaper (adding decimals) and cut wallpaper (subtracting decimals) to obtain the right fit for each inside wall.

Scaffolding tools and inquiry-based lessons are included to deepen conceptual understanding and address misconceptions. For example, to help students make the connection between fractions and decimals, symbolic models like the one in figure 1 are used. The scaffolding tool enables students to see that 8.6 is the same as 8 ones and 6 tenths and that 3.2 is the same as 3 ones and 2 tenths. It shows the big ideas that decimals are a way of writing fractions within the base-ten system, that the decimal point indicates the unit’s position, and that adding and subtracting with decimals is based on the concept of adding and subtracting the numbers in like position values.

When students submit an answer that indicates they did not add or subtract according to proper place value alignment, an inquiry based lesson is triggered to help explore the misconception. The student is asked to solve three addition problems, each using the same values but lined up differently. The player is prompted to solve each problem in turn so answers can be compared (see figure 2). It becomes clear very quickly that lining up numbers in different ways makes a big difference in the answers they get. The lesson goes on to further explore place value alignment and give students the opportunity to look at place value alignment when adding decimals (see figure 3).

FIGURE 1
FIGURE 2
FIGURE 3
FIGURE 7
The player would like to complete the second row from the bottom. First, the fractions must be added, then the sum must be subtracted from 2 (the size of the row) to determine which brick to use to fill the row.
FIGURE 8
Now the player brings up the sketchpad tool. The equivalent fraction tool (on the bottom line) is now used to operate on the fraction bars representing the bricks on the number line, helping the student envision fractions as magnitudes.
FIGURE 9
Here, the player has found a way to use equivalent fractions to get to like denominators.
FIGURE 10
Like denominators made the addition easy. Notice that to find the brick that can complete the row, the player must again use their fluency with equivalent fractions to find a brick that fits (in this case, the 7/6 brick).
FIGURE 11
Here is the completed row.

Throughout the many games and activities on FogStone Isle, visuals and tools are used both to reinforce conceptual understanding and to relate new concepts to earlier learning – gradually building up the student’s intuitive understanding of mathematical structure. Constant, detailed adaptation to each student keeps them interested and challenged.

For more information on some of the underlying science behind our approach, and how that factors into our games, see:

https://cignition.com/neuroscience/

and

https://cignition.com/workingmemory/

Start Transforming How Kids View And Learn Math Today

Active, Engaging, Effective Learning

FogStone Isle engages late elementary through middle school students in creatively crafting their own island, with math intrinsic to all activities.

The program constantly adapts to each students’ level, adjusting the difficulty of the problems and the concept being learned as they progress through the game.

Our team of teachers, neuroscientists, and game designers build on research-supported approaches, working to teach current curriculum while also preparing kids for future learning, illustrating the conceptual progressions that tie mathematical topics together.

Below, learn more about a few of our activities, with examples of the underlying conceptual learning supported.

Fraction Equivalence

FIGURE 1

FIGURE 2

FIGURE 3

FIGURE 4

FIGURE 5

FIGURE 6

Fraction Addition

FIGURE 7The player would like to complete the second row from the bottom. First, the fractions must be added, then the sum must be subtracted from 2 (the size of the row) to determine which brick to use to fill the row.

FIGURE 8Now the player brings up the sketchpad tool. The equivalent fraction tool (on the bottom line) is now used to operate on the fraction bars representing the bricks on the number line, helping the student envision fractions as magnitudes.

FIGURE 9Here, the player has found a way to use equivalent fractions to get to like denominators.

FIGURE 10Like denominators made the addition easy. Notice that to find the brick that can complete the row, the player must again use their fluency with equivalent fractions to find a brick that fits (in this case, the 7/6 brick).

FIGURE 11Here is the completed row.