In the 9-12 classroom, the students once again start with what they know, the golden bead material, and learn a new application. Beginning with learning how to square numbers, students apply knowledge of multiplication in a new way—now they are starting to see that the area of a square equals the measurement across times its measurement up. The students work with this concept with many familiar materials, such as the stamp game and pegboard and are then introduced to a more abstract concept of squaring—finding the square root using these materials. Instead of simply abstracting the concept using the algorithm without materials to show the way, students at this level learn how to obtain the root of a square first by doing all the steps they later learn in the written algorithm.
The next level is taking the known—the ability to find the area of a cube and the trinomial cube (first introduced at the pre-primary level)—and reaching a new level called “cubing.” Students again build the cube of a number first, see what and why the are doing, then learn the notation for these steps. As with square roots, cube roots are introduced first by doing with the materials and then learning the notation for the steps the students already know.
The other areas of the curriculum include learning how to use a compass and ruler to divide lines and make graphs and charts. Understanding ratio and proportion using a calculator for percentages and discovering interest using these same skills. The students can also gain an awareness of compounding interest by once again using the familiar materials when teaching this concept. The classification of numbers is taught by bringing in history to give the students an understanding of how many cultures throughout many centuries were able to function without an understanding of negative numbers or even 0. Games are used when teaching the laws of numbers so that the students gain a true understanding of why subtracting a negative number is similar to adding that number. Algebra is explored by showing the students that they have been doing this mystery number game since they added their first numbers. When the “X” of algebra is shown to simply be a balancing act—that each side of the equation must be balanced, but that there are rules in the game—the students are able to see algebra as both easy and fun.
Finally, the mystery and patterns of numbers are discussed when the students become aware of using bases other than ten when counting and when learning about the Fibonacci Sequence and the Golden Rectangle. Within this area of the curriculum the students become aware that some numbers can be found in nature, and that some numbers somehow end of in recognizable patterns.