Base Four Blocks or Thinking Outside the Cube
On a recent visit to a local thrift store, I discovered a cardboard box labelled, "Multi-Base Arithmetic Blocks." This in itself was interesting because I figured they were some sort of base ten blocks set until I noticed the additional text, "Additional Base 4." Because it was in a display case, I asked the clerk if I could look at the box. To my delight, I found a wooden set of blocks very similar to base ten blocks, but using a base of four instead.
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| The box containing the multi-base arithmetic base 4 blocks. |
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| A pile of base 4 arithmetic blocks. |
Even the uninitiated know of other base number systems such as binary (base 2), hexadecimal (base 16) and the Babylonian numbering system (the latter being what we use for seconds and minutes with a base of 60). Many people, however, have difficulty conceptualising numbers in other base systems, and this is probably due, in part, to the fact that we use the same numerals for base number systems under 10 (the decimal system). In the binary system, for example, 0's and 1's are used. In the base 4, or quaternary system, the numbers: 0, 1, 2, and 3 are employed.
After the initial excitement and purchasing the block set for a low price of $5.99, I wanted to find out more about these blocks and their history and find out what other additional bases were available. The information proved to be difficult to find which was surprising with the amount of information available on the Internet. Apparently, the manufacturer, Tiger Toys Ltd of Petersfield in England, is no longer in business. One of the only references to Tiger Toys was a short article about a 2011 reunion of Tiger Toys employees who had worked for the company in the 1960's. Searches on EBay resulted in no results for anything similar to these arithmetic blocks. From other sources, it seems that these blocks were available in different bases up to ten, and that different base number systems were routinely taught in schools in the past. With no additional information, it was time to focus on the more interesting aspect of the base four blocks, actually using them.
If you are familiar with expanded numbers in the base ten system, you probably recognise that the number 4567 can be represented as (4 × 10
3) + (5 × 10
2) + (6 × 10
1) + (7 × 10
0). Notice that the powers of ten format clearly shows us the base number used, in this case 10. To represent numbers in the base 4 system, the powers of 4 are used. The number, 1223
4 (the subscript 4 indicates the base 4 numeral system) for example is represented as (1 × 4
3) + (2 × 4
2) + (2 × 4
1) + (3 × 4
0).
Converting between different base numeral systems takes a little effort, but it is fairly straight-forward. In the case of converting from a base four system to a decimal system, one could just multiply out the number in expanded form. In our previous example, (1 × 4
3) + (2 × 4
2) + (2 × 4
1) + (3 × 4
0), multiplying out would result in (1 × 64) + (2 × 16) + (2 × 4) + (3 x 1) = 64 + 32 + 8 + 3 = 107. The reverse of this is converting a decimal number to a base four number. In the case of 107, keep dividing the number by 4 until you end up with a quotient less than 4. 107 ÷ 4 = 26 R 3; 26 ÷ 4 = 6 R 2; 6 ÷ 4 = 1 R 2. Starting from the last quotient then using the remainder values, you get 1, 2, 2, 3 or 1223 which was our original number in the base four system.
If that confused you, then the base four blocks are for you! Let's model the number 3333
4 using the base four blocks (this is the largest number you can model with these blocks without getting "creative").
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| The base 4 number, 3333, represented with base 4 blocks. |
Students should be able to easily see the divisions in the blocks and find out that the small cubes are worth 1, the rods worth 4, the flats worth 16 and with a little help maybe, the large cubes are worth 64. If you have a set of base ten blocks, students can exchange their blocks for the base ten blocks. For example, trading 1 large block, 2 flats, and 1 rod (64 + 32 + 4 = 100) for a 100 flat gets rid of many of their blocks to start with. Grouping together the two remaining large blocks (128) and two unit cubes (2) enables the student to exchange for a 100 flat and three 10 rods. The final blocks add up to 25 (1 flat, 2 rods, 1 unit) and can be exchanged for two 10 rods and 5 units from the base 10 set. All told, students will end up with 100 + 100 + 30 + 20 + 5 = 255.
Working within the base four system with the blocks is just as easy as working with base 10 blocks. The only different rule is that piles of blocks must be exchanged for larger blocks in groups of 4 rather than groups of 10.
So, what is the point of learning another base number system. This could easily result in some confused students if they don't already have strong skills in place value and the decimal numbering system, but for those students who have it mastered and need some more challenge, teaching other base number systems and getting them to work with them can have certain advantages. The biggest advantage, in my estimation, is in the computer programming field. An understanding of hexadecimal, binary, octal (base 8), Base64 and other numbering systems give students a huge advantage. Even though base 4 numbering systems aren't really in practical use, extending skills from one numbering system to another is always a valuable brain-building activity.
Feel free to comment on this post, especially if you have worked with other multi-base arithmetic blocks or know where to purchase multi-base arithmetic blocks in different base numbers.
