The core of the Digi-Block Program is a system of small rectangular-shaped blocks and empty holders. These materials enable children as young as four to discover, for themselves, the important relationship between ones and tens – a concept crucial to understanding how arithmetic operations work.
Here, the “ones” or single blocks pack into the small holders to make a larger block. The “smart box” design only allows the holders to close when there are 10 single blocks inside. This unique feature allows children to know that there are always exactly 10 blocks inside the new, larger block once it is closed.
Since there can not be any more or any less, we are able to call it what it is: a “block-of-10″. With Digi-Block, there is no need for recounting or “trading.”
A key distinguishing feature of Digi-Block is that the block-of-10 looks exactly the same as the single block, except it is 10 times as large in volume. The identical shape makes it natural to count blocks-of-10 with the same digits (1,2,3 …) as we count single blocks.
This is a key step in understanding place value. For example, in the number 47, the first digit represents four blocks-of-10 and the second digit represents seven single blocks.
Ten blocks-of-10 can then be packed into the medium holders to create a block-of-100. This block looks the same as the block-of-10, except it is 10 times as large in volume. Again, the “smart box” holders will not close unless there are exactly 10 blocks-of-10 inside the block-of-100.
The progression is repeated with 10 blocks-of-100 packed into the large holder making a Block-of-1000.
In principle, we can continue packing indefinitely to build larger and larger blocks. No matter how large or small the blocks get, they stay the same in two important ways:
- They do not close to make a larger block until there are 10 one-size-smaller blocks inside
- When a new block is formed, it looks exactly like the others, except for the size
The block-of-1000 stands over a foot-and-a-half tall and weighs 18 lbs. This makes quite an impression on elementary school students! Size and weight practicality leads us to end the actual block manufacturing here; however, students can now extend the pattern to visualize larger and larger blocks indefinitely. Taking chalk out to the schoolyard and having students draw what a block-of-10,000, a block-of-100,000 and even a block-of-1,000,000 look like is a common activity at schools using Digi-Block.
In addition to facilitating the understanding of number sense and place value, the blocks also accurately model the addition, subtraction, multiplication and division algorithms in a manner so simple that we have seen first graders do large division problems with the blocks! See the Tutorials section to learn more about block basics and using the blocks for arithmetic operations.
Typically a difficult topic with students, Digi-Block makes decimals a direct extension of the whole number system students learned with Digi-Block in younger grades. Not only do the decimal blocks clarify for students what a decimal is, through modeling with the decimal blocks students can see how addition, subtraction, multiplication and division operations are performed with decimals – the same way as with whole numbers! This is a powerful clarification for students who struggle with understanding decimals. See the Tutorials section for examples of operations with decimals.
Taking the block-of-1000 we just packed, we will now begin unpacking. If we open the block-of-1000 and unpack it one time, we will have 10 blocks-of-100. Unpack one of the blocks-of-100 and we now have 10 blocks-of-10. Unpack one block-of-10 and we have 10 single blocks.
If a single block could be unpacked, what would we find inside?
Ten smaller blocks of course!
While working with the blocks in the early elementary grades, students imagined larger and larger blocks that go on indefinitely to make larger and larger numbers. With decimals, students now extend their imagination to smaller and smaller blocks, which can also go on indefinitely.
With the decimal blocks, students only need to make the transition from thinking about packing larger and larger blocks to thinking about unpacking to create smaller and smaller blocks.
In practice there are limits to the size of holders that can be manufactured or easily used in the classroom; therefore, Digi-Block decimal blocks do not use holders. However, the tenth blocks are a one-tenth scale of the single blocks and the hundredth blocks are a one-tenth scale of the tenth blocks and one-hundredth scale of the single blocks, keeping the integrity of the pattern. Though the decimals do not pack, they do show the visual relationship of the larger blocks to the tenths and hundredths.
With the deep understanding of whole numbers gained through The Digi-Block Program in earlier grades, decimals can easily be introduced as an extension of the whole number system where numbers now get smaller indefinitely. The role of the decimal point, a symbol that causes so much strife to so many students, is only to act as an indicator for the ones unit. Combined with their strong knowledge of place value, students now have a clear understanding of the meaning of decimals, leading to greater comprehension of operations with decimals.
The Digi-Diamond – Four Views of a Number
When single blocks are loose (or unpacked), they embody our intuitive “counting view” of number that we use when we count by ones (“one, two, three,” etc.). When the single blocks are packed into the larger blocks, they represent our image of numbers when organized by ones, tens, hundreds and so on. We call this the “place-value view”. In the diagram below, the counting view of “47″ is represented by forty-seven single blocks laid out in a line. When the same blocks are packed-as-much-as-possible, we get four blocks-of-10 and seven single blocks. This is the place value view of “47″.
By means of the simple processes of packing and unpacking, students gain a concrete understanding of the link between the intuitive idea of counting and the abstract notion of place value. This link is a critical component of number sense.