Here is an activity designed to help students really think about and understand the grouping by ten that forms the foundation of our decimal number system.
Access this activity at the National Library of Virtual Manipulatives site.
Begin with decimal places = 0, base = 10, and, depending on your grade level, set the columns to 2, 3, or 4. Begin clicking on the single cube in the ones place, and watch the display of the number on the right. You will notice that the numbers disappear after 9. Ask students why they think this is so.
After they offer explanations, show them how you can group ten single cubes to make one group of ten by clicking outside the group and enclosing the ten cubes in a rectangle. Then show how you can move the group of ten to the next column to the left, and watch the number on the right re-appear. Ask students how the digits in the written number correspond to what they see with the blocks.
You can repeat this with the tens, hundreds, and thousands columns, grouping by tens and moving the new group to the place value on the left. Show students that, just as you “composed” a ten or a hundred or a thousand, you can “de-compose” a group and move them to the place value on the right – and that a group always decomposes to ten of the place value to the right. Ask them where they have seen or used something like this before (this is the concept they use when subtracting with regrouping).
Next, ask your class why they think we always group by tens. Let them think about what it is about the number ten that so attracted the people who developed our current number system. Lead them to see that, although we don’t know for sure, it is probably because we have ten fingers and people traditionally used their fingers (and other body parts) to count objects. Then tell them that Aliens have landed on Earth who have only 5 fingers and they want to teach us their number system. Change the base on the Virtual Manipulative to 5, and repeat what you did for base 10. Watch the numbers on the right and see when they disappear. Group by fives and move groups over just as you did for grouping by tens.
You can repeat this activity with base 4, 3, and 2 (binary). Discuss which digits are needed for the different base systems and why. Make up a homework assignment where students have to convert numbers between different base systems. For advanced students, have them figure out how to add and subtract with regrouping in a different base system (they can experiment with the activities for addition and subtraction). Ask them to teach the class what they have discovered!
For a cool project, have students create a poster of a common scene where all the numbers have been replaced by numbers in base 2, 3, 4, or 5, and have them show the conversions to base 10. By posting these around your room, students will be reminded all year long about the reasoning behind our place value system and how we work with base 10 numbers.