When children first move beyond tens and ones and venture into the hundreds place, it can be confusing to see the abstract symbols (numbers with place value) if they have not had enough concrete experience with counting. The same can be said about thousands and beyond, and even more so when exploring to the right of the decimal point.
Here is a progression of activities that will help students understand the size of large (and very small) numbers and help them solve problems with adding and subtracting those numbers with and without regrouping.
Buy some bags of beans or lentils in the supermarket. They’re an inexpensive manipulative – you can get thousands for only a few dollars. Or use craft materials – beads, sequins, etc. – that are small and inexpensive.
Tell students that your class has been hired by a factory to count and package the beans for shipping to the store. You will need to package the beans in bags of one hundred. Have the beans (several hundred) in a large container in front of the class, and ask the students to work together to think of an efficient way to count the beans. Also ask for predictions of how long they think it will take to count them all. Discuss as a class, and settle on the idea of grouping by tens as an efficient way to make one hundred. The class will work in small groups: some groups will be “counters”, who are responsible for counting ten beans into small cups, some will be “deliverers”, who will wait for a group to have 10 small cups of 10 beans each and then deliver them to the “packagers”, who will spill the ten cups into a bag. Ask the class how long they think it will take working in this way. Then start up your factory. Time how long it takes and compare to the students’ initial and revised predictions. Discuss why working in this way might have been more efficient than just counting all the beans one by one.
Your finished product should be bags of hundreds, up to nine cups of tens that did not make it into a bag, and up to 9 beans that did not make a full cup of ten. Discuss how to write the number of beans and why it is written in that way.
Next, discuss that the class next door is doing the same activity and they have counted 125 beans. How many do we have all together? What if they counted 207? 398?
Suppose you ended up with 5 bags of hundreds, 2 cups of tens, and 4 individual beans, or 524 beans. Ask the class to think about how they grouped the beans as they worked, and ask if there are any ways we can show 524 other than the 5 bags, 2 cups, and 4 beans. For example, if we take one of the hundreds and pour the beans back into 10 cups, and take 1 of those cups and pour it out, then instead of 5 hundreds, 2 tens, and 4 ones, we now have only 4 hundreds, 11 tens, and 14 ones. Ask the students if there are still the same number of beans.
Now have the groups each come up with 5 different ways that we can show 524 beans with hundreds, tens, and ones. Then come together, have each group justify their choices, and discuss.
If you have been working with subtraction with regrouping, and students are still struggling, the next activity is an opportunity for that aha! moment that comes from connecting and understanding:
Ask students the following question: Our 524 beans are all packed, but the manager just called me and said he needs 246 beans quickly to make some soup for his boss, who loves bean soup. Which grouping of the 524 beans makes it easiest to subtract 246?
Lead students to discover that the different ways you expressed 524 shows the same kind of reasoning we use when we regroup, where we break up groups of hundreds and tens in order to make subtraction easier. Work on a variety of scenarios and how they would work with the actual beans, pouring bags of hundreds back into cups and pouring out cups to get individual beans.
For older students, extend this activity to thousands, grouping the hundreds into larger bags of one thousand. A few large bags of beans from the supermarket can yield several thousand beans.
Moving beyond the decimal point will require a larger manipulative that can be partitioned into ten equal parts. Squares or rectangles of paper or other materials can work for this.
The importance of understanding place value cannot be overstated, as it is THE central organizing principle of our number system, and is the reason that all the operations, properties, and standard algorithms that we teach can work.