Discovering the magnitude of large numbers through tactile examples and relations to what we do understand

Powers of ten or order of magnitude of tens, especially in the higher numbers, are a challenging concept for people to visualize, not to mention children. A reminder, the powers of ten are our place value system. 

10⁰=1 one,  10¹=10 tens,
10²=100 hundreds,
10³=1,000 thousands  
10⁴=10,000 ten thousands…

The cool part of the powers of ten is the exponent corresponds with how many zeros are in the number. Most people can understand the jump from ten to one hundred. Even the jump from one hundred to one thousand. However, when we start shifting from a thousand to a million or a million to a billion, not to mention a billion to a trillion, the value of these jumps greatly increases. Exponentially in fact. 😉

(Side note for a rabbit hole: the history of the numbering system for large numbers is pretty interesting and not completely standardized. Many nations have different ways of name larger numbers. The use of scientific notation has helped to standardize the use and communication by varying nations for both large numbers and minute numbers.)

What is a good way to help with the understanding of these numbers? One way is to explore the growth of exponential numbers. We can easily start with numbers. There is a book by Demi, One Grain of Rice: A Mathematical Folk Tale, that tells a story of a negotiation between Rani and the raja. She, understanding the value of doubles, request one grain of rice to be doubled every day for thirty days. Each day we see the rice doubled, and by the end of the thirty days she has over a billion grains of rice. 

This can be explored with Cuisenaire rods and is a wonderful introduction to exponents. You can start with one, the white cube, and then continue to double as long as the student has lost interest or you run out of rods. Each time you go along, you can write the calculated number, for example, four doubled would be eight, and the number of times you doubled the number. This last one is the exponent. You can also express the calculated number by writing what is happening in the doubling. 

8 = 2 x 2 x 2

We do this kind of work on a white board where we lay the rods on the board and write on the board nearby. This could also be done in a math journal, where you are drawing what you are exploring with the rods, and writing the equivalency in mathematical expressions alongside your drawing. I like to space these and give the opportunity for memory recall by exploring on the white board the first day, asking my children how much they remember of what we explored the first day and what patterns or concepts we learned, and then placing that work into their math journals. This uses some tools – memory recall, spacing, reflection – that we know from science increases learning and long-term retention. 

To introduce exponents, you can say, “I’m tired of writing ‘2 x 2 x 2 x 2….’ I know a shortcut. Would you like me to show it to you? We can just write the number we are multiplying here. We call this the base. In our case, we are multiplying 2. We can write how many times we are multiplying that number as a smaller number in the upper right of our base. We call this number the exponent. When we write in smaller writing at the top, it’s called superscript. ‘Super’ means up or higher, and ‘script’ means writing. 

This is also a great time to introduce towers and ‘L’s’. Towers are derived from crosses using the Cuisenaire rods. If we travel back to area model multiplication, we can place the rods to model multiplication into a rectangular form. For instance, if we are looking at ‘5 x 7,’ in our rod set, the five rod is a yellow color and the seven rod is a brown color. This particular expression is saying we have five of seven, or five sevens. For this expression, we would choose five brown rods, and place them into a rectangle. We can see we have five on one side of the rectangle, and seven on the adjacent side. 

Knowing the commutive property of multiplication, we know this expression works even if we change the order. If we have 7 x 5, we have seven of fives, or seven yellow rods. If we were to place this rectangle on top of the previous, we would see that they are indeed the same area. This is where the crosses come in. 

We can take a single yellow rod, and place it on a single brown rods in a cross formation. This shows us that we have five, represented by the yellow on one side and seven, represented by the brown, on the other side. The cross using the rods is a 2-digit multiplication representative. 

Towers are an extension of the crosses. They represent multiplication. If you have a tower of a red, a yellow, and a purple rod – 2, 3, and 4 respectively for our set – then this represents ‘2 x 3 x 4’. We work our way into exponents when we have a single color tower. In the case of our doubling play with the twos, all red. 

This is where we can introduce the L’s. It turns out in Cuisenaire rods that the height of the tower, is the same as the height of the rod that can represent the exponent. For example, if we have 2⁴ where two is our base and four is our exponent, we can build a tower of four reds that crisscross each other as they move up. In place of this tower, we can make an ‘L’ where the two lays flat on the surface and the four stands on its end to represent how high the tower is. 

Another way that you might explore this kind of growth is through an exercise that I once saw in The Math Book by Clifford A. Pickover, a favorite book of mine by the way. This book is a history of math in pictures, theorems, and activities. One entry was asking how many times can we fold a piece of paper before the thickness of that said paper reached the moon. This also explores exponential growth by doubling or powers of two. 

Transmuting this understanding of doubling to magnitudes of order for tens, we can now deepen our understanding of place value. In the same way that we played with our twos by doubling, we can play with our tens to express place value by building towers of ten and discussing how it represents our place value system. This pairs well with playing with Gattegno’s place value chart. 

This still, nevertheless, may not give us a good appreciative of how large these numbers grow. I have an activity in my Place Value guide that uses rice to give a better understanding. We start with our first place value of one and isolating a single grain of rice. Then we count our ten grains of rice to be in its one group. Perhaps then we count out one hundred grains of rice, or we wizen up like many a mathematician and look for an easier way to count. I suggest using weight if you have a scale that can do so. If not, then count the one hundred and weight it. You can make ten groups of ten and group them together to further illustrate the powers. 

Like our other work, I’d likely do this on a white board, but you can also do this on a table or even the floor, labeling each group by its number, the number of tens it is multiplying, its place value label, and the exponent. It can be spaced and placed in a math journal after a little memory recall of the activity the following day. 

One, 10, 10 x 1 or 10¹
Hundred, 100, 10 x 10 or 10²
Thousand, 1,000, 10 x 10 x10 or 10^{3 *}

Definitely if you counted a hundred grains of rice, switch to weighing. If you do not have a scale, find some capacity that fits 100 grains, a teaspoon, tablespoon. Weigh out 100 grains, then ask how much the weight should be if we had 1,000 grains. We calculate this by simply multiplying the number by ten. If our 100 grains were 2.4 grams, we would expect that 1,000 grains would be 24 grams. You can also use the standard American measurement of ounces and pounds, but that adds an extra layer of calculations because of conversions. This is great for a student that needs this practice. 

Needless to say, eventually you are going to run out of rice quickly. You can however look for bulk goods or bags of things that have the weight equivalency. We buy 25 lb. bags of oats and beans. Those could be used. If you have a pet that you purchase bulk food for, you can use this. You can also then start to talk about how much area of a 25lb bag each equivalency would fill. 

For instance, take the dimensions of a 25lb bag of rice if you can find one (if not substitute another 25lb bag, and calculate how many bags you can fit into a football field. Then multiply that by the amount of rice you would have in a 25lb bag. How much rice is this? You will soon discover that the difference between a million and billion is enormous. A billion is ONE THOUSAND millions. A trillion is exponentially higher, literally. A trillion is one thousand billions or a million millions. 

This exercise is not likely to be accurate, especially if move from a 25lb bag of rice to something that is available that weighs the same. It will, however, give a better idea of the differing amounts between the higher magnitudes of ten. 

I’ve also seen someone explore this through time. I think this gives a clear illustration of the magnitude of numbers. 

One second is, of course, one second. Ten seconds are ten seconds. One hundred seconds are one minute and forty seconds. One thousand seconds are sixteen minutes and forty seconds. 

One million, though, is much higher. One million seconds is 11 days, 13 hours, 46 minutes, and 40 seconds. Here we move from minutes to days, skipping hours all together.

What do you think a billion will be? One billion seconds is 31 years! 8 months, 7 days, 1 hour, and 40 seconds. Now we are moving from days to years! 

Lastly, a trillion, the value that the US budget currently is in, one trillion seconds is 31 millennium, 688 years, 8 months, 25 days, 1 hour, 46 minutes, and 40 seconds. 

The jump from a million to a billion, 11 days to 31 years!!! is a large one, but the jump from a billion to a trillion, from 31 years to 31 millennium is breathtaking. This gives us something to think about when we are looking, at say, that the first trillionaire might be in our lifetime, or at something like the distance in space. 

These exercises help us build an understanding of place value and magnitudes of order. They make something that seems abstract and possibly beyond understanding, to something more comprehensible. When we can see the jumps in the physical or relate it to something we know more clearly, we are able to better understand these mathematical concepts. 

* This is also a great way to explore why any number raised to the ‘0’ power will be the value of one. Write a few exponent expressions you do know, i.e., 3², 3³, 3⁴, and then work your way down with the pattern, thriving each time if using three as the base, or halving each time if using 2 as the base. . Doing several of these will show that we always end with N⁰ as 1.