Place Value, or Thank You Hindus and Arabs, or The Beauty of the Base-Ten Numeration System

Evolution of the Hindu-Arabic Number System

I have been doing a lot of thinking about place value lately. Yes, I need a life outside my standard state-issue gray cubicle. Nonetheless, I have become caught up in the beauty of the Hindu-Arabic number system. We also know it as the Base-Ten numeration system. It is beautiful and elegant. Let me elucidate (I love that word!)

One of the best t-shirts I have ever seen had this quote on the front, “There are only 10 kinds of people in the world; those who understand binary and those who don’t.” Some of you might be laughing right now while others are scratching their heads wondering what is wrong with me. Okay, here’s the joke. Binary is a base-2 numeration system. It has two digits – 0 and 1. Place value in binary is determined by powers of 2. So, in the units place you can have 0 and 1. Then you move to the 2’s place, then the 4’s, the 8’s, the 16’s, the 32’s and so on. Maybe a chart will help.

So, here we have a base two place value chart with the top line thrown in for us decimal thinkers. Let’s say we wanted to represent the number 2, as in the above t-shirt. It would look like this:

128’s	64’s	32’s	16’s	8’s	4’s	2’s	1’s
27	26	25	24	23	22	21	20
						1	0

Yup, one two and zero ones. Now does the -shirt make sense? How about if we wanted to represent 45? Well, that would look like this:

128	64’s	32’s	16’s	8’s	4’s	2’s	1’s
27	26	25	24	23	22	21	20
		1	0	1	1	0	1

See how that works? Look it over a little – you’ll see it.

What if we wanted to add 101101 (45) and 100100 (36)? Hmmm… Let’s try it using the standard United States addition algorithm:

composing-	1		1	1
45 in base 10		1	0	1	1	0	1
36 in base 10            +		1	0	0	1	0	0
answer	1	0	1	0	0	0	1

So, what did we do? Here goes! 1 one + 0 ones = 1 one, so that is in the 1’s place

0 twos + 0 twos = 0 twos, so that is in the twos place. We really ought to say (0 x 2¹ ) + (0 x 2¹ ) = 0 in the 2’s place, but that will get messy. So, we’ll stick with combining our binary system with the base ten numerals, okay? Are you with me so far?

Okay! 1 four + 1 four = one eight, 0 fours, 0 twos, 0 ones so we compose an 8 and regroup it to the 8’s column.

1 eight + 1 eight = one sixteen, 0 eights, 0 fours, 0 twos, 0 ones, so we compose a 16 and regroup, moving it to the 16’s column.

And so on. We finally get the answer of 1010001. There! That was easy, right? Oh, you want to know what the answer is in base 10? Okay! 64+16+1 = 81. Right?

Doing all of this makes me not want to live in a base two world. After all, who wants to still owe $11110011110000 on his car? Do the conversion.

We could also live under the Roman system. The Romans had no place value system. So, let’s suppose you are a Roman general. Your adjutant comes and says, “General! The emperor is sending you LXVI divisions of soldiers with MDXC soldiers in each one! He thinks you will need CLXIV soldiers in each of the XL areas of Rome to defend against the Huns! He wants to know if he needs to send more!” Time to head to the villa in France, I think.

We are so lucky that the Hindus first and then the Arabs developed and adopted a system based on the digits 1 through 9 and then realized the need of the number zero to be used as a placeholder in the place value system.

According to Van de Walle, there are five big ideas that explain why the place value system in base ten is so important.

Big Idea One – Sets of ten (and tens of tens) can be perceived as single entities or units. For example, three sets of tens and two singles is a base-ten method of describing 32 single objects. This is the major principle of base-ten numeration.

In other words, the system is based on groupings of tens. Ten ones are composed into 1 ten. Ten tens are composed into one hundred. Ten hundreds are composed into one thousand, ten ten billions are composed into one hundred billion. That is a lot easier than dealing with groups of twos. Or sevens. Or MDC’s. But the Romans didn’t even have place value, so that doesn’t count. The Utah Core State Standards want students to develop that understanding in standards like 1.NBT.2 (First Grade Number and Operations in Base Ten, Standard Two) which reads:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
a. 10 can be thought of as a bundle of ten ones — called a “ten.”
b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

Big Idea Two – The positions of digits in numbers determine what they represent and which size group they count. This is the major organizing principle of place value numeration and is central for developing number sense.

In other words, there are only ten digits and they are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The value each of those digits possesses depends on the position it holds in the place value system. If a nine is in the ones place it has the value of 9 ones, or single units. If a nine is in the ten thousands place it has the value of 9 ten thousands, or 90 thousand. The Utah Core State Standards emphasize this concept in standards such as K.NBT.1, which reads:

Work with numbers 11–19 to gain foundations for place value.

Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (such as 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

The standards continue to develop this big idea up through fourth grade.

Big Idea Three:  There are patterns in the way that numbers are formed. For example, each decade has a symbolic pattern reflective of the 0-9 sequence (e.g., 20, 21, 22 …29).

When students begin to count they soon encounter this pattern. The decade numbers (20, 30, 40, 50, 60. 70, 80, 90) all follow the same pattern. They begin with the number of tens (say, 40) and then move to that number of tens +1 (41), +2 (42) and so on. They don’t think of it as addition, of course, it is just counting. But once you know the pattern you can literally go on forever. The same is true through the rest of the base ten system, and it never varies. Except in the teen numbers. Why on earth do we say eleven instead of ten and one? Who thought of seventeen? That is a weird number. So kindergartners are taught the exception to the rule, those teen numbers, and then the rest of the system follows that very rigid pattern. Again, this big idea is emphasized in the standards in grades K-4.

Big Idea Four: The groupings of ones, tens, and hundreds can be taken apart in different but equivalent ways. For example, beyond the typical way to decompose 256 of 2 hundreds, 5 tens, and 6 ones, it can be represented as 1 hundred, 14 tens, and 16 ones but also as 250 and 6. Decomposing and composing multi-digit numbers in flexible ways is a necessary foundation for computational estimation and exact computation.

This is a powerful idea and is one of the main concepts the standards refer to when they require students to use place value understanding to round, estimate, and compute numbers in all four arithmetic operations. This idea gives students the facility to do such things as computing the answer to 457 + 263 by decomposing the numbers into their place value parts and writing them in expanded form: (400+50+7) + (200+60+3). They can then use the associative and commutative properties to move the groupings and the numbers into combinations that are more easily computed. For example, (400 + 200) + (50 +60) + (7 +3) . The next step is to compute the combinations 600+110+10= 720. Having done so then allows them to know that they can compute that same problem, when written in the standard algorithmic form, from left to right in place value groups, such as:

Of course, being flexible in number composition and decomposition because of their place value understanding  also allows students to do operations such as this:

457 + 263 – I can take the three from 263 and add it to the 57 in 457. That makes my problem 460 + 260. I can then do the computation, or I can take it farther by subtracting 40 from 260 and adding it to 460. That makes my problem 500 + 220. Now I have a very easy problem. I can tell the answer is 720. Of course, these are learning steps. Eventually we want students to be fluent with using the standard algorithm including using regrouping. This big idea, however, helps students maintain their place value understanding and move toward that fluency.

One final note on Big Idea Four – there are many adults who still use the place value understandings above to do mental computations. Do you? Are you one of those people who will want to buy two pairs of socks that cost $4.95 each who will do the computation mentally similar to this? “I know that $4.95 is 5¢ less than $5.00. $5.00 times 2 is $10.00. Then I subtract the two 5¢ I added in and know that the socks will cost $9.90. Plus tax.”

We’re almost done.

Big Idea Five: “Really big” numbers are best understood in terms of familiar real-world referents. It is difficult to conceptualize quantities as large as 1000 or more. However, the number of people who will fill the local sports arena is, for example, a meaningful referent for those who have experienced that crowd.

Young children, even into the primary grades in school, don’t have a grasp of how large certain numbers are. Ask them how many nails are in a box and they are likely to tell you “A thousand!” or “A million!” or a “Billion kazillion!” They need real life referents to put value to large numbers. So, when you are watching the Super Bowl, you teachers may want to take a snapshot of the screen with your smartphone and show it to your students the day after the game. Ask them if they can tell how many people attended the game by looking at the picture. After a few guesses you may want to start writing the real number on the board. You can find it on the web. Easily. Such activities can give students a referent of how big 100,000 actually is. Parents, you can do it while you watch the game.

So, place value. It is a big topic. It is a huge part of number sense. Students need to understand this part of the base ten numeration system in order to be successful at operations. It is also extremely helpful in higher mathematics. How do you teach it? Here are some resources you might want to use:

• Be sure to read the standards (http://schools.utah.gov/CURR/mathelem/core.aspx) Understand what the core is asking you to do.
• You can get more information about what the standards really mean by reading the Number and Operations in Base Ten progression document. You can find that at: http://schools.utah.gov/CURR/mathelem/Instructional-Resources.aspx .
• Learn Zillion has some excellent materials and videos on this topic and all other topics in the core standards. Take a look at this one on understanding place value. I think it is very good.
• The Teaching Channel also has some nice videos on the subject. This one called Beyond Fingers is great for kindergarten teachers.
• Take a look at the Core Guides created by Utah teachers for more information about the standards as well as suggestions for instruction and assessment and background knowledge. Scroll the page down to see them. They are arranged by domain.
• A group of state mathematics specialists that I lead is developing a professional development module on place value. I will post the URL here when it is done. It is VERY close. Final editing stage.

Place value – it is really a beautiful and extremely important concept. Take your time teaching it.

Citations:
Van de Walle, J. A., Karp, K.S., & Bay-Williams, J. M. (2013). Elementary and Middle School Mathematics: Teaching Developmentally, Pearson Education