8 Trading and Place Value

You will need: Arcade Prize Tokens (Material Card 3) and Base Blocks (Material Cards 4-16).

Exploration Activity

Use your cut out Arcade Prize Tokens to follow along with the scenario below. Do not use a calculator!

The High Five Arcade has a special system for earning prizes. As you play games you can earn “prize tokens” in the values of one, five, twenty-five, one hundred twenty-five, and six hundred twenty-five prize points. Your goal is to earn 1000 points to win the big prize!

After playing games for an hour you have earned the following:

• 19 one point tokens
• 19 five point tokens
• 6 twenty-five point tokens

Carrying around so many prize tokens is getting annoying, so you go to the counter and exchange them for the fewest number of prize tokens possible. How many of each token type do you have after you exchange them? (You may have a shortcut way to do this, but use your cut-out Arcade Prize Tokens and physically exchange them. It will help you later!)

On the next game you win the grand prize and earn one of each prize token type. You go back to the counter to again exchange your prize tokens for the fewest number possible. Now, how many of each token type do you have?

You can now claim the 1000 point big prize! After you claim your prize, how many of each token type do you have left? (You may use a calculator for this part.)

Notice how whenever you had five of one type of prize token you were able to exchange it for one of the next size up. These tokens model a base five numeration system.

Base Ten Numerals

The Hindu-Arabic system is a base ten system. Each place value is made up of ten of the previous place value. Consider the number 2,385. You probably read that in your head as “two thousand, three hundred, eighty-five.” You are automatically recognizing the place value of each piece. The digit 2 is in the thousands place, the digit 3 is in the hundreds place, the digit 8 is in the tens place, and the digit 5 is in the ones place. This means there are 2 things of size one thousand, 3 things of size one hundred, 8 things of size ten, and 5 things of size one. Here are the first five place values in the base ten system.

Notice how as you move from the right to the left each new place value is ten times the previous one. This is why our place values are all powers of 10.

In elementary classrooms, whole numbers are often represented with base ten blocks.

Thousands	Hundreds	Tens	Ones
Cubes	Flats	Longs	Units

Notice how each piece, when you join ten of them in a row, creates the next piece: 10 joined units make a long, 10 joined longs make a flat, and 10 joined flats make a cube.

Pull out your cut out Units (Material Card 4) and Base Ten Blocks (Material Card 13) and answer the following questions.

You may have found a shortcut when solving exercise 1. For example, consider part (b). If you add the values of the blocks, you get [latex](24\times 1)+(11\times 10)+(9\times 100)=1034.[/latex] This means you will have 1 cube, 0 flats, 3 longs, and 4 units. The reason you can do this trick is because you already know base ten. Your future students do not have this advantage! In order to put you in their shoes, we need to consider other number bases. As you are working through this section be thinking of what questions your future students might have and how you can answer them.

Base Seven Numerals

Examine your cut out Units (Material Card 4) and Base Seven Blocks (Material card 8) and answer the following questions.

When 162[latex]_{\text{seven}}[/latex] is read aloud we say ” one six two base seven.” Do not say “one hundred sixty-two base seven.” That doesn’t make sense. The 1 is in the 49’s place, not in the hundreds place, so it is not a hundred. The 6 is in the 7’s place, not the tens place, so it is not sixty. Form the habit now of reading all numerals that are not in base ten this way. It will save you from a lot of confusion later!

Now that you are getting comfortable with the blocks, let’s look at how they help us to count. Start with one unit. That is the number one. Now put down another unit. That is the number two. This continues until we have reached our trading point (which in base seven is seven). Then all seven units can be traded for a long. Here are the first ten numbers represented in base ten.

Number	Representation in Base Seven Blocks	Base seven numeral	Pronunciation
One		[latex]1_{\text{seven}}[/latex]	one base seven
Two		[latex]2_{\text{seven}}[/latex]	two base seven
Three		[latex]3_{\text{seven}}[/latex]	three base seven
Four		[latex]4_{\text{seven}}[/latex]	four base seven
Five		[latex]5_{\text{seven}}[/latex]	five base seven
Six		[latex]6_{\text{seven}}[/latex]	six base seven
Seven		[latex]10_{\text{seven}}[/latex]	one zero base seven
Eight		[latex]11_{\text{seven}}[/latex]	one one base seven
Nine		[latex]12_{\text{seven}}[/latex]	one two base seven
Ten		[latex]13_{\text{seven}}[/latex]	one three base seven

Now that we have a visual for how base seven works, we can extend to even higher place values. So far we have place values

One way to visualize the value of our numerals is to look at them in expanded form. For example, the base seven numeral 52463[latex]_{\text{seven}}[/latex] can be separated into its place values and written as

[latex](5\times 7^4) + (2\times 7^3) + (4\times 7^2) + (6\times 7) + (3\times 1)[/latex] or

[latex](5\times 2401) + (2\times 343) + (4\times 49) + (6\times 7) + (3\times 1)[/latex].

Notice how this form really shows what each separate digit of the numeral represents.

Base Five Numerals

Remember our introductory activity with the Arcade Prize Tokens? It took five of each coin to make the next size. That’s a base five system. If we use the more traditional blocks to visualize base five, we have the following pieces.

How did you do with exercise 11? Were you able to list all twenty five numerals on the first try? If so, that’s great! You are starting to really understand place value. If not, that’s okay. This is how your future students will feel when you write ” …98, 99, 100, 101,…” That jump from 99 to 100 seems bizarre and out of the blue to them too. Keep working and eventually it will click.

Recall how we extended from our blocks to higher place values in base seven. This works in any number base. For base five our blocks represent:

Now that you have worked with three different number bases, it is time to compare them and look for commonalities.

So, there are only five symbols that will ever appear in a base five numeral: 0, 1, 2, 3, 4. In base seven, we need seven symbols: 0, 1, 2, 3, 4, 5, 6. In base ten, we need ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This pattern works for any number base!

You may have thought something was funny about your answers to exercise 14 parts (e) and (f). We should take a closer look at these two bases.

Base Two Numerals

In base two, our place values will be based on powers of two. So our blocks each represent the value

Units – 1
Longs – 2
Flats – 4
Cubes – 8

This means the number eight expressed as a base two numeral is [latex]1000_{\text{two}}[/latex]. Apparently, we will need a fifth place value very early on in counting. It will help if we make blocks for the next few place values. Look at the blocks below and make sure you see how each one is two times the one to its right.

Cube Cube	Flat Cube	Long Cube	Cube	Flat	Long	Unit
64	32	16	8	4	2	1
[latex]2^6[/latex]	[latex]2^5[/latex]	[latex]2^4[/latex]	[latex]2^3[/latex]	[latex]2^2[/latex]	[latex]2^1[/latex]	[latex]2^0[/latex]

Base two seems strange because the numerals grow in length very quickly. It is still the exact same numbers you are used to; they are just in a different language from your native base ten.

For more practice with base two see section 3-5 Supplementary Topics.

Base Twelve Numerals

What happened on part (c) of the last exercise? If you wrote [latex]811_{\text{twelve}}[/latex], it probably looked wrong (and it is) but what else can you do? There are eleven units left after exchanging. We need a way to say “eleven things are in this place value.” Consider the symbols we already have: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The list is only ten symbols, but base twelve needs twelve symbols. Since base twelve is not a standard base, there are no standard symbols for “ten things are in this place value” and “eleven things are in this place value.” We will use [latex]\text{T}[/latex] and [latex]\text{E}[/latex], respectively. We will use these same symbols in bases eleven and thirteen as well. If you are in base thirteen and need “twelve things are in this place value” use [latex]\text{W}[/latex].

If you can handle bases two and twelve, then you are a place value whiz! As long as you are comfortable with at least one base that is not base ten, you should have the depth of knowledge required to answer whatever base ten questions your students come up with!

Summary

Consider the table below and answer the questions in Exercise 21.