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In part one of this article, you read about representing and adding numbers using base ten blocks. Once these two skills are mastered, it is time to move onto many a child’s nightmare: subtraction. Subtraction, as you may have heard, is essentially addition in reverse. It can be an arduous task on paper, but it can be quite easy with base ten blocks.

Recall that there are four different base ten blocks: cubes (ones), rods (tens), flats (hundreds), and blocks (thousands). Groups of ten base ten blocks can be regrouped or traded for equivalent amounts of other base ten blocks; for instance, ten cubes can be traded for one rod because both are worth ten. For subtraction, it is useful to know how to trade down rods, flats, and blocks. Trading down means converting larger place value blocks into smaller place value blocks. For instance, one flat can be traded for ten rods since they are both worth 100.

Before describing the subtraction procedure, let’s go over some vocabulary . . .

Minuend – The amount from which you are subtracting.

Subtrahend – The amount that you are subtracting.

Difference – The answer.

In the equation, 234 – 187 = 47, the minuend is 234, the subtrahend is 187, and the difference is 47. Most people don’t bother with the terms minuend and subtrahend, but they are useful in describing the subtraction procedure using base ten blocks.

To begin, represent the minuend with base ten blocks. Try to keep the blocks in order from largest to smallest as this will help to transfer knowledge and skills to paper and pencil methods later on. Remove from the minuend piles, enough blocks to represent the subtrahend. If there aren’t enough blocks available, trade some of the larger place value blocks until there are enough smaller place value blocks to remove. The resulting piles after the subtrahend is removed represents the difference.
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Base ten blocks are an excellent tool for teaching children the concept of addition because they allow children to touch and manipulate something real while learning important skills that translate well into paper and pencil addition. In this article, I will describe base ten blocks and how to use them to represent and add numbers.

The numbering system that children learn and the one most of us are familiar with is the base ten system. This essentially means that you can only use ten unique digits (0 to 9) in each place of a base ten number. For instance, in the number 345, there is a hundreds place, a tens place and a ones place. The only possible digits that could go in each place are the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. In this example, the place value of the ones place is 5.

Base ten blocks turn the base ten concept into something children can see and touch.

Base ten blocks consist of cubes, rods, flats, and blocks. Cubes represent the ones place and look exactly like their name suggests – a small cube usually one centimeter by one centimeter by one centimeter. Rods represent the tens place and look like ten cubes placed in a row and fused together. Flats, as you might have guessed, represent hundreds, and blocks represent thousands. A flat looks like one hundred cubes place in a 10 x 10 square and attached together. A block looks like ten flats piled one on top of the other and bonded together.

In order to use base ten blocks to add numbers, students should be familiar with how to represent numbers using base ten blocks. To see what base ten blocks look like, and to try them out, go to the National Library of Virtual Manipulatives:

http://nlvm.usu.edu/en/nav/frames_asid_154_g_1_t_1.html

To represent a number using base ten blocks, make piles of base ten blocks to represent each place value. If your number was 2,784, you would make a pile of 2 blocks, a pile of 7 flats, a pile of 8 rods, and a pile of 4 cubes. It is useful to arrange the piles in a row in the same order that they appear in the number as that will be useful later on when children learn the paper and pencil algorithm.
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Do you ever feel like you have some great ideas, but when you sit down to write them, they’re not so great? Or even worse, you can’t really get a sense of what the ideas were? In one of my graduate student coaching groups we have been discussing the difficulty of translating partly formed ideas into words on paper. One technique that makes use of a normally underutilized part of our brain is called “Mind Mapping.”

What is a Mind Map? Tony Buzan, who created the word “Mind Map” and has written extensively on it, describes it as a powerful graphic technique that makes use of the way our brains naturally work. He says it has four characteristics.

1. The main subject is crystallized in a central image
2. The main themes radiate from the central image as branches
3. Branches comprise a key image or key word printed on an associated line.
4. The branches form a connected nodal structure

How Do You Mind Map? Mind mapping is best done in color. If you have some markers or colored pencils, and a sheet of white paper, you’re ready. If you don’t, just use what you have. Start with the central idea that you are trying to wrap your mind around. It could be the big picture (e.g. your next chapter) or a smaller idea (e.g. the next few paragraphs.) Write it down in one or two words at the center of the paper, and draw a circle around it. If there is a symbol or picture that you can put with the words, sketch that in. The idea is that you are activating the non-verbal side of your brain. The quality of what you draw is not important, since you will be the only one seeing it. The same is true for the ideas you come up with. Don’t edit, just put in what comes to mind.

There are no rules for the way to proceed from here. I tend to break rules, anyway. The way my mind works, I start thinking of related ideas, categories, and ideas, which I write in little circles surrounding the circle in the middle. I then use lines to connect them. Tony Buzan likes to draw curved lines emanating from the center, and write the related or associated ideas on the lines. The result looks like a tree emanating from a central spot. My technique looks more like a bunch of lollipops.
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Adults use rounding and estimation in their everyday lives. They approximate the temperature, the cost of items, the time, and even their age. Consider this conversation:

“How much did it cost to fix your car?”

“Six hundred bucks!”

Without any words such as: about, approximately, around, roughly, or nearly, it can be assumed that the second person rounded the actual cost. Before they had their car fixed, they probably received an estimated cost of the repair from the shop. Adults experience rounding and estimation skills in their daily lives. Children need to learn these important skills partly because they often hear estimation and use estimation, but more importantly, it helps to solidify math learning by teaching them the idea of reasonableness.

Even though rounding and estimating are related, there is a significant difference. Rounding involves converting a known number into a number that is easier to use. Estimation is an educated guess of what a number should be without knowing the actual number. In the conversation above, it is unlikely that the second person remembered the actual price of the bill; they likely rounded the number at the time, so they could better remember it.

Children usually learn rounding as an explicit skill, often with the purpose of estimating the answers to math questions. They commonly use estimation to check the reasonableness of an answer by either estimating ahead of time or after they have completed the question. Students run into difficulty when estimating because they don’t have the intuitive sense that adults do to break the rules.

For the uninitiated, the idea of rounding is fairly simple – decide where to round the number (e.g. the hundreds place), either keep the digit at the rounding place the same or round it up, and replace the digits to the right with zeros. The decision to keep the digit the same or to round it up is based on everything that comes after the digit. If it is less than half, the digit remains the same; if it is greater than half, the digit is increased by one; if it is exactly half, the digit remains the same if it is even and increases by one if it is odd. For example, to round 638 to the nearest hundred, you would base your decision on the “38″ portion of the number. Since it is less than half (50), the digit in the hundreds place remains the same, and the 38 is changed to zeros, so the rounded number is 600. If the question is to round 7500 to the nearest thousand, you would round up to 8000. 8500 also rounds to 8000, but 8501 rounds to 9000. Hopefully, this illustrates that rounding follows a strict set of rules that often cause difficulties for children in estimation.
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