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Posts Tagged ‘mathematics’

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.

To give you an idea of how following the rounding rules can be problematic in estimation, consider the question 7359 divided by 82. The first difficulty is deciding what place to round to. Let’s say that the student decides to round to the nearest hundred in the first number and the nearest ten in the second number, thus the question is now 7400 divided by 80. At this point some students might resort to a calculator, others to long division, and others might stare confusedly at their paper. An adult with more intuitive sense might look at the numbers and recognize that if she rounded 7359 to 7200, it would be fairly simple to divide by 80 (because 72 divided by 8 is easy).

Many people develop an ability to estimate both by following the rules and by breaking the rules of rounding. Many children need to be taught these skills, so there is a genuine purpose to their estimation rather than just another question to answer. Estimation should be thought of as a tool to quickly determine whether an answer is reasonable or not. One way of teaching estimation for this purpose is by allowing students to break the rounding rules and find an easy question that they can do in their head. In the question 3564 – 2801, rounding to the nearest hundred results in 3600 – 2800, but 3700 – 2700 is much easier to handle, and it is not so far off the real answer. If the purpose of estimating was to get as close to the real answer as possible, you might as well use a calculator to check your answer instead.

Parents can help develop students’ estimation skills by regularly asking real questions. For instance, ask them how long they think it will take to get to hockey practice (time), have them add up the cost of the groceries as you are shopping (money), get them to count the number of people in one area of the mall and have them estimate how many people are in the whole mall (multiplication or addition). Educators should make estimation a regular part of the problem solving process. In a science investigation, students make hypotheses and predictions, so why not make an estimate in a math problem? Students can develop their estimation skills by answering questions on worksheets and comparing their estimated answers to the actual answers.

Remember these rules for estimation: (i) KISS – keep it simple silly, (ii) break the rounding rules if necessary, (iii) ensure students see a purpose for estimation, (iv) give students a lot of practice and experience with estimation and rounding, (v) include estimation in problem solving and other daily math work. The main rule for parents and teachers: support your students and be flexible!

It may not sound like a difficult task, but constructing hexagons and other polygons can be a frustrating and daunting task for children and adults. A sketch of a square is fairly simple to make as the corners are familiar right angles that most people have no trouble creating. Every other regular polygon from equilateral triangles to dodecagons and beyond can be a challenge without a highly developed ability to recognize and construct a variety of angles. Thankfully, there is a slick technique for constructing all sorts of regular polygons based on the fact that all regular polygons fit neatly inside of a circle.
For the uninitiated, a regular polygon is a closed figure with equal length sides and equal angles. A pentagon with three centimetre sides and 108 degree angles is a regular pentagon. Regular polygons are the figures that are most commonly used to represent each family of polygons.
To experience the most success with this method, it is recommended that you use a full circle protractor. A half circle protractor will work just fine except the procedure changes slightly. The basic procedure for the full circle protractor is to place the protractor on a piece of paper, make a bunch of dots, and join the dots. The trick is dividing the 360 degrees of the circle by the number of vertices in the regular polygon, and making dots at the resulting interval. In a hexagon, for example, there are six vertices, so divide 360 degrees by six to get sixty degrees. Starting at zero degrees, make a mark every sixty degrees around the full circle protractor; there will be dots at 0, 60, 120, 180, 240, and 300 degrees. Join the dots, and voila; you have a perfect regular hexagon. With a half circle protractor, it is necessary to establish a center point first, so when you rotate the protractor to complete the dots on the other side, it can be lined up properly with the zero point and the center point.
The really nice thing about using a 360 degree circle to construct regular polygons is that it works for all of the regular polygons that one would encounter in an elementary or primary school. This is because 360 is divisible by 24 different numbers including 3, 4, 5, 6, 8, 9, 10, and 12. To construct an equilateral triangle, for example, first divide 360 by three to get 120. Make dots at 0, 120, and 240, join the dots, and enjoy a perfectly drawn equilateral triangle. Squares are constructed by marking dots at 90 degree intervals, pentagons at 72 degree intervals, octagons at 45 degree intervals, nonagons at 40 degree intervals, decagons at 36 degree intervals, and dodecagons at 30 degree intervals. “But what about a heptagon?” you may ask. Even numbers that don’t divide evenly into 360 can be approximated using this method. For example, a heptagon (seven sided polygon) can be approximated quite well using 51 degree intervals. It will be hard to tell with the naked eye that you were one or two degrees off.
One limitation of this method is that there is only one size of circle available, so all of the polygons come out quite large. With a little ingenuity, this limitation can be overcome. One simple solution is to cut out a circle of paper and place it on top of the round protractor. Any paper circle smaller than the round protractor can be used. Make the dots around the edge of the paper circle lining them up with the scale on the protractor. The paper circle becomes an intermediate protractor that can be used just as the regular protractor, but it will make a smaller polygon.
Another limitation is that your students might not be at the point where they can divide or find multiples of large numbers. In this case, you could tell your students at which numbers to make the dots, or create paper protractors with just the intervals marked on them for each polygon.
This is the quickest and most efficient method I have seen for constructing regular polygons. It takes little time to teach and little time to learn, and it makes the construction of regular polygons a simple and painless activity for students. And if you need a bit of a challenge, try the 180 sided polygon with two degree intervals. I’ll bet you never guessed you could make one of those so easily!

In the first two parts, representing, adding, and subtracting numbers using base ten blocks were explained. The use of base ten blocks gives students an effective tool that they can touch and manipulate to solve math questions. Not only are base ten blocks effective at solving math questions, they teach students important steps and skills that translate directly into paper and pencil methods of solving math questions. Students who first use base ten blocks develop a stronger conceptual understanding of place value, addition, subtraction, and other math skills. Because of their benefit to the math development of young people, educators have looked for other applications involving base ten blocks. In this article, a variety of other applications will be explained.

Multiplying One- and Two-Digit Numbers

One common way of teaching multiplication is to create a rectangle where the two factors become the two dimensions of a rectangle. This is easily accomplished using graph paper. Imagine the question 7 x 6. Students colour or shade a rectangle seven squares wide and six squares long; then they count the number of squares in their rectangle to find the product of 7 x 6. With base ten blocks, the process is essentially the same except students are able to touch and manipulate real objects which many educators say has a greater effect on a student’s ability to understand the concept. In the example, 5 x 8, students create a rectangle 5 cubes wide by 8 cubes long, and they count the number of cubes in the rectangle to find the product.

Multiplying two-digit numbers is slightly more complicated, but it can be learned fairly quickly. If both factors in the multiplication question are two-digit numbers, the flats, the rods, and the cubes might all be used. In the case of two-digit multiplication, the flats and the rods just quicken the procedure; the multiplication could be accomplished with just cubes. The procedure is the same as for one-digit multiplication – the student creates a rectangle using the two factors as  the dimensions of the rectangle. Once they have built the rectangle, they count the number of units in the rectangle to find the product. Consider the multiplication, 54 x 25. The student needs to create a rectangle 54 cubes wide by 25 cubes long. Since that might take a while, the student can use a shortcut. A flat is simply 100 cubes, and a rod is simply 10 cubes, so the student builds the rectangle filling in the large areas with flats and rods. In its most efficient form, the rectangle for 54 x 25 is 5 flats and four rods in width (the rods are arranged vertically), and 2 flats and five rods in length (with the rods arranged horizontally). The rectangle is filled in with flats, rods, and cubes. In the whole rectangle, there are 10 flats, 33 rods, and 20 cubes. Using the values for each base ten block, there is a total of (10 x 100) + (33 x 10) + (20 x 1) = 1350 cubes in the rectangle. Students can count each type of base ten block separately and add them up.

Division

Base ten blocks are so flexible, they can even be used to divide! There are three methods for division that I will describe: grouping, distributing, and modified multiplying.

To divide by grouping, first represent the dividend (the number you are dividing) with base ten blocks. Arrange the base ten blocks into groups the size of the divisor. Count the number of groups to find the quotient. For example, 348 divided by 58 is represented by 3 flats, 4 rods, and 8 cubes. To arrange 348 into groups of 58, trade the flats for rods, and some of the rods for cubes. The result is six piles of 58, so the quotient is six.

Dividing by distributing is the old “one for you and one for me” trick. Distribute the dividend into the same number of piles as the divisor. At the end, count how many piles are left. Students will probably pick up the analogy of sharing quite easily – i.e. We need to give everyone an equal number of base ten blocks. To illustrate, consider 192 divided by 8. Students represent 192 with one flat, 9 rods and 2 cubes. They can distribute the rods into eight groups easily, but the flat has to be traded for rods, and some rods for cubes to accomplish the distribution. In the end, they should find that there are 24 units in each pile, so the quotient is 24.

To multiply, students create a rectangle using the two factors as the length and width. In division, the size of the rectangle and one of the factors is known. Students begin by building one dimension of the rectangle using the divisor. They continue to build the rectangle until they reach the desired dividend. The resulting length (the other dimension) is the quotient. If a student is asked to solve 1369 divided by 37, they begin by laying down three rods and seven cubes to create one dimension of the rectangle. Next, they lay down another 37, continuing the rectangle, and check to see if they have the required 1369 yet. Students who have experience with estimating might begin by laying down three flats and seven rods in a row (rods vertically arranged) since they know that the quotient is going to be larger than ten. As students continue, they may recognize that they can replace groups of ten rods with a flat to make counting easier. They continue until the desired dividend is reached. In this example, students find the quotient is 37.

Changing the Values of Base Ten Blocks

Up until now, the value of the cube has been one unit. For older students, there is no reason why the cube couldn’t represent one tenth, one hundredth, or one million. If the value of the cube is redefined, the other base ten blocks, of course, have to follow. For example, redefining the cube as one tenth means the rod represents one, the flat represents ten, and the block represents one hundred. This redefinition is useful for a decimal question such as 54.2 + 27.6. A common way to redefine base ten blocks is to make the cube one thousandth. This makes the rod one hundredth, the flat one tenth, and the block one whole. Besides the traditional definition, this one makes the most sense, since a block can be divided into 1000 cubes, so it follows logically that one cube is one thousandth of the cube.

Representing and Working With Large Numbers

Numbers don’t stop at 9,999 which is the maximum you can represent with a traditional set of base ten blocks. Fortunately, base ten blocks come in a variety of colors. In math, the ones, tens, and hundreds are called a period. The thousands, ten thousands, and hundred thousands are another period. The millions, ten millions and hundred millions are the third period. This continues where every three place values is called a period. You may have figured out by now that each period can be represented by a different colour of place value block. If you do this, you eliminate the large blocks and just use the cubes, rods, and flats. Let us say that we have three sets of base ten blocks in yellow, green, and blue. We’ll call the yellow base ten blocks the first period (ones, tens, hundreds), the green blocks the second period, and the blue blocks the third period. To represent the number, 56,784,325, use 5 blue rods, 6 blue cubes, 7 green flats, 8 green rods, 4 green cubes, 3 yellow flats, 2 yellow rods, and 5 yellow cubes. When adding and subtracting, trading is accomplished by recognizing that 10 yellow flats can be traded for one green cube, 10 green flats can be traded for one blue cube, and vice-versa.

Integers

Base ten blocks can be used to add and subtract integers. To accomplish this, two colours of base ten blocks are required – one colour for negative numbers and one colour for positive numbers. The zero principle states that an equal number of negatives and an equal number of positives add up to zero. To add using base ten blocks, represent both numbers using base ten blocks, apply the zero principle and read the result. For example (-51) + (+42) could be represented with 5 red rods, 1 red cube, 4 blue rods, and 2 blue cubes. Immediately, the student applies the zero principle to four red and four blue rods and one red and one blue cube. To finish the problem, they trade the remaining red rod for 10 red cubes and apply the zero principle to the remaining blue cube and one of the red cubes. The end result is (-9).

Subtracting means taking away. For instance, (-5) – (-2) is represented by taking two red cubes from a pile of five red cubes. If you can’t take away, the zero principle can be applied in reverse. You can’t take away six blue cubes in (-7) – (+6) because there aren’t six blue cubes. Since a blue cube and a red cube is just zero, and adding zero to a number doesn’t change it, simply include six blue cubes and six red cubes with the pile of seven red cubes. When six blue cubes are taken from the pile, 13 red cubes remain, so the answer to (-7) – (+6) is (-13). This procedure can, of course, be applied to larger numbers, and the process might involve trading.

Other Uses

By no means have I explained all of the uses of base ten blocks, but I have covered most of the major uses. The rest is up to your imagination. Can you think of a use for base ten blocks when teaching powers of ten? How about using base ten blocks for fractions? So many math skills can be learned using base ten blocks simply because they represent our numbering system – the base ten system. Base ten blocks are just one of many excellent manipulatives available to teachers and parents that give students a strong conceptual background in math.

The base ten blocks skills described above can be applied using worksheets from http://www.math-drills.com. The worksheets come with answer keys, so students can get feedback on their ability to correctly use base ten blocks.