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!

Whether you as a teacher realize it or not, you are the best model of behavior in your classroom. A large part of your proactive behavior plans should include your own behavior you demonstrate to the students every day.

You must set expectations for your students, demonstrate the behaviors, and be vigilant to correct the kids. Don’t waver on your expectations; inconsistencies will only confuse the students and cause you more problems.

If you stay calm, collected, and in control, your students will exhibit the same behaviors. The same is true about enthusiasm; if you are excited about your lesson and truly believe in its importance, the kids will respond in kind. Conversely, the kids will know when you are tired, bored, don’t want to be there, or are ‘winging it.’

If you are late to class, or don’t start on time, the kids will pick up on it and be more likely to do the same. The same is true about the way you dress, the way you act, the language you use, and your ‘body language’.

If you want your students working from ‘coast to coast’, or from bell to bell, you need to set the expectation of activity all hour. Start with a warm up, and ensure the kids are doing it. Keep them busy on activities with transitions between each. Don’t let there be any down time. Work them to the end of the period, and have them pack up when you say so, not whenever they want to.

If you want your students to quietly read in class, but you are spending that time working on other things, it sends the message that you don’t value the activity personally. Modeling the skill for the kids reinforces your belief that it is important. It shows you as a lifelong learner who values the skills you’re teaching them.

The same is true for writing, or labs, or math problems. Students rarely have the chance to see real people performing schoolwork - for many, the only examples (and role models) are their classmates. Work along with your students.

Now this doesn’t mean you have to do this the entire time. You must also supervise, coach, monitor, and actively support their learning. But you can spend at least a few minutes ‘at their level’.

Be a positive role model for your students. Don’t just explain and show the behavior; be the example day in and day out.

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The word “cognition” is defined as “the act of knowing” or “knowledge.” Cognitive skills therefore refer to those skills that make it possible for us to know.

It should be noted that there is nothing that any human being knows, or can do, that he has not learned. This of course excludes natural body functions, such as breathing, as well as the reflexes, for example the involuntary closing of the eye when an object approaches it. But apart from that a human being knows nothing, or cannot do anything, that he has not learned. Therefore, all cognitive skills must be TAUGHT, of which the following cognitive skills are the most important:

CONCENTRATION

Paying attention must be distinguished from concentration. Paying attention is a body function, and therefore does not need to be taught. However, paying attention as such is a function that is quite useless for the act of learning, because it is only a fleeting occurrence. Attention usually shifts very quickly from one object or one thing to the next. The child must first be taught to focus his attention on something and to keep his attention focused on this something for some length of time. When a person focuses his attention for any length of time, we refer to it as concentration.

Concentration rests on two legs. First, it is an act of will and cannot take place automatically. Second, it is also a cognitive skill, and therefore has to be taught.

Although learning disability specialists acknowledge that “the ability to concentrate and attend to a task for a prolonged period of time is essential for the student to receive necessary information and complete certain academic activities,” it seems that the ability to concentrate is regarded as a “fafrotsky” — a word coined by Ivan T. Sanderson, and standing for “things that FAll FROm The SKY.” Concentration must be taught, after which one’s proficiency can be constantly improved by regular and sustained practice.

PERCEPTION

The terms “processing” and “perception” are often used interchangeably.

Before one can learn anything, perception must take place, i.e. one has to become aware of it through one of the senses. Usually one has to hear or see it. Subsequently one has to interpret whatever one has seen or heard. In essence then, perception means interpretation. Of course, lack of experience may cause a person to misinterpret what he has seen or heard. In other words, perception represents our apprehension of a present situation in terms of our past experiences, or, as stated by the philosopher Immanuel Kant (1724-1804): “We see things not as they are but as we are.”

The following situation will illustrate how perception correlates with previous experience:

Suppose a person parked his car and walks away from it while continuing to look back at it. As he goes further and further away from his car, it will appear to him as if his car is gradually getting smaller and smaller. In such a situation none of us, however, would gasp in horror and cry out, “My car is shrinking!” Although the sensory perception is that the car is shrinking rapidly, we do not interpret that the car is changing size. Through past experiences we have learned that objects do not grow or shrink as we walk toward or away from them. You have learned that their actual size remains constant, despite the illusion. Even when one is five blocks away from one’s car and it seems no larger than one’s fingernail, one would interpret it as that it is still one’s car and that it hasn’t actually changed size. This learned perception is known as size constancy.

Pygmies, however, who live deep in the rain forests of tropical Africa, are not often exposed to wide vistas and distant horizons, and therefore do not have sufficient opportunities to learn size constancy. One Pygmy, removed from his usual environment, was convinced he was seeing a swarm of insects when he was actually looking at a herd of buffalo at a great distance. When driven toward the animals he was frightened to see the insects “grow” into buffalo and was sure that some form of witchcraft had been at work.

A person needs to INTERPRET sensory phenomena, and this can only be done on the basis of past experience of the same, similar or related phenomena. Perceptual ability, therefore, heavily depends upon the amount of perceptual practice and experience that the subject has already enjoyed. This implies that perception is a cognitive skill that can be improved tremendously through judicious practice and experience.

MEMORY

A variety of memory problems are evidenced in the learning disabled. Some major categories of memory functions wherein these problems lie are:

Receptive memory: This refers to the ability to note the physical features of a given stimulus to be able to recognize it at a later time. The child who has receptive processing difficulties invariably fails to recognize visual or auditory stimuli such as the shapes or sounds associated with the letters of the alphabet, the number system, etc.

Sequential memory: This refers to the ability to recall stimuli in their order of observation or presentation. Many dyslexics have poor visual sequential memory. Naturally this will affect their ability to read and spell correctly. After all, every word consists of letters in a specific sequence. In order to read one has to perceive the letters in sequence, and also remember what word is represented by that sequence of letters. By simply changing the sequence of the letters in “name” it can become “mean” or “amen”. Some also have poor auditory sequential memory, and therefore may be unable to repeat longer words orally without getting the syllables in the wrong order, for example words like “preliminary” and “statistical”.

Rote memory: This refers to the ability to learn certain information as a habit pattern. The child who has problems in this area is unable to recall with ease those responses which should have been automatic, such as the alphabet, the number system, multiplication tables, spelling rules, grammatical rules, etc.

Short-term memory: Short-term memory lasts from a few seconds to a minute; the exact amount of time may vary somewhat. When you are trying to recall a telephone number that was heard a few seconds earlier, the name of a person who has just been introduced, or the substance of the remarks just made by a teacher in class, you are calling on short-term memory. You need this kind of memory to retain ideas and thoughts when writing a letter, since you must be able to keep the last sentence in mind as you compose the next. You also need this kind of memory when you work on problems. Suppose a problem required that we first add two numbers together (step 1: add 15 + 27) and next divide the sum (step 2: divide sum by 2). If we did this problem in our heads, we would need to retain the result of step 1 (42) momentarily, while we apply the next step (divide by 2). Some space in our short-term memory is necessary to retain the results of step 1.

Long-term memory: This refers to the ability to retrieve information of things learned in the past.

Until the learning disabled develop adequate skills in recalling information, they will continue to face each learning situation as though it is a new one. No real progress can be attained by either the child or the teacher when the same ground has to be covered over and over because the child has forgotten. It would appear that the most critical need that the learning disabled have is to be helped to develop an effective processing system for remembering, because without it their performance will always remain at a level much below what their capabilities indicate.

Strangely, though, while memory is universally considered a prerequisite skill to successful learning, attempts to delineate its process in the learning disabled are few, and fewer still are methods to systematically improve it.

LOGICAL THINKING

In his book “Brain Building” Dr. Karl Albrecht states that logical thinking is not a magical process or a matter of genetic endowment, but a learned mental process. It is the process in which one uses reasoning consistently to come to a conclusion. Problems or situations that involve logical thinking call for structure, for relationships between facts, and for chains of reasoning that “make sense.”

The basis of all logical thinking is sequential thought, says Dr. Albrecht. This process involves taking the important ideas, facts, and conclusions involved in a problem and arranging them in a chain-like progression that takes on a meaning in and of itself. To think logically is to think in steps.

Logical thinking is also an important foundational skill of math. “Learning mathematics is a highly sequential process,” says Dr. Albrecht. “If you don’t grasp a certain concept, fact, or procedure, you can never hope to grasp others that come later, which depend upon it. For example, to understand fractions you must first understand division. To understand simple equations in algebra requires that you understand fractions. Solving ‘word problems’ depends on knowing how to set up and manipulate equations, and so on.”

It has been proven that specific training in logical thinking processes can make people “smarter.” Logical thinking allows a child to reject quick and easy answers, such as “I don’t know,” or “this is too difficult,” by empowering him to delve deeper into his thinking processes and understand better the methods used to arrive at a solution.

Chemistry is generally divided into two broad branches: organic chemistry and inorganic chemistry. Other types of chemistry include physical chemistry, biochemistry, and analytical chemistry, with each field branching off into several specific subfields. Here’s a brief description of the most common branches of chemistry.

Organic Chemistry

Organic Chemistry has to do with the study of compounds that contain carbon (and sometimes hydrogen). Even though carbon is only the fourteenth most common element on the planet, it produces the greatest number of different compounds on Earth. Not surprisingly then, much of the study of chemistry involves organic chemistry.

The most studied groups of organic compounds are those that contain nitrogen. These organic compounds are important because they are often linked to the amino group. When the amino group combines with the carboxyl group, amino acids are born. Amino acids are important because they are as the building blocks of proteins.

Inorganic Chemistry

Inorganic chemistry involves the study the properties and reactions of compounds that do not contain carbon and which are not organic. Inorganic chemistry studies all non-living matter, such as minerals found in the Earth’s crust. There are many branches of inorganic chemistry, including geochemistry, nuclear science, coordination chemistry, and bioinorganic chemistry.

There is much overlap between organic and inorganic chemistry. For instance, organometallic chemistry studies the use of compounds that are capable of creating a covalent bond between carbon and metal.

Physical Chemistry

As its name implies, physical chemistry has to do with the physical properties of materials. Physical properties that are studied may include the electrical and magnetic behavior of materials, as well as their interaction with electromagnetic fields.

There are several subcategories of physical chemistry. These include thermochemistry, electrochemistry, and chemical kinetics. Thermochemistry studies the changes of entropy and energy that naturally occur during chemical reactions. Electrochemistry is concerned with the study of interconversions of electric and chemical energy of matter, as well as the effects of electricity on chemical changes. Chemical kinetics involves the study of chemical reactions. Specifically, chemical kinetics studies the equilibrium it reached between products and their reactants.

Biochemistry

Biochemistry is a branch of chemistry concerned with the composition and changes of living matter. Biochemists commonly focus on the physical properties and structures of biological molecules. Common biological molecules include carbohydrates, proteins, lipids, and nucleic acids. Biochemistry is sometimes referred to as physiological chemistry and biological chemistry. Biophysics, molecular biology, and cell biology are research fields closely related to biochemistry.

Analytical Chemistry

Unlike the other main types of chemistry, analytical chemistry doesn’t deal specifically with specific elements. Analytical chemistry is concerned mainly with the various techniques and laboratory methods used to determine the composition of materials. Qualitative and quantitative analysis are the two most basic methods used in analytical chemistry. Qualitative analysis has to do with identifying all the atoms and molecules in a sample of matter, with attention paid to trace elements. Quantitative analysis also involves determining the atomical and molecular structure of matter, but includes also measuring the exact weight of each chemical constituent.

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 .math-drills.com. The worksheets come with answer keys, so students can get feedback on their ability to correctly use base ten blocks.