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. .math-drills.com has thousands of worksheets with answer keys that you could use for this purpose.

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!

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.

In the example, begin by representing 234 with 2 flats, 3 rods, and 4 cubes. The goal is to remove 187 or 1 flat, 8 rods, and 7 cubes from these piles. Removing one flat is simple enough, but 8 rods and 7 cubes are difficult to remove if there are only 3 rods and 4 cubes! To solve this problem, trade in one flat for 10 rods, and one rod for 10 cubes. The result would be 1 flat, 12 rods, and 14 cubes. Removing the subtrahend - 1 flat, 8 rods, and 7 cubes - at this point would leave no flats, 4 rods, and 7 cubes. The difference is whatever is left after removing the subtrahend, so the difference is 47.

For beginners, it would be wise to start with subtraction that does not require trading. For example 1954 - 1831 would require no trading because there are enough blocks in the minuend to remove the subtrahend. For more advanced students, questions that include zeros can present a bit of a challenge. For example, 4000 - 3657 would require several trading steps all starting with four blocks. .math-drills.com has several thousand free math worksheets including subtraction questions with no regrouping (trading). One of the nice features of this website is that answer keys are provided, so students can get feedback on their results.

With enough experience, students learn subtraction on a conceptual level and are better equipped to apply it to pencil and paper methods later on. Students who only learn the paper and pencil method don’t always develop a conceptual understanding of subtraction and are less able to identify errors in their work.

Base ten blocks are not limited to just addition and subtraction of whole numbers. In part III of this series, several other uses of base ten blocks will be explored.

It’s parent-teacher conference time! Some are positive experiences where teachers are able to make great connections with parents. And yet other meetings are foretold by apprehension and met with strife. Over the years, you will encounter the gamut of positive and negative experiences, and everything in between. However, there are strategies you can use to make the best of any situation.

It is extremely important to make a good first impression (even if you already know the parents). Make eye contact with them, and greet the parents with a firm handshake. No weak grips! If you’ve never met the parents, stand up to introduce yourself. Welcome them with a smile. Remember that you are building relationships, and setting the tone for the conference.

A good way to open the conference is to ask how the student is doing in other classes. Ask about their other grades, and start building an overall picture. You will often find the student’s strong and weak areas, and you may even find surprises. I’ve found students who were failing every class but mine. And I’ve found the opposite too. A good overall picture can really give you a new perspective on your students.

Always try to say something positive. Even in the cloudiest of situations, you should find some ray of sunshine. And if you do have bad news to share, opening with good news can help ease the transition.

Be objective with bad news. Give truthful and accurate facts, and keep from making speculations. Make sure you have your facts straight! Work with parents, and try to offer suggestions. Most parents will look to you for ideas. Plan what you’ll say ahead of time. If you’ve taken the time to get to know your students well, you’ll find the conferences easier.

Positive parents are what we all expect and hope for. They come in with an open mind, are pleasant, and are willing to both listen to your comments and help with solutions to problems that do occur. These are often very short conferences at the middle and high school levels. The parents have heard the stories all before, and with good reason; students whose parents regularly attend conferences have higher grade averages and fewer instances of behavior problems than those students whose parents rarely interact with school personnel.

The truth be known, many parents are intimidated by teachers. Many do worry that their concerns and critiques will be turned around and used against their kids. Even though teachers find this entire concept laughable and preposterous, it does, nonetheless, cross many parents’ minds.

So, what do you do with a hostile parent? Diffuse the situation by being patient and listening. Sometimes its hard to just listen while parents are going off on you. They may be right or wrong, misinformed or even plain out of line. It is only a mistake to interrupt them, especially if they are on a roll. Stop yourself, focus on what they’re saying, even take notes to show you’re listening, and let them burn themselves out. Sometimes the hostile parents are looking for an audience, and sometimes they just need to vent. By giving them the time to ‘get it all out of their system’, you allow them to calm down so you both can reasonably discuss the situation.

Be sure to stand when they leave, again this is being courteous and polite. Thank them for attending. And let them know you’ll contact them if anything changes. Parents generally want to be kept informed about their kids, both the good and bad.

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For this article, and more on teaching and education, be sure to check out our website:

.starteaching.com

Frank Holes, Jr. is the editor of the StarTeaching website and the bi-monthly newsletter, Features for Teachers. Check out our latest issue at:

.starteaching.com/Features_for_Teachers_2feb2.htm

You can contact Frank at:

editorstarteaching.com

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!

Everyone gets those situations in life where an emergency has come up, and you don’t have the time (or sometimes the ability) to get a good lesson plan in to school for your students. Maybe you have a family emergency or a disrupted travel plan and you just cannot get into school to leave detailed lessons. That is why it is essential for you to have an emergency lesson plan available and handy.

The emergency lesson plan should be able to be used at ANY point in the year. It doesn’t have to fit in with what you’re currently doing (nor should it - it is to be used when you cannot leave normal sub plans). The lesson should be related to your normal curriculum, but it could be a supplement or an enrichment activity.

Get a folder (or a three-ring binder), and label it appropriately on the outside cover. There are even folders you can purchase (some schools even make these available to teachers) labeled ’sub folder’ or ‘emergency plans’. Also make sure you have an appropriate spot for your emergency folder on or in your desk area. Some schools will ask you to keep an emergency plan in the office. In either case, make sure it is easily accessible by a substitute teacher.

Think about keeping class activities to 10 to 15 minute increments. This way the sub will have better control of your kids. Students have difficulties adjusting to changes in their routines, and you don’t want to have to return to discipline referrals.

Keep the information organized and easily accessible for a sub. Remember, the sub won’t know where you normally keep things, and they can’t read your mind. Spell out exactly what you want done, where it can be found, and what you want done with it when they’re finished.

Make sure you have made enough copies of any worksheets so the sub doesn’t have to. And be sure to leave answer keys. Many subs will actually even grade your assignments for you if you ask them in your plans.

Get this done early in the year, and you can save yourself many headaches later, not to mention worries about what will happen in your room if you are unable to be there.

EMERGENCY LESSON PLAN IDEAS:

Language Arts: Include short writing activities involving students opinions. Thus they don’t have to have ‘background’ information, and they can write from their own experiences. Parts of speech review can include mad-libs or easy, fun worksheets.

Math: Leave a calculator activity. These could even be puzzles or partner games. Or give review problems.

Science: Copy a science article and have students read carefully and answer questions. Make speculations and use the scientific method. Or have students create the plans for a lab activity.

Reading: Leave students a copy of a short story or article, and questions to answer. You could even set up a ‘test-taking’ exercise, and discuss appropriate answers and strategies.

Social Studies: Map activities are great for emergency plans. You can even set up a one-day unit on any area/region of the world, including your own town or city.

Everyone gets those situations in life where an emergency has come up, and you don’t have the time (or sometimes the ability) to get a good lesson plan in to school for your students. Maybe you have a family emergency or a disrupted travel plan and you just cannot get into school to leave detailed lessons. That is why it is essential for you to have an emergency lesson plan available and handy.

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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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For this article, and more on teaching and education, be sure to check out our website:

.starteaching.com

Frank Holes, Jr. is the editor of the StarTeaching website and the bi-monthly newsletter, Features for Teachers. Check out our latest issue at:

.starteaching.com/Features_for_Teachers_2feb2.htm

You can contact Frank at:

editorstarteaching.com

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.

Starting your class on the right foot each day is very important to both you and the students. There are certain expectations you will have, be they required materials (texts, folders, gym clothes), basic supplies (pencils/paper), or behaviors (on time, in seats, working on opening activities). You are going to want these expectations met every day.

We designed a simple set of 5 rules to start out every class. These are easy to remember and easy to keep track of. Several of our teachers use a variation of the 5 rules to start their classes, and you may feel free to adapt these to your class. These are the rules I use in English class:

Rule 1: Students must be in their seats when class begins. In some schools, classes begin (and are dismissed) by a bell. Some classes begin at a specific time. Still other classes are started by a particular signal from the teacher.

Rule 2: Students must have a writing instrument. Again, different teachers have different expectations, be it pencil or pen or whatever. For me, it doesn’t matter as long as it s dark enough to read. I only balk at silver, gold, white, or any other light or fluorescent color (hot pink or yellow for example).

Rule 3: Students must have their folder out on their desk. Each of our classes requires students to keep important papers, notes, and other course artifacts. Some teachers allow students to keep these, and others provide a location in the room for folders.

Rule 4: Students must have all required materials for class that day. To reduce the number of times students ask me about what they need for the day’s class, I will either write the materials list on the board or put it on the class announcements on our TV (watch for the article on creating a class cable TV network our upcoming March issue).

Rule 5: Students must be working on the class warm up activity. In English class, students write out Daily Oral Language (DOL) sentences, practicing proofreading skills. On the edge of each day’s entry are the numbers 1 through 5, making it easy to grade. All you have to do is circle the appropriate number.

Again, we give each student a daily grade of points (1-5). Some teachers have only four rules and one rule is worth 2 points. You can change up and set your own rules and create an easy to grade set of points to fit your own classroom.

After a few weeks of practice, the checking of daily points becomes a student job. One student from each group (the RECORDER) gets the weekly responsibility to check the students’ daily points and circle the proper number. The teacher is freed up for other activities, and you only need to spot check through the room. This way I can record the daily points only once every two weeks and they are already tallied up for me.

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For this article, and more on teaching and education, be sure to check out our website:

.starteaching.com

Frank Holes, Jr. is the editor of the StarTeaching website and the bi-monthly newsletter, Features for Teachers. Check out our latest issue at:

.starteaching.com/Features_for_Teachers_2feb2.htm

You can contact Frank at:

editorstarteaching.com

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:

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.

Another useful skill to practice is trading base ten blocks. Each block can be traded for 10 flats, each flat for 10 rods, and each rod for 10 cubes. Going the other way, 10 cubes can be traded for one rod, 10 rods for one flat, and 10 flats for one block.

One simple use of base ten blocks that translates well to a paper and pencil method of addition is to add by regrouping. To add two or more numbers, start by representing each number with base ten blocks. Put all of the cubes from both numbers in the same pile; do this with the rods, flats, and blocks as well. Next, trade any groups of 10 cubes for a rod. Trade any groups of 10 rods for a flat; then trade any groups of 10 flats for a block. To read the resulting number, count the number of base ten blocks left in each pile and read the number.

To illustrate this procedure, picture the addition question, 568 + 693. After representing both numbers with base ten blocks and combining the piles of like base ten blocks, you should have a pile of 11 cubes, a pile of 15 rods, and a pile of 11 flats. Trading 10 of the cubes for 1 rod means you now have 1 cube, 16 rods and 11 flats. Trading 10 of the rods for one flat results in 1 cube, 6 rods, and 12 flats. Trading 10 of the flats for one block gives you your final piles of 1 cube, 6 rods, 2 flats, and 1 block. The answer to the addition question, therefore, is 1,261.

If you don’t have base ten blocks, you can use the virtual base ten blocks or make paper versions. If you need addition questions (with the answers included), you can access thousands of free math worksheets at .math-drills.com

In future articles, I will describe more uses for base ten blocks including subtraction and multiplication, and I will continue the series with other manipulatives that can help your child or student learn math.

More than likely, when you learned how to add, you started on the right and moved to the left. If you were adding whole numbers, you added the ones, “carried” if necessary, and repeated for the tens, hundreds and so on. This works well on paper, and it is the most efficient paper and pencil method; however, adding in the other direction has several desirable advantages: the left to right method promotes a better understanding of place value, it can be done mentally with much greater ease, and it does not require that numbers be lined up in a column. Students can learn left to right addition, so they have another method to choose from when presented with addition problems.

Left to right addition involves adding the largest place values first. As you move from left to right, you keep a cumulative total, so it is simply a number of smaller addition problems. To give you an idea of how it works and what it sounds like, consider the example, 677 + 938.

Begin by adding the left most place values. In the example this is 600 plus 900 equals 1500. Add the values in the next place, one at a time, to the previous sum, and keep track of the new sum each time. In the example, 1500 + 70 is 1570, 1570 + 30 is 1600. For students who are more proficient at this algorithm, they don’t necessarily think “plus 70″ or “add 30.” Their thought process, if said out loud might sound like, “600, 1500, 1570, 1600, . . .” Continue adding the values in each subsequent place until finished. The final steps in the example are 1600 + 7 is 1607, 1607 plus 8 is 1615. The sum is 1615.

As you can imagine, students need to be proficient at single digit addition and have an understanding of place value before attempting left to right addition. When they are first learning it, they might try repeating sums as they go along (e.g. 1500, 1570, 1570, 1570, 1600, . . .) to help them retain the newest sums. They might also cross out digits as they are adding. There is no rule about having to add in this way mentally. Students could write down the sums as they proceed.

Left to right addition promotes a better understanding of place value than right to left addition. In right to left addition, single digits are carried or regrouped with little emphasis placed on what the value of those carried digits are. In the example, 1246 + 586, students add 6 + 6 to get 12; they write down the 2 and carry the 1 when they should be carrying the ten. In the next step, they add 8 + 4 + 1 to get 13; they write down the 3 and carry the 1 when they should be adding 80 + 40 + 10, writing the 3 in the tens place (i.e. 30) and carrying the hundred. Essentially, right to left addition excludes vocabulary related to place value. Left to right addition, on the other hand, promotes an understanding of place value as each digit is given its correct value. In the example, the one in the thousands place is one thousand, the two in the hundreds place is two hundred, and so on.

Left to right addition is well-suited to mental addition since the sum is cumulative with no steps in between; in other words, there is nothing for the student to keep in mind except for the cumulative sum. In right to left addition, several numbers must be remembered as the student proceeds. To illustrate this, consider the simple example, 64 + 88. In left to right addition, the sum is simple to find: 60, 140, 144, 152. Only one number had to be remembered at any point. In right to left addition, 4 + 8 is 12, so there are already two numbers to remember: the two in the ones place and the regrouped ten. The next step is to add 60 + 80 + 10 to get 150. At this point, the two must be recalled and added to the 150 to get 152. Although this sounds simple, it becomes more complicated with more digits.

Right to left addition does not require numbers to be lined up in a column, but it is often taught that way because the method tends to ignore place value and relies on a student’s ability to line up the place values to compensate. Many errors that students make in right to left addition occur because they don’t have a strong knowledge of place value, and they forget or don’t realize that like place values need to be lined up. They might, for instance, add a digit in the tens place to a digit in the hundreds place. Another scenario is a sloppy recording of numbers where a digit is mistakenly added to the wrong column. In left to right addition, the emphasis is on finding a certain place value in each number rather than relying on the place values being aligned. Students, of course, need to be able to recognize place value before they can be successful at this method. For instance, they should be able to recognize that the ones in the numbers: 514, 1499, and 321 are in the tens, thousands, and ones places respectively. If they can’t, further teaching on place value is required before addition can be taught effectively.

Although left to right addition has several advantages, it isn’t suggested that you scrap everything else. Learning a wide variety of addition methods allows you latitude in problem solving situations. By teaching students this method, you give them another option when they are tackling addition questions.