Not So Far-Out Math

Prove or Disprove: 1/2 +1/3 = 2/5

2011-12-04T12:43:00.002-05:00

Prove or Disprove: 1/2 + 1/3 = 2/5

This prompt was handed to my classes last week. They were told that they could use words, numbers, pictures, charts, pieces of paper....in order to prove or disprove the above. They could not just say "yes" or "no". It was quite fun watching the wheels turn! Try it at home.

Word Problem Blues

2011-10-09T12:55:00.003-04:00

Please click on the images to see a larger version.

One of these things is not like the other. Both were answers on a recent HW sheet, and both have similarities. However, only one really answers the question asked. I think that you will be able to tell which is which!

When I see a student answer a word problem with only a number, I often remind students that teachers and test graders are among the least intelligent life forms on the planet. Are we talking 21 9/25 tons of elephant poo? 21 9/25 pounds of cheese? Is 21 9/25 even a legitimate answer to the question that was asked? Without a sentence, I simply do not know.

Split-It!

2011-09-05T19:49:00.005-04:00

09-05-2011

I had a student ask me recently for some help with an EDC component that I call split it! Actually, I had several students ask on behalf of their parents.

There was a homework problem that asked for students to show strategies to split a number like 7,758.

What I look for is a way to make splitting the number easy by using simple mental calculations. Usually, the first step is to decompose the number into its place values. So, 7,758 becomes

7,000 + 700 + 50 + 8 . Then, I ask that a student rewrite any number that they cannot readily split. Usually, these numbers start with odd numbers. We'll look at 7,000.

7,000 could be rewritten as 6,000 +1,000 (both of which are easy to split). After that, it's all down hill.

The whole number could be rewritten as 6,000 + 1,000 + 600 +100 + 40 + 10 +8, and finding half of these numbers should be EZ! 3,000 +500 +300 + 50 +20 +5 +4 = 3,879!

A Few Thoughts About Operations With Decimals

2011-06-05T12:17:00.005-04:00

Lately, our class work has centered on getting kids ready for middle school standards. In doing so, we have started working on multiplying numbers that include decimals.

I have posed many questions like: "How much would it cost to purchase 7.3 lbs. of nails @ $11.29?

I have encouraged students to do a few things to insure that their answers make sense, which roughly translates into getting the decimal located in the correct position. First, I have asked that students make an estimate that uses only whole numbers. In the case of the nails, we would round 7.3 to 7, and we would round $11.29 down to $11. So our estimate would be $77. We know this estimate is a bit low as we rounded both numbers down, but it is plenty good to let us know where to place the decimal. I have also encouraged students to think about the problem as if it was written as a mixed number times another mixed number. If they do so, it is easy to think about the tenths being multiplied by hundredths, and that has to produce a denominator of thousandths. That is what your math teacher did NOT tell you when he/she said to count up the digits to the right of the decimal in the problem and match that number in the answer.

So, our estimate is $77, and the multiplication of 73 X 1129 yields a product of 82417. So it should be pretty obvious that the only place to put the decimal so that you get an answer close to $77 is after the two...$82.417, and since we have no coin worth a thousandth of a dollar, we round to the nearest hundredth and get $82.42 .

Trying to put the math in front of the "
tricks",

T-Cubed

Do You Like Pyramids or Would You Rather Go to Prism?

2011-02-13T14:10:00.005-05:00

In our math classes we are currently working on identifying attributes of three dimensional figures (space figures), and most of that work focuses on pyramids and prisms.

We do need to become familiar with some basic 3DG vocab like: bases, faces, edges, and vertices. We also need to be able to find the surface area of said figures. So, I found a couple of links to basic info that might help.

http://www.math.com/school/subject3/lessons/S3U4L2GL.html#

http://www.mathsisfun.com/geometry/pyramids.html

It's Not Just a Multiple Choice Test!

2011-01-09T19:54:00.007-05:00

Years ago, most standardized math tests were using rather low complexity questions. That's not to say that the questions were easy. Some may have been very difficult, but knowing what to do was pretty straight forward.

A test might have asked simply asked for the sum of 3/8 + 1/6, and then given four possible answers.

Today, the state of Florida is putting a heavier emphasis on the cognitive complexity level of questions on tests like the FCAT.

As an example, a test might ask how much pie was eaten if the shaded portion of the first pie represents the pie before dessert, and the shaded portion of the second pie represents the amount after dessert.

This turns a very simple problem into a more cognitively challenging problem.

So, what fraction in its lowest terms represents how much pie was eaten?

A. 4/12 B. 7/12 C. 1/4 D. 1/3

On a scale of one to dumb....

2010-12-11T11:06:00.003-05:00

Quite recently, I received a comment on my blog from one of the coolest students that I ever taught. Ferney J. sent me a comment about my posting of the "top math jobs", and in her comment she asked if math was needed for a career in criminal justice.

At that moment, several things popped into my pea-sized brain. First, I really was happy to "hear" form Ferney. I have been wondering how life in Cally had been for her. So Ferney, if you read this, please send an email address (maybe a school email address).

Next, I realized how idiotic the list of top math jobs that I posted was. An actuary was listed as one of the top jobs. Be real. I think that should have been on the list of the "most dreaded" math jobs!

There are millions of careers that require a good math knowledge, and criminal justice is one of them. In fact, the FBI has several special units for folks that have great mathematical abilities. More importantly, everyday quality of life is improved with a good math schema!

So on a scale of one to dumb, my choice to post some random Internet math job statistics was DUMB!

Ferney, thanks for continuing to teach this old dog new tricks!

Fractions, Pizza, Percents, and More

2010-12-05T12:59:00.010-05:00

Lately, the class has been involved in answering questions involving the addition and subtraction of fractional amounts.

Some of the typical equations might look like:

2 1/4 + (1 1/3 -5/6) or

4 3/10 - (2 3/5 + 7/10) or

(3 3/4 + 2 1/8) - 2 2/7

While you might expect for me to be interested in the correct answer as my primary goal, I am actually much more concerned that kids are looking at the amounts and using appropriate strategies based on each unique circumstance.

We have studied several models that allow students to quickly create common denominators by thinking of such things as equivalent fractions on a clock, equivalent percents, and/or a good old common denominator. The trick is to know when to use each model.

The first problem is a great opportunity to use a clock model as all of the fractional amounts can be expressed as twelfths. Students should be familiar with clock fractions from the game "Roll around the Clock". 2 3/12 + ( 1 4/12 - 10/12)

The second problem is perfect for using percents as all of the amounts are very easily converted into percents that are easily added and subtracted. 430% - (260% + 70%)

The third problem is probably best solved by finding a common denominator as the fraction 2/7 is not easily represented on a clock, because 12 hours and/or 60 minutes cannot evenly be split into seven whole number pieces. Also, without a calculator, finding and using the percent that is equivalent to 2/7 is not practical. (3 6/8 + 2 1/8) - 2 2/7.... 5 7/8 - 2 2/7... 5 49/56 - 2 16/56...= 3 33/56

Of course, this post leaves out many steps, but the most important step is choosing the best strategy with which to work.

When Does -3 -3 +3 = 12,000,000

2010-11-28T20:25:00.004-05:00

Boise State's kicker missed a field goal in regulation to win their game against Nevada this weekend. The same kicker then went on to miss another in overtime. That's the -3 -3 part.

Nevada's kicker made his attempt in overtime for the win.

The loss may mean a loss in $12,000,000 for Boise State as they have no hope in playing in the NCAA Championship game now, and will most like play in the Humanitarian Bowl and receive a much smaller pay check.

See, math is COOL and CRUEL!

Fractions: How Do I Add Thee, Let Me Count the Ways

2010-11-21T12:53:00.010-05:00

When it comes to adding fractions, everyone knows that there is only one good way to do it. Right?

So, if I were to ask you to add something simple like 1/2 + 1/4, you would say that we have to find the LCD and change 1/2 into 2/4, add, and get a sum of 3/4. Right?

Or could I change 1/2 into 50% and 1/4 into 25% and then add and get 75% which is equivalent to 3/4? Possibly?

Or could I say that on a clock, 1/2 is the same as 6/12 and 1/4 is the same as 3/12 and then add and get 9/12 which is the same as 3/4? Any chance?

Admittedly, the problem above is very simple. Also, the problem was designed so that three or more methods could be used easily to solve the problem. Not all problems are so friendly! In fact, much of what we do in class centers on finding methods that work for the numbers presented.

Please think about how 1/2 + 1/7 would be different. Do you know both percents? Do you know what 1/7 of a clock face looks like? I don't!

A Really Cool Fraction Model

2010-11-13T15:01:00.008-05:00

http://illuminations.nctm.org/ActivityDetail.aspx?ID=11

Please click on this link and tell me what you think about this interactive fraction model. You will have to click on the right hand bottom "length" icon and choose "region" to see the circular model, and the circular model is the best. You also want to choose "Wide Range" on the top tabs to see smaller fractional pieces. Try it. It's cool, even if I am a math geek!

I love to see the regions change size as the denominator is changed. Can you imagine what it would look like if you could show millionths?

I really think that it takes models like these to really "get" fractions. I know that many students would agree that fractions can be frustrating. Perhaps, visual aids like this can help.

Math Jobs Anyone?

2010-11-07T11:06:00.006-05:00

In a financial way, what can a knowledge of math bring your way? Below is a listing of "math" careers. Can you think of others?

The following were recently were listed as the five "best" jobs.

They were :

software engineer
actuary
computer systems analyst
computer programmer

mathematician

This list was the result of the comparison of two hundred fifty jobs classified according to :
income
future outlook
physical demands
job security
stress
work environment

A List of Professions:

The following list briefly describes work associated with some mathematics-related professions :
actuary-- assemble and analyze statistics to calculate probabilities of death, sickness, injury, disability, unemployment, retirement, and property loss; design insurance and pension plans and ensure that they are maintained on a sound financial basis

mathematics teacher-- introduce students to the power and beauty of mathematics in elementary, junior high, or high school mathematics courses

operations research analyst-- assist organizations (manufacturers, airlines, military) in developing the most efficient, cost-effective solutions to organizational operations and problems; this includes strategy, forecasting, resource allocation, facilities layout, inventory control, personnel schedules, and distribution systems

statistician-- collect, analyze, and present numerical data resulting from surveys and experiments

physician-- diagnose patient illnesses, prescribe medication, teach classes, mentor interns, and do clinical research; students with a good mathematics background will find themselves being admitted to the best medical schools and discover that mathematics has prepared them well for the discipline, analysis, and problem- solving required in the field of medicine

research scientist-- model atmospheric conditions to gain insight into the effect of changing emissions from cars, trucks, power plants, and factories; apply these models in the development of alternative fuels

computer scientist-- interface the technology of computers with the underlying mathematical principles of such diverse applications as medical diagnoses, graphics animation, interior design, cryptogrraphy, and parallel computers

inventory strategist-- analyze historical sales data, model forecast uncertainty to design contingency plans, and analyze catalog displays to make them more successful; analyze consumer responses

staff systems air traffic control analyst-- apply probability, statistics, and logistsics to air traffic control operations; use simulated aircraft flight to monitor air traffic control computer systems

cryptologist-- design and analyze schemes used to transmit secret information

attorney-- research, comprehend, and apply local, state, and federal laws; a good background in mathematics will help a student get admitted to law school and assist in the understanding of complicated theoretical legal concepts

economist-- interpret and analyze the interrelationships among factors which drive the economics of a particular organization, industry, or country

mathematics professor-- teach mathematics classes, do theoretical research, and advise undergraduate and graduate students at colleges and universities

environmental mathematician-- work as member of interdisciplinary team of scientists and professionals studying problems at specific Superfund sites; communicate effectively across many academic discilplines and be able to summarize work in writing

robotics engineer-- combine mathematics, engineering, and computer science in the study and design of robots

geophysical mathematician -- develop the mathematical basis for seismic imaging tools used in the exploration and production of oil and gas reservoirs

design -- use computer graphics and mathematical modeling in the design and construction of physical prototypes; integrate geometric design with cost-effective manufacturing of resulting products

ecologist -- study the interrelationships of organisms and their environments and the underlying mathematical dynamics

geodesist -- study applied science involving the precise measurement of the size and shape of the earth and its gravity field

photogrammetrist -- study the applied science of multi-spectral image acquisition from terrestrial, aerial and satellite camera platforms, followed up by the image processing, analysis, storage, display, and distribution in various hard-copy and digital format

civil engineer -- plan, design, and manage the construction of land vehicle, aircraft, water, and energy transport systems; analyze and control systems for land vehicular traffic; analyze and control environmental systems for sewage and water treatment; develop sites for industrial, commercial and residential home use; analyze and control systems for storm water drainage and storage; manage construction of foundations, structures and buildings; analyze construction materials ; and surface soils and subterranean material analysis

geomatics engineer -- once known as "surveying engineer", includes geodetic surveying : takes into account the size and shape of the earth, in order to determine the precise horizontal and vertical positions of geodetic reference monuments----ad infinitum...it goes on for 30 lines and make civil engeneering look down right boring!

I know my favorite,and I get to do it each day!

I also know my least favorite, a cryptologist. I hate Soduko!

What about you?

...Favorite Math Book? Well, uh...let me see...

2010-10-31T12:53:00.003-04:00

Well, I am about to ask a rather stupid, or at least risky, question. What is your favorite math book? I fully expect to get a few responses from math teachers, but I hold out little hope that the general populace will have much of an opinion.

As for me, two of the pictures above show my favorite math books, and the other is my least favorite. Can you guess which is which?

What Kind of a Math-Person are You?

2010-10-24T13:19:00.004-04:00

What kind of a math person are you? That's a pretty tough question to answer without some help. So, I will supply some help.

Pick just one of the following problems, and think about how you would go about the math. Would you need tools (paper & pencil, a calculator, just your brain)? If you worked mentally, would you see pictures in your head? If so what would they be? Would you see numbers being carried, borrowed, or traded? Do you see numbers stacked vertically?

Well, here goes...

27 + 53

53 - 27

2,456 divided by 12

27 X 53

What do you think our kids should be learning in math?

2010-10-16T20:34:00.005-04:00

If I were in charge of the math world, I would have classrooms full of calculators, check books, receipts, and bank statements. It seems pretty clear that most "math energy" in our world is spent balancing our assets against our material wishes. It is also clear that many students graduate from high school without these basic skills.

I also think that textbook publishers have a pretty warped view of what real world math is all about. If Bobby, Suzy and Sam have to share any more pizzas, I think that I may become ill (it sounds real world, but it really isn't), ditto for 2n X 48z cubed = the square root of pi.

Having said this, I do deeply value building a deep understanding of basic number sense and mental math abilities. If this ability is developed early, higher math certainly becomes more manageable, and more mundane daily math tasks become less of a mystery.

What do you think?

I Don't Want to be a Hater, but..........

2010-10-10T13:19:00.004-04:00

What is the thing that you most hated about math or a math class?

For me, the thing that I could never understand was teachers assigning 53 similar problems for homework! I can remember sitting at home from the age of about 11 to the age of 17 thinking that the teachers must be quite thick if they did not realize that if I could do two problems, I could do 53 problems. We also never went over more than two problems in the classroom. So, I am still confused and angry. I mean, I could have played a million more minutes of soccer under the street light!

Peace,

T-Cubed

What Fun is Football Without Math?

2010-10-03T14:03:00.006-04:00

How does math improve your life?

Maybe it doesn't, but as I sat and watched the lizards get pummeled by the Tide, I really enjoyed understanding all of the math involved in football. It must really stink not to understand things like, "11 carries for 42 yards" or "allowed two TDs in nine red-zone trips".

Today, as I watch the NASCAR race, thousandths of a second or thousandths of an inch could make all the difference in the world, or Jimmie Johnson could just win like always...

So, I challenge you to answer the question. How does math impact your life daily?

For 2010-2011; What math should your kiddo know?

2010-08-18T08:47:00.003-04:00

Perhaps the most important, and also the least understood, information that a parent can have relates to the exact information that a student should be learning in their classroom. This is extremely important this year as Florida has adopted new standards in many subjects. For me the New Generation Math Standards should drive my instruction each day. For students and parents, these standards should provide a guide as well.

Really, every parent should expect a teacher to cover all of the concepts listed in these standards, and they should ask questions if they think something is being overlooked. The really cool thing is that these standards are not top-secret. In fact by logging on to http://www.floridastandards.org/Standards/FLStandardSearch.aspx anyone on the planet can find out what should be going on in the classroom. Having said that, this does not mean that all children will master all of these standards. Some kids will fly. Some kids will struggle, and most will do both, but at least the standards make the big picture a bit more transparent (not a pun, but it is great for parents). So, please look at these standards often.

What is a "Normal" Cat?

2010-04-25T13:41:00.003-04:00

Please click on the thumbnail images to make them larger.

So, just what is a normal cat? How could a student explain what normal is? Is it easier if you have numbers? Why or why not? What do you do with nonnumerical or categorical data? Can a cat have a mean (average) fur color? If not, how would you describe "normal" fur color. What kinds of charts or graphs would help? Is a sampling of 14 cats enough to represent all of the cats in the world? If not, how many should be sampled?

Is my cat's belly the most disgusting thing that you have ever seen?
Inquiring minds want to know!

Dude, this is like totally circular!

2010-03-20T22:27:00.003-04:00

Please click on the thumbnail image to make it larger.

Use your visual glossary and please identify the following:

All of the radiuses (AKA Radii)

All of the diameters

All of the the chords

The centerpoint

Calculate the circumference (the photo shows the length of the radius).

Coordinate Grids Using Negative Numbers

2010-02-21T13:17:00.003-05:00

Please click on the thumbnail image to see a larger version :-}

In 5th grade coordinate grids go to four quadrants! Please see if you can locate points C and D so that you could create a square by connecting all of the coordinates.

Some hints:

Always start at the origin

Think (left or right, then up or down...plot a point)

Negative numbers on the X axis ask you to move left

Negative numbers on the Y axis ask you to move down

I reached point B by moving right 30 from the origin and then moving down to the -10 on the Y axis. I plotted a point at the intersection of the two line segment found at (30, -10).

Best of luck!

Guess My Rule....If you know Geometry!

2010-02-07T15:47:00.003-05:00

A long time ago, little Tommy was sitting in his classroom, and the teacher asked, "Tommy, can you use the word "geometry" in a sentence?". Tommy said, "Sure! One night an acorn fell asleep, and the next day it wake up and said,'Gee I'm a tree!' ".

All kidding aside, one of the simplest, and yet deepest, ways to explore your knowledge of geometry is to play a game called Guess My Rule. It's simple. You put three or four polygons that follow a certain rule inside a circle and a couple that do not follow the rule outside of the circle. Then, you have someone "Guess My Rule".

So, what do think my rule is? Please don't be cute and say "polygons with less than five sides". While, the shapes inside the circle do have less than five sides (the ones on the outside do as well), my rule is a bit more complex :-}

Oh, a hint you wish? Well, I always investigate polygons by looking at their sides and angles!

Have fun!

How Do You Read That Decimal?

2010-01-10T12:43:00.005-05:00

The picture above tells a thousand stories! Well, actually it might tell a tenth of a story or a hundredth of a story or a thousandth of a story depending on where the decimal point is and what digits follow the decimal point!

I cringe every time that I hear TV reporters read decimals, especially weather forecasters, because they always say things like the barometric pressure is "29 POINT 92 inches", and these guys are scientists? Yikes! I hate to think that scientists are not concerned with things like place value. 29.92 should be read as twenty-nine AND ninety-two hundredths. Maybe students do understand that 29 point 92 is almost 30, but maybe they don't. Anyway, reading decimal values using correct place value is a good place to start.

SO,

.7 is seven tenths
.70 is seventy hundreds
.700 is seven hundred thousandths

Simple, say the number like a whole number, and then say the SMALLEST place value. (and thousandths are waaaayyyyy smaller than tenths-don't believe me? ---try cutting your pizza into 1,000 equal pieces versus ten equal pieces:-} )

However, we are not limited to looking at decimal numbers and only reading them as decimals. Huh? Well, .7, .70, .700 are all also 70%, and they can all be written as fractions as well.

So, which way should we read a decimal? In short, in whatever way makes the most sense for the problem. The work above shows three ways to read each decimal, percent or fraction. If I was trying to enter 1/8 on a cheap calculator, I probably would have to use .125. If I wanted to describe the same quantity to a person, I would probably say 12 1/2%, as I have learned that most folks have a pretty good "percent schema". If I was trying to add 1/8 to a simple fraction like 1/4, I'd leave both as fractions (1/8 + 2/8 = 3/8 woohooo).

So, for a fifth grader what is important to know?

.5 or .50 or .500...is the decimal way to say 1/2
So, .675 = 67 1/2 hundredths or 67 1/2 % or 67.5 %
.33333333 is the decimal way to say 1/3
.66666666 is the decimal way to say 2/3
1 whole = 100% 2 wholes = 200% .....
So, 3 1/2 = 350% or 3.5

Be flexible, and remember that the word POINT is dead and buried!

Fractions on a Clock Face.

2009-11-22T09:47:00.004-05:00

Please click on the thumbnails to enlarge the photographs.

One great way to get students to see that they can list several names for one fraction is by exploring what we could name each section of a clock face that has had the hands placed on different numbers. We always leave one hand at 12, and our clocks have equal length arms so we never know know whether we are looking at an hour hand or a minute hand.

Starting with the bottom clock, we see that one hand is on the 12 and one hand is on the 2. Most students first see this as 2 out of 12 hours, and they write the fraction 2/12. Many also see that this could be 10 out of 60 minutes, or 10/60. From that point, many notice that neither of these fractions is stated in its lowest terms. Most then "chop-chop" 2/12 (divide by 2/2) to get the fraction 1/6. Many then can say that 16 2/3% of the clock face has been "covered up" or "rotated through".

The next picture up shows one hand on the 12 and one on the 8. This shows 8/12 or 40/60. Again, neither fraction is in lowest terms. Many students will just see this as 2/3, as 4/12 looks like 1/3, but to prove it, one could divide 40/60 by 10/10 to get 4/6. Then one could divide 4/6 by 2/2 to get 2/3. We have mathematical PROOF! Most students now see that 8/12 is the same as 2/3, and that is the same as 66 2/3%. SWEET!

The goal in all of this is to be able to add fractions with unlike denominators. The third picture up illustrates the addition of 1/4 + 2/3. 1/4 is seen to be equivalent to 3/12 and 2/3 was proven to be equivalent to 8/12. So, we get 3/12 + 8/12 = 11/12.

We might also get 25% + 66 2/3% = 91 2/3% but 91 2/3 / 100 is not in its lowest terms, and most middle school teachers would simply freak-out if the answer is presented in a percent. So, even though the % is correct, I would emphasize finding the answer as a fraction in its lowest terms.

One last note, please remember that any fraction can be stated as another equivalent fraction simply by multiplying or dividing the fraction by the number 1, and any fraction where the numerator and denominator are the same (N/N) equals 1.

So, 32/48 divided by 16/16 , still has the value of 32/48, but it is more universally recognized as 2/3. 400/500 = 4/5 not because "you can drop the zeros", but because 400/500 divided by 100/100 = 4/5!

Similarly, 3/4 = 9/12, because 3/4 X 3/3 = 9/12. 5/6 = 10/12, because 5/6 X 2/2 =10/12!

I hope you are not confused, but if you are, please send me your comments :0}

Fractions Are Instructions to Divide! Sir!

2009-11-09T17:01:00.003-05:00

Please click on the thumbnail to make the photo larger.

Just how do you go about solving a question like: What is 3/12 of 36 penguins? There seems to be an almost endless list of strategies that don't work well. However, there is one strategy that works VERY well.

In my class, I have the kids repeat, military style, "Fractions are instructions to divide by the denominator and then multiply by the numerator! Sir!"

Almost all kids can remember this"direct order". Many do not, however, know what needs dividing up, or into how many groups. In the problem above, the 36 penguins need to be divided into 12 even groups, and that puts 3 penguins in each group. 3 would be a fine answer to the question, "What is 1/12 of 36 penguins?", but is not a good answer for 3/12. This is where the "and multiply by the numerator" comes into play. We need to take into account 3 of the 12 equal groups of penguins, or 3 X 3 = 9 penguins.

The work also shows an array of 36 "penguins", and it would be quite correct to say that 3/12 could be understood as 3 out of every 12. This is a great model for small numbers, but I would not want to split up 3600 penguins into an array!

So, "Fractions are an instruction to divide by the denominator and then multiply the answer by the numerator! Sir!"

Fraction Equivalent Strips

2009-11-02T15:41:00.002-05:00

Please click on the thumbnails to enlarge the photos.

Please take a look at the fraction strips above. The goal of the above was to match about 20 landmark fractions to their equivalent percents. The goal was not rely on memory, but to strategically think about the placement of these fractions. For example, most students immediately know that 1/2 = 50%. They can use that knowledge to "discover" that 1/4 = 25%, and then to find that 1/8 = 12.5%. Even the more difficult fractions like 1/6 become easy to place when there are landmark percents already present. 16 2/3 % is easy to place if 10% and 20% are already listed.

Fraction Action in Fifth Grade

2009-10-29T19:19:00.008-04:00

Please click on the thumbnails to enlarge the pictures. If you wish, you may leave a comment by clicking on the "comment counter" at the very bottom of the page.

This story about fractional numbers is told from the bottom picture up.

In our EDC notebooks, we have been tracking equivalent fractions. We started on day 1 with the fraction 1/16. All students were able to understand that there were 16 pieces in the top row and if we were to shade one of those boxes it would equal 1/16 of the whole. On day 2, most students immediately realized that when we added another 16th, we created an area equal in size to the the 8ths row immediately below. When we got to 4/16, also 2/8, students saw that this was equal to 1/4 of the whole. It was at that point that students started to fill in percentages for the fractions that we had created. Most students knew that 1/4 = 25%, and they concluded that 1/8 was half as large so it must = 12.5%. A few students then realized that 1/16 must equal half of 12.5% or 6.25%. This knowledge of fractions and their equivalent percents will be a key component in adding, subtracting and comparing fractional pieces. Most students know several landmark percents like 25%, 50%, and 75%, and they can easily shade an object like a circle graph correctly with those amounts. Soon, students learn to represent other amounts like 30% (3/10) with accuracy based on their knowledge of the landmarks mentioned. Putting fractional amounts into percentages just makes life easier.

The third picture up comes from our current investigation, and it shows how students come to see "out of" statements as fractions. One out of three is the fraction 1/3, which seems easy enough but is VERY important conceptually.

The next two pictures are the absolute most important concepts that we will cover in this unit. Students shade in portions of a 10 X 10 grid and report the percent (parts out of 100) that it took to cover the portion. One picture shows one fourth of a grid shaded (25 blocks and 25%). the shading is a bit "artistic" , and it probably led to the misconception about shading in 1/8 of the grid, which should be 12 1/2 blocks or 12.5% . The percent was listed correctly, but the grid does not have 12 1/2 blocks shaded. Still, I can tell that the idea of 1/8 being half of 1/4 is understood.

The top picture is the Holy Grail of understanding fractions. All of the fractions listed are considered landmark (IMPORTANT) fractions in our number system. Understanding the percentages that go with each of these is a very critical conceptual step.

So, know the fraction. Be The fraction!

Know the percent. Be the percent!

Revisions: Another Great Example!

2009-10-14T07:50:00.004-04:00

Please click on the thumbnail image to enlarge the photo.

If you wish to comment, please click on the word "Comments" at the bottom of this post.

Doing revisions correctly takes a bunch more work than just stating a new answer. Only by clearly stating why a prior answer was wrong and providing new evidence (NEW WORK), can an instructor be sure that the concept involved has really been mastered.

In this example, every distractor (answer choice) is examined. Clear reasons are stated as to why there is only one correct answer. This is enough to convince the person in charge (ME) that the concept of what a square number really is is now understood.

PTB Prime-Time-Battle

2009-10-08T16:03:00.003-04:00

Please click on the thumbnails to enlarge the photos. To leave a comment please scroll to the comment section at the bottom of the blog.

Prime-Time-Battle is a really fun EDC game that we play daily. This games gets students to try and find the prime factorization of numbers, as all numbers can be expressed as the product of prime numbers (OK, not 1 or 2). Besides building automaticity with multiplication facts, PTB prepares kids for much of the learning that they will see in many advanced courses. In advanced math courses, many times LCM and GCF are found through comparing prime factors. It also seems to just be a bunch of fun, as many kids see it as a a chance to compete in friendly math endeavor!

Have I Got A Story For You!

2009-10-02T15:35:00.012-04:00

Please click on the thumbnail to enlarge the photo.

To leave a comment, you have to click on the word "comment" in the comment counter at the bottom of the blog :-} I smell candy!

All of these samples of student work show how creative and efficient students can be when asked to solve basic multiplication and division problems. All of these students used very efficient strategies to carry out work that is usually a very scripted set of steps (Traditional algorithms allow for very limited flexibility in mathematical thinking).

Equally important, the students were able to create their own word problems that could be answered by solving the numerical problems. This is always a very tricky situation for students of this age, (in truth, it is for students of any age) as students are so used to solving numerical problems that are written for them. Many start by imitating problems that they have read in math books. However, when students start to create problems that are really relevant to their lives, they gain a very deep insight into what multiplication and division really are.

These are works in progress, and there are some very simple errors, especially in the division problems (usually because of faulty subtraction), but the kids are showing a pretty deep understanding of what the answers really mean. In most math books, the remainders of division problems are just written as a fraction or a decimal amount, but in the real world, remainders mean something. Remainders can be valuable or inconsequential, and contrary to popular opinion, not all remainders can be split into equal pieces.....kids, puppies, and crunchy potato chips included!

In photo number 1, there is a subtraction error. 835-690 = ?

In photo number 2, one factor pair, 26 X 20, was left out.

In photo number 3, the remaining $17 could, and probably should, become 50 cents extra per person, although giving the money to charity is admirable :-}

In photo number 4, 3135-2300 = ?

As you can see, there is a ton of input to manage, but the concepts are coming together!

Double-Trouble or Split-It!

2009-09-27T15:45:00.002-04:00

Recently we have added a "game" to our EDC notebooks. Double-Trouble and Split-It are games that are designed to get students to strategically think about doubling and splitting numbers. These skills are crucial if a student is to build great mental number sense.

For example:

When I ask kids to split 1,700, many kids balk, because they see 17 groups of 100 as an odd number of hundreds. Most kids are not seeing this as 8 1/2 groups of 100. However, most students can readily replace 1,700 with 1,600 + 100, and they can easily split these numbers to get to 850! This still may not seem important, but since the product of 17 X 10 is so easily calculated, it seems logical that 17 X 5 (half as big as 17 X 10) should be easily found if 1,700 can be easily split.

In a similar way, being able to double numbers with ease leads to being able to double small factor pairs into larger factor pairs. If 7 X 2 = 14, then 7 X 4 is twice as big. So, 7 X 4 = 28. Then, 7 X 8 = 56, and 7 X 16 = 112...likewise, 17 X 2 = 34, 17 X 4 = 68...17 X 4 = 68, 17 X 40 = 680, and 17 X 80 = 1,360 ...all done through doubling.

When doubling a number, many students find it easiest to double the largest place value first and then work through the smaller place values.

Doubling a number like 1, 486 might look like 1,000 + 1,000 = 2,000...400 +400 = 800 ...80 +80 = 160...6 + 6 = 12...added together, the sum is 2,972.

These are VERY important concepts! Please practice. It's the kind of thing you can do in the car.

Division and Multiplication A Match Made in Heaven

2009-09-22T15:21:00.004-04:00

Please click on the thumbnail to enlarge the photo.

This student shows why a great understanding of basic multiplication is so vital to solving division problems. By knowing the simple factor pairs shown (15 X 1 =15 15 X 2 =30...) and knowing the relationship of these to larger landmark factor pairs (15 X 10=150 15 X 20=300), this division problem becomes an absolute snap to complete correctly. I highly advise students to write the first several multiples, and a few larger landmark multiples, of the divisor when solving almost any division problem.

2009-09-17T15:24:00.002-04:00

Please click on the thumbnail to enlarge the photo.

Boxes of Markers is a student sheet designed to introduce the idea of portioning items evenly as a form of division. Many students simply skip count by the number of students , in this case 23, in order to find out how many groups of 23 can be made by multiples of 70 markers. It might look like 23, 46, 69;at which point the kids realize that they have run out of markers. Most can see that each "skip count" represents a student with a marker. So, three skip counts would equal 3 markers per student.

The work above is from a student that immediately recognized this as a division problem that could be solved using easy multiples of 23 until the number of markers was exhausted. This really is the goal of this type of exercise, as it makes division and multiplication forever seen as related activities. Once the student that did this work adds sentences that explain the numerical answers, the work will be at standard to say the least!

Revisions Are Vital to Mastery!

2009-09-16T15:19:00.002-04:00

Please click on the thumbnails to enlarge the photos.

Students in our class are able to complete "revisions" of tasks to insure that concepts that were missed are cleared up or at least made more clear. Students do have to prove that they understand why they were wrong, and they have to show new work that proves that they really have reached an understanding of the concept.

In the top photo, the student explains what went wrong in the problem in the bottom photo. In this case there was a fairly major misconception about the formula used to compute the amount of money to subtract ($1,000 X the date or $25,000) from the original balance ($712,000). This misconception made the problem impossible for the student to solve. However, it would not be correct to assume that the student in question did not know how to subtract. Therefore, this thorough revision provides equitable remediation. Revisions are hard work, but they are well worth the effort!

It's Not Your Father's Math Anymore!

2009-09-14T15:40:00.006-04:00

Please click on the thumbnails to enlarge the photos.

The work above shows some more interesting strategies for subtracting and for showing possible combinations. On a recent test, students were asked to subtract $25,000 from $712,000. Many students used the traditional algorithm with mixed results. Today, the class decided that decomposing the numbers, less the 0s, and using positive and negative values made solving the problem easier. Most students realize that using negative values is way easier than it first sounds. Look at the middle photo.

The possible combinations problem involved making sandwiches with three major ingredients, a bread (Wheat or Rye), a meat (turkey, ham, or chicken), and a cheese (Swiss or American). Most students as comfortable using a tree diagram to show the combinations, but many students fail to understand exactly what a combination is. In this case, a combination is a type of bread with one type of meat and one type of cheese. Once that became clear, students could easily see (via their tree diagrams) that each bread type could have six possible combinations of ingredients. Look at the top photo.

Why Do We Do the Things We Do?

2009-09-09T15:48:00.004-04:00

Please click on the thumbnails to enlarge the photos.

This edition of my blog emphasizes two things that are hard for most 5th graders and adults to do well. The first is subtraction (be honest adults, you know it's easy to make mistakes), and the second is metacognition, or thinking about your thinking.

This task asked that students solve a fairly easy subtraction task two ways and then to give a rationale for the method that they prefer. Many kids wrote that they preferred one method over another, "because it was easier for them to do" or because, "that's the way that they did it last year in class". However, some students were able to make statements that relate to the value of the numbers, the distance between the numbers, a preference for addition over subtraction, a need for a visual strategy, or a like or dislike of negative numbers!

If you are an adult, try to answer that same question. I bet it will not be easy :-}

Rounding Using A Number Line

2009-09-08T15:35:00.004-04:00

Please click on the thumbnails to enlarge the photos.

Today we worked rounding numbers using a number line to provide "visual evidence" that conclusively indicates which end of the number line the number is closer to. This process helps students see the relative distance of numbers, and leads students to really think about place value. This, in my mind, is much more valuable than the traditional method of looking at digits and playing the "4 or less is down and 5 or more is up" game.

Counting Puzzles Stimulate Great Thinking

2009-09-04T15:24:00.004-04:00

Please click on the thumbnails to enlarge the pictures.

These pictures show two recent student sheets that we have been working on in class. Both are designed to stimulate thinking about counting patterns. Often, students fall back on methods of solving these questions that are far from efficient. For example, a student might simply list all of the multiples of 25 in order to get to 300 and then state, after counting the written numbers, that it takes 12 people counting by 25 to get to 300. However, many students quickly latch onto the idea that it is much easier just to figure out how many 25s are in 100 and then triple the amount to get to 300.

On the "What's in Between" page, students have to sort through a multitude of important concepts in order to find the answers to these puzzles. For example, students must be able to find the mid-point of two numbers in order to find a reference point to know if a number is a number is closer to the numbers on the ends of the number line. Some students get really confused with larger numbers like 7,900 and 8,100. These same students would also have no trouble finding what comes exactly in the middle of 79 and 81. Sometimes, "ignoring the zeros" can be a great strategy. Students also have to be flexible and efficient when they are forced to find numbers that are multiples of two different numbers. If a puzzle said that the mystery numbers were said if you count by 125s and were also multiples of 500, then only certain numbers would qualify. I would try to think about multiples of 500 first, as that means only numbers that end in two or three zeros would qualify. See if you can find the puzzle with answers that are not quite correct.

Basic Operations Voted Most Efficient Week 2!

2009-09-01T15:41:00.007-04:00

Please click on the thumbnail to enlarge the photo.

Our first homework sheet proved to be of great use, as many simple and very efficient strategies emerged!

Problem number one was a subtraction problem, and the most popular strategy was the use of an open number line to add up from $267 to $323. The class then gave a go at "straight subtraction" which uses positive and negative values. Many in the class found this to be even easier to navigate, and most thought that it was faster too. Fast and accurate; you cannot beat that!

The second problem shows a very efficient execution of left to right adding. Adding the greatest place value first can help reduce large errors in calculations. I'd much rather be off by a dollar than a million dollars!

The final problem was almost universally solved by using a "generic rectangle" to multiply 16 by 12. The partial products are usually accurate, easy to calculate, fast, and easy to add up. Some students used multiplication clusters like 16 X 10 and 16 x 2 to solve the problem, and that is another fast and accurate strategy that is great for this factor combination!

These students ROCK!

Homework With Meaning!

2009-08-28T14:13:00.007-04:00

You may enlarge the thumbnail image by clicking on it.

These two homework problems may seem too easy for a 5th grade student, but on examination of the thinking displayed, it is clear that a deep understanding of important concepts, like place value, are well understood by this student.

The first problem was quickly identified as a "subtraction" problem, but it was solved using simple addition. This young mathematician added up from 267 to 323 on an open number line. This is kind of like what folks did in the good old days before cash registers had built in electronic calculators. They would count back the change using landmarks along the way. This student also used landmarks (easily recognized and "easy to work with numbers"), as she first "jumped from 267 to 270. This allowed her to easily add on to 270 in order to get to 300. From 300, the jump to 323 was a piece of cake. Finally, all of the jumps were totalled, and the distance between $267 and $323 was found correctly.

Also of note are the sentence restating the prompt (question) and the matching equation, These also signify a real sense of math understanding.

On the second problem, the student added from left to right, and to me that is great! Adding the largest place values first makes it less likely to make a mistake of great magnitude. Using the traditional algorithm makes it more likely to make a mistake in the larger place values, and that's a real drawback to sticking with traditional algorithmic thinking, unless you REALLY understand the method well.

EDC and More!

2009-08-27T14:38:00.007-04:00

Please click on the thumbnails to enlarge the pictures.

Today we spent time completing another day of our EDC curriculum, and we began to diagnose the needs of students. It was great to see that many students do remember the strategies that they have learned over the last five years. It was so great to see students paying such close attention to detail on our fourth day of EDC. I wrote, in black pen, some of the important concepts that we discussed during today's session on one student's awesome work :-}

Simple Beginnings Have Powerful Meaning

2009-08-26T16:59:00.005-04:00

Click on the thumbnails to enlarge.

These are pictures of the first couple of days work in a our calendar math series, "Every Day Counts". All of these examples show student work that is remarkably complete and detailed. The concepts may seem simple, but in reality all of this work involved very deep conversations about place value and other important concepts that relate to number sense.

A big "Booooyyyaaahhhh" goes out to the three students that produced this work.

2009-2010 School Year Approaches!

2009-08-13T19:23:00.010-04:00

Well if you are reading this, you have found my "famous" math blog. This year, I plan to include, dare I say this, almost daily pictures of the work that is being produced in our math classes. I think that by doing this, students and parents will get a very clear picture of just what is really going on in class. There will be way less writing by me (stop clapping), and way more for you to DISCUSS.

To do this, I really need parents to sign and turn in all of the Internet permission slips. I will never post a child's name. I might call Suzy "Student S." :-}, or I might just leave it at "This student". You guys please tell me what you are OK with. I will never embarrass a student by leaving a negative comment about work.

Oh, just another note about the photos of work that I attach. What you will see first is a small "thumb nail" image at the top of the blog. By clicking on the thumb nail, you will see the full size piece of work. Sometimes, I forget to select the correct size for the image, and the work will be too large to view. If I do this, please let me know.

I am blessed and honored to have your kids in my class. They WILL create remarkable work!

Line Plots, Cats, and 5th Graders

2009-04-19T22:24:00.005-04:00

If you were presented with this data about cat height, what could state about what was typical of the cats in this group. Could you find the mean, the mode, the range, and the median? Could you tell what fraction and percent of the cats were taller than 5.5 inches but less than 8 inches? (Click on the picture if you need a full-size view.)

I just measured my cat and she is 10 inches tall at the shoulder. She is a smallish seven-year-old female cat. Does that make you think anything about what kind of sample this might have been?

If you have a cat, could you measure their height and reply? We'll stick to inches since the chart was created in that unit:-}

Spinner Games and Probability

2009-03-22T20:20:00.005-04:00

If the above picture was a spinner game, and you were given the choice of "two scoring options" to choose (you would get a point if you landed on a number that met the requirements of either of your two options chosen), what would you choose?

A. a prime number
B. an odd number
C. a multiple of 5
D. an even number

Many students would look at choice A, and state that it has a 4:8 or 50% probability, as 4 of the 8 numbers are prime.

B would have a 5:8 or 62.5% probability, as there are 5 odd numbers

C would have a 4:8 or 50% probability, as there are 4 multiples of 5

D would have a 3:8 or 37.5% probability, as there are 3 even numbers

Man students would just look for the highest probabilities. Perhaps, they would choose A and B, but if you study those choices closely, you may see that several sections would be possible "losers". So, what two options would you choose?

Hints for Taking the FCAT Test!

2009-03-08T12:44:00.003-04:00

1. Your teachers, principal, family, friends, and classmates have faith in you!

2. You have all the tools you need in order to be successful. I've taught a bunch of kids, and you guys have the goods!

3. Do not stress out. This is a time to show off.

4. If a question is too difficult, skip it! It is far better to miss one item than the 23 items that come after it :-}

5. Read each question carefully. Be sure to read with "a voice" that you can hear in your head. Rereading with a voice has helped me solve so many problems over the years.

6. Make some notes to back up your mental math strategies. If you show some work, your chances of being correct go up a bunch. I have seen that most errors occur through rushed simple operations like adding or subtracting. So, don not rush (you know who I am talking to ;-} )However, please remember that you do not have time to "write a book" about each problem (again, you know who I am talking to ;-} ) .

7. Look at your final answer and see if it makes sense. Elephants do not weight 17 pounds!

PEACE!

Seven Things.............

2009-02-12T21:27:00.019-05:00

Seven Things You Don’t Really Need To Know About Me!

(You can click on the images at the top for a more detailed and spectacular view of these rare artifacts! The ones that are embedded do not show all the available glorious details.)

Well, there was this time when I had to have Mrs. Shannon come rescue DA at the ER at 12:00 midnight, because I had a kidney stone! Oh, I already emailed that huh?

Well then, there was this fall from my road bike that peeled me like a tater! Oh yeah, I emailed and sent pictures.

OK, I’ve got it. When I was born, I cost exactly $196.80, and I can prove it!

Now, if you knew that, you must work for the CIA or something.

Keeping things chronologically oriented, I have always been better at math than I was at more cultured things like the fine arts, and again, I can prove it.

Thanks to my dear old mom’s careful interrogation and subsequent documentation, I know that the picture (both are from first grade) contains one black bear, one deer under a tree, one raccoon hiding in a hole in a log, one bunny, one bird flying someplace, and a partridge in a pear tree! You try and figure it out.

I have never acted in a play unlike Ms. Lipsky, but my family did kind of resemble a TV family that traveled on a big groovy bus, and again, I have the evidence!

Hyde Grove Elementary Patrol Boy on the loose!
Nice Lamb-Chops Pops!

I often spent part of the summer with my grand mom and granddad in Vidalia, Georgia. They had the biggest house in town, and granddad always had the latest Caddy, except for the time that he bought a Rolls Royce! They were pure southern gentry right down to their bigoted cores. Granddad built almost all of the roads in S.E. GA. (well, he owned the company, but his job seemed to be riding around in his 30 foot long land-yacht checking on the crew and admiring or criticizing their work). Grand mom stayed home, made homemade candies, crocheted quilts, told the maids what to do, played bridge, played the stock market and counted her money. Being the son of a preacher man that marched with Dr. King, opened the first integrated church in Montgomery, Alabama, and started an integrated summer camp in Pensacola, Florida, I did my dead-level best to set my grandparents on the right path. When I was seven, I asked my grand mom why the very nice maid was not eating lunch with us; after all, she made the lunch. I got the stare and then was told that “the help” eats on the back porch. Without a word, I got up with my food and started to stroll away from the table. Grand mom asked where I was headed. Needless to say, I had a very nice lunch on the porch. The scene of the crime is pictured below. The Chamber of Commerce used their house on their brochure cover.

Outlaw biker days! From 6 to 15-years-old, I do not remember a day that I did not ride a motorcycle. At 15, you were legally allowed to pilot a motorcycle that was “producing less than five brake horse power”. Somehow, I convinced my dad that my 120 MPH Honda café racer only made 4-5 horsepower. He was not, and never will be, mechanically inclined! However, the joke would be on me as, on December 30, 1979 I left planet earth after my motorcycle rear-ended a turning car. I was taken to St. Vincent’s ER and “pronounced”—“called”—and about to be toe-tagged when an older ER nurse said that my vitals did not show due to a build up of fluid around my heart and in my lungs. She “bagged me” with a salt and ammonia mixture and saved my life. I spent the next five days in ICU. My folks say that I kept asking the same question over and over again, “Has the Gator Bowl started?” (that was the year that Clemson beat Ohio State and Woody Hayes punched Clemson linebacker Charlie Bauman in the mouth ending Hayes career). I also complained about bugs running up and down my IV drip (there were no bugs). When I left the hospital, I had blurred vision for a while and I tended to repeat myself, I tended to repeat myself. I still love cycles, and I have owned about 20, but it took me 15 years to get on one after the wreck. I do not ride today, as the costs outweigh the benefits. My 15th birthday card was a bit prophetic?

Number VI: I really could run a 4.45 forty coming out of high school!

Final bit of totally useless knowledge: My whole family calls me by a different name than my CCE family. I was known to the world by my middle name, Rives, right up until I enrolled at UNF. It’s pronounced Reeeeevs. However, most of the time when people read my name it turned into Rivers, Riiiiiiiiiives, Rivas, Rico, boy with the blonde hair, or “did they misspell this?”. Being shy and unwanting of extra attention, I cringed every time I got a new teacher, because they would butcher my name and then argue that they were right, because “the silent e makes the vowel long”. Combine the middle name with the equally challenging last name, Ruark, and you simply cannot say them both without sounding like you have a mouth full of marbles. So finally, at UNF when a prof started to say “Ruh, uh Rii, uh Ria”, I broke in and said, “It’s Tom. Teee Ohhh M.” Oh, Ma Bell still can’t get with the program either. To wit,

Peace!

Polygon Attributes and Names

2009-02-01T12:39:00.003-05:00

One of the most important concepts in studying two-dimensional geometry centers on classification of polygons based on attributes. If you click on the picture, you will see two polygons. One has three sides and one has four sides.

Just counting the number of sides allows us to name the polygons (a triangle and a quadrilateral---If you said, in your head, "square" you are about five steps ahead so slow down :-} ).

Looking at the bottom of the page, notice that the student labeled the vertices (angles) of the polygons. This will allow that student to describe many more attributes of each polygon. In the triangle, line segment BC is congruent to line segment AB. A triangle with at least two congruent line segments is an isosceles triangle. Also, the student should note that angle ABC measures 90 degrees, a right angle. So, this triangle can also be called a right triangle. Putting both attributes together, we get an isosceles right triangle! None of this is possible without close examination of the sides and angles.

In the quadrilateral, please note that the opposites sides are parallel and congruent. This makes our quad a parallelogram. If you notice that all sides are congruent, our parallelogram also becomes a rhombus. If you notice that this parallelogram has four 90 degree angles, then our parallelogram becomes a rectangle. If you notice that our parallelogram has all congruent angles and sides, then we have a regular polygon called a square!

Students are often asked to classify polygons from "least restrictive" name to "most restrictive" name. For our quadrilateral names would include (in order):
polygon
quadrilateral
parallelogram
rectangle
rhombus
square
(rectangle and rhombus could be flip-flopped)

If students were asked to provide attributes of our quadrilateral, they may state:
Our quad has four 90 degree angles
Our quad has opposite sides that are parallel
Our quad has opposite sides that are congruent
Our quad has four lines of symmetry
Our quad has perpendicular line segments...

Please remember that attributes lead to names; they are not the same.

Peace!

My Fraction is Nothing Compared to Your Fraction!

2009-01-19T18:38:00.004-05:00

Comparing fractions has become pretty sophisticated by the 5th grade. Not long ago, this student was probably asked by a teacher in an earlier grade to compare two fractions. Like most kids, this student probably made an attempt to draw out the comparison. In prior drawings, there may not have been any real expectation of having equal size pieces (like really equal), or even any expectation that the wholes shown be the same size.

Now, this student uses two traced circles (same sized wholes) and really demonstrates how to create accurate models by using a clock for guidance. 2/3 becomes 8/12, and 2/5 becomes 24/60. That's pretty sweet, and it insures equally sized pieces :-}

The decimal notation is another big step up too, and so is the mentioning of what would be half in each denominator (1/2 in thirds is 1 1/2 , but in fifths it's 2 1/2).

Survey Says.....

2008-12-14T19:46:00.005-05:00

After watching the Heisman awards show the other night, I started thinking about my classes recent attempts at verifying the validity of surveys. The way that I see it, the Okie QB should not be holding the hardware right now if Superman got the most first place votes, and I am not a Gator fan at all, as the picture should prove! -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

At any rate, our class has determined that a valid survey should represent exactly 100% of the group that was surveyed. That seems to make sense, as, if there were five people asked to respond, you should have five responses, not 4 or 12 ;-}-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

However, we determined that this all becomes a little trickier if we blend the way that the results are recorded into a mush of fractions, percents, and decimals. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Now the challenge ensues. See if you can figure out if the following survey could be valid. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

All of the teachers at CCE were asked to provide one response each to the following question; How could CCE best improve the CCE experience for our students?------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

0.22 of the teachers replied that Mr. P. and Mr. D. should not be allowed to wear Steelers football clothing.--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1/5 of the teachers replied that Mrs. H. should hold no more than 12 tech meetings a week.-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

30% of the teachers stated that Mrs. Phillips should lobby for overtime pay for time spent scoring diagnostic tests. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

0.125 of the teachers said that Coach Hall should have a Character Ed. class for the teachers.------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1/8 of the teachers said that Mrs. Phillips, the math teacher, should write all lesson plans for all math teachers.------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

3% of the teachers said that lunch should be catered by P.F. Changs.------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1/4 of the teachers said that there should be no more surveys ever.-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Please look for combinations that "simply must go together". Are there any amounts that are equivalent? Would it help to know that 0.125 could be read as one hundred twenty-five thousandths or 12 and one half hundredths or 12 1/2% ? ----------------------------------------------------------------------------------------------------------------------------------------

Well, is it mathematically valid? -----------------------------------------------------------------------------------------------------------------------------------------------------------------

Peace and Happy Holidays!

T-Cubed ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Sorry for all of the dashed lines. This post scrunches all of my hard returns together leaving no space between the lines.

Please have some flexibility!

2008-12-07T21:04:00.003-05:00

READING DECIMAL NUMBERS

Imagine playing a game with the following three decimal number cards:

0.4 = Four tenths

0.45 = Forty-five hundredths

0.375 = Three hundred seventy-five thousandths

You might be asked to put these in order from least to greatest. You might be asked to add these all up to find a final sum. You might be asked to shade in a 10 X 10 grid in a way that represents the amounts shown. In fact, students will be asked to all of the above.

A few simple things will always makes matters work out better:

First, since out students are so tuned in to comparing fractional amounts through conversion to equivalent percents, why not convert decimals to percents? But how?

Well, it's easy enough for most kids to see that 0.4 (four tenths) could be written as 4/10, and most know that we could multiply 4/10 X 10/10 to get 40/100. Once we have a denominator of 100, we have a percent (parts per hundred). So, 0.4 = 40%.

For 0.45 (forty-five hundredths) we just think 45/100, and we have our percent. 45%

Now for the challenging one, not really, if we look at .375 (three hundred seventy-five thousandths) we could write 375/1000. If we divide 375/1000 by 10/10 we get 37.5 /100 or 37 1/2 percent. so, if you were shading in a 10 by 10 grid, you would simply shade thirty-seven and one half blocks.

If I was asked to add these decimals, I would rewrite them so that they all have the same number of digits on the right hand side of the decimal point. 0.4 = 0.40 = 0.400 = 0.4000000000000000000000000000000000000000000.

0.45 = 0.450 etc.

I'd end up with:

0.400
0.450
+0.375

From there, I would add the largest place value (the tenths) .4 +.4 +.3 = 11/10 or 1.1
Next, I would add the hundredths .00+.05 +.07 = 12/100 or .12
Finally, I would add the thousandths .000 + .000 +.005 = 5/1000 or .005

1.1
.12
+ .005 = 1.225 or 1 22/1000 or 1 22.5/100 or 122 1/2 %

Have Fun :-}

I am right a fraction of the time!

2008-11-23T18:38:00.008-05:00

Be you a student, a teacher or an interested guardian, please ask yourself if you could come up with any more profound statements about the fractions 2/3 and 3/4. Also, please challenge yourself to create accurate models of each as well. To the mathematically uninterested, this may not seem like a grand accomplishment, but to a well-trained math geek, this is poetry!

Yep, all of this was done by a ten-year-old! Could you have done this at ten? Yes, I am quite proud of my students.

Well, chime in now. What do you think?

Peace and have a great holiday!

T-Cubed

I am about to make Mr. Pinchot really mad!

2008-11-09T18:13:00.004-05:00

No, the cat will not anger Mr. Pinchot, unless it makes him think of the Mt. Nittany (spelling?) Lions that his beloved Penn State uses as a mascot! You see Penn State lost to Iowa and ruined their perfect record and their chances of an NCAA football national championship. However, none of that is going to make Mr. Pinchot as angry as he will be when he finds out that I actually published a math piece using the sophisticated math terms "chop-chop" and "double-double"! He absolutely hates these terms even though they are based on sound math principles.

The term "chop-chop", as all of my students know, refers to creating an equivalent fraction by reducing the numerator and denominator at the same rate. Usually, the "chop" implies dividing by 2/2, but we could "chop" by any fraction with a value of 1 whole, n/n, 10/10, 4/4...We would create a fraction that has an equivalent value.

For example: 12/16 divided by 2/2 = 6/8, and 6/8 divided by 2/2 = 3/4 ! Oh, I say chop-chopping is a fine way to create new and equivalent fractions!

You could probably guess that he also hates "double-double", which is just multiplying by 2/2 (the whole number 1). This, again, creates a new and equivalent fraction! For example part deux; 3/4 X 2/2 = 6/8, and 6/8 X 2/2 = 12/16..., but we don't have to settle for using 2/2 all the time. We could go crazy and use 13/13 or 3.5/3.5 to multiply by. We could take 1/2 X 3.5/3.5 and get 3.5/7, and that IS equivalent to 1/2. Boooyahhh!

The key is to remember that what we are really doing is multiplying or dividing by 1!

Please be sure to ask Mr. P. about Penn State, and please tell him that we have been chop-chopping a bunch!

Some of the digits? No, Sum of the digits!

2008-11-02T13:46:00.003-05:00

Hello again. Once more, a student has shown how working "smarter" makes for a better and more efficient math student!

In this example of our last homework, this student relied heavily on the "rules for divisibility" in order to make answering the divisibility questions a snap. In general, I am opposed to anything that smacks of being a math "trick" as an aid to completing math problems. However, the rules of divisibility are all based on sound math principles, and these rules do lead to more efficient and more accurate work. We rely heavily on these rules each day as we try to find common factors for the numerator and denominator of fractions. Right now, 5th grade students (and really 4th grade students) should know the "easy to use" divisibility rules for 1, 2, 3, 4, 5, 6, 9, and 10. The rule for 8 is also useful, but the rule for 7 is really too complex to use at this stage.

If you don't know the rules, please contact any worthy math teacher, student or the Internet for help:-}

I also have to comment that the Prime-Factor tree diagram shown is also a huge step towards understanding higher math. I also love PFTs , because they are lateral thinking exercises, as there are many paths to the correct answer. Again, ask a wise student to explain. The kids love doing these "puzzles" too!

Thanks for reading,

T-Cubed

Problems In Our "Times"

2008-10-26T20:43:00.005-04:00

On a recent test, students were asked to find an expression the would be equivalent to 25 X 41 .

I'd be willing to bet that you read that as 25 "times" 41.

The students were given four choices as a correct answer:

A) (20 x 40) + (5 X 1)
B) (20 X 40) + (40 X 1)
C) (25 X 40) + ( 25 X 1)
D) ( 25 X 40) + (26 X 1)

Now, before you put on your number crunching hat, consider the rereading the "times" sign in the original problem. If you read the X as "groups of", you become aware that we are looking for some combination that is equivalent to 25 groups of 41 or 41 groups of 25.

Do you see a choice that makes 41 groups of 25? I do! Decompose the 41 into 40 and 1. Does that help? The answer is C, and I'll C U L8TR.

T-Cubed

May seem so simple, but......WOW!

2008-10-19T20:47:00.005-04:00

Sometimes the simple things in life are better! This student used simple multiplication problems to help solve his/her division situations. I am so pleased to see students working "smarter not harder"! Obviously, he/she really understands that division depends on a real understanding of the relationship between multiplication and division. After all, in division you start with a known product and a known factor, and you simply try to figure out logically what the unknown factor is. 745 divided by 16 is the same as 16 X n = 745 .

So, why not start out with 16 X 1 = 16 ? It's easy and it leads directly to figuring out 16 X 10 = 160 , and 16 X 2 = 32 , and 16 X 20 = 340 , and 16 X 40 = 680... (being able to double and halve is sooooo important!). By the way, I'd probably throw in a 16 X 5 = 80 , as it is just half of 16 X 10!

At any rate, this work really does show great "smart" thinking! Even with the error in problem number two, this really is great mathematical thinking. Can you find and fix the error in problem number two? I bet the student that did this will see it quickly and be able to revise this problem easily!

T-Cubed

Multiplication and Division RELATED? :-}

2008-10-12T21:14:00.011-04:00

The two scans above, starting with the top scan (uh, not the pug), represent some of the most efficient strategies around when it comes to multiplication and division. Perhaps, the best thing about these strategies is that most of the calculations can be carried out mentally and do not rely on the rote memorization of many math "facts". Truly, this is how most mathematicians think, smarter not harder!

Parents, students, fellow countrymen, and all other living beings, please challenge yourself to see if you can find the beauty in this kind of thinking. You may just find your inner math geek after all!

As a footnote, I wish that I could say that I "hired" a student to write this for me, but that would not be honest. I wrote this myself! Bad hand writers of the world unite! We shall rule.

PLEASE COMMENT!

T-Cubed

Revsions: Not easy to do, but worth it!

2008-10-05T13:16:00.003-04:00

Remember, you can click on the images to show them at full size.

These revisions are very well done. They really show an understanding of what was done incorrectly, and the new work shows that the concepts in question are, indeed, now understood.

I especially liked the explanation in the "money" problem that sated that the student in question "Didn't add back the 3 (cents) that I rounded to." This shows a pretty complete understanding of the error that was made. I can also accept the "blonde moment" explanation for multiplying by 7 instead of 6, as that is just a lapse in concentration, and it's funny!

I hope this shows that really revising problems is a bunch of work, and that it is not something to rely on as a way to overcome a lack of participation in the class or at home. Rather, revisions are a way to refocus on a few concepts or errors that were not quite clear at test time.

What do you think?

Click on "comments" below. Type your comments. Type in the funky letters in the funky letter box. Click on anonymous. Click publish. wait for me to approve and post your comment :-}

Homework That ROCKS!

2008-09-28T21:08:00.000-04:00

The homework shown above (you can click on each image and it will enlarge) really shows what we are looking for in terms of effort in and out of the classroom!

The first thing that I notice is that ALL of the work is completed on separate paper. Bigger space means more ability to think outside of the tiny "box" provided on the page itself, and this is not optional. All students should be completing work on separate paper.

The second thing that I look for is work that was revised, enhanced, or merely recorded during our class review sessions. These sessions take about 20 minutes, and they are only valuable if the strategies shown are recorded. Again, this is not an option. Even if you had the same answer, you can record alternative strategies.

The last thing that I noticed was the level of detail in the review work. This student was so focused on our discussion that she recorded her answer in the same "surfer voice" that I was using during our review ("The length of Mrs. Phillip's porch is totally like 16 feet!") .

This work is totally rad and awesome! WORD!

Mr. R.

AKA T-Cubed

I should have mentioned that leaving a reply is pretty easy. All you have to do is click on the "comments" tab, type your reply (please don't use your name if you are a student), and then click on "anonymous" before you hit "publish". The post will come to me, and if it is appropriate, I will post it to the page.

Beijing Olympic Subtraction Example: Cool!

2008-09-22T22:59:00.000-04:00

Take a look at this work from our Beijing Olympic Subtraction problem, and see if you are on the right track.

The work shown is very well constructed and efficient.

Please let me know what you think:-}

Splitting Up Work Makes Life Easy!

2008-09-21T14:07:00.000-04:00

I wish that I could make this uploaded work a bit more clear, but it still is great to emphasise my point.

This work comes from our last Mini-Quiz, and it is a great example of how organizing your work in a very simple way can lead to a great outcome. K did an amazingly simple but powerful thing when she drew lines to separate her problems. She made it easier for her brain to focus on the tasks individually. Separate spaces for separate thoughts. Simple but very important.

I also noticed something else really cool. K did not use ultra-small writing. While I like saving trees as much as the next teacher, I know that simply writing bigger and using more space often creates better results.

I also love love love love love love the expanded notation in the addition problem, the open number line in the subtraction problem, the generic rectangle in the multiplication problem, and the misconception corrected in the division problem. This work ROCKS!!!!!!

What is subtraction anyway? Difference or...?

2008-09-01T21:24:00.000-04:00

We will be starting our investigations in the area of number sense tomorrow, and our unit starts with a look at subtraction. You know, where you find the difference between two numbers.

What if I changed that to read "find the distance between two numbers"? Does that sound any easier?

Here's my point: If I ask a kid to subtract 95 from 100, I hate to see the kids that are driven by traditional thinking as they line up their numbers in columns and subtract to find the distance. Wouldn't it be easier to just think about how far it is from 95 to 100? You can use an open number line if you wish as well, for all of you visual learners out there. Hopefully, you'll just think about the distance and know that the distance is five.

Many teachers that I talk to say "That's great for easy small numbers, but I don't want my kids to rely on that for bigger numbers!" I think that you might want to rethink that stance if you've taken it already.

Using an open number line or "counting up" works well with any numbers, and it prepares kids to make estimates far better than traditional thinking. We all know that "in the real world" folks will use calculators to deal with big numbers anyway. So why not build some audacious number sense now.

$1,000,000 - $257, 665 can be solved as follows:

257,665 +35 = 257,700 +300 =258,000 +2,000 = 260,000 +40,000 = 300,000 +700,000=

Bingo 1,000,000

So, if I add the numbers in bold that I "added on" I get a sum of $742,335 which is the distance between the two numbers. The beauty is that I did it in my head for the most part:-} Another beautiful thing is that there are an infinite number of ways to "add on". Kids choose the number combinations that they like. I would, of course, make sure to lead them into using landmark numbers (usually numbers that end in 0, 00, 000...) as targets to add up to.

Peace, and try it...

T-Cubed P.S. Do you know the man in the picture?

United Streaming, Fay, and Other Far-Out Stuff

2008-08-24T11:07:00.000-04:00

Recently, a very intelligent cohort of mine called me to tell me that she had found, on United Streaming, an hour's worth of five-minute snippets of real folks telling how they used math in their jobs. She described the very varied types of individuals profiled. All of the interviews sounded great! I cannot wait to watch these interviews.

Then, T.S. Fay toppled a huge Oak tree onto my house and I lost phone service, cable, Internet, and electricity. At that moment, 3:00 A.M. 08-21-08, I started to think about math! Some of my thoughts: Is there still a $5,000 hurricane deductible? How much do carpenters charge per hour? How much does plywood cost per sheet? Exactly how hot will it get in my house without AC? How long will my generator run on one five-gallon tank of fuel?

Math may not always be pretty, but it is always around us!

Peace,

T-Cubed

1st Thought Was?

2008-08-14T22:18:00.000-04:00

I am going to write out a math situation, and then I am going to ask you to do two things. First, I want you to tell me what words you think are the key(s) to solving the problem. Second, I want you to tell me what your first physical step (or first couple of steps) would be to solve the problem (grab a calculator, line up the digits...). ELA FOLKS, THERE IS NO WRONG ANSWER HERE!

Gidgi and Kay-T want to go to a concert with there best friend Roger. The tickets are $37.50 each. Together they have saved $92.75. Do they have enough money? Do they have too much money? Will they get change back? Will they need more money? Will Roger enjoy the show? Will this blog ever end?

OK, I'll Explain The Garb!

2008-08-14T22:01:00.000-04:00

Many people have inquired about my state of dress in my photo. Well, here's how it went down. I went to Kingsley Plantation, a former cotton plantation (owned at one point by an African-American woman!). I was visiting the slave quarters that were originally constructed of tabby, a mixture of oyster shell, sand, water, and lime. One of the small buildings was being restored to its original condition which is quite different than the weathered look that the buildings have today. Part of the restoration process involves letting new tabby dry at a slow rate, and that involves wrapping the drying tabby in burlap.

To be short, I found some burlap on the ground. I made my daughter wear it for a gorgeous shot. She took a photo of me, and then the Park Ranger screamed at us for bothering the burlap. Really, it was just resting on the ground.

That's it!

Doubling: A Great Skill & A Million Different Ways To Get There!

2008-08-11T20:02:00.000-04:00

Doubling numbers is a powerful math skill. Doubling can help solve a multitude of problems be they addition, subtraction, multiplication, division...

I'll try doubling 13. 13, 26 (two 10s + two 3s), 52 (two 20s + two 6s...40 +12 or 40 + 10 +2), 104, 208, 416, 832, 1664, 3328 (had to think about that one for a moment!), 6656, 13,312, 26,624...

Try to think about why some numbers are easier to double than others. What strategies make the tougher ones manageable?

Try doubling 16 until you run out of MENTAL strategies....What have you got to lose?

Peace,

T-Cubed

Getting Ready For A Brave New Math World

2008-08-11T19:45:00.000-04:00

As I sit here, a few days before the kids roll into class, I am thinking about math issues that 10-11 year-old kids would really engage with. I am asking math geeks from all over the planet to send me one idea that they think that the kids would really get a kick out of (preposition, see, I am not an English teacher!).

I know they love to gather info on the size of feet, circumference of heads..., but I really want to start out the year with a meaningful challenge that will generate buzz. I want them to text someone about how cool their work seems.

We are working on building number sense (4 basic operations). So, I would love suggestions in those areas.

Money? "Green" issues? Tech milestones (how much will I have to pay if I go over my texting limit by 1000 texts?)?

Help?

By the way how do you subtract 269 from 412? (with no paper, it took me 8 seconds) What you got for me?

More about 25 please!

2008-06-23T11:24:00.000-04:00

Wow! I am sitting in a cyber-cafe in the mountains thinking about 25!

What is it larger than?
What is it half of?
What is half of 25?
What if it were a percent, what would it look like?
What landmark numbers is 25 close to? How close?
What's a landmark number?
If it was money, what could I buy?

Now you try.........

Peace,

T-Cubed

25 huh? Tell me more!

2008-06-12T08:59:00.000-04:00

A challenge follows. Please take a few minutes and jot down everything that you know about 25.

I'll be back (to be read in an Austrian accent with malice).

Peace,

T-Cubed

Subtraction by Addition?

2008-06-11T23:35:00.000-04:00

Ask somebody to think of subtraction as the distance between two numbers. Then ask them to find the distance between 464 and 782 by adding up (you may want guide them into using landmark numbers...usually numbers that end in 0 or 00 or 000...). 782 - 464 becomes: 464 + 6= 470 470 + 30 = 500 500 + 282 = 782 In summation, we found the distance to be 6 + 30 + 282. Adding from left to right gives place value sums of 200 (2 is the only hundreds place value digit)+ 110 (the sum of 30 +80) + 8 (6 + 2) or 318. This may look complicated, but all can be done as simple mental math, and the concept works well with all number combinations.

Please remember that you can start thinking this way using really small numbers to help understand. We all know that 10-4 = 6, but if you think about the distance between 4 and 10, it becomes easy to see how we can add-on to 4 until we get to 10 (4 + 6 =10)

Peace,

T-Cubed

Adding and Thinking From Left to Right!

2008-06-11T23:31:00.000-04:00

Add a number from left to right! $1,248 + $6,921 becomes 7000 + 1100 + 60 + 9 or $8169 (read as 81 hundred sixty-nine dollars) Place value-place value-place value.................................... Start with the place value that has the most value. If you make a mistake, it will likely be in a lower place value and not in a higher place value. Would you rather be off by a few bucks or a few thousand bucks?

Peace,

T-Cubed