The most important aspect of any middle school mathematics classroom is to get the students thinking. Moving through the regular curriculum will help them learn processes and terminology. The teacher must also be planning ways to engage the students in lateral-and logical- thinking activities. This could be thought of as the Sustained Silent Reading portion of a mathematics class; the activities that encourage the students to apply what they have learned, and also think outside-the-box in coming up with unique solutions. All middle school classes across the curriculum have room for puzzles and questions that have the students thinking differently.

Many curriculum today are powered by an imposed schedule where the driver is the number of outcomes covered prior to state testing.  Teachers need to control their own schedules based on what their students need, rather than meeting artificial deadlines.  Time for presenting lateral and logical thinking has got to be found.  In order to maintain the United States’ lead in innovative thinking, students must have practice in applying logic and lateral thinking within the school day.

A great way to do this is to seek puzzle resources on the web. Some sites that fit the bill are www.folj.com, www.mycoted.com, and www.lateralpuzzle.com.  Presenting these to a classroom can be simple.  Put the puzzle on presentation software; either PowerPoint, ActivStudio, or your districts preferred program.  Find ten minutes in the day. Capture the students’ attention.  Give them paper.  Have them think individually about the problem, and write or draw a response.  Have them pair with their neighbor in discussing the logic of their response, then have the students share in some way.  The result will be an engaging activity that will help all students move forward in their lateral and logical thinking.

Let’s take a look at lateral thinking. How about a puzzle called “Manhole Covers”, found at the www.mycoted.com website. “Creativity & Innovation; Science & Technology” is the catchphrase of mycoted.com, making it a great place to look for different ways to look at problem solving. I have looked at several other examples of lateral thinking puzzles – some I got, some I didn’t get. Some I got because I looked at other examples. In the manhole covers problem, you are looking at why manhole covers are round rather than square. My solution was that they are very heavy, and if they are round they fit into the hole in the ground the first time. You don’t have to match corners with a circle, because there are none! Once I had my solution formulated, I looked at the posted solution. To my surprise, two were offered. (Spoiler alert!) One was that round manhole cannot fall through the hole like a square one can when held perpendicular and diagonal to the hole. The secondary solution was that the workers were able to roll the round covers, rather than carry them. Perhaps, as my solution was also in the “manholes are heavy” solution range, it would pass muster. Perhaps not.

The point is, lateral thinking problems are not solved by knowing formulas, or having quick facility with basic facts. They are an activity which finds itself in Bloom’s Application level, where the thinker would determine the varying attributes of a round and a square manhole cover, then determine how those attributes relate to the manhole itself. Having said that, an argument can be made that the question could also be on the Analysis level, as the thinker has to analyze the factors involved, and use the information to come up with a response.

Some advanced elementary students and many middle school students are ready to solve a problem such as Manhole Covers. Out of 27 fifth graders, probably three quarters of them would come up with some solution. When I do the problem with my students, I will supply them with paper and have them do a think-pair-share activity with lots of paper and pencils so they can draw as they think. Toward the end of the class, the students would share their solutions with the entire group so they could learn from each other. Following the solution to Manhole covers, I would ask the students to look at other lateral thinking problems to enhance their ability to think outside the box.

An example of a slightly different puzzle is the “eggs in a basket” question. In this question, there are six eggs in a basket. Six people each take one, yet there is one egg left in the basket. This one was not as obvious to me, and the answer really requires thinking outside the box. **(Spoiler alert)** I did not come up with the correct answer, which is that the last person to take an egg takes it in the basket. This did not make as much sense to me. In the “manhole problem”, it is identified as both a logic problem as well as a lateral-thinking problem. For me, logic may be more understandable.

No matter what the puzzle type, giving your students one unique problem a day will help them expand their thinking and questioning techniques, as well as giving them a life-long interest in finding out-of-the-box solutions!

For an initial geometry vocabulary post, I am defining two- and three-dimensional figures because they are mentioned in the first paragraph of the NCTM standard regarding geometry. “Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships”

Two-dimensional figure – A two dimensional figure is one that has length and width, but no depth, giving it measurements in two directions. Some examples of two dimensional figures are the square, circle, rectangle, parallelogram, and rhombus. This definition is from my prior knowledge.

From mathisfun: A 2 dimensional shape. Has width and breadth, but no thickness. These are plane shapes.

Three-dimensional figure – A three-dimensional figure is one that has length, width, and depth, giving it measurements in three directions. Some examples are the sphere, cube, rectangular prism, and the triangular pyramid. This definition is from my prior knowledge.

From mathisfun: An object that has height, width and depth, like any object in the real world. Example: your body is three-dimensional

References: www.mathisfun.com

All through my education, when I would be assigned a project on a subject of my choice, it would turn out to be featuring maps and globes in some way. It was a thread running though many years of classes. When I first became an elementary enrichment teacher, I became more interested in maps. Google maps and Google Earth were introduced in these years and became a staple with my special groups. Geocaching added another, more adventurous side to this subject, and I visited caches from Canada to Switzerland and many points in between, finding them with family members and friend, enjoying the technical and social aspects of the activity. Although I enjoy my GT position, I am gently nudging myself toward a change, and it may be toward math. Geometry, and therefore, maps, are a key part of math, and that is a major reason for me taking the PLS course, Geometry for the Middle School Teacher Online. The other reason is to just simply learn more about geometry. I want to broaden and deepen my understanding of this interesting and useful subject area.

Sarah

Here are the definitions of several terms that relate to statistics and investigating a set of numbers. The definitions were taken from Drexel University’s Math Forum, which is a valuable resource for teachers and students.
Mean – The mean is otherwise known as the average, and is found by adding all items in the set and dividing by the number of items.
Dr. Math says “The MEAN is the arithmetic average, the average you are probably used to finding for a set of numbers – add up the numbers and divide by how many there are: (80 + 90 + 90 + 100 + 85 + 90) / 6 = 89 1/6.”

Mode – The mode is the most frequently occurring number in the set. In a set of 4, 5, 6, 7, 7, 7, 9, 10, 7 would be the mode.

Dr. Math says “The MODE is the value that occurs most often. In this case, since there are 3 90’s, the mode is 90. A set of data can have more than one
mode.”

Median – The median is the number that is numerically in the middle of the set of numbers.
Dr. Math says, “The MEDIAN is the number in the middle. In order to find the median, you have to put the values in order from lowest to highest, then find
the number that is exactly in the middle.”

Measures of central tendency – These are all the methods of finding a middle point in a set of numbers. They include the median, the mode, and the mean. All three have benefits and detriments in developing a clear understanding of the data.
Measures of variability – This is a study of variables that affect the measures of central tendency. One piece of data that affects the results of the mean, mode, and median is the range. A way to overcome the affect of the range is to determine the standard deviation. This is an often used measure of variability as it gives more information about the set of data.
A site that has a plethora of information on various statistical terms and methods is Connexions. Another site that contains links to many sites that relate to teaching statistics to elementary students is Cyber-sleuth Kids. Statistics is an interesting topic that provides a way of gaining vital information about the world around us.

A sample is a group that represents the entire population.  In that case, the sample should represent likely voters in the area.  One is that it might not be random unless it is selected without bias.  Another is that it might not represent the population as a whole.

We know that a survey, if the results are to be considered valid, must be taken with many considerations in mind.  In taking a survey of voters, it is necessary to have your random sample be taken from those who are likely to vote in the upcoming election.  Taking a poll that included those not registered to vote would not be valid.  If you were taking a survey and used numbers from the telephones numbers from that area, that sample would not contain those who don’t have telephones, and would therefore not be valid.  If the sample is known to be from the telephone directory, it would still be necessary to make sure that the survey taker included others living in the same household but were not listed in the telephone book.  The others living in the household would not own house phones because they are using the telephone that the one listed is supplying.

In another case, envelopes were dropped in three neighborhoods in Los Angeles to test honesty.  The envelopes that seemed to contain money had the lowest rate of return.  The two other sets of envelopes had a much greater rate of return.  This seems to prove that people are less honest when money seems to be available.  Some people retuned the envelopes that had blank papers in them or fake money in them after opening them.  I believe that those envelopes would not have been counted as returned, because the openers would have realized that it was some sort of a test, and that had contaminated the process.  The envelopes had to be returned unopened to prove honesty in the finder. There are some aspects of the survey about which I have questions.  One question would be why were only ½ as many envelopes that were empty and labeled “research study” placed on the street?  Why were the words “research study” written on the outside of the envelopes of the empty envelopes?  Was that necessary, or does it contaminate the study?  Another question is why the envelopes were dropped in different areas of the city.  If the idea was to test honesty, with the variable being the different envelopes, wouldn’t dropping them in different areas based on income be an unnecessarily complicating factor.  It seems that the survey could have been better managed.

The final lesson that I take from the activities on samples and surveys is that it is very important to design a survey with a strong understanding of samples. The survey must be designed to accurately represent the target population.

You know how sometimes you read something and don’t get it?  It was that way with Pascal’s triangle with me. . . not to mention binomials!  Okay, so I buckled down and didn’t actually do any formal math for three days.  I’m using the theory that talking about it and thinking about it can be better than actually doing it.  Now I’ve taken a second look, and the cloud is lifting!  I’m ready to attack this area of probability!

Now the next issue – I am asked to create a school-based scenario where Pascal’s triangle can be used to figure out the probability of  certain outcomes.  Since I’ve looked at many well-written blog posts with very interesting scenarios, it seems as if all the good ones are taken. Does it sound like I am whining?  Maybe. . . . so we head to the cafeteria.  The students at my school who buy milk to accompany their lunch have three choices. They can buy white milk, chocolate milk, and, believe it or not, strawberry milk!  Research has shown that there is a 30% chance that a student will buy strawberry milk.  Now we have an interesting scenario!  Let’s look at a group of friends, five students who all buy strawberry milk 30%, or .30 of the time, and buy other flavors 70%, or .70 of the time.  Since there are five students in the group, we look at the 5^th row of Pascal’s.   We see that the numbers on the 5^th row of Pascal’s Triangle are 1, 5, 10, 10, 5, 1.  Putting this in table form . . . .

Number of students choosing the strawberry milk	0	1	2	3	4	5
Number of ways	1	5	10	10	5	1

We can see that there is 1 way that no children will have the strawberry milk and also 1 way that all 5 children will choose the strawberry milk.  We see that there at 10 ways that 2 children or 3 children will choose the strawberry milk.

The probability that none of the five students choose the strawberry milk can be expressed as 1(0.7 x 0.7 x 0.7 x 0.7 x 0.7).  To the nearest tenth, the probability is 16.8%.

There are 10 ways two students will choose strawberry milk.  The probability of this can be expressed as an equation:  10(0.3 x 0.3 x 0.7 x 0.7 x 0.7) or 30.9%.

Using Pascal’s Triangle to determine the number of ways the students can choose strawberry milk helps expedite the computation of this important information without creating a tree diagram.

Independent Event – this is an outcome that is not dependent on the previous events in the situation. An example is flipping a coin – no matter how many times you flip it, the chance of getting a head or a tail is 50%. From Math is Fun: An outcome that is not affected by previous outcomes. Example: tossing a coin. Heads or tails is not affected by previous tosses.

Tree Diagram – this is a way of logically demonstrating the outcome of multiple events, making it easier to determine the chances of the outcomes.

Outcomes – results of events. The outcome of flipping a coin would be either heads or tails. The outcome of a pizza with two possible toppings would be one of four different choices. From icoach.com – A possible result of a Probability experiment is called an outcome.

Replacement – with regard to probability, replacement means that an item is put back, as in a card into a deck, making the next outcome the same as the previous one.

Without Replacement – again, with regard to probability, without replacement means that the probability is changed for succeeding events, reduced by one chance each time.  For example, if you draw a card from a deck, you can state that it is 1/52.  If you do not replace it, the chance of the next card can’t be more than 1/51.

Here are a few geometry terms and their meanings:

A face is one flat side of a polygon.  According to www.math.com, it is a flat surface of a three-dimensional figure.

A vertice is a corner of a polygon.  It is actually the plural of the term “vertex”, which is an angle where two or more lines come together.

An edge is the line on one side of a three dimensional figure.  www.excelmath.com says ” on a three-dimensional figure where two faces meet, may be flat or curved.”

Two-dimensional means having length and width, such as a square or rectangle.  According to www.excelmath.com’s glossary, it is a “figure having only lenght and width”.

Three-dimensional means having length, width and depth, such as a cube or a tetrahedron. Again according to www.excelmath, it is a  “figure with length, width, and height, also called solids.”

In regular tessellations, which create beautiful mosaics, regular polygons surround a point exactly, with no gaps.  The polygon angles must add up to 360 degrees in order to surround the point with no gaps.  In a mosaic with all hexagons, the point could be represented by the math term 6-6-6, showing the number of sides of each of the polygons that surround the point.  In a design of squares and octagons, the mosaic is described by 4 – 8 – 8.  In a design of triangles and hexagons, the term is 6-3-3-3-3.  In a design of squares, hexagons and dodecahedrons, the designation is 4-5-12.

Number of sides	3	4	5	6	8	9	10	12
Mirror angle – ext.	120	90	72	60	45	40	36	30
Angles of polygon –int.	60	90	108	120	135	140	144	150

The above chart provides a handy guide to determine if polygons can fit together to form a point surrounded by the polygons.  In other words, if the interior angles can be added together to a sum or 360, the point will be surrounded by polygons.  In a 5-6-8 polygon, the calculation is 108 + 120 + 135 = 363.  Since the sum is greater than 360, there would be an overlap of 3 degrees in the mosaic.  In a 5-5-10 mosaic, the math sentence is 108 + 108 + 144 = 360, proving that the point in this mosaic is fully surrounded by polygons with no overlap.  In a 5-5-10 mosaic, several of the points can be surrounded by one decagon and two polygons (in the example on page 272 of Mathematics, a Human Endeavor by Jacobs, 3^rd Edition points A, B, D, and E are surrounded).  There will be a point that is not surrounded by two pentagons and one decagon.  (Point C in the example.) In this case, the mosaic is prevented from continuing by a point that has three pentagons surrounding it.  This can be demonstrated by a 5-5-5 point, 108 + 108 + 108 = 324.  It needs to be 360 degrees to be complete.  This point would need a regular polygon with an interior angle of 36 degrees to be complete, and this doesn’t exist.  This proves that it is not necessarily possible to make a mosaic from regular polygons.  Even if some points are completely surrounded by regular polygons, in some cases not every point in the mosaic is surrounded by 360 degrees of polygons.

In teaching about mosaics to elementary students, asking “good” questions will greatly enhance their learning.  I would ask the students to “discover” for themselves the above chart.  Some questions that would enhance their learning are “What combination of polygons will meet at a point without overlap or gaps?”; “Can you predict which ones will work before trying it with pieces?”; “What designs can be made using mosaics?”; “Is there a limited number of mosaic patterns possible?”

The study of mosaics is fascinating and a great way to learn about polygons and angles.  Mosaics are beautiful and make an endless number of designs.

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