All formal definitions are from http://encyclopedia.thefreedictionary.com

 Histogram – (a bar graph representing a data set) – a statistical graph that represents the frequency of values of a quantity by vertical rectangles of varying heights and widths

http://quarknet.fnal.gov/toolkits/new/histograms.html

Mean – (the average of a number set) – mean is the sum of the observations divided by the number of observations

Example: 2 + 7+ 9 = 18; 18/3 = 6

http://www.mathsisfun.com/definitions/mean.html

 Median – (the middle number of a data set when arranged from least to greatest) –  the number separating the higher half of a sample, a population, or a probability distribution, from the lower half.

 http://www.mathsisfun.com/definitions/median.html

Mode – (the most frequent number in a data set) – The number or range of numbers in a set that occurs the most frequently

 http://www.mathsisfun.com/definitions/mode.html

 Frequency Distribution – (the tally of numbers from a sample) – a list of the values that a variable takes in a sample. It is usually a list, ordered by quantity, showing the number of times each value appears.

               Frequency Distribution for High Temperatures

 Temperature  Tally   Frequency

 51                   ////          4

 50                   ////          4

 49                   //////        6

 48                                  0

 47                   ///           3

 46                   ///           3

example data from: http://www.mnstate.edu/wasson/ed602lesson3.htm

 Surveys done by telephone are not random samples because they exclude those people who don’t have a telephone or have unlisted numbers. Updating the information in our text to today’s telephone use, even more people would be eliminated because many people have dropped having a land line and are using their cell phones exclusively instead. They often do not list their cell phone numbers in traditional directories. Perhaps the only way a random sample of people having telephones could be conducted would be to direct the telephone companies to randomly link the survey to the telephone bill since everyone who owns a telephone, has to pay for service.

 The experiment done by the two psychologists to test people’s honesty was interesting.  The envelopes labeled “research study had the highest rate of return: 91%  ( 68/75)  The lowest rate of return was recorded for envelopes seeming to contain money: 68% (102/150) The reason the empty enveloped were marked, ” This is a research study…” encouraged people to follow through with mailing it rather than throwing it away because it looked empty.

 The envelopes were dropped near mailboxes, so it would be logical that people would pick the envelopes up and deposit them without even looking at them closely so one would assume the rate of return would be high. Because the rate of return of envelopes with blank paper (80%) and fake money (68%) were returned and the study was designed to see if people would take money that didn’t belong to them, they may have been counted even though it is dishonest to open another person’s mail.  When it came to returning empty envelopes all demographic groups had high rates of return:  poor – 88%, middle – 96% and wealthy – 88%

Blank paper envelopes had similar results: poor – 86%, middle – 74% and wealthy – 80%

Returns for those with fake money: poor – 56%, middle – 66% and wealthy – 82%. There was a greater discrepancy especially between the poor and wealthy groups. Because there seemed to be coins inside, it may be a greater temptation to actually open the letter. Perhaps some of the poor people didn’t recognize the money as fake so they decided to keep it.  When it comes to money, it’s a greater temptation to actually open the letter for some than others.

 http://ptri1.tripod.com/

After extending the triangle and viewing the sums of the numbers in the rows, I found it to be a geometric sequence. ( 2, 4, 8, 16, 32)

 The activities dealing with the probability of teenage accidents would be one to share with those who are preparing to get their driver’s licenses!

3. Number of teenagers having an accident      0   1    2    3   4   5

             Number of ways                                   1   5  10  10   5   1

4. The probability that none of the five teenagers having an accident is 0.6⁵ = 0.07776 or 8%.

5. The probability that exactly one teenager can have an accident is 5 x (0.4 x 0.6⁴) = 0.2592 or 26%.

6-7. There are 10 ways exactly two teenagers can have accidents so the probability would be 10 x (0.4² x 0.6³) = 0.3456 or 35%.

8-9. There are 10 ways exactly three teenagers can have accidents so the probability would be 23%.10 x  (0.4³ x 0.6² = 0.02323)

10-11. There are exactly 5 ways exactly four teenagers can have accidents so the probability would be 7.7%.  5 x ( 0.4⁴ x 0.6 = 0.0768)

12.The probability that all five teenagers will have accidents is 1%.  (0.4⁵ = 0.01024).

 Using the triangle for solving problems that would be relevant for elementary children can be a challenge. I wanted to find some elementary examples that I could use as models for developing some good questions. I found a simple, clear example:

http://www.krysstal.com/binomial.html

This example involve choosing 2 books out of a possible 5. It showed how to line up number of selections and number of ways like we explored in our lesson.

Another good way to find an answer to a “line item” was found at http://ualr.edu/lasmoller/pascalstriangle.html:

“One can in fact find the number of ways of choosing k items from a set of n items simply by looking at the kth entry on the nth row of the triangle. So, to see how many different trios you could form using the 45 members of your jazz band, you would look at the 3rd entry on the 45th row. (The “1″ at the top of the triangle is considered the “0″th row, and the first entry on each row is labeled the “0″th entry on the row.

So let’s see if this can work with choosing letters from one’s name. If my name is Kris and I wanted to choose two letters, how many ways could be chosen?

 Using the table: 0   1   2   3   4 

                           1   4   6   4   1    

I should get 6 different ways: KR   KI   KS   RI   RS   IS  

(other combinations would be the same letters, just reversed)

If I use the “kth entry on the nth row of the triangle” I would also get 6

[the 2nd entry of the fourth row (1 4 6 4 1)]

I think students would have fun figuring out the letter combinations using their own names!

 Another good question they would enjoy solving would be choosing a number of different treats from a list of eight to create Halloween Goodie Bags. For example:

You have 8 different types of candy and you want to put 3 different treats in each goodie bag, how many different goodie bags would be made?

 Using the table      0     1     2     3     4       5     6     7     8

                              1     8    28   56   70   56   28    8     1

or the triangle row:      1     8    28   56   70   56   28    8     1

 Using this model, they could create their own problems.

At my grade level (grades 3-5) I wouldn’t go into the probability extension like we processed in the lesson. If they can do examples like the ones above, they will be prepared for figuring out the binomial probability later in middle and high school.

Formal definitions from: www.teacherlink.org/content/math/interactive/probability/glossary/glossary.html

Tree Diagram – (a picture of all the possible combinations) – A method of visualizing and listing an experiment’s sample space

http://www.studyzone.org/testprep/math4/d/possiblecombinationl.cfm

Fundamental Counting Principle – (a multiplication method for finding all possible combinations) – A method used to calculate all of the possible combinations of a given number of events

This is an excellent site for a short video explaining how to use the fundamental counting principle:

http://www.nutshellmath.com/textbooks_glossary_demos/glossary_content/fundamental_counting_principle.html

 Nutshell Math does have a sample list of math glossary terms that can be accessed without signing up for a free trial. Each glossary term is explained simply and clearly.

 Probability – (the chance of something occurring) – The likelihood that an event will occur

 http://www.mathsisfun.com/definitions/probability.html

 Independent Event – (events not dependent upon one another) – Events in which the outcome of one event does not affect the outcome of the other event

 Now you know that the probability of a heads-up landing when you flip a coin is 1/2.

 What is the probability of getting tails if you flip it again?
It is still 1/2.
 

The two events do not affect each other. They are independent.

Dependent Event – (events that rely on previous outcomes) – Events in which the outcome of one event affects the outcome of the other event.

images and explanation for independent and dependent from: http://www.learningwave.com/lwonline/probability/dependent_independent.html

Formal Definitions from http://www.mathwords.com/index_geometry.htm

 Prism – (a three-dimensional solid with rectangular sides and congruent polygon ends) A solid with parallel congruent bases which are both polygons

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

 Pyramid – (a polyhedron with a regular polygon base and triangular sides) – A polyhedron with a polygonal base and lateral faces that taper to an apex

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

Polyhedron – (a three dimensional solid with congruent polygon faces) – A solid with no curved surfaces or edges. All faces are polygons and all edges are line segments

http://en.wikipedia.org/wiki/Polyhedron

 Interior Angle – (the inside angle created by two sides of a polygon) – An angle on the interior of a plane figure

 http://www.mathsisfun.com/geometry/interior-angles-polygons.html

Polygon – (a closed figure with straight sides) – A closed plane figure for which all sides are line segments

http://en.wikipedia.org/wiki/Polygon

 Identifying polygons that surround a point is the first step in the process. Exercises 1-4 on page 271 demonstrate a variety of patterns: 6-6-6, 4-8-8, 3-3-3-3-6, and 4-6-4-12.

The key to creating a mathematical mosaic that successfully fills a plane is the sum of the angles surrounding any given point in the mosaic. The sum must be 360⁰. In exercise 5 on page 272 the sum of the angles is 363⁰ (108 + 120 + 135). Sometimes, when looking at just a few polygons and one or two points, a configuration may look plausible, like exercise 6-9. A 5-5-10 looks like it would create a mosaic because the sum of the angles is 360⁰ (108 +108 + 144) but when putting several units together, not all points create 360⁰. Point C in the diagram has three pentagons adding to 324⁰ leaving a gap. Although there are tessellations having two numerical representations (called demi-regular tessellations), there are never any gaps and the patterns always continue.

 Questions I would ask my students:

1. Create a semi-regular mathematical mosaic using two or three regular polygons of your choice. Name the numerical representation of your mosaic.

2. Using only triangles and hexagons, create a tessellation. Explain why or why not your tessellation is a mathematical mosaic.

3. Study these mathematical mosaics. How are they alike and how are they different?  (Images from http://www.coolmath4kids.com/tesspag1.html)

http://tinyurl.com/33ntc9

This links to an Australian educational site that lists over 20 sites dealing with geometry for elementary students in grades K-5. The sites include all aspects of geometry. It includes lessons and well as fun ways to review.

One of the sites titled Geometry Interactives (http://classroom.jc-schools.net/basic/mathgeom.html) links to fun math games using geometric terms. I can see using this site often!

Educational Videos and Lessons for K-12 School Kids

http://www.neok12.com/Geometry.htm

These videos are about all aspects of geometry. They are great for additional instruction and review. I like the way the information is clearly and visually presented.  Many are from http://yourteacher.com.  Additional videos about geometry are located there for the more advanced student allowing for differentiation if needed.

Math is Fun!

http://www.mathsisfun.com/geometry/plane-geometry.html

This is the index for all the explanations and activities for plane geometry. I have used some of these for my vocabulary blog. The explanations are simple enough to use with elementary children.  For solid geometry I use the following:

http://www.mathsisfun.com/geometry/solid-geometry.html It has links to “nets” used in constructing polyhedra.

Johnny’s Math Page

http://jmathpage.com/JIMSGeometrypage.html

This site is full of great interactive practice using geometry from pattern blocks to polyhedra to Cartesian Grids.  Each site topic provides lots of opportunities for exploration, sorting, and designing patterns. I checked out several of the sites and they were all excellent. The Twelve Points of View in the Polyhedra topic is a great way to practice spatial geometry which is a weak area for many students. (http://www.fi.uu.nl/toepassingen/00247/toepassing_rekenweb.xml?style=rekenweb&language=en&use=game)

(Formal definitions from http://thefreedictionary.com)

Arithmetic Sequence – (a sequence of numbers where each term is found by adding a set number) – a sequence of numbers such that the difference of any two successive members of the sequence is a constant. The difference between each pair of successive terms in the sequence is called the common difference.

Examples:   3,  5,  7,  9 …          1,  6,  11,  16,  21 …

Exponent – (the number of times the term is multiplied by itself) – a number placed as a superscript to another number indicating how many times the number is to be used as a factor

Example:

http://www.mathsisfun.com/definitions/exponent.html

math.pppst.com/exponents.html

Geometric Sequence – (a sequence of numbers where each term is found by multiplying by a set number) – a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio.

Examples: 2, 4, 8, 16 …            5, 15, 45, 135…

Function – (the rule applied to input number resulting in a unique output number) – the expression of dependence between two quantities, one of which is given (the independent variable, argument of the function, or its “input”) and the other produced (the dependent variable, value of the function, or “output”).

the symbol used to represent a function

www.solving-math-problems.com/math-symbols.html

http://www.mathsisfun.com/sets/function.html

For this function the ordered pairs (input (x), output (y)) would be:

( 1, 1)  (2,4) (3,9) (4,16)

Ordered Pair – (a pair of numbers indicating position on the x and y on a coordinate grid) – a collection of two not necessarily distinct objects, one of which is distinguished as the first coordinate (or first entry or left projection) and the other as the second coordinate (second entry, right projection). The common notation for an ordered pair with first coordinate x and second coordinate y is (x,y).

http://www.mathsisfun.com/definitions/ordered-pair.html

 http://blog.isallaboutmath.com/category/triangular-number/

Square numbers should not be confused with the term “square” used when calculating area.  Area is always represented as the number of squares of an interior space (square units.) However, it is never represented using an exponent which indicates to multiply the term by itself. The only time the area measured in square units matches a square number as a product is when the sides are of equal length.  Real-life applications of square numbers that come to mind are based on the Pythagorean Theorem (a² + b² = c²). Carpenters, architects, and surveyors today could use the same method the Egyptians did to create to “square” a foundation by using a = 3 and b = 4 and c = 5.  Distance can also calculated by using the theorem.  Another example is the use of the term gross to represent 12 X 12 (a dozen dozen) = 144. This is commonly used when ordering items in units of 12. A dollar is actually 10 groups of 10 pennies and can be represented as 10 X 10 or 10². Other real-life examples are more easily found with using exponents in general. Place value is based on the powers of 10; units, 10¹, 10², 10³…These applications are used in all fields of science. Even in the field of computer science is memory capacity is measured in megabytes (10⁶) and gigabytes (10⁹)!

 I teach the multiplication tables out of order. I start with the 1s, 2s and 5s because by the time one commits these to memory along with the concepts of zero multiplied by any number is zero and 2 x 3 = 3 x 2, over half of all the basic facts have been learned. I then introduce the squares through 10 X 10 by cutting squares from grid paper to indicate a 2 by 2 square, and a 5 by 5 square. We discover the other squares by cutting a 3 by 3, a 4 by 4, etc. The students seem to remember the “square facts” much more quickly by doing this activity.

[Formal Definitions from http://www.thefreedictionary.com/]

Inductive Reasoning – (analyzing specific patterns to draw a conclusion or generalization) – reasoning from detailed facts to general principles

http://chuplink.uwichill.edu.bb/fd12a/diagrams2.gif

Deductive Reasoning – (using a generalization or established rule for a specific situation) – reasoning from the general to the particular (or from cause to effect)

http://chuplink.uwichill.edu.bb/fd12a/diagrams1.gif

Logic – (using reasoning) – The relationship between elements and between an element and the whole in a set of objects, individuals, principles, or events

http://encefalus.com/wp-content/uploads/2008/10/humor-penguin-logic.jpg