Module 7 Vocabulary

 

 

Word	Informal Definition	Formal Definition	Illustration	Links
Mean	Another word for average.	In a data set, the sum of all the data points, divided by the number of data points		Image from: e How Formal definition from: Math.com
Median	The middle number	The median of a set of numbers is the value for which half the numbers are larger and half are smaller. If there are two middle numbers, the median is the arithmetic mean of the two middle numbers		Images from: Statistics 4 u and Static Formal definition from: Mathwords
Range	In {4, 6, 9, 3, 7} the lowest value is 3, and the highest is 9, so the range is 9-3 equals 6.	In statistics, the difference between the largest and the smallest numbers in a data set.		Formal definition from: Math.com Informal definition from: Math is fun Image from: Static
Survey	A method of gathering information about a group	To gather information by individual samples so as to learn about the whole thing.		Image from: Georgetown Formal definition from: Math is fun
mode	The number that occurs the most often in a list.	the number (or numbers) that occurs most frequently in a set of data.		Image from: Tustin Formal definition from: Math.com

 

 

Samples and Surveys… Some Simple Random Thoughts….

     A sample is simply a smaller group meant to be representative of a larger population, the group of people or items that have a specific characteristic that is being studied. However, the process of selecting an appropriate sample can be tricky. There are a number of different methods that are used and it is important to examine what method was used to choose a sample along with the size and composition of the sample group.

In order to draw valid conclusions from a survey, you need to look at how the sampling was done.  In random sampling, each member of the population has an equal chance of being included in the survey.

 

      The validity of the survey would then depend on whether the sample size was large enough to represent the population but small enough to accurate assess.

In systematic sampling, the population is arranged and then samples are taken at regular intervals. For example, a survey of pets on a particular street, might survey every fifth house rather than randomly selecting five. Random selection depends so much on chance that it can result in less accurate data, especially in smaller samples. This hold true whether  it is simple random sampling or a modified version such as systematic.        

 

Convenience sampling, also known as grab or opportunity sampling, is something many of us have experienced first-hand. This type of sampling involves getting samples that are near at hand. Have you ever been in a mall and someone has approached you to ask if they could ask you a few questions? Or have you ever been called and asked if you were the main grocery shopper in the house? These are examples of convenience sampling where it is much more difficult to validate the sample as truly representing the larger population. So any time we are given survey results, it really is necessary to ask ourselves, “Who were the survey respondents? How big was the sample? Is the sample truly representing the population?”   

When designing an experiment such as the one referenced in the text concerning envelopes, it is also important to make sure that the variables are simple and controlled. So although this study was large, the element of using three different envelopes and make two of them very similar, appearing to contain paper, and one to look like it contained money, there was no control over the variables.  The people who found envelopes and opened them probably returned fewer envelopes because they could see that there was really nothing of importance in them. It is not possible to truly test people’s honesty when only the envelopes with the fake money would provide a temptation to the people who found them.   The obvious pattern that exists in the data is that more wealthy people returned 82% of the envelopes which seemed to contain money.  The middle class returned 66% and the poor returned 56%. I would guess that the poor and middle class people opened the envelopes and perhaps finding that the money was fake decided not to bother returning the envelope. It cannot be assumed that they were merely dishonest. 

Exploring Pascal’s Triangle

1.    

 from                      http://mathforum.org/workshops/usi/pascal/patterns_pascal.html                      

2.            column of sums-each row is the previous row doubled

• 2
• 4
• 8
• 16
• 32
• 64
• 128
• 256
• 512
• 1024
• 2048

3.

Number of teenagers having an accident	0	1	2	3	4	5
Number of ways	1	5	10	10	5	1

 

4.            0.6 x 0.6x 0.6 x 0.6 x 0.6=0.07776= 7.8%

5.            5x (0.4 x 0.6 x 0.6 x 0.6 x 0.6)= 0.2592= 25.9%

6.            There are 10 ways exactly 2 teenagers can have an accident.

7.            10x (0.4 x 0.4 x 0.6 x 0.6 x 0.6)= 0.3456 = 34.6%

8.            There are 10 ways exactly 3 teenagers can have accidents.

9.            10x (0.4 x 0.4 x 0.4 x 0.6 x 0.6)= 0.2304 = 23%

10.          There are 5 ways exactly 4 teenagers can have accidents.

11.          5x(0.4 x 0.4 x 0.4 x 0.4 x 0.6)= 0.0768= 7.7%

12.          0.4 x 0.4 x 0.4 x 0.4 x 0.4= 0.01024= 1%

 

http://dimacs.rutgers.edu/~judyann/LP/lessons/12.days.pascal.html

This is a link to a great lesson that uses Pascal’s Triangle to figure out the number of gifts received in the song” The Twelve Days of Christmas”. The visual activity along with the song would be really motivating for elementary and middle school students. It’s a fun way to put Pascal’s Triangle to use in a unique context.

http://library.thinkquest.org/TQ0312134/pascalproblems.html

This site has a printable version of an activity using Pascal’s triangle. There are also activities for a great number of math concepts that I want to check out.

 

http://library.thinkquest.org/J002490/patterns.html

The website here is all about Pascal and the patterns found in the triangle. There is a nice flash animation included.

http://library.thinkquest.org/11506/learn.html

The linked site include five lessons on probability along with exercises for practice.

 

The following problem is one that I could easily use in my class.

 

You take a 10-question multiple choice exam where each question has

five possible answers. You guess at every question.

Find the probability that: ( rounded to hundredths place)

 

A) you get exactly 2 questions correct

B) you get no questions correct

C) you get at least one question correct

D) you get  9 questions correct

E) you get between 4-6 answers correct

Module 6 Vocabulary

Vocabulary Word	Informal Definition	Formal Definition	Illustration	Links
dependent event	One outcome or chance depends on another	Events for which the outcome of the second event depends on the outcome of the first event		Formal definition from: Harcourt math   Picture from: Learning Wave
independent event	The outcome of the second event does not depend on the outcome of the first event	An outcome that is not affected by previous outcomes. Example: tossing a coin. Heads or tails is not affected by previous tosses.	Tossing the coin and rolling the number cube are independent events	Formal definition from: math is fun Picture and informal definition from: Harcourt math
tree diagram	Represents all the possibilities	A diagram that shows all possible outcomes for an event		Picture and formal definition from: Harcourt math
Pascal’s Triangle	To build the triangle,  start with the three 1’s  at the top and put 1’s down the sides.  To get the numbers in the middle, you add the two numbers right above.   To get the  4, you add 1+3.   To get the 10, you add 4+6.	Pascal’s triangle is a geometric arrangement of the binomial coefficients in a triangle.		Formal definition from: word iq   Picture and informal definition from: Cool Math
probability	Probability is the chance that something will happen – how likely it is that some event will happen. Sometimes you can measure a probability with a number: “10% chance of rain”, or you can use words such as impossible, unlikely, possible, even chance, likely and certain.	Probability is the measure of how likely an event is.     The chance that an event will occur expressed as the ratio of the number of favorable outcomes to the number of possible outcomes between 0 and 1.		Informal definition and picture from: math is fun       Formal definition from: Math Goodies

Mathematical Mosaics

Below are my answers to the exercises from the text dealing with regular polygons and math mosaics.

Page 271-272

1. 5-5-5
2. 4-8-8
3. 3-3-3-3-6
4. 4-6-10
5. A 5-6-8 mosaic is not possible since the angles of 108 + 120 + 135 = 363⁰ and the sum must be 360⁰ to surround a point.
6. A 5-5- 10 mosaic could exist because the sum of the angles is 108 + 108 + 144 = 360⁰.
7. In the figure, points A, B, D, and E are surrounded by two pentagons and a decagon.
8. The mosaic doesn’t continue because point C is not surrounded by a decagon, leaving a gap.
9. Although it seems as though a set of regular polygons that surround a point should make a mosaic,  2 triangles, a square, and a dodecahedron surround a point ( 60 + 60 + 90 + 150 = 360) but the pattern cannot be extended indefinitely. Two other examples that cannot be extended indefinitely even though the polygons surround a point are, 1 triangles, 2 squares, and a hexagon ( 60 + 90 + 90 + 120 +360), and 2 pentagons and a decagon ( 108 + 108 + 144 = 360). Although the previous answer to number 6 made it seem that a mosaic of 5-5-10 might be possible, actually attempting to extend the pattern does not work.

 

Considering the nature of regular polygons and mosaics, there are a number of open ended math questions that can be developed. Keeping in mind the level of my 8th grade resource room math students the following questions should challenge them to explore these concepts. Here is one example I might use:

1.

My father is trying to create a unique mosaic for the backyard patio using regular polygons. He has looked at a number of examples such as:

                          

 

                

What do these examples have in common? What polygons are used to make each pattern? Can you create a new mosaic pattern using any of these regular polygons?

 

2. Look at the examples above again. Can you think of any ways to represent each mosaic mathematically?  Explain your reasoning.

 

3. Look at the following illustrations

 

A    B        C           D                      E

    

 

Can you classify these illustrations into 2 groups? What characteristics did you use to classify the illustrations? Can you define a mosaic based on your classifications?

Resources for Geometry Classes

 

Resources for Teaching Geometry

Link	Useful Information
aaamath	This site has links to activities and includes interactive lessons. The graphic on the right is of an interactive game to identify characteristics of polygons. There are additional activities for area, perimeter, and volume. There are links to various math topics by grade level.
Baltimore County Public Schools	This site provides lists of links by grade level to useful activities, lesson plans, and resources.
Creative Teaching	This blog links to great teaching resources- lessons, flashcards, puzzles, and even focuses on vocabulary.
Johnny’s Math Page	Links are provided to a variety of interactive math tools such as tangrams, geoboards, pattern blocks, plane shapes, angles, polyhedra, Cartesian grids, and geometry fun. The link at the right is from geometry fun for an interactive activity creating various types of symmetry.

Sequence of Squares

 

 The sequence starts with a row of 1 dot. The second figure adds another row with 2 dots for a total of 3. The third figure adds a row with 3 dots for a total of 6.To figure out the arrangement for the next 5 terms I just keep adding an additional row with one more dot.

1

1+2=3

1+2+3=6

1+2+3+4=10

1+2+3+4+5=15

1+2+3+4+5+6=21

1+2+3+4+5+6+7=28

1+2+3+4+5+6+7+8=36

1+2+3+4+5+6+7+8+9=45

1+2+3+4+5+6+7+8+9+10=55

 The numbers that result in the right hand column are triangular numbers.

 The first ten triangular numbers are:

1              3              6              10           15           21           28           36           45           55

 Adding each successive pair of triangular numbers gives these results:

4              9              16           25           36           49           64           81           100

 The sums of the triangular number pairs are square numbers.

 By looking at the figures in the textbook and looking at the sequences generated above, what I noticed is that each term in the sequence of squares contains two triangular numbers. The figures clearly illustrate how each square can be divided and rotated to reveal two successive triangular numbers. The first figure for example contains the first 2 triangular numbers, 1 and 3. The second figure contains 3 and 6.

From Wikipedia: http://en.wikipedia.org/wiki/Triangular_number

 

Most simply, the sum of two consecutive triangular numbers is a square number, with the sum being the square of the difference between the two. Algebraically,

 

Alternatively, the same fact can be demonstrated graphically:

6 + 10 = 16			10 + 15 = 25

Squared numbers are used in many places around the home. Any area that needs to be totally covered involves using squared numbers. This includes finding how much paint to use to complete a job, buying materials to make curtains or clothes, buying carpeting, flooring, wall paper, creating a garden or planting bed, and buying shelf paper to cover a certain area.

In working with multiplication I would point out to students that each multiplication table has a fact where the number is multiplied by itself. I would then introduce the term squared numbers as a special name for these facts. I would give students visuals of squares with different measurements and ask them to write the number fact that matches the figure. When I feel confident that students have a concept of a squared number I would introduce the notation t² as a short way to identify squared numbers.

Module 4 Vocabulary

Module 4 Vocabulary

 

Vocabulary Word	Informal Definition	Formal Definition	Picture	Links
arithmetic sequence	A list of numbers that follow a pattern. Example: 1, 4, 7, 10, 13, 16, 19, … starts at 1 and jumps 3 every time.	A sequence such as 1, 5, 9, 13, 17 or 12, 7, 2, –3, –8, –13, –18 which has a constant difference between terms.		Definition from http://www.mathwords.com/ Informal definition and picture from: Math is Fun
exponent	The exponent of a number shows you how many times the number is to be used in a multiplication. It is written as a small number to the right and above the base number.     73=7×7x7	A mathematical notation that gives the number of times a number is  multiplied by itself. In the expression a7, the exponent is 7. ‘a‘ is the base. In 24, 4 is an exponent.  2 is  multiplied by itself 4 times. 24 = 2 × 2 × 2 × 2 = 16		Informal definition from: Math is Fun Formal definition from: Exponent Picture from: learningupgrade.com
function	A function can be thought of as a machine where putting in one value results in only one value.	A function is a special relationship between values: Each of its input values gives back exactly one output value. It is written “f(x)” where x is the value you give it. Example: f(x) = x/2 (“f of x is x divided by 2″) is a function, because for every value of “x” you get another value “x/2″. So: * f(2) = 1 * f(16) = 8 * f(-10) = -5		Definition from: Math is Fun   An online function machine: Function Machine Picture from: Function
geometric sequence	A list of numbers or objects which are in a special order found by multiplying the previous term. Example: 2, 4, 8, 16, 32, 64, 128, 256, … is a geometric sequence (each number is 2 times the number before it)	A sequence such as 2, 6, 18, 54, 162 or which has a constant ratio between terms. The first term is a1, the common ratio is r, and the number of terms is n.		Definition from http://www.mathwords.com/   Informal definition from: Math is Fun Picture from: Geometric sequence
ordered pair	two ordered numbers; two numbers with a specific first number and a specific second number. The symbol is: (1st number, 2nd number).	On the coordinate plane, the pair of numbers giving the location of a point (ordered pair).		Definition from http://www.mathwords.com/ Informal definition and picture from: Ordered pair

Module 3 Vocabulary

 

 

Number Sense
A person’s ability to use and understand numbers:* knowing their relative values, * how to use them to make judgements, * how to use them in flexible ways when adding, subtracting, multiplying or dividing * how to develop useful strategies when counting, measuring or estimating.

retrieved from: http://mathisfun.com

 

 

Operations

A mathematical process.The most common are add, subtract, multiply and divide (+, -, ×, ÷ ).If it isn’t a number it is probably an operation. Example: In 25 + 6 = 9, the operation is add

retrieved from: http://www.mathsisfun.com/definitions/operation.html

 

 

Deductive reasoning-

 

 

When you move from things you know or assume to

be true – called ‘premises’ – to conclusions that must follow from

them, you are using deductive reasoning.  The most famous example of deduction is:

 

  Socrates is a man.

  All men are mortal.

  Therefore, Socrates is mortal.

 

The first two statements are premises, and the third statement is a

conclusion. If the first two statements are

true, then the conclusion must be true. 

 

The conclusion follows deductively, or is ‘deduced’, from the

premises. IF the premises are true, THEN the conclusion must be true. 

 

 

retrieved from: http://mathforum.org/library/drmath/view/55620.html

All graphics retrieved from yahoo video search.

Marbles in the Matchbox

As I was working through the problem, Marbles in the Matchbox, the main fact that I used was that the labels on all three boxes were wrong. I knew that one box contained 2 red marbles, one contained 2 white marbles, and one contained 1 red and 1 white marble. I also knew that I could only open each box far enough to see 1 marble. So if I picked up the box labeled ” 2 red” and saw I red marble, I can deduce that the box contains 1 red and 1 white marble since the “2 red” label is wrong and the only other box is 2 white which is also obviously wrong. If I opened that same box and saw a white marble, the second marble could be either white or red since I know that the “2 red” label is wrong and seeing  1 white marble doesn’t eliminate the other two choices.

If I opened the box labeled “2 white” and saw a white marble I could conclude that the box contains 1 white marble and 1 red. The “2 white” label is wrong and I know that there is at least 1 white so 2 red is also incorrect. That only leaves the option of 1 white and 1 red. If I opened that same box and saw a red marble, I could not eliminate either of the remaining boxes so the second marble in that box could be red or white.

Choosing the box labeled “1 red, 1 white” and seeing a red marble would allow me to conclude that the second marble is red. The label is wrong so I cannot have a white marble and since there is at least 1 red marble I cannot have the choice of 2 white. If I opened the same box and saw a white marble then I would conclude that the second marble was white. Again the “1 red ,1 white” label is incorrect and if I see a white marble there cannot be 2 red marbles inside.

The box that is best to look inside is the one labeled “1red,1white” because no matter what color the marble I see is, I can determine the correct label for the box. Opening that box and seeing a red marble would show me that the box labeled “1 red,1 white” contained 2 red marbles because that is there only remaining possibility. I would then be left with labels reading “2 red” and “2 white”. The box labeled “2 white” cannot contain 2 white marbles so it must contain 1 red and 1 white since the 2 red marbles have already been found in the box labeled “1 red,1 white”. Thus the box labeled “2 red” must actually contain 2 white marbles since that is the only remaining possibility.