CIEP 104 Final

What two numbers does probability exist between?

0 and 1

Write the first 8 lines of Pascal's triangle. Show 5 patterns.

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1

Why should probability and fractions be taught together?

Probability and fractions should be taught together because they're similar and they both represent parts of a whole.

Describe a regular prism. Describe a regular pyramid

Regular prism: A solid with 2 bases and rectangle faces Regular pyramid: A solid with 1 base and triangle faces.

What is Galileo's famous quote about mathematics, the alphabet, God and the creation of the universe? How is Galileo's idea related to the big idea—math works?

"Mathematics is the alphabet that God used to create the universe" -Galileo This is related to the idea that "math works" because math is found naturally in our environment. Also, manipulatives prove that math works.

Give the first 10 terms of the Fibonacci sequence. Name 3 ways it is related to nature.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55 3 ways it's related to nature: The rate at which rabbits reproduce, the number of petals on a flower, and the spirals in a pine cone.

Give 3 examples to show that the sum of consecutive odd numbers create square numbers. Explain why "Happy Numbers" are an example of productive computation. Show that 68 is a "Happy Number". Show that 4 is not a "Happy Number".

1. 1+3=4 2. 1+3+5=9 3. 1+3+5+7=16 Happy numbers are an example of productive computation because figuring out if a number is happy or not requires the use of addition, multiplication, etc. 68 is a happy number because when squaring and then adding its digits, you end up with 1. 4 is not a happy number because when squaring and then adding the digits, you end up with an endless loop.

Describe the three steps in teaching a set of numbers and give an example of a question you might ask at each step for one of the following sets of numbers. (counting numbers, integers, rational numbers).

1. Introduction: Where is the number on the number line?, how do I show what it looks like? 2. Compare: equal not equal, greater than less than, etc. 3. Computation

Give five reasons Dr. Schiller described to teach numeration systems.

1. support for understanding place value in their own numeration system; 2. productive computation practice (multiple outcomes); 3. understanding of math as a cultural phenomenon; 4. an opportunity to teach mathematics across the curriculum; 5. critical thinking; 6. opportunities to look for patterns; 7. puzzles; 8. deeper understanding of mathematics; and 9. math fun.

Write the symbols for the mathematical notation for a mosaic that is created by a regular dodecagon, a regular hexagon and a square.

12-6-4

What are complementary events?

2 events that add up to 1. Ex: The probability of my birthday being on a Monday (event 1) and not on a Monday (event 2).

Use Gauss's insight to add the counting numbers from 1 to 300.

301 x 150 = 45,150

Create an area flash card for 4 x 7. Show how you can use that card to demonstrate the distributive principle: 4 x 7 = (4 x 5) + (4 x 2). How can you use the distributive principle to help students learn their multiplication facts? Explain how to use area flashcards for 3-d, 2-d, and 1-d (developmental model of instruction).

3D: use actual blocks they can put together and separate, 2D: use graph paper, 1D: use numbers to show how to find area

Give 5 partitions for the positive integer 7. What famous mathematician is associated with partitions? What cultural challenges did he face in order to study mathematics? Explain how a teacher can use the idea of partitions to develop number sense. What are Cuisenaire rods and how can a teacher use this manipulative?

6+1 5+2 3+4 3+3+1 3+2+2 Srinivasa Ramanujan was the famous mathematician associated with partitions. He was very poor and also, it was culturally hard for him to go to England and study mathematics. Cuisenaire rods are effective manipulatives for teaching about numbers.

What is theoretical probability based on?

A formula

Explain why a multiple choice quiz is an independent event (fundamental counting principle) and a matching quiz is an example of a dependent event (permutation).

A multiple choice quiz is an independent event because you can have repetition in multiple choice-- your answer for one question does not affect your chance of picking the correct answer for the next question (such as the fundamental counting principle). A matching quiz is an example of a dependent event because once you've chosen an option, you can't choose it again-- your answer for one question affects what you can answer for the next one (similar to a permutation).

Define a square. Are all squares rectangles? Are all rectangles squares?

A square has equal sides and angles and is a quadrilateral All squares are rectangles, but not all rectangles are squares due to the above definition

Are all congruent figures similar? Are all similar figures congruent?

All congruent figures are similar, but not all similar figures are congruent

Are all squares similar polygons? Are all right triangles similar polygons?

All squares are similar polygons because they will all have the same angles and their sides will always be proportional. All squares are similar but not all are congruent. Not all right triangles are similar because their sides may not be proportional

What is experimental probability based on?

An experiment

Explain the difference between arithmetic and geometric sequences. Why is each important to the elementary school teacher? Give an example of an arithmetic sequence with a common difference of 5, -5. Give an example of a geometric sequence with a common ratio of 5, 1/5

Arithmetic sequences involve addition and subtraction whereas geometric sequences involve multiplication and division. They're important to elementary teachers because they teach students patterns in a 1-D model Arithmetic: 5, 10, 15, 20... Ex: Times table Geometric: 1, 5, 25, 125, 625... Ex: Place value

When was the metric system invented? Name one of the mathematicians involved in designing the metric measurement or SI system. Explain how metric measures are derived via the properties of natural objects. What are 5 reasons that the metric system is an improvement over the English or Customary system? What is the prefix for thousand? thousandth?

Around the time of the American and French Revolutions Joseph Louis Lagrange meter: 1/10,000,000 of the distance from the north pole to the equator liter: the amount of water that would fill a cubic decimeter (a cube that is 1/10 of a meter x 1/10th of a meter by 1/10th of a meter.) gram: the weight of a cubic centimeter of water Celsius: 100 degrees water boils; 0 degrees, water freezes The metric system is an improvement over the English or customary system because it uses base 10 rather than base 12, metric units are decimal based so they are easily converted by moving the decimal over, more practical for trade (all nations have same standard), standard naming convention, easy to use. Thousand: kilo Thousandth: milli

Name 3 important ideas in assessing quality of graphs. Describe three ways the public can be misled by graphs.

Assessing quality: should start at zero, have equal intervals, and have intervals that make sense (percents should go up to 100) Misled: if a graph doesn't start at zero, doesn't have equal intervals, or does not have a good range for the intervals

Draw a diagram of 1234 in base ten and base five.

Base 10: use 10x10 cubes Base 5: use 5 x 5 cubes

Here is a tally. Express the amount in base 4, 5 and 6 / / / / / / / / / / / / / / /

Base 4: / / / / / / / / / / / / R: / / / Base 5: / / / / / / / / / / / / / / / Base 6: / / / / / / / / / / / / R:/ / /

Why is your birthday on a different day of the week each year? Find the range of dates for Thanksgiving (4th Thursday in November), Labor Day, Mother's Day, Father's Day.

Because 365 and 366 are not evenly divisible by 7. For a day to always fall on the same day of the week, the number of days in a year would have to be evenly divisible by 7. Thanksgiving can be Thursday November 22, 23, 24, 25, 26, 27, or 28 Labor day can be Monday September 1, 2, 3, 4, 5, 6, or 7 Mother's day can be Sunday May 8, 9, 10, 11, 12, 13, or 14 Father's day can be Sunday June 15, 16, 17, 18, 19, 20, or 21

As the basketball coach, you would like to develop a holiday tournament with the 5 schools in your community. How many games will there be? Explain why this is a combination problem. Show the combination notation that you would use to solve the problem.

Combination notation: 5C2= 5x4/2x1. This is a combination problem because order doesn't matter.

Give an example of productive computation for geometry.

Complementary, supplementary, reflexive angles (geometry, computation)

Who invented the coordinate plane and why is it valuable? What other important events were happening at the same time? How is Descartes invention used to identify locations around the world? What is the point (0,0) in Chicago?

Descartes invented the coordinate plane. It is valuable because it provided the first systematic link between Euclidean geometry and algebra. (0,0) is at State and Madison.

What are the divisibility rules for multiples of 2,3,4,5,6,7,8,9,10, 11 and 12?

Ending digit: 10 (5 or 0) 5 (5 or 0) 2- divisible by last digit 4- divisible by last 2 digits 8- divisible by last 3 digits Add digits: 3 and 9 Product: 6 (2&3) 12 (3&4) Truncating: 7 and 11 -When truncating 7, double. Divisible if you end with 7 or 0, not divisible if anything else. -When truncating 11, keep the same. Divisible if you end with 11, not divisible if anything else.

Make a chart showing the values of the powers of 10 from 10 cubed to 10 to the power of -3 in exponential, factor, fraction and standard forms.

Exponential: Factor: Fraction: Standard: 10 ^ -3 1/10x1/10x1/10 1/1000 0.001 10 ^ -2 1/10 x 1/10 1/100 0.01 10 ^ -1 1/10 1/10 0.1 10 ^ 0 1 1/1 1.0 10 ^ 1 10 10/1 10 10 ^ 2 100 100/1 100 10 ^ 3 10 x 10 x 10 1000/1 1000

What is the formula for probability?

Favorable outcomes/total outcomes

Name three models we used in class to teach fractions.

Fraction rulers, fraction towers, infinite pine tree.

In what three ways can probability be expressed?

Fraction, percent, and decimal

Write and solve a proportional problem for each category: · Adding and subtracting fractions · Similar figures · Measurement conversions · % · Scale

Fractions: 1/2 = ?/4 Similar figures: Two triangles, similar but not congruent Measurement conversions: 1 foot = ? in %: 50 % of 16 is...? Scale: 1 mile = ?

What is the importance of studying permutations and combinations?

Good practice for multiplication, introduces probability,

Sketch a graph of an arithmetic sequence. How can you use the coordinate plane to teach multiplication? Why would this be considered productive teaching?

Graph would be linear; one can see in the equation that you multiply each number by the same number (the common difference)/on the graph, one can see that the x always increases by the same amount it is productive because you can teach equations, multiplication, graphing, and arithmetic sequences

Two rectangles are similar. One is 14 x 10. Give the dimensions of the other. About how many similar rectangles are there? How can math problems about similar figures be considered productive multiplication practice?

Has to be in proportion: 7x5, infinite numbers of similar rectangles, geometry and computation

What is the difference between a histogram and a bar graph?

Histogram represents continuous (quantitative) data, bars are right next to each other. Examples: age and money Bar graph represents non-continuous (qualitative) data. Example: what is your favorite color?

Explain how you would use concept development to teach a lesson.

I would use concept development by giving non-examples of regular polygons and then giving examples of them and asking which shapes in a group are regular polygons and which are not

Give 4 upper case alphabet letters that are topologically equivalent to a straight line.

I, L, C, S

Explain how events can be independent or dependent

Independent: 2 events are independent if the result of the second event is not affected by the result of the first event Dependent: 2 events are dependent if the result of the second event is affected by the result of the first event

Describe two types of mathematical reasoning.

Inductive and deductive reasoning. Inductive reasoning- bottom-up logic; narrow to general. Someone who uses inductive reasoning makes specific observations then draws a general conclusion. Deductive reasoning- top-down logic; general to conclusion. Deductive reasoning is a specific conclusion follows a general theory.

What is the difference between interpolation and extrapolation in graphing?

Interpolate is guessing the value of a variable between known values Extrapolate is guessing the value of a variable beyond those that are known

In what way are operations with rational numbers intuitive? Counter-intuitive? Describe a language issue with rational numbers instruction? What are three uses of rational numbers? How does it impact the addition/subtraction algorithm?

Intuitive: Multiplying fractions Counter-intuitive: When you multiply fractions, the fractions get smaller. When you divide fractions, they get bigger. Language issue: Saying "one quarter" versus one over 4, or saying "one half" versus one over 2 causes confusion 3 uses of rational numbers: parts of a whole, ratios, division Changes the way we add fractions (have to change the denominator)

Explain why it is important to use "advanced organizers". Replicate the advanced organizer Dr. Schiller used for elementary school mathematics. Can an outline or a table be considered an advanced organizer? Explain why students should preview their textbooks at the beginning of the year.

It is important to use advanced organizers because they increase student productivity and organization. Advanced organizer Dr. Schiller used for elementary school mathematics: Number and Operations sheet with Algebra and Measurement on left and right and Data Analysis & Probability and Geometry on top and bottom. An outline or a table can be considered an advanced organizer. Students should preview their textbooks at the beginning of the year because it helps them see what they will be learning in class and sparks interest.

The real numbers (not imaginary—the square root of -1) consist of 5 different sets. Name each set and represent it using a number line. Explain why each set is necessary. Approximately how many irrational numbers do we have?

Natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Natural numbers: counting numbers. 1,2,3,4,5,6... Whole numbers: 0,1,2,3,4,5,6... Integers: -3,-2,-1,0,1,2,3... Rational numbers: Fractions too, negative and positive Irrational numbers: square root of 2, pi, e There are an infinite amount of irrational numbers.

51. Use a square to draw a picture of the 7 tans of a tangram. Label each tan as a fraction of the large square.

Large triangles make up one half together, medium triangle makes opposite corner, middle part goes parallelogram, small triangle, square, small triangleLarge triangles=1/4, medium triangle, parallelogram, square=1/8, small triangles=1/16

What is the need for leap year? Give all the leap years between 1776 and 1812 in the Gregorian calendar, in the Julian calendar. How can you tell if any year is a leap year?

Leap years are needed to keep our modern day Gregorian calendar in alignment with the Earth's revolutions around the sun. Leap year is also needed to make up for the .25 not accounted for in 365 days. Leap years between 1776 and 1812: 1776, 1780, 1784, 1788, 1792, 1796, 1804, 1808, 1812 Leap years are divisible by four, not 1800 though because century years did not count unless divisible by 400 Gregorian: Gregory knocked off 10 days. "That's too much" leap centuries/non-leap centuries Julian calendar: Every 4 years is a leap year

What is the difference between line and rotational symmetry?

Line folds it exactly in half (mirror), rotation is within its outline around a central point

What important ideas do you need to teach about lines?

Lines never end, line notation, parallel, perpendicular, intersecting (if they do intersect, they can be perpendicular or not)

Write 2,497 in Roman numerals.

MMCDXCVII

List ten math concepts that require multiplication.

Measurement: area, volume Place Value Probability Prime Factorization Arithmetic sequences Slope Figurate numbers Binomial expression Geometric sequence Order of operations Proportions

What are the measures of central tendency? What are the measures of variability?

Measures of central tendency: mean, median, and mode Measures of variability: range and standard deviation

51. Describe 3 manipulatives you can use to teach integers. Use open and shaded circles to show -3 + 4.

Number line, cards, and integer pennies

What is the difference between teaching fraction computation using a number theory base and an algebra base?

Number theory base: uses common denominators, proportions, etc. Algebra base: any number can be expressed in the infinite number of ways (list out other fractions and pick the one that will work best to solve)

Why is Pascal considered the "father of the computer age"? Who is "The Prince of Mathematics"? Who discovered a formula for find the number of partitions for positive integers?

Pascal is considered the father of the computer age because he created the language all computers run off of. Gauss is the prince of mathematics Ramanujan discovered a formula for finding the number of partitions for positive integers

Use pattern to find the first 8 terms of pentagonal, hexagonal, heptagonal, octagonal, nonagonal and decagonal sequences.

Pentagonal: 1, 5, 12, 22, 35, 52, 70, 92 Hexagonal: 1, 6, 15, 28, 45, 66, 91, 120 Heptagonal: 1, 7, 18, 34, 55, 81, 112, 148 Octagonal: 1, 8, 21, 40, 65, 96, 133, 176 Nonagonal: 1, 9, 24, 46, 75, 111, 154, 204 Decagonal: 1, 10, 27, 52, 85, 126, 175, 232

What happens to the perimeter of a square if the length of its side is doubled? To the area of a square if the length of its side is doubled? What happens to the volume of a cube if the length of the side is doubled? To the surface area?

Perimeter: doubles Area: quadruples Volume: multiplied by eight Surface area: quadruples

Show how you would calculate the number of permutations in a word like "bookkeeper'. Explain how you might use permutations to connect math and a list of spelling or vocabulary words.

Permutations- order matters. 10! / 3! x 2! x 2!

What is an Archimedean solid? What the mathematical meaning is of truncated?

Platonic solid, truncate is an Archimedean solid An Archimedean solid is one of the 13 semiregular polyhedral- more than one polygon as a face, but appear in the same order Truncated means the corners of a platonic solid have been "removed" and are replaced by another polygon

Use the fundamental counting principle to find the number of unique three digit id numbers that can be generated if the first digit is a multiple of 2, the second digit is a power of two and the third digit is a number to the second power. Give an example of an id number that could be generated.

Potential answers: 444, 829, 621, etc.

What are the first 8 prime numbers? The first 8 composite numbers?

Prime: 2, 3, 5, 7, 11, 13, 17, 19 Composite: 4, 6, 8, 9, 10, 12, 14, 15

What is productive computation practice? Why should the elementary teacher use it? Be prepared to show that these hypotheses are true by a systematic trial of 5 numbers: · Any number can be expressed as a sum of 3 or fewer triangular numbers. · Any number can be expressed as a sum of 4 or fewer square numbers. · Any number can be expressed as a unique sum of powers of two. · All counting or natural numbers can be categorized as happy or not happy.

Productive computation practice is accomplishing at least two goals with one activity. i.e. practicing computation and learning about straight angles, reflective angles, happy numbers, etc. Any number can be expressed as a sum of 3 or fewer triangular numbers: 1, 3, 6, 10... 14= 10+3+1 Any number can be expressed as a sum of 4 or fewer square numbers: 1, 4, 9, 16... 14= 9+4+1 Any number can be expressed as a unique sum of powers of two: 1, 4, 8, 16... 14= 8+4+1+1

Describe an activity to help students understand the value of a point.

Put patterned blocks around a point to find angle measurements

You receive the following information from your students' performance on the PARCC test: mean = 120; s.d. = 12. Calculate the range. What % of your students is within 1 standard deviation of the mean? Draw a normal curve with this data

Range: 72 68 %

Describe 3 types of transformations. What is the importance of the point of vertex in transformations?

Reflection- flips over an axis Rotation- turns the shape Translation- slides Vertices are important because you use them to complete transformations. For example, in a translation if it says to move up 4 spaces, you would move each vertex up four spaces to find where the new shape is.

Translate 2,344 into Roman, Mayan, Babylonian and Egyptian.

Roman: MMCCCXLIV Mayan: eye dash 2 dots, three dashes 4 dots Babylonian: 600 <, 60 v, 10 <, 1 v Egyptian: Lotus flower= 1000 Coiled rope= 100 Heel bone= 10 Staff= 1

Write this number in English: 23,000,000,000,400,000,000,078. How will you be able to help students distinguish between one million, one billion and one trillion? (How Much is a Million?) Why is it important?

Schwartz says it would take 23 days to count to one million. Schwartz says it would take 95 years to count to one billion. Schwartz says it would take almost 200,000 years to count to one trillion. Jacobs says it would take nearly 12 days to count to one million.

Design a tree network with a diameter of 4.

See Jacobs p. 625-626

Test the hypothesis that the circle creates the largest area/perimeter (circumference). What shape represents the greatest volume for a given surface area? Why are most items packaged in rectangular solids rather than spheres?

Table Shape: 1x5, 2x4, 3x3, circle Perimeter/circumference (pid): 12, 12, 12, 12 Area (pirsquared): 5, 8, 9, about 12 Sphere gives most surface area, but would roll off shelves (not most practical)

What is the fundamental theorem of arithmetic? Give an example of it. What mathematician stated it?

The Fundamental Theorem of Arithmetic states that every number is a prime number or a product of prime numbers. Example: prime factorization. Carl Gauss started this theorem.

The calendar is an important part of humankind's need for quantitative thinking. Expand upon this statement. For what momentous calendar reform is Pope Gregory famous? What questions might people have about such reform?

The Gregorian calendar: he took out ten days in October 1582 to make up for the lost 10 days (vernal equinox was not falling on March 21, confusing when Easter should be). People may have questioned how we lost 10 days, how to change the dates of important holidays because of this

How is Euler related to the study of topology? Describe the real world problem that led to the development of topology. Design one network that has at least 5 vertices and 5 edges that can be traveled and one that cannot be traveled.

The Seven Bridges of Konigsberg problem is significant because it was one of the problems that led to the development of topology. It was deemed by Leonhard Euler as impossible to solve. Euler discovered that if there were more than two odd vertices, the network could not be traveled.

Give sample questions to show how you can be a more productive mathematics teacher.

Use 3 tangrams to make a square and calculate the sum Find the interior angle measure of a hexagon by using triangles Basically any question/direction that includes more than one math concept

What was the value of the invention of zero?

Without zero, there wouldn't be an appropriate amount of space between 1 and -1. Problems such as 1-1 and 2-2 would not have a solution. Furthermore, zero lets you hold a space and move on, or in other words, it allows you to keep places.

Define one year. The earth takes 365 days 5 hours 48 minutes and 46 seconds to make one complete revolution around the sun. What is the difference between the exact time and 365.25 days? What difference would this make in 10 years? 100 years? 1000 years?

The earth takes 365 days 5 hours 48 minutes and 46 seconds to make one complete revolution around the sun. This means that the difference between the exact time and 365.25 days is 11 minutes and 14 seconds. 10 years: 112.5 minutes or about 2 hours 100 years: 1,125 minutes or about 20 hours (1 day) 1,000 years: 11,250 minutes or about 200 hours (10 days)

Why does the elementary school teacher need to know about standard deviation? Sketch a picture for a normal curve. Include lines to illustrate the mean, the standard deviations from the mean and the % expected in each standard deviation. If a teacher grades "on the curve" what % of the class will get a C? a B or D? an A or F?

The elementary school teacher needs to know about standard deviation to evaluate test scores. normal curve- lines go: -3, -3, -1, 0, 1, 2, 3 -3--2=2% (F) -2--1=14% (D) -1-0=34% (C) 0-1=34% (C) 1-2=14% (B) 2-3=2% (A)

What is the 4 colored map problem and why should teachers know about it?

The four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no adjacent regions have the same color.

How many time zones are in the world? Why do we need time zones? Why is the International Date Line located in the Pacific Ocean? What happens when you cross it?

There are 24 different time zones in the world. We need time zones because the earth is rotating. The IDL is located in the Pacific Ocean because very few people live in that area so few people are affected by it. The date changes when you cross it.

51. Explain why this formula will give you the measure of the angles of interior angles of a regular polygon: (n-2) x 180/n. Show a pictorial way to find the sum of the interior angles of a polygon.

This formula works because it finds how many triangles are in a regular polygon (always two less than the number of sides), multiplies it by 180, which is the total measure of angles in a triangle, and divides it by the number of sides to find each angle's measure. This works because this is how you would find the interior angles for one equilateral triangle. A pictorial way of showing this is to divide a regular polygon into triangles to see how many there are, find the total angle measurement based on that, and then divide by the number of sides.

Give an example of a non-decimal measurement unit. Name benchmarks you can use to help students estimate measurements in both U.S. Customary and metric measurements.

Time (hours, minutes, etc.)

How do you find the probability of successive events?

To find the probability of successive events, multiply each event.

Fully describe a Platonic solid. Explain why there are only 5.

To fully describe a platonic solid, you must tell: -the number of faces -the shape of the faces -the number of edges -the number of vertices and the degree at each vertex Tetrahedron: -4 faces -faces are triangles -6 edges -4 vertices -180 degrees at vertex (three triangles meet at each vertex) Octahedron: -8 faces -faces are triangles -10 edges -6 vertices -240 degrees at vertex (four triangles meet at each vertex) Icosahedron: -20 faces -faces are triangles -30 edges -12 vertices -300 degrees at vertex (five triangles meet at each vertex) Cube: -6 faces -faces are squares -12 edges -8 vertices -270 degrees at a vertex (three squares meet at each vertex) Dodecahedron: -12 faces -faces are pentagons -30 edges -20 vertices -324 degrees at a vertex (3 pentagons meet at each vertex) There are only five because any other regular polygon would exceed 360 degrees at a vertex, which isn't possible

What is topology? Name five things that are involved in the study of topology.

Topology is the study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures. Five things that are involved in the study of topology: mazes, puzzles, tree diagrams, magic tricks, and map coloring.

Create a tree diagram for independent events. (Jacobs p 403). Create a tree diagram for dependent events (Jacobs p 415).

Tree diagram for independent events: Suppose that an ice cream store sells two drinks, sodas and milkshakes, in three sizes, small, medium, and large, and four flavors, vanilla, chocolate, strawberry, and cherry. If you order one drink, how many choices do you have? 3x4x2= 24 Tree diagram for dependent events: Eight horses are entered in a race in which bets are placed on which horses will finish first, second, and third. If the race is run and there are no ties, in how many orders can the first three horses come in? 8x7x6= 336

Fully describe a given polygon (hexagon, heptagon...) lines of symmetry, rotational symmetry, rotation, angles, sum of angles

Triangle: Lines of symmetry- 3 Rotational symmetry- 3 Mirror angle- 120 (360/3) Interior angle- 60 (180/3) Sum of interior angles- 180 Mosaic- 3-3-3-3-3-3 Square: Lines of symmetry- 4 Rotational symmetry- 4 Mirror angle- 90 (360/4) Interior angle- 90 (360/4) Sum of interior angles- 360 Mosaic- 4-4-4-4 Pentagon: Lines of symmetry- 5 Rotational symmetry- 5 Mirror angle- 72 (360/5) Interior angle- 108 (540/5) Sum of interior angles- 540 Mosaic- not possible Hexagon: Lines of symmetry-6 Rotational symmetry- 6 Mirror angle- 60 (360/6) Interior angle- 120 (720/6) Sum of interior angles- 720 Mosaic- 6-6-6 Heptagon: Lines of symmetry- 7 Rotational symmetry- 7 Mirror angle- 51.4... (360/7) Interior angle- 128.57 (900/7) Sum of interior angles- 900 Mosaic- not possible

Patterns with figurate polygonal numbers: · illustrate the first 4 terms of the following sequences (2-D); · give the next three terms (1-D); · write the rule for each; · give the formula for each; · show how the sum of square and triangular numbers are related; and · show how the sum of triangular numbers and oblong numbers are related. i. triangular (sum of consecutive counting numbers) ii. square (sum of consecutive odd numbers)

Triangular numbers: 1, 3, 6, 10, 15, 21, 28. Rule: Add the next whole number. Formula: n/2 (n+1) Square numbers: 1, 4, 9, 16, 25, 36, 49. Rule: Add the next odd number. Formula: n squared

51. Why is a deck of cards useful for math at any grade level? Give three reasons. Describe three games you can play with a deck of cards.

Variables, easy to differentiate, and positive/negative numbers with black and red cards. Lowest median, highest mean, greatest range.

Why should we change "combination locks" to "permutation locks"?

We should change "combination locks" to "permutation locks" because combination infers that order doesn't matter, but order does matter in combination locks, as it does with permutations.

Why is a balance an essential manipulative in every classroom? Show how you would set up this problem on a balance. I can buy 4 sandwiches and two $1.00 drinks for the same price as 2 sandwiches and eight $1.00 drinks. How much does each sandwich cost?

We want to say that equations are balances, not equal. 4 times 2 balances 8, not 4 times 2 equals 8. There are lots of different ways to get the same answer. Good 3-D manipulative. Each sandwich costs $3.00

How do we use probability everyday?

When selecting a gift for a friend's birthday, when buying a lottery ticket, when studying for a test.

How can mazes be solved?

You can always get out of a maze by using topology. Always follow along a wall of a maze in order to get out.

Illustrate the Pythagorean Theorem. Solve problems using the Pythagorean Theorem.

a squared + b squared = c squared PYTHAGOREAN SHEET (MATH PARTY)

Explain the following division problems by using measurement, visual representation, decimal, complex fraction and multiplication of reciprocal: a) 1 ½ divided by ½ b) ¼ divided by ½

a) 3 b) 1/2

a) What is the proper name of a soccer ball? What is the sum of the angles at each vertex of a truncated icosahedron? b) What is the sum of the angles at each vertex of a truncated icosahedron? c) In degrees, what is the difference between a sphere and a truncated icosahedron?

a) truncated icosahedron(mosaic notation: 6-6-5) b)120+120+108=348 degrees at each vertex c)360-348=12 degrees

Sketch a tangram. If the tangram is equal to 1, give the fractional value of each tan. Sketch a second tangram. If the tangram is equal to 16, give the value of each tan.

draw

Explain the everyday use of n1, n2 n3 in measurement.

n1 measures length n2 measures area n3 measures volume