Final study guide math Schiller
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 God used to create the universe" - We can use our 3D, 2D, and 1D to show math works
What is the difference between teaching fraction computation using a number theory base and an algebra base?
# theory is common denominator Algebra base > any number has an infinite number of expressions and use whatever is most helpful
-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+3=4, 1+3+5=9, 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.
Multiply 123 x 200 using the alternative multiplication algorithms: Egyptian, Russian Peasant, and Lattice (Napier rods). Show all work.
- Egyptian: start at 1 for 123 and 200 then double each until you can add numbers to get 123 and add the corresponding numbers (answer is 24,600) - Russian Peasant: Half 200 and double 123 until you reduce 200 to 1, then take note of the odd numbers under 200 and add together the corresponding numbers under 123 to get your answer (which is 24,600) - Lattice (Napier rods): draw a chart with 9 boxes and cut each box in half diagonally, multiply each digit and put the tens place (or a 0 if its a single digit) on the top of the diagonal and the ones place on the bottom, after all the boxes are filled add each diagonal
Show that math is a cultural phenomenon by using a table to describe the five different numeration systems we have studied.
- Hindu-Arabic: the one we use - Babylonian: uses upright V for one and a < for 10, a space multiplies the symbols before it by 60 - Egyptian: | is worth one, n is worth 10, a coiled rope/swirl is 100, a "lotus flower" is 1000 - Mayan: dots are worth 1, lines are worth 10, the table is on a base 20 system (bottom is one, next one up is 20, next one up is 400, next one up is 8,000) and you multiply across to get each place value - Roman: | is one, V is five, X is ten, L is 50, C is 100, D is 500, M is 1000
How long does Schwartz say it would take to count to one million? one billion? one trillion?
- One million: 23 days - One billion: 95 years - One trillion: 190 years
Why is Pascal considered the "father of the computer age"? Who is "The Prince of Mathematics"?
- 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"
Give five reasons Dr. Schiller described to teach numeration systems.
- Support for understanding place value in their own numeration system - Productive computation practice for multiple outcomes - Understanding of math as a cultural phenomenon - An opportunity to teach mathematics across the curriculum - Critical thinking - Opportunities to look for patterns - Puzzles - Math fun
Write and solve a proportional problem for each category: · Adding and subtracting fractions · Similar figures · Measurement conversions · % · Scale
- add 1/2 and 1/4 - Proportions: 5/10 = x/20 - 3 ft = 1 yard, 9 ft = x number of yards - 60% of 42 -- 60/100=x/42 - 1 inch = 1 ft on a map -- ? ft = 26 inches
Create a 3 x 3 magic square using an arithmetic sequence with a common difference of 3 that begins with -4. What is the magic sum? a. Create a 3 x 3 magic square using algebraic notation to show why the square is "magic".
-1. 14. 11 20. 8. -4 5. 2. 17 sequence: -4, -1, 2, 5, 8, 11, 14, 17, 20 > magic sum is 16 a. n-9. n+6. n+3 n+12. n. n-12 n-3. n-6. n+6
Who invented the coordinate plane and why was it valuable? What other important events were happening at the same time?
-Descartes -Valuable because it provided the first systematic link between Euclidean geometry and algebra
List ten math concepts that require multiplication.
-Measurement: perimeter, area, volume, surface area -Probability: Successive (Compound) Events -Prime factorization: Fundamental theorem of arithmetic: Carl Gauss -Arithmetic sequences and graphing linear equations—times tables -Figurate Numbers -Binomial expansion -Geometric Sequences (exponential growth) --place value -Order of operations -Proportions (Scale, Percent, Common denominator, Sampling, Similarity, Measurement conversions) -Alternative Algorithms (grid, lattice -John Napier; Russian Peasant)
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; (1-D) -give the formula for each; (super 1-d) -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) n/2 (n+1) ii. square (sum of consecutive odd numbers) n2 iii. oblong (sum of consecutive even numbers) n((n+1)
Give the first 10 terms of the Fibonacci sequence. Name 3 ways it is related to nature.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 - Flowers, shells, weather patterns
first five lines of pascals triangle
1 11 121 1331 14641
Write the first 8 lines of Pascal's triangle. Show 5 patterns.
1- Symmetrical with repeating pattern 2- Add all the rows together they create the power of two going down 3- Multiples of 2 in Pascal's triangle 4- Hockey stick 5- Petal
triangle number sequence- illustrate the first 3 terms of the sequence and give the next 3
1. 3.: 6.:.: 10,15,21
Write the results in scientific notation. (4 x 10^6) x (3.1 x 10^5).
1.24x10^12
Write the symbols or the mathematical notation for a mosaic that is created by a dodecagon, a hexagon and a square.
12-6-4
write the mathematical notation for a regular dodecagon, hexagon, and sure
12-6-4
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
120 + (12x3)= 156. 120- (12x3)= 84 The range is 84 to 156, so 72 68% is the average, with 34% 1 standard deviation above and 34% 1 standard deviation below the mean. 34+34=68 Normal Bell Curve would start at 84 (2%) to 96, (14%) to 108, (34%) to 120, (34%) to 132, (14% to 144), (2%) to 156.
Give and example of a number that is a 2-step palindrome.
28 28+82= 110 110+011 = 121
Express 45 as a unique sum of powers of 2
2^0=1 2^1=2 2^2=4 2^3=8 2^4=16 2^5=32 32+8+4+1=45
Use Gauss's insight to add the counting numbers from 1 to 300.
301(150)= 45,150
Evaluate 6P3 and 6C3. Compose a problem for each.
6P3: 6!/(6-3)!= 120 - A horse race has 6 horses. How many different ways can 1st, 2nd, and 2rd occur? 6C3: 6!/(3!(6-3)!) = 20 - You have won first place in a contest and are allowed to choose 3 prizes from a table that has 6 prizes numbered 1-6. How many different combinations of 3 prizes can you choose?
Evaluate 6P4 and 6C4
6P4= 6X5X4X3=360 6C4=(6X5X4X3)/(4X3X2X1)
What 4 consecutive even numbers equal 302?
74, 75, 76, 77 1x 1x+1 1x+2 1x+3
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?
A balance allows students to visualize what the equal sign means in an equation. Problem: 4x+2(1)=2x+8(1) -> 2x=6 so x=3 On a balance, put 4 blocks and 2 chips on one side and 2 blocks and 8 chips on the other. Then, show how two blocks can be taken away and two chips can be taken away from each side. Then, show that taking away one of the remaining blocks lets you take away half of the chips. The balance should then have one block on one side and 3 chips on the other.
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).
What is the sequence of teaching each set of real numbers?
A rational number is a number that can be expressed as b/q where q is not equal to zero. An irrational number is a number that cannot be expressed as a simple fraction. You should teach each set of real numbers together.
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.
Describe the secrets of pedagogy presented by Dr. Schiller. Give an example of each one.
A. Secret to impressing children: Puzzles, riddles and magic tricks B. Secret to better teaching: What do you see? C. Secret to first day classroom management: Learn names 1. Algebra name plate 2. Theater/Math Game: Agree/Disagree D. Secret to classroom management after Day 1 1. Partners 2. Math Parties E. Secret to Super Teacher: 1. Productivity: Multiple outcomes (example: learn names, review substitution, practice computation and teach statistical analysis) 2. Accelerate Learning: Connections with other content (example: Gauss, Germain) 3. Use tables to help students organize their thinking (example: numeration systems description; homework assignments) 4. Classroom Routine: (hypothesis testing, puzzles, magic tricks) F. Secret to Super Math Teacher: Use 3-D, 2-D 1-D (developmental) teaching model.
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 an Archimedean solid? What is the mathematical meaning of truncated?
An Archimedean solid is one of the 13 semi-regular polyhedral - more than one polygon as a face, but appear in the same order a) What is the proper name of a soccer ball? Truncated icosahedron b) What is the sum of the angles at each vertex of a truncated icosahedron? 120+120+108= 348 degrees c) In degrees, what is the difference between a sphere and a truncated icosahedron? 360-348=12 degrees
Explain why is it 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.
An advance organizer is information presented by an instructor that helps the student organize new, incoming information. An outline and a table can be considered an advanced organizer. Students should preview textbooks to gain an understanding of what they will be studying and to prepare themselves for the type of reading they will be doing throughout the school year.
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 ration of 5, 1/5
Arithmetic is addition and subtraction while geometric is multiplication and division, they are important because they teach patterns in a 1D model Arithmetic sequences: -5, 0, 5, 10, 15, etc Example of geometric sequence: 1/125, 1, 5, 25, 125, etc
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 RevolutionsJoseph Louis Lagrangemeter: 1/10,000,000 of the distance from the north pole to the equatorliter: 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 waterCelsius: 100 degrees water boils; 0 degrees, water freezesThe 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: kiloThousandth: milli
Here is a tally. Express the amount in base 4, 5 and 6 / / / / / / / / / / / / / / /
Base 4: //// //// //// remainder: /// = 33 base four Base 5: ///// ///// ///// = 30 base five Base 6: ////// ////// remainder: /// = 23 base six
Draw a diagram of 1234 in base ten and base five.
Base ten: a cube, two squares, 3 rectangles (divided into 10), and 4 small squares Base 5: a cube, 2 squares, six rectangles (divided into 5), and 4 small squares
Name 3 important ideas in assessing quality of graphs. Describe three ways the public can be misled by graphs.
Baseline zero, equal intervals, should go to 100%
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 28Labor day can be Monday September 1, 2, 3, 4, 5, 6, or 7Mother's day can be Sunday May 8, 9, 10, 11, 12, 13, or 14Father's day can be Sunday June 15, 16, 17, 18, 19, 20, or 21
Why should probability and fractions be taught together?
Because the formula for probability is a fraction
Sketch a box and whiskers graph. Sketch a stem and leaf graph. Sketch graphs of perfect correlation. What is the range of measures of correlation?
Box and whiskers: box with a line on each end (which are the whiskers), the left end is the minimum, the right end is the max, the left end of the box is Q1/lower quartile, the middle of the box is Q2/median, the right end of the box is Q3/upper quartile Stem and leaf: a t chart with stem on the left and leaf on the right, the numbers correlate as follows - (0,5) (1, 6;7) (2, 8;3;6) (3, 4;5;9;5;5;8;5) (4, 7;7;7;8) (5, 5;4) (6,0)
Why is a deck of cards useful for math at any grade level? Give three reasons.
Can be used to show patterns, to practice counting, and to learn multiplication and division
How would you teach variables using "concept development"?
Concept development can be used in an activity that asks students to identify examples of polygons and non-polygons.
Give an example of an activity you could use to support primary students' understanding of algebra in the following concepts: · · Coordinate Geometry · Patterns · Simplification · Variables (wild cards, face cards)
Coordinate Geometry: graphing points on a number line by graphing individual points across a number line Patterns: square number sequence by using square manipulatives to build up visual representations of square numbers Simplification: taking a word problem that consists of multiple parts and breaking it down into smaller sections such as "a penny and a dime and two dimes and one penny" becoming "a penny and a dime" then "two dimes and one penny" Variable: using deck of cards as substituted variables in a problem such as 4 of hearts and 6 of diamonds equals 10 of clubs, meaning 4+6-10
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?
Elementary school teachers need to know about standard deviation for standardized testing. A or F = 2% each or 4% total, B or D is 14% or 28% total, C is about 68%
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.
What is the fundamental theorem of arithmetic? Give an example of it. What mathematician stated it?
Every number greater than 1 is either a prime or a product of prime, stated by Carl Gauss (an example would be prime factorization of 20)
Make a chart showing the values of the powers of 10 from 10^3 to 10^-3 in exponential, factor, fraction and standard forms.
Exponential, Factor, Fraction, Standard 10^-3, (1/10)(1/10)(1/10), 1/1000, 0.001 10^-2, (1/10)(1/10), 1/100, 0.01 10^-1, 1/10, 1/10, 0.1 10^0, 1, 1/1, 1 10^1, 10, 10/1, 10 10^2, 10x10, 100/1, 100 10^3, 10x10x10, 1000/1, 1000
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 name of our numeration system? Use a table to describe it.
Hindu-Arabic 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
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 moneyBar 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
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: when we add they get bigger and subtract they get smaller Counterintuitive: When multiplying fractions we expect them to get larger but they get smaller (vice versa with division)
How is Descartes' invention used to identify locations around the world? What is the point (0,0) in Chicago?
It is used to identify different point around the world on a grid-like system. - State and Madison
Sketch a graph of a geometric or exponential sequence. Explain why an elementary teacher needs to understand geometric sequences to teach place value.
It's a curved line. It is important in learning operations.
What is the difference between line and rotational symmetry?
Line folds, rotational turns
What important ideas fo you need to teach about lines?
Lines never end, line notation, parallel, perpendicular, intersecting
Fully describe a given polygon
Lines of symmetry, rotational symmetry, mirror angle (360/# of sides), interior angle, sum of interior angles, mosaic
Name ten math inventions
Logarithms, complex numbers, binary sequence, zero, negative number, imaginary numbers, calculus, coordinate plane, Euclidean geometry, and algebra
Describe 3 manipulatives you can use to teach integers. Use open and shaded circles to show -3 + 4.
Manipulatives can be pennies, marbles, and paper clips. ... + oooo = o so -3+4=1
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
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.
Non decimal measurement units are minutes, hours, days, weeks, months, inches, feet, pounds Customary: top knuckle on thumb, thumb tip = 1 inch, elbow to wrist = 1 ft Metric: width of index finger = 1 cm, length of 10 football fields = 1 m, thickness of a dime = 1 mm
Sketch graphs of the power sequences.
Page 157 of Jacobs textbook
Describe an activity to help students understand the value of a point.
Patterned blocks around a point to find angle measurements.
How can you use pattern to prove the rules for multiplication of integers?
Patterns can be used to justify the common rules for adding and subtracting integers: -The product of a positive and negative integer is negative -The product of a negative and positive integer is negative -The product of two positive integers is positive -The product of two negative integers is positive 3-1 = 3-2 = 3-3 = 3-4 = 3-5 = 3-6
Give examples of perfect, abundant, and deficient numbers.
Perfect: when the factors not including the number itself equal it (ex. 6) Abundant: the factors add up to more than the number itself (ex. 12) Deficient: the factors add up to less than the number itself (ex. 5)
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 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. · Any number can be expressed as sums and/or differences of unique powers of 3. · 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, for example practicing computation and learning about straight angles, reflective angles, and happy numbers, etc. - 14 = 10+3+1 - 14 = 9+4+1 - 14 = 8+4+2 - 14 = 27-9-3-1 - Happy numbers versus not happy is when you square and add the digits of a number, it is happy if you eventually get 1, it is not happy is you end up going in a loop
What are the real numbers? Why is each set needed?
Real numbers are rational and irrational numbers. Both are needed because they satisfy and understand all concepts of numbers.
Describe 3 types of transformations. What is the importance of the point or vertex in transformations?
Reflection: flips Rotation: turns Translation: slides
Describe a regular prism. Describe a regular pyramid.
Regular prism: 2 bases, rectangles on sides (gets closer to a cylinder) Regular pyramid: regular base with triangles at sides (gets closer to a cone)
Translate 2,344 into Roman, Mayan, Babylonian and Egyptian.
Roman: MMCCCXLIV Mayan: ..... / .......| / .... Babylonian: <<<VVVVVVVVV VVVV Egyptian: 2 lotus flowers, 3 coiled ropes, 4 "n", and ||||
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 (pi squared): 5, 8, 9, about 12 Sphere gives most surface area, but would roll off shelves (not most practical)
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?
The Earth takes approximately 365.25 days to travel around the sun, so every 4 years, we account for the .25 by adding an extra day. That year is called a Leap Year. Between 1776 to 1812 in the Gregorian calendar, 1780, 1784, 1788, 1792, 1796, 1804, 1808, 1812. Not 1800 because it was a century. The Revised Julian calendar adds an extra day to February in years that are multiples of four with no exception, 1780, 1784, 1788, 1792, 1796, 1800, 1804, 1808, 1812. You can tell if it is a Leap Year by dividing the year by 4, if it is a century divide by 400.
What causes seasons? Define the beginning of each season in terms of daylight. How does this show that the earth, like an equation, is in balance?
The Earth's tilt on its axis at 23.5 degrees causes seasons. Summer: Summer in Northern Hemisphere: more daylight Summer in Southern Hemisphere: less daylight Winter: Winter in Northern Hemisphere: less daylight Winter in Southern Hemisphere: more daylight Spring/Fall: Equinox in the Spring and in the Fall: equal amounts of daylight/darkness in both hemisphere
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.
What is the digital rot of 4,517? What is the value of digital roots to the elementary school teacher?
The digital root of a number is the single digit you get by adding all of the digits of the original number together. If the result is a multiple, you keep adding until a single digit. 4+5+1+7=17=1+7=8
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 hours100 years: 1,125 minutes or about 20 hours (1 day)1,000 years: 11,250 minutes or about 200 hours (10 days)
What is the sum of this series of fractions: ½ + ¼ + 1/8...? What is the sum of this series of fractions: ½ + 1/3 + ¼ + 1/5...?
The first approaches 1 but never gets there, the second approaches infinity
Explain why this formula will give you the measure of the angles of interior angles of a regular polygon: (n-2)(180/n). Show a pictorial way to find the sum of the interior angles of a polygon.
The formula works because it finds how many triangles are in a regular polygon, multiply it by 180, which is the total number of angles, and divides it by the number of sides to find each angle's measure. A pictorial way to show this is to divide a regular polygon into triangles to see how many there are, find the total angle measurement based on that then divide by the sides
What is the 4-color map problem?
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.
Sketch a graph of an arithmetic sequence. How can you use the coordinate plane to teach multiplication? Why would this be considered productive teaching?
The graph is linear and the equation can be created by multiplying each number by the same number, allowing the ability to teach multiplication. It is productive because you can teach equations, multiplication, graphing, and arithmetic sequences.
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?
The perimeter doubles if the length of the side is doubled. The area quadruples if the length of its side is doubled. The volume gets multiplied by 8 if the length of the side is doubled. The surface area quadruples if the length of its side is doubled.
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. - Use permutations to find the number of unique 3 digit id numbers if all odd digits can be used but not repeated. Give an example of an id number that could be generated
There are 18 possible answers. Potential answers: 221, 444, 889, 641, etc - 5x4x3= 60 (ex. 135)
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 time zones and we need them because of the rotation of the Earth. The International Date Line is located in the Pacific Ocean, so it is not on land and people do not have to switch to a new day. When you cross the International Date Line it switches to a new day.
Fully describe a Platonic solid. Explain why there are only 5.
To full describe a platonic solid, you must tell the number of faces, shape of the faces, number of edges, number of vertices, and degree of each vertex. There are only five because any other regular polygon would exceed 360 degrees at a vertex, which isn't possible. Tetrahedron: 4 faces, faces are triangles, 6 edges, 4 vertices, 180 degreed at vertex Octahedron: 8 faces, faces are triangles, 12 edges, 6 vertices, 240 degrees at vertex Icosahedron: 20 faces, faces are triangles, 30 edges, 12 vertices, 300 degrees at vertex Cube: 6 faces, faces are squares, 12 edges, 8 vertices, 270 degrees at vertex Dodecahedron: 12 faces, faces are pentagons, 30 edges, 20 vertices, 324 degrees at vertex
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.
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?
Twenty-three sextillion, four hundred billion, seventy-eight. Students can learn to distinguish by counting zeros and using scientific notation.
Give an example of productive computation for geometry.
Two rectangles are similar. One is 14 x 10. GIve the dimensions of the other. About how many similar rectangles are there?
Explain the following division problems by using measurement, visual representation, decimals, complex fraction and multiplication of reciprocal: a) 1 ½ divided by ½ b) ¼ divided by ½
Use a line for measurement, circles for visual, write it as 1.5 divided by 0.5 or .5 divided by .25 for decimals, write it as 1.5/.5 or .5/.25 for complex fraction, and multiplication of reciprocal is straightforward
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 or 202 would not have a solution. Furthermore, zero lets you hold a space and move on, or keep place.
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.
Give an example to show what the following essential ideas of probability mean: a) Probability can be expressed in three ways b) We use probability c) Probability exists between 0 and 1. Create a visual to show the range of probability. Use fractions, decimals and percents. d) Events can be independent or dependent. e) The formula for probability is: Favorable Outcomes /Total Outcomes . f) To find the probability of successive events, multiply each event. g) Complementary events are two events that add up to one. For example: The probability of my birthday being on a Monday (event 1) and not on a Monday (event 2). h) Theoretical probability is based on a formula. i) Experimental probability is based on an experiment.
a) 1/2 05 50% b) probability of weather c) On a number line from left to right: 0 0 0% -- 1/2 0.5 50% --- 1 1 100% d) independent is throwing 2 dice, dependent is pulling cards and not putting them back e) ex. red gum balls over total of 20 would be 3/20 f) get 1 on both dice -- 1/6 x 1/6 = 1/36 g) on Monday 1/7, not on Monday 6/7, 1/7+6/7 h) fav/total outcome i) toss dice 10 times and get data
Use pattern to find the first 8 terms of pentagonal, hexagonal, heptagonal, octagonal, nonagonal and decagonal sequences.
a. First is 0, then 1, then up by 3, then 6, 10, 15, 21, and lastly 28 b. Pentagonal: 1, 5, 12, 22, 35, 52, 70, 92 c. Hexagonal: 1, 6, 15, 28, 45, 66, 91, 120 d. Heptagonal: 1, 7, 18, 34, 55, 81, 112, 148 e. Octagonal: 1, 8, 21, 40, 65, 96, 133, 176 f. Nonagonal: 1, 9, 24, 46, 75, 111, 154, 204 g. Decagonal: 1, 10, 27, 52, 85, 126, 175, 232
write the rule and formula for the triangular number sequence
add the next counting number and (1n/2) n+1
Name three ways we used fractions in class?
fraction towers, rods, or a ruler
what are the three uses of rational numbers
parts of whole, rations, division
give a example of a perfect number, abundant number, and deficient number and prove it
perfect-6=3,2,1 abundant-12=1,2,3,4,6=16X deficient-15=1,3,5=8X
What are the real numbers and why is each set needed?
real numbers include rational and irrational numbers- they are not imaginary and are needed for the number system
Fully describe the Hindu-Arabic numeration system
started in India, base of 10, unique numbers was translated
write this number in English-2,002,002,002,400,000,000,000
two sextillion two quintillion two quadrillion two trillion four hundred billion
Use algebra to show a number, a number times 2; a number greater than 2, one more than a number, the next number.
x = a number 2x = a number times two x>2 a number greater than two x+1 a number one more than a number x+1 is also the next number
