CIEP104 Closure Questions
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 with which God has written the universe" We can use math in 3-d, 2-d or 1-d form to show how the universe works.
Name ten math inventions
#0 computer Logarithms Negative numbers Cell phones Calculators Fractions Square numbers Irregular numbers Compass protractor
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? Create a 3 x 3 magic square using algebraic notation to show why the square is "magic".
-4 -1 2 5 8 11 14 17 20 Magic number = what is in the middle 8 Magic sum 8x3 = 24
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 flowers, golden ratio --> shells, bunnies reproducing
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 If you add the numbers from left to right in each row of the triangle, this forms a pattern of numbers to the power of 2. If you assume the numbers in each row are together and are one number, this forms a pattern of numbers to the power of 11. hockey stick pattern flower pattern When you color in multiples of a number, they form patterns of upside down triangles.
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?
1. Natural Counting · 1, 2, 3, 4, 5, ... 2. Integer · ..., -2, -1, 0, 1, 2, 3, ... 3. Rational · numbers that have a pattern · 3/2 = 1.5 · 8/4 = 2 · positive or negative 4. Irrational · Have no pattern · Pi · Have no ratio 5. Whole Numbers · 0, 1, 2, 3, 4, 5, ... Rational Numbers → Division Irrational Numbers → measurements (square root) Integer → subtraction (negative #s) Whole → subtraction (zero) Counting → adding and multiple We have an infinite amount of irrational numbers.
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 results in scientific notation. (4 x 10^6) x (3.1 x 10^5).
12.4 x 1011 → 1.24 x 1012
You receive the following information from your students' performance on the PARCC test: mean = 120; s.d. = 12. Calculate the range. Multiply by 3 because there are 3 standard deviations in either direction. 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 or 72. 68% 34% are 1 std dev above, and 34% are 1 std dev below the mean. So 34+34=68 normal bell curve, start at 84 (2%) to 96, (14%) to 108, (34%) to 120, (34%) to 132, (14%) to 144, (2%) to 156.
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?
17th century. French Revolution Lagrange . The meter was originally defined as 1⁄10,000,000 of the distance between the North Pole and Earth's equator as measured along the meridian passing through Paris. The liter is the amount of water that would fill 1 dm3. The gram is the weight of 1cm3 of water. The Celsius temperature scale was derived from the properties of water, with 0 °C being defined as its freezing point and 100 °C being defined as its boiling Base ten/ Decimal-based Common prefixes Measure of length(meter), capacity(liter), weight(gram) and Temperature (Celsius) were derived from each other Derived from Nature It is used globally - almost every country has adopted the metric system U.S. children are at a disadvantage because they have to learn two measurement systems Greek prefixes indicate units larger than the base measure: kilo- (thousand) Latin prefixes indicate units smaller than the base measure: milli- (thousandth),
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?
24 the Earth rotates So that it is not on land and people don't have to switch as often you can't just be walking and poof a new day switch dates/days
What are the divisibility rules for multiples of 2,3,4,5,6,7,8,9,10, 11 and 12?
2: If the last digit is even 3: if the sum of the digits is divisible by 3 4: if the last two digits are divisible by 4 5: if the last digit is 0 or 5 6: if the number is divisible by 2 and 3 7: if it can be truncated 8: if the last 3 digits are divisible by 8 9: if the sum of the digits is divisible by 9 10: if the last digit is 0 11: if it can be truncated 12: if the number is divisible by 3 and 4
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.
2n (2,4,6,8)= 4 options 2^n (0,1,2,3)= 4 options n^2 (0,1,2,3) = 4 options 4x4x4=64 options ex: 429
Use Gauss's insight to add the counting numbers from 1 to 300.
301 x 150 = 45,150
Here is a tally. Express the amount in base 4, 5 and 6 / / / / / / / / / / / / / / / → 15 tallies
33four 30five 23six
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.
365 not a multiple of 7 (dates shift over one/two days each year) . Labor Day: September 1 - September 7 (first monday in september -> dates for the first week) Mother's Day: May 8 - May 14 (second sunday in may -> dates for the second week) Father's Day: June 15 - June 21 (third Sunday in june -> dates for the third week) Thanksgiving: November 22 - November 28 (fourth Thursday in November -> dates for the 4th week) we labor for our mothers and fathers and we're thankful for them (1,2,3,4-- courtesy of Mikayla and Dani) way to remember them in ORDER OF WEEKS
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
7 x 5 or 28 x 20 Infinite Geometry and multiplication You are using the factors of the original problem to make new similar shapes.
How many squares in a checkerboard?
8^2 + 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2 + 1^2 If it is a 10 by 10, simply start with 10^2 + 9^2 + ... Same thing if it was a 7x7 → 7^2 + 6^2 + ...
Explain why a multiple choice quiz is an independent event and a matching quiz is an example of a dependent event.
A multiple choice quiz is an independent event because the answers you choose throughout the quiz are not reliant on the answers of the previous questions. A matching quiz is an example of a dependent event because the answers you choose are reliant on the answers of the previous questions.
Are all congruent figures similar? Are all similar figures congruent?
ALL congruent figures ARE similar All similar figures are NOT congruent
Are all squares similar polygons? Are all right triangles similar polygons?
ALL squares ARE similar polygons Right triangles are NOT always similar
Use Pascal's triangle to calculate the following probabilities in a family of 5 children: ( g + b) ^5 All boys All girls At least one girl At least one boy More girls than boys More boys than girls
All boys → 1/32 All girls → 1/32 At least one girl → 31/32 At least one boy → 31/32 More girls than boys → 16/32 More boys than girls → 16/32
Who was Benjamin Banneker? Why will it be important to include him in your future math classroom?
American mathematician he argued against slavery
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, ⅕
An arithmetic sequence is a number sequence in which each successive term may be found by adding the same number. (common difference) A geometric sequence is a number sequence in which each successive term may be found by multiplying by the same number. (common ratio) Arithmetic sequences are important for multiplication tables. Geometric sequences are important for place value. 5 → 1, 6, 11, 16, 21, 26 -5 → 26, 21, 16, 11, 6, 1 5 → 1, 5, 25, 125, 625, 3,125 ⅕ → 3,125 , 625, 125, 25, 5, 1
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 should we change "combination locks" to "permutation locks"?
Because the order of the numbers being inputted matters, just like they do in permutations. The order does not matter in combinations.
Give 4 upper case alphabet letters that are topologically equivalent to a straight line.
C, G, I, J, L, M, N, S, U, V, W, Z
Give sample questions to show how you can be a more productive mathematics teacher.
Clock and angles (supplementary and complementary angles) What are the supplementary and complementary angles at 5 o'clock? Two angles: one angle is 60 degrees and another is 30 degrees Are they complementary? What are the angle and reflexive angle at 3 o'clock?
Describe the components of a developmental lesson. Give an example of each one. Explain how you could teach the concept of adding developmentally.
Components: 3D, 2D, and 1D 3D: adding blocks together 2D: drawing smiles to add together 1D: adding numbers together
Who invented the coordinate plane and why is it valuable? What other important events were happening at the same time?
Descartes The coordinate plane is valuable because the system allows us to describe algebraic relationships in a visual sense, and also helps us create and interpret algebraic concepts 17th century (1600s): countries were navigating around the world to obtain new land. This could mean that this invention made it easier
How can you make problem solving more fun in your future classroom?
Drama in the classroom: 1. Pantomime 2. Echo Reading 3. Reader's Theater The Power of Two Fairy Tales for the Math Stage 4. Scenes Hey, Benjamin Banneker, Come Play with Us
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;
Draw triangular numbers using dots in the shape of a triangle; square numbers using dots in the shape of squares. Triangular:1 3 6 10 15 21 28; Square: 1 4 9 16 25 36 49 Triangular: Add the next counting number (+2, +3, +4, +5...); Square: add the next odd number (+3, +5, +7, +9...) Triangular: n/2 (n+1); Square: n^2; n= the number in the sequence (1st, 2nd, 3rd...) Sketch square numbers using circles, 1 4 9 16 25 (add the next odd number). Square numbers are made up of triangular numbers added together (draw the picture). 1+3=4, 3+6=9, 6+10=16, 10+15=25...
Create a tree diagram for independent events. (Jacobs p 403). Create a tree diagram for dependent events (Jacobs p 415)
For independent events: use the fundamental counting principle. Ex: 3x3x3 For dependent events: use permutations. Ex: 3! Or 3x2x1
What is the fundamental theorem of arithmetic? Give an example of it. What mathematician stated it?
Gauss, every integer greater than 1 either is prime itself or is the product of a unique combination of prime numbers. 10= 5x2, 11 is prime
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)?
Green → 4 block x 5 blocks = 20 blocks Orange → 4 blocks by x 2 blocks = 8 blocks, 28 blocks total. Distributive principle can help students learn multiplication facts because it breaks up harder multiplication facts with easier multiplication facts 1-d: just the problem 2-d: picture/graph paper 3-d: manipulatives that represent area
Why is Pascal considered the "father of the computer age"? Who is "The Prince of Mathematics"? Who discovered a formula for finding the number of partitions for positive integers?
He invented the first calculator and Pascal's Triangle which contains many patterns of numbers. Carl Friedrich Gauss Srinivasa Ramanujan
What is the name of our numeration system?
Hindu-Arabic Numeration System
In what way are operations with rational numbers intuitive? Counter-intuitive?
IN- they work the way you think they will work (¾ > 2/4 ) CN - it doesn't work the way you think it is going to work (¼ or ⅓)
What was the value of the invention of zero? Use a set of equations that you can use to show children why it is impossible to divide by zero.
If there was no zero, then problems like -1+1 would not work, Made large numbers possible and zero was also invented to show place value Division by zero is undefined if you think about the relationship between multiplication and division. 12/6=2 so 6x2 is 12, 12/0=X so Xx0= 12 but that's not possible because any number multiplied by 0 is 0.
Describe two types of mathematical reasoning.
Inductive → looking at patterns Deductive → Using a formula
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).
Introduction Comparison Computation
What is the 4 colored map problem and why should teachers know about it?
It is possible to color the states of a map with 4 different colors and never have the same colored states touch. it takes us from inductive to deductive no one could prove it (inductive) and some one did prove it (deductive)
Who invented logarithms? What is the value of logarithms?
John Napier. You could multiply by adding
How is Descartes invention used to identify locations around the world? What is the point (0,0) in Chicago? How many blocks north of Chicago's (0,0) point is Devon Avenue? How many blocks west is Kenmore Avenue? Does your home community use a coordinate grid system for location?
Latitude and Longitude State Street and Madison Street 64 blocks north (6400) 10 blocks west (1000) [this varies]
Find the product of 52 and 43 using lattice multiplication and using the Russian peasant method.
Lattice multiplication: [see doc] Russian: Write each number at the head of a column. Double the number in the first column, and halve the number in the second column. If the number in the second column is odd, divide it by two and drop the remainder. If the number in the second column is even, cross out that entire row. Keep doubling, halving, and crossing out until the number in the second column is 1. Add up the remaining numbers in the first column. The total is the product of your original numbers. answer: 2236
What is the difference between line and rotational symmetry?
Line: you can fold it in half Rotational: twist/turn it
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.
MADG, An advanced organizer is information presented by an instructor that helps the student organize new, incoming information, preview the textbook in the beginning to figure out their interests.
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 or partitions to develop number sense. What are Cuisenaire rods and how can a teacher use this manipulative?
Mathematician: Ramanujan challenges: limited food options, no travel partitions: 1+1+1+1+1+1+1 1+1+1+1+1+2 1+1+1+2+2 1+2+2+2 1+1+1+3 1+1+2+3 1+3+3 2+2+3 1+1+1+4 3+4 1+2+4 1+1+5 2+5 1+6 7 why it's important: shows the variety of ways a number can be made Cuisenaire rods can be used to show the different partitions.
MADG
Measurement, algebra, data analysis & probability, geometry
List ten math concepts that require multiplication.
Measurement: perimeter, area, volume, surface area Probability: Successive 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) Distributive Principle Algebra: substitution Divisibility rules 9 Numeration systems Cross multiplication Methods of Counting (Combinatorics)
Illustrate a "math magic trick" with two different numbers. Then write two proofs that show that the trick will always work—one proof with boxes and circles, and one with algebraic symbols.
Number +3 x2 +4 /2 - original = 5
Describe 3 manipulatives you can use to teach integer.
Number Lines with Blocks Cards Integer Pennies or Two Color Chips
A merchant has a fox, a rabbit, and a head of lettuce and sits on the edge of a river. He has a small raft capable of carrying only himself and one item at a time, but without his supervision the fox will eat the rabbit, and the rabbit will eat the lettuce. How can he successfully transport all goods from one side of the river to the next without losing the lettuce or rabbit? The dilemma, of course, is true regardless of which side of the river they are on and there is no other way across.
On side A are him and the fox, rabbit, and lettuce. He takes the rabbit across first from side A to side B. He returns to side A alone. He takes the fox across from side A to side B. He takes the rabbit back across from side B to side A. He leaves the rabbit on side A and takes the lettuce from side A to side B. He returns to side A alone. He takes the rabbit from side A to side B. On side B are him and the fox, rabbit, and lettuce.
How long does Schwartz say it would take to count to one million? one billion? one trillion?
One Million → 23 days One Billion → 23,000 days One Trillion → 23,000,000 days
What important ideas do you need to teach about lines?
Parallel and Intersecting, perpendicular
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?
Pay rent (it controls deadlines) The calendar shapes our planning. How many days until xyz? This is due on this date, etc. advanced the calendar 10 days (skipped 10 days) holidays, birthday, anniversary
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?
Perimeter/Circumference of 24. Shape: 2x20 /// Area: 20 5x7/////Area: 35 circle///Area: 48 sphere If it was a sphere then it would roll away, a rectangle package stays put. It has large flat sides.
Name three models you can use to teach fractions. What is the difference between teaching fraction computation using a number theory base and an algebra base?
Pie chart, Rulers, fraction towers Algebra Base → proportions ½ = 2/4 = 3/6 Number theory → common denominator (GCF/LCM)
Draw a diagram of 1234 in base ten and base five.
Place value cubes Cut them up in each, 10 by 10, or 5 by 5
Explain how you would use concept development to teach a lesson.
Present a worksheet/instruction tool using the layout: This is... These are not... Which of these are...?
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
Explain how the following essential ideas of probability are connected to either identification of rational numbers; comparison of rational numbers; or computation of rational numbers.
Probability can be expressed as a fraction, decimal or % Probability it will rain today is 50% We use probability every day. Probability exists between 0 and 1. The formula for probability is: Favorable Outcomes / Total Outcomes Events can be independent or dependent. To find the probability of successive events, find the probability of each event, then multiply the fractions. (Jacobs p 471 # 21-24 Jacobs p 509 18-20) 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) Theoretical probability is based on a formula Experimental probability is based on an experiment
Why should probability and rational numbers be taught together?
Probability is a fraction, decimal, and percentage
What is the importance of studying permutations and combinations?
Probability: In order to calculate probability, you need to know the total number of possible outcomes.
Describe a language issue with rational numbers instruction? What are three uses of rational numbers? How does it impact the addition/subtraction algorithm?
Quarter = ¼ half = ½ Division parts of a whole ratio if we are doing Ratio → you don't know what the whole is ¼ + ½ = 2/6 if we are doing parts of a whole → we have to find common denominator ¼ + ½ = ¾
Translate 2,344 into Roman, Mayan, Babylonian and Egyptian.
R: MMCCCXLIV M: [vertical] ___, ** __ __ __. **** B: <<<||||||||| |||| E: lotus lotus rope rope rope heel heel heel heel staff staff staff staff
Why do we need integers? Rational numbers? Irrational numbers? What are the Real Numbers?
Rational Numbers → Division Irrational Numbers → measurements (square root) Integer → subtraction (negative #s) Whole → subtraction (zero) Counting → adding and multiplying
Describe 3 types of transformations. Why is the importance of the point or vertex in transformations?
Rotations → turns Translations → slides Reflections → (it) flips ****Importance: shapes change or transform in relation to a point or vertex of the shape. For example, a square may rotate around/in relation to the top left vertex
Describe the secrets of pedagogy presented by Dr. Schiller. Give an example of each one. Explain how you used Secrets A, B, C, E and F in your first lesson plan.
Secret to impressing children: Puzzles and magic tricks (card trick), Secret to better teaching: What do you see? (algebra name tags), Secret to first day classroom management: Learn names (algebra name tags), Secret to classroom management after Day 1: Partners (switch every 2-3 weeks), Secret to Super Teacher: productivity (multiple outcomes), Secret to Super Math Teacher: Use 3-D, 2-D 1-D (developmental) teaching model.
Define a square. Are all squares rectangles? Are all rectangles squares?
Square: a plane figure with 4 equal straight sides and 4 right angles, Have two sets of parallel lines, 4 lines of symmetry ALL squares ARE rectangles All rectangles are NOT squares
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.
That's how many triangles you can make = (n-2) Draw in triangles in the shape [see doc for pic]
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 365 days, 5 hours, 59 minutes, 60 seconds - 365 days, 5 hours, 48 minutes, 46 seconds The difference is 11 minutes, 14 seconds 10 years → 112.5 min ~ 2 HOURS 100 years → 1,125 min ~ 20 HOURS ~ 1 DAY 1,000 years → 11,250 ~ 200 HOURS ~ 10 DAYS Every year we lose 11.25 minutes. This comes from the difference of 11 minutes and 14 seconds each year.
What is the need for leap year? Give all the leap years between 1776 and 1812 in the Gregorian calendar. Give all the leap years between 1776 and 1812 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. Every 4 years, we resolve the situation by adding an extra day. We call it leap year. 1780, 1784, 1788, 1792, 1796, 1804, 1808 No 1800 because it is a century year and not divisible by 400. The Revised Julian calendar adds an extra day to February in years that are multiples of four, and this repeats forever with no exceptions 1776, 1780, 1784, 1788, 1792, 1796, 1800, 1804, 1808, 1812 Divide by 4. If it is a century year, divide by 400.
What happens to the perimeter of a square if the length of its side is doubled? 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 The area quadruples The volume gets multiplied by 8 The surface area quadruples
Describe an activity to help students understand the value of a point.
Tic-tac-toe Transformations
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, also known as the mathematics of distortion. 1. Mazes 2. Puzzles 3. Tree Diagrams 4. "Magic" Tricks 5. Map Coloring
Use pattern to find the first 8 terms of pentagonal, hexagonal, heptagonal, octagonal, nonagonal and decagonal sequences.
Triangle 1,3,6,10,15,21,28,36 Square 1, 4, 9, 16, 25, 36, 49, 64 Pentagonal 1, 5, 12, 22, 35 ,51, 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 is an Archimedean solid? What the mathematical meaning is of truncated? What is the sum of the angles at each vertex of a truncated icosahedron? In degrees, what is the difference between a sphere and a truncated icosahedron?
Truncated = Cut off the vertex Archimedean: regular polygons that form a solid (Truncated icosahedron → soccer ball) Two hexagons and a pentagon 6-6-5 (around a point) 120+120+108= 348 Hexagon angle + hexagon angle + pentagon angle 360- 348=12 degrees is the difference 360 is total degrees for sphere 348 is total degrees for truncated icosahedron
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, and seventy-eight (look at chart in Jacobs) Place Value → multiply by one 1,000 Children need to see a pattern Distinguish million, billion and trillion by using days
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? (#74)
Show that math is a cultural phenomenon by using a table to describe the five different numeration systems we have studied.
[order: name, origin, date, base, place value, zero, characters, # unique characters, large #s) Hindu Arabic, India, 500CE, 10, Y, Y, 0 1 2 3 4 5 6 7 8 9, // 10, Y Babylonian, Mesopotamia, 3000 BCE, 60, Y, N, sticks and boomerang symbols, 2, Y Mayan, Mexico/Mesoamerica, 300CE, 20, Y, Y, 0 looks like an eye/loaf of bread, 1 is a dot, 5 is a line, etc// 3, Y Egytpian, Egypt, 3400 BCE, 10, N, N, 1 is staff, 10 is heel bone (upside down U), 100 is coil of rope, 1000 is lotus flower, 10,000 is pointing finger, // 7, N Roman, Rome, 500 BCE, 10, N, N, 1 is I, 5 is V, 10 is X, 50 is L, 100 is Cm 500 is D, 1000 is M// 7, Y
Fully describe a Platonic solid. Explain why there are only 5..
[order: platonic solid, face, vertices, edges, degrees] Cube: 6, 8, 12, 270 Tetrahedron: 4, 4, 6, 180 Octahedron: 8, 6, 12, 240 Icosahedron: 20, 12, 30, 300 Dodecahedron: 12, 20, 30, 324 Platonic solids must be constructed with regular polygons and If the sum of the angles coming together to form a vertex is greater than 360 degrees, then a vertex cannot be constructed with them.
Fully describe a given polygon (hexagon, heptagon...)
[order: shape, sides, lines of sym, # of folds in a rotation system, measure of mirror angle {360/n}, measure of interior angle, polygon mosaic, total degrees] Triangle: 3,3,3,120,60,Y,180 Square: 4,4,4,90,90,Y,360 Pentagon, 5,5,5,72,108,N,540 Hexagon: 6,6,6,60,120,Y,720 Heptagon: 7,7,7,51.4, 128.5,N,900 Octagon: 8,8,8,45,135,N,1080
What is the 4-color map problem?
any separation of a plane into regions (called a map) needs no more than 4 colors to fill it in so that no two of the same colors border
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.
colors Numbers Suits . range, mean, median, mode head up (addition game) "Salute" peace place value → largest number
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?
earth's tilt Summer: summer in northern hemispheres → more daylight summer in southern hemisphere → less daylight Winter winter in northern hemisphere → less daylight winter in southern hemisphere → more daylight Spring/Fall two equinoxes in spring and fall → equal daylight/night in both hemispheres just know that there are two times of year when there is an equinox which means light and dark have equal amounts of time in the day so it's like an equation (it's balanced on either side) sometimes you have more sunlight, sometimes you have less sunlight, equinox is equal (sunlight)
Make a chart showing the values of the powers of 10 from 103 to 10-3 in exponential, factor, fraction and standard forms
exponential: 10^3 to 10^-3 factor form: 10x10x10 to 1/ (10x10x10) fraction form: 1000/1 to 1/1000 standard: 1000 to .001
How can mazes be solved?
following either the right or left wall all the way through the maze
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.
founder of topology → the Bridges of Konigsberg The Seven Bridges of Konigsberg, how to travel over all of them without skipping or repeating only have 1 odd vertex if you want to travel or pair of odd vertices
Name 5 problem solving strategies.
guess and check draw a diagram look for a pattern make the problem smaller work backwards make a table
Explain why Wordsworth Riddles from Riddle Math are an example of productive instruction.
involves critical thinking, language arts, algebra, and number and operations
Sketch a graph of an arithmetic sequence. How can you use the coordinate plane to teach multiplication? Why would this be considered productive teaching? Explain how you can use the idea of a race to help children understand linear graphing.
linear line It is productive teaching because you can teach the students multiple ideas with only one graph. Teach students about the coordinate plane to teach multiplication and to teach slope Connect dots and multiply numbers and calculate the slope 10 foot race problems
What are the measures of central tendency? What are the measures of variability?
mean, median, mode range, 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.
minutes, hours, days, weeks, months, inches, feet, pounds Metric width of index finger= 1cm length of 10 football fields = 1 km thickness of a dime = 1 millimeter Customary top knuckle on your thumb to your thumb tip = 1 inch elbow to wrist = 1 ft
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. 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.
multiple outcomes You do not have enough time to teach everything you want More creative Makes connections with the students from one thing to another . For the # 25: 1, 3, 6, 10, 15 and 10+15=25 . For the #35: 1, 4, 9, 16, 25 and 1+9+25=35 For the # 25: 1, 2, 4, 8, 16 and 1+8+16=25 For the # 22: 1, 3, 9, 27, 81 and Happy numbers- take the sum of the squares of each digit and then if they are happy they will equal 1 or if they are unhappy it will go in an endless loop
Explain the everyday use of n1, n2 n3 in measurement.
n1: Length (1D) n2: Length x Width (2D) (Area/ Surface Area) n3: Length x Width x Height (3D) (Volume)
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?
standardized testing C → 68% B or D → 28% A or F → 4%
Give five reasons Dr. Schiller described to teach numeration systems.
support for understanding place value in their own numeration system; productive computation practice (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; deeper understanding of mathematics; and math is fun.
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?
use two triangles and 4 circles to show that they're balanced to 2 circles and 8 triangles
Design a tree network with a diameter of 4.
| / \ / \ \ \
Construct a line segment to show the range of probability as a fraction, decimal and percent.
|0% - 50% - 100%| |0 - .5 - 1| |0 - 1/2 - 1|