CIEP 104 Final
4. Add the counting numbers from 1 to 300.
1+300=301, 150 pairs between 1-300, so the sum is 45,150 (multiply 301 times 150)
48. • Any number can be expressed as a sum of 3 or fewer triangular numbers.
1. take any number: for example, 14 it can be rewritten as a sum of 3 triangular numbers: 1+3+10
11. Describe the components of a developmental lesson. Give an example of each one. Explain how you could teach the concept of adding developmentally.
1D (with numbers, words, letters), 2D (with pictures), 3D (with objects)
48. • Any number can be expressed as a sum of 4 or fewer square numbers.
2. take any number: 22 Can be rewritten as sum of 4 square numbers: 1+1+4+16
48. • Any number can be expressed as a unique sum of powers of two.
3. take any number: 54 Can be rewritten as unique sum of powers of two: 32+16+4+2
57. Define a square. Concept Development Rectangles 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
91. Probability exists between 0 and 1. Create a visual to show the range of probability. Use fractions, decimals and percents.
Can never have negative or over one for probability
76. Write and answer a problem that is an example of productive computation for geometry.
Complementary, supplementary, reflexive angles (geometry, computation)
24. Sketch graphs of the power sequences.
Even: curve up like a u Odd: negative, flattens out by zero, increases into positive
107. 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.
On Jacobs page 42
91. Experimental probability is based on an experiment.
Toss the dice to see what numbers come up most often (do an experiment)
109. Explain how you can incorporate the fine arts into your math instruction.
Use songs, music, drama (theater games, skits to show problems)
77. The calendar is an important part of humankind's need for quantitative thinking. Expand upon this statement.
We depend on it for religion, economics, social, and political parts of life.
28. Write the results in scientific notation. (4 x 10^6) x (3.1 x 10^5).
multiply first numbers, then add exponents 3.1 times 4=12.4 12.4 x 10^11
49. Explain the everyday use of n1, n2 n3 in measurement.
n1: length, line n2: area n3: volume
54. What important ideas do you need to teach about lines?
They never end, their notation, parallel, perpendicular, intersecting (if they do intersect, they can be perpendicular or not)
70. 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
93. 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?
When we add fractions, they get bigger, when we subtract them they get smaller When I multiply fractions, the numbers get smaller (changes the way we think about multiplication) One half instead of one out of two (has nothing to do with two), also when we say "quarter" Parts of a whole, ratios, division (3 uses of rational numbers) Changes the way we add fractions (have to change the denominator)
12. What is the importance of studying permutations and combinations?
good practice for multiplication, introduces probability,
91. We use probability every day.
it probably won't rain today, so I won't bring an umbrella
17. show how the sum of square and triangular numbers are related
two consecutive triangular numbers equal the next square number
102. Give an example of an activity you could use to support primary students' understanding of algebra in the following concepts:
• Coordinate Geometry Tic tac toe • Equations & Graphing hands on algebra (pennies and variables with a balance) • Inequalities & Graphing Sort and match the picture graphs, match statements to graphs • Patterns • Simplification Pennies, dimes, and nickels so they're not all together • Variables (wild cards, face cards) Card games
34. Make a chart showing the values of the powers of 10 from 103 to 10-3 in exponential, factor, fraction and standard forms.
Exponential: writing it out with the exponents (10^3, 10^2, 10^1, 10^0, 10^-1, 10^-2, 10^-3) Factor: 10x10x10, 10x10, .......1/10x1/10x1/10 Fraction: 1000/1, 100/1, 10/1, 1, 1/10, 1/100, 1/1000 Standard: 1000, 100, 10, 1, .1, .01. 001
91. The formula for probability is:
Favorable Outcomes /Total Outcomes
26. Sketch a graph of a geometric or exponential sequence. Explain why an elementary teacher needs to understand geometric sequence to teach place value.
Graph: starts out very close together then quickly increases; geometric sequence is related to place value because you multiply by the same number each time, so you go from the tens places to the thousands and so on
73. Two rectangles are similar. One is 14 x 10. Give the dimensions of the other. About how many similar rectangles are there? Explain why math problems about similar figures be considered productive multiplication practice.
Has to be in proportion: 7x5, infinite numbers of similar rectangles, geometry and computation
103. 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.
Has two colors (integers), base cards as variables, easily sortable Mean, median, and mode games
41. What is the name of our numeration system? Use a table to describe it.
Hindu-Arabic Place of Origin: India Date: 500-800 CE Base: 10 Place Value: Yes Symbol for Zero: Yes # of Unique Characters: 10 Capable of large numbers: Yes
105. Describe two types of mathematical reasoning.
Inductive: look for patterns, but never totally sure about patterns Deductive: best, have proof (magic trick problems)
104. Describe 3 manipulatives you can use to teach integers. Use open and shaded circles to show -3 + 4.
Integer pennies, deck of cards, number line Closed shaded for negative (3 circles), 4 open circles, pair them up to see what is left
101. 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).
Introduce, compare, compute Show me -1 (introduce) Which is larger? Are they equal or not equal? (compare) Adding problem like number 104 (compute)
5. Explain why Postage Stamp Riddles from Riddle Math are an example of productive instruction.
It has multiple objectives: critical thinking, language arts, algebra, numbers and operations
108. Name 5 problem solving strategies. Be prepared to solve one of the problems completed in class and identify the strategy used.
Make a diagram, make a table, make the problem smaller, work backwards, use a model
7. 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
84. What are the measures of central tendency? What are the measures of variability?
Measures of central tendency: mean, median, and mode (used to find the central value in a given set of numbers) Measures of variability: range, standard deviation (used to find how spread out the data is)
106. What is the 4 colored map problem and why should teachers know about it?
Only need four colors to color any map (no boundary touches the same color) Started out as inductive only seeing patterns, until someone actually proved it (then deductive)
81. 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?
Seasons are caused by the earth's tilt: summer is tilted toward the sun, winter away In summer, we get the summer solstice, which is the longest amount of daylight, while winter has the winter solstice that has the least amount of daylight. In between there are the spring and fall equinoxes, which has equal day and night length. The earth is in balance because the amount of daylight balances out as we have a longest day and a shortest day and then two equal days in between.
80. For what momentous calendar reform is Pope Gregory famous? What questions might people have about such reform? Discuss the difference between the Julian and the Gregorian calendar.
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 Julian calendar: assumes a year is 365.25 days, not making up for the 11 minute difference-centuries not divisible by 400 would still be leap years
37. 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 language in which God has written the universe"; math works because it is how the universe functions, we can find the reasoning behind certain phenomena because of math
42. Show that math is a cultural phenomenon by using a table to describe the five different numeration systems we have studied.
*Hindu-Arabic in previous question Babylonians: Place of Origin: Babylonia Date: 3000-2000 BCE Base: 60 Place Value: Yes Symbol for Zero: No # of Unique Characters: 1 (V turned different ways) Capable of Large Numbers: No Mayan: Place of Origin: Mayan civilization Date: 300 CE Base: 20 Place Value: Yes Symbol for Zero: Yes # of Unique Characters: 3 (dot, line, and oval shape with lines for zero) Capable of Large Numbers: Yes Egyptian Place of Origin: Egypt Date: 3400 BCE Base: 10 Place Value: No Symbol for Zero: No # of Unique Characters: 7 (1: rectangle, 10: upside down U, 100: coil of rope, 1,000: lotus, 10,000: finger, 100,000: tadpole, 1,000,000: person praising) Roman Place of Origin: Rome Date: 500 BCE Base: 10 Place Value: No Symbol for Zero: No # of Unique Characters: 7 (1=I, V=5, X=10, L=50, C=100, D=500, M=1,000) Capable of Large Numbers: No
17. Square 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;
- 1, 4, 9, 16, 25, 36, 49, 64 -rule: add the next odd number -formula: n^2
17. Triangular 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;
-1, 3, 6, 10, 15, 21, 28 -rule: add next counting number -formula: (n+1)n/2
19. 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 three nature examples: flower petals, seeds, galaxies, etc
44. 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.
91. Probability can be expressed in three ways (as a fraction, decimal or %)
1/2, .5, 50%
51. Write the symbols or the mathematical notation for a mosaic that is created by a dodecagon, a hexagon and a square.
12-6-4
48. • All counting or natural numbers can be categorized as happy or not happy.
4. Ex: 19 Split and square both: 1^2+9^2=1+81=82 and continue on If you eventually get 1, it is happy If you continue to cycle through computing and never reach 1, it is unhappy
59. Are all congruent figures similar? Are all similar figures congruent?
All congruent figures are similar, but not all similar figures are congruent
58. 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 Not all right triangles are similar because their sides may not be proportional
53. Describe an activity to help students understand the value of a point
An activity to help students understand the value of a point is to have them work on a graph to find the transformations of shapes by using the vertexes. You could even use just one point and have them learn to move it using directions like "up two, over 3 to the right". put patterned blocks around a point to find angle measurement
38. Give 5 examples for everyday use of the geometric sequence 2n.
Biology: a zygote continually splits into two, doubling its number of cells every time Barcodes: filled in black for square numbers to find the total price of the item Music: each note is 2^n more than the last Games: putting teams against each other and then the winners against each other, so each round goes down in the sequence of 2^n Probability: using a chart to show the options of different combinations of getting heads or tails (increases by 2^n each time)
89. Sketch a box and whiskers graph. Sketch a stem and leaf graph.
Box and whiskers: find the median, then the first quartile, third quartile, put a box around the three and then two lines coming out to the end of the data numbers on either side of the box Stem and leaf: make a table, put tens on one side and ones on the other
21. Who invented the coordinate plane and why was it valuable? What other important events were happening at the same time?
Descartes, it brought algebra and geometry together (you could visually see an equation) Shakespeare writing plays, Galileo (scientist), Fermat and Pascal (developed probability theory)
99. 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?
Equal sign means balance, easier to show balance than equal Draw a balance, put 4 sandwiches and two 1's on one side, and then 2 sandwiches and eight 1's on the other, match up pairs to take away on either side to see what is left
62. 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 triangle Large triangles=1/4, medium triangle, parallelogram, square=1/8, small triangles=1/16
56. What is the difference between line and rotational symmetry?
Line folds it exactly in half (mirror), rotate within its outline around a central point
25. Name ten math inventions
Logarithms, computation rods ("Napier's Bones"), "chess arithmetic", decimal point, coordinate plane, partitions, Cuisenaire rods, pi, buoyancy, pulleys
91. Theoretical probability is based on a formula.
Make a table to see what may come up
71. Give an example of non-decimal measurement units.
Non-base ten: inch, foot, yard, mile
78. 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?
One year is about 365.25 days (.25=6 hours) The difference between the exact time and this is about 11.25 minutes In 10 years, this difference would be 112.5 minutes-100 years would be 1125 minutes (20 hours, 1 day)-1000 years would be 11250 minutes (200 hours, 10 days)
74. 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
65. 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
43. Translate 2,344 into Roman, Mayan, Babylonian and Egyptian.
Roman: MMCCCDLIV Mayan: 400 goes in 5 times, so there is a line on top (one line=5, one dot=1) 20 goes in 17 times, so there are three lines with two dots above them in the middle Left over is four, which is just four dots on the bottom ______ . . ______ ______ ______ .... Babylonian: 60 goes in 39 times, with 4 leftover <<<VVVVVVVVV VVVV Egyptian: two lotus flowers, three coils, four upside down U's, and four rectangles
36. Explain how a teacher can use "balance" to help students understand equations
Saying "balance" instead of "equals" in an equation helps students see that math really does work because, on a scale, four and five on one side would balance out nine on the other
75. 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)
61. 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.
a) What is the proper name of a soccer ball? 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
13. Why should we change "combination locks" to "permutation locks"?
in a combination, the order does not matter in which you pick the items (or numbers), but it does for a number lock, which means one would use a permutation to find the total number of possibilities
55. Fully describe a given polygon (hexagon, heptagon...)
mirror angle=120 (360/3), interior angle=60 (180/3), mosaic=3-3-3-3-3-3, total degrees of interior angles=180 Square: 4 sides and angles, 4 lines of symmetry, 4 fold of rotational symmetry, mirror angle=90, interior angle=90, mosaic=4-4-4-4, total interior angles=360 Pentagon: 5 sides and angles, 5 lines of symmetry, 5 folds of rotational symmetry, mirror angle=72, interior angle=108, mosaic not possible, total interior angles=540 Hexagon: 6 sides and angles, 6 lines of symmetry, 6 folds of rotational symmetry, mirror angle=60, interior angle=120, mosaic=6-6-6, total interior angles=720 Heptagon: 7 sides and angles, 7 lines of symmetry, 7 folds of rotational symmetry, mirror angle is about 51.43, interior angle is about 128.57, mosaic not possible, total interior angles=900
87. 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?
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)
91. 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)
probability of getting "a" and of not getting "a", only two events that add up to one
82. 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 time zones, earth is rotating, affects the fewest number of people (not in a country), changes dates
9. What are the divisibility rules for multiples of 2,3,4,5,6,7,8,9,10, 11 and 12?
2: ends with 0, 2, 4, 6, 8, 3: if the sum of the digits is also (333=9) 4: divisible by 2 twice or the last 2 digits are divisible by two 5: ends with 0 or 5 6: divisible by two and three 7: double the last digit and subtract it from the rest of the original, must be divisible by 7 (826 82-12=70) 8: if last three digits are divisible by 8 9: sum of the digits divisible by 9 10: ends with 0 11: 19151 --> 1915-1=1914 --> 191-4=187 --> 18-7=11 12: divisible by 3 and 4
48. • Any number can be expressed as sums and/or differences of unique powers of 3.
3^0(1), 3^1(3), 3^2 (9), 3^3 (27): 23=27-3-1
6. 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.
3x2x3=18 4-8-9
91. To find the probability of successive events, multiply each event.
8 girls: 1/2x1/2x1/2....and so on
39. Show how you could make 9, 1/5 using only ones.
9: 1+1+1+1+1+1+1+1+1 1/5: 1/1+1+1+1+1
29. Describe a developmental lesson to teach square numbers.
A developmental lesson to teach square numbers would be to start 3D by using cubes to make squares and see the patterns of how they increase, then to use a grid to color in squares (2D), then to show the patterns of square numbers using just numbers
1. Describe the secrets of pedagogy presented by Dr. Schiller. Give an example of each one. Explain how you used Secrets A, B, and F in your first lesson plan activity.
A: magic trick "think of a number" B: nametag (what do you sees?) F: use developmental model: 3D: objects (pennies and paper clips), 2D: pictures (nametag), 1D: numbers (magic trick with only numbers)
86. 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
32. Draw a diagram of 1234 in base ten and base five.
Base 10: 1 thousand cube, 2 square hundreds, 3 sticks of 10, 4 small 1 cm cubes Base 5: 1 large cube (would be 125), 2 squares (each 25), 3 sticks (each 5), and 4 1 cm cubes
3. What is the fundamental theorem of arithmetic? Give an example of it. What mathematician stated it?
Carl Gauss, prime numbers are the building blocks, every number is a unique product of prime numbers (35=5x7, 24=8x3:can be split further into prime numbers)
14. 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)?
Draw square that is 4 x 7, split in half so that you now have 2 and 2 by 7. This helps learn multiplication because they can visually see how four is equal to two and two and can see it as area. 3D: use actual blocks they can put together and separate, 2D: use graph paper, 1D: use numbers to show how to find area
94. Name three models we used in class and/or at Swift to teach fractions. What is the difference between teaching fraction computation using a number theory base and an algebra base?
Fraction rulers, tangrams, fraction towers, ruler game board and card game Number theory base: use common denominators 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)
72. 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?
French Revolution Lagrange helped 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 in a 1 dm x 1dm. x 1 dm container. The gram is the weight of the water in a 1 cm x 1cm x 1cm container. 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 point c) 1. Measure of length, weight, and temperature are all derived from each other 2. Derived from the properties of the earth 3. Base ten 4. Global use 5. Prefixes all the same Prefix for thousand: kilo Thousandth: milli
23. 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
85. What is the difference between a histogram and a bar graph?
Histogram: continuous sequence, bars next to each other Bar graph: not a continuous sequence
60. Explain how you would use concept development to teach a lesson.
I would use concept development by giving nonexamples of regular polygons and then giving examples of them and asking which shapes in a group are regular polygons and which are not
91. Events can be independent or dependent.
Independent: putting objects back Dependent: leaving objects out (don't put them back)
33. 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".
Just the magic square puzzle we did at the beginning of class one day: don't need to know the common difference of 3 part, just that we know the square can be of any common difference square goes: n-3, n+2, n+1 n+4, n, n-4 n-1, n-2, n+3
79. What is the need for leap year? Give all the leap years between 1776 and 1812. How can you tell if any year is a leap year?
Leap year is 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
22. 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?
Only need to know: The coordinate plane helped create latitude and longitude; point (0,0) in Chicago is State and Madison;
35. 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?
Partitions: 1+1+1+1+1+1+1, 1+1+1+1+1+2, 1+1+1+1+3, 1+1+3+4, 1+1+5 Ramanujan is associated with partitions He faced poverty growing up and little access to books, his mother was worried about him traveling to England because of their religion, he did face troubles there because of his vegetarianism A teacher can use partitions to show how many different numbers can used to make another number. Cuisenaire rods can be used to represent numbers visually (3D) and to see how they fit together to create another
47. What was the value of the invention of zero? Write a set of equations that you can use to show children why it is impossible to divide by zero.
Place value: we can have infinite numbers 3+2=5, 2+3=5 0x3=0, 3x0=0 3/0=0: not possible
8. 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
92. Why should probability and fractions be taught together?
Probability is a fraction: favorable/total Understand probability more than fractions (choose the spinner)
48. What is productive computation practice? Why should the elementary teacher use it?
Productive computation practice is teaching multiple math concepts using one activity It should be used because it allows students to see how multiple concepts are interconnected and to use them with one another and it is helpful for teachers to be able to cover multiple concepts with one activity
98. What sets of numbers compose the real numbers. Why do we need each number set?
Real numbers can be seen on the number line Counting numbers (multiplication, addition), then whole numbers (includes zero, subtract), integers (subtract), rational numbers (division), irrational numbers (measurement (pi, square root of two))
52. Describe 3 types of transformations. Why is the importance of the point or vertex in transformations?
Reflection-flips over an axis Rotation-turns the shape Translation-slides The vertex is important because you use them to complete a transformation. For example, in a translation, if it says to move up four, you would move each vertex up four to find where the new shape is.
64. Fully describe a Platonic solid. Explain why there are only 5.
Tetrahedron (four triangles, three meet at a vertex): 6 edges, 4 vertices, 4 faces, 180 degrees at vertex Octahedron (eight triangles, four meet at a vertex): 10 edges, 6 vertices, 8 faces, 240 degrees at vertex Icosahedron (5 triangles meet at a vertex): 30 edges, 12 vertices, 20 faces, 300 degrees at vertex Cube (six squares, 3 at a vertex): 12 edges, 8 vertices, 6 faces, 270 degrees at a vertex Dodecahedron (3 pentagons meet at a vertex): 30 edges, 20 vertices, 12 faces, 324 degrees at a vertex There are only five because any other regular polygon would exceed 360 degrees at a vertex, which isn't possible
30. What is the value of the dog-sitter problem? Write a similar problem.
The dog-sitter problem helps to differentiate geometric and arithmetic sequences by seeing how different they look on a graph. A similar problem would be find which mover would save you more money if one charges $2 per item of furniture, while the other doubles the price for each item of furniture, starting with $0.1. You have 14 pieces of furniture to move.
31. Here is a tally. Express the amount in base 4, 5 and 6 / / / / / / / / / / / / / / /
The last lines are supposed to be the ones that cross over the others (ex: only the fourth lines in a group would cross over; only the fifth, and so on) Base 4: l l l / l l l / l l l / l l = 32 Base 5: l l l l / l l l l / l l l l = 24 Base 6: l l l l l / l l l l l / l l = 22
2. Explain why it is important to use "advanced organizers". Replicate the advanced organizer Dr. Schiller used for elementary school mathematics. Explain why an outline or a table is an advanced organizer. Explain why students should preview their textbooks at the beginning of the year.
They are important because they increase student productivity and organization. Students should preview textbooks to help show what they will be learning in the class and prepare them for the material. An outline or a table is one because it organizes information the students will learn. (example is yellow sheet with the three circles saying measurement, geometry, algebra, data analysis, and then numbers and operations outside)
10. Explain why a multiple choice quiz is an independent event and a matching quiz is an example of a dependent event. Use a tree diagram to show that the number of different three-scoop (chocolate, vanilla and strawberry) cones is 27 as an independent event (flavors may be repeated) and that the number of different three-scoop (chocolate, vanilla and strawberry) cones is 6 as a dependent event (flavors may not be repeated.)
multiple choice is an independent event because your answer for one question does not affect your chance of picking the correct answer for the next question matching is dependent because your answer for one question affects what you can answer for the next one
18. . Use pattern to find the first 8 terms of pentagonal, hexagonal, heptagonal, octagonal, nonagonal and decagonal sequences.
pentagonal: 1, 5, 12, 22, 35 (n(3n-1)/2) hexagonal: 1, 6, 15, 28, 45 (2n^2-n) Heptagonal: 1, 7, 18, 34, 55 pattern like pascal's triangle: all ones down the column, next column is counting numbers, then triangular, then tetrahedral, then by 10s when you write them out
14. Write the first 8 lines of Pascal's triangle. Show 5 patterns.
symmetrical, second diagonal is counting numbers, then triangular, then tetrahedral, hockey stick (follow diagonal down, curve up-sum of the numbers) ones all down the sides, add up the two numbers above it
88. Why does the elementary teacher need to know about standard deviation? • 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
to evaluate test scores add/subtract 12 for every line going away from 120 in the center same percentages
90. Give examples to show that measurement drives progress.
x-rays, CAT scans, computers, cars, glasses
