Math CIEP Final
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 1 8 28 56 70 56 28 8 1 1. First column is all ones 2. Second column is counting numbers 3. Square numbers where 4. Triangle numbers 5. Fibonacci sequence where Powers of 11, powers of 2, multiples of any number
Why should probability and fractions be taught together?
Probability is a fraction, decimal, and percentage
Here is a tally. Express the amount in base 4, 5 and 6 / / / / / / / / / / / / / / /
/ / / / / / / / / / / / / / / = 33four Base 4= | | | /// / / / / / / / / / / / / / / / = 30five Base 5= | | | Base 6= | | /// = 23six
Name 3 important ideas in assessing quality of graphs. Describe three ways the public can be misled by graphs.
1. Percentage on graph should show 100% (if doesn't, can be deceiving) 2. Start at 0 (also can be deceiving) 3. Intervals should be equal (or deceiving) 4. Look at in terms of height, area, volume 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
Write the results in scientific notation. (4 x 10^6) x (3.1 x 10^5).
1.24 x 10^13 1.24 x 1012. multiply first numbers, then add exponents 3.1 times 4=12.4 12.4 x 10^11
Make a chart showing the values of the powers of 10 from 103 to 10-3 in exponential, factor, fraction and standard forms.
10^3 = 1000 10^2 = 100 10^1 = 10 10^0 = 1 10^-1 = 0.1 10^-2 = 0.01 10^-3 =0.001
Write the symbols or the mathematical notation for a mosaic that is created by a dodecagon, a hexagon and a square.
12-6-4
9. What are the divisibility rules for multiples of 2,3,4,5,6,7,8,9,10, 11 and 12? http://www.savory.de/maths1.htm
2 - if the last digit is 0, 2, 4, 6, or 8 3 - if the sum of its digits is also divisible by 3 4 - if the last 2 digits make a number that is divisible by 4 5 - if the last digit is 5 or 0 6 - if the number is also divisible by 2 and 3 7 - truncate and double 8 - if the last three digits make a number that is divisible by 8 9 - if the sum of the digits is divisible by 9 10 - if the number is also divisible by 5 11 - truncate 12 - if the number is also divisible by 3 and 4 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
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
51. What is the difference between a theorem and a theory? How are math and the scientific method related?
A theorem has already been proven by deductive reasoning, whereas a theory is an idea that is proposed but is yet to be proven. The scientific method has to do with inductive reasoning, which can be used in mathematical terms as well. Both math and science use inductive reasoning to make observations, discover regularities, and
Are all congruent figures similar? Are all similar figures congruent?
All congruent figures are similar, but not all similar figures are congruent
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 if you divide 365 by 7 you get 52 R 1. The range is always 7 because the date can be any day during that one week given the 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
51. Describe 3 manipulatives you can use to teach integers. Use open and shaded circles to show -3 + 4.
Cards, pennies, and the number line. Closed shaded for negative (3 circles), 4 open circles, pair them up to see what is left
What are the measures of central tendency? What are the measures of variability?
Central tendency: mean, median, mode Variability: range, standard deviation, variance
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?
Dimensions: 1x1 - 2x2 - 4x4 Perimeter: 4, 8, 16 (multiplied by 2) Area: 1, 4, 16(multiplied by 4) volume: 1, 8, 64, (volume multiplied by 8) Surface area: 6, 24 (multiplied by 4)
Give the first 10 terms of the Fibonacci sequence. Name 3 ways it is related to nature.
First 10 terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 , 55 The rate at which rabbits reproduce, the number of petals on a flower, the spirals in a pine cone.
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.
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?
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) 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
What is the difference between a reinforcement worksheet and an instructional worksheet?
Instructional: new Reinforcement: strengthening a pre-existing skill
What is the difference between line and rotational symmetry?
Line symmetry: A line through a shape and on either side of the line, it is the same. Rotational Symmetry: Shape is rotated so that it appears the same 2 or more times. Line symmetry is when the image is reflected on either half of the line, like a mirror Rotational symmetry is when an image is rotated around a central point
What important ideas do you need to teach about lines?
Line, line segment, ray, angle, parallel, intersecting, perpendicular They never end, their notation, parallel, perpendicular, intersecting
Name ten math inventions
Logarithms, computation rods ("Napier's Bones"), "chess arithmetic", decimal point, coordinate plane, partitions, Cuisenaire rods, pi, buoyancy, pulleys
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. Mathematics is the way to express everything, even how nature works. So then, we have a clear explanation for everything. "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
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)
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?
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
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 Forming large numbers,cannot divide 3+2=5, 2+3=5 0x3=0, 3x0=0 3/0=0: not possible
Give an example of a non-decimal measurement unit.
Pounds to ounces Feet and yards Hours and minutes Non-base ten: inch, foot, yard, mile
Explain what the following essential ideas of probability mean: a) Probability can be expressed in three ways (as a fraction, decimal or %) b) We use probability every day. 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.
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
Who invented the coordinate plane and why was it valuable? What other important events were happening at the same time?
Rene Descartes. Gave us a way to map. Mathematically brought geometry and algebra together—known as coordinate geometry Shakespeare writing plays, Galileo (scientist), Fermat and Pascal (developed probability theory)
Define a square. Concept Development Rectangles Are all squares rectangles? Are all rectangles squares?
Square - plane figure with four equal straight lines and four right angles Rectangle - plane figure with four straight lines and four right angles, opposite sides are parallel All squares are rectangles, not all rectangles are 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
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): 12 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
Describe an activity to help students understand the value of a point.
The one where you trace the shape and then move the block around the vertex. 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".
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)?
The value is that kids have a visual representation, which makes the activity 2D and more engaging. They can also count the squares themselves to double check their answer. Distributive principle=repeating addition. Take something hard and use it in a way they will understand from things they already know. 3-D: The cards? 2-D: the lines and shapes used 1-D: the numbers used
Why should we change "combination locks" to "permutation locks"?
This is because combination locks require order, but in a real combination order does not matter. It should be changed to permutation locks because with permutations, order does matter.
51. 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.(See Jacobs p 42)
This is super straightforward. Just look at page 42 in Jacobs. Number +3 x2 +4 /2 - original = 5
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
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?
We depend on it for religion, economics, social, and political parts of life. The Gregorian calendar: he took out ten days in October 1852 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
What is the importance of studying permutations and combinations?
We need it for probability cause it'll give us total number of outcomes. We need them for real life experiences. Like locks (permutations). P=order matters. C=order does not matter. Easier to deal with.
Explain the everyday use of n1, n2 n3 in measurement.
n^1= teaching length and perimeter n^2= is area n^3=teaches volumes
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
Give an example of an activity you could use to support primary students' understanding of algebra in the following concepts: • • Coordinate Geometry • Equations & Graphing • Inequalities & Graphing • Patterns • Simplification • Variables (wild cards, face cards)
• 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
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?
1+1+5 2+3+1+1 3+2+2 1+1+1+1+1+2 1+1+1+1+1+1+1 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 be 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
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 + ...
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.
Describe the components of a developmental lesson. Give an example of each one. Explain how you could teach the concept of adding developmentally.
1.) 3d, concrete, fraction rulers 2.) 2d, pictorial, drawing of a pizza cut up into fractions 3.) 1d, abstract, writing fractions on the board You could teach the concept of adding by using fraction rulers to add fractions together 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
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
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.
2x1= 2 2x2=4 2x3=6 2x4=8 2^1=1 2^2=4 2^3=8 1^2= 1 2^2= 4 3^2=9 4x3x3=144 FCP: Multiplying outcomes. 4 dresses, 2 hats, 1 scarf= 4x2x1
Answer any of the questions in Jacobs Chapter Review: Chapter 7: Set I p 436.
3. 5x3x3=45 4. 7x6x5= 210
Add the counting numbers from 1 to 300.
300 + 1 = 301 299 + 2 = 301 298 + 3 = 301 150 pairs of numbers between 1-300 thus 150 x 301 = 45,150 1. Cut number in half 2. Multiply half the number by number plus 1
52. What is the 4 colored map problem and why should teachers know about it?
4-color map problem - coloring in a map, shapes cannot be same color as any of the shapes it is touching, if you use 4 colors, none of the same colors will share borders.Started out as inductive only seeing patterns, until someone actually proved it (then deductive)
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.
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. Secret to impressing children: Magic tricks and puzzles • The very beginning of the lesson with the "I can guess your sum" magic trick. o B. Secret to better teaching: What do you see? • Having the children look at the name tags and pointing out all the different things they saw before getting into the actual lesson o C. Secret to first-day classroom management: Learn names • The lesson on making the algebra name tags, or a game. o D. Secret to classroom management after Day: Partners and math parties o E. Secret to Super Teacher: multiple outcomes, connection questions, tables, establish routines. Asking connection questions. o F. Secret to Super Math Teacher: Use 3-D, 2-D 1-D teaching model. • Our 3-D model in our lesson plan was the the physically moving aroung the money on the nametags to show the differences in the worth • Our 2-D model in our lesson plan was the color differences in the the letters on out nametags • Our 1-D model in our lesson plan was when we actually wrote out the fractions and did the problems.
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
Who was Benjamin Banneker? Why will it be important to include him in your future math classroom?
American mathematician he argued against slavery
What is an Archimedean solid? What the mathematical meaning is of truncated? a) What is the proper name of a soccer ball? What is the sum of the angles at each vertex of a truncated icosahedron? b) What is the sum of the angles at each vertex of a truncated icosahedron? c) In degrees, what is the difference between a sphere and a truncated icosahedron?
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 a)truncated icosahedron b)120+120+108=348 degrees at each vertex c)360-348=12 degrees
28. 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 sequences involve addition and subtraction whereas Geometric sequences involve multiplication and division. They are important for elementary teachers because they teach students patterns in a 1-D model and also pictures and activities. Arithmetic sequence example: 5, 10, 15, 20. Aritmetic sequence are the times tables; geometric sequences are place value—1, 10, 100 etc.Geometric sequence example: 1, 5, 25, 125, 625
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
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?
EX. area is 24, i can have a shape that is... 1x24(perimeter=50) 2X12(perimeter=28) spheres represent greatest volume for given surface area most items are packaged that way because they are more easily stackable
Fully describe a given polygon (hexagon, heptagon...)
Equilateral Triangle: 3 sides, 3 angles, 3 lines of symmetry, 3 folds of rotational symmetry, 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 Octagon: 8 sides, 8 lines, 8 folds, 45 degree mirror angle, 135 interior angle, no mosaic, sum is 1080 degrees
What is the difference between interpolation and extrapolation in graphing?
Extrapolation is an estimation of a value based on extending a known sequence of values or facts beyond the area that is certainly known. In a general sense, to extrapolate is to infer something that is not explicitly stated from existing information. Interpolation is an estimation of a value within two known values in a sequence of values. Polynomial interpolation is a method of estimating values between known data points. When graphical data contains a gap, but data is available on either side of the gap or at a few specific points within the gap, interpolation allows us to estimate the values within the gap.
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
Why is Pascal considered the "father of the computer age"? Who is "The Prince of Mathematics"? Who discovered a formula for find the number of partitions for positive integers?
He created the language all computers run off of. Gauss is considered the prince of mathematics. Ramanujan.
30.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
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
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
12. Create a tree diagram for independent events. (Jacobs p 403). Create a tree diagram for dependent events (Jacobs p 415)
ID: (small medium large),(vanilla, chocolate, cherry, strawberry), (soda, shake). 3x4x2=24 D: 5 horses (A,B,C,D,E) in a race=5! 5x2x3x2x1 = 120 For independent events: use the fundamental counting principle. Ex: 3x3x3 For dependent events: use permutations. Ex: 3! Or 3x2x1
Sketch a tangram. If the tangram is equal to 1, give the fractional value of each tan. Sketch a second tangram. If the tangram is equal to 16, give the value of each tan.
If the tangram is 1, then each of the 7 tans (pieces )will be a fraction large triangles = 1/4 each medium triangle = 1/8 small triangles = 1/16 each square = 1/8 parallelogram = 1/8 Row 1: Medium triangle, row 2: small triangle, square, small triangle, parallelogram, row 3: 2 large triangles If the tangram is 16, then each of the 7 tans (pieces )will be a whole number. large triangles =4 each medium triangle = 2 small triangles = 1 each square = 2 parallelogram = 2
Illustrate the Pythagorean Theorem.
Illustrate the Pythagorean Theorem.
51. Describe the difference between inductive and deductive thinking for a mathematician. What are the limitations of inductive reasoning?
Inductive is the method of drawing general conclusions from a limited set of observations. It is reasoning from particular to general. Deductive reasoning is a method of using logic to draw conclusions from statements that we accept as true. It is from general to particular. Inductive is limited because scientists and mathematicians need more data and evidence. Science and math are always changing. Inductive reasoning could be wrong, and people need more evidence to prove their hypotheses and to eliminate irregularities in their findings.
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)
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?
Invented during French Rev. by Lagrange. Celsuis temp derived from properties of water, 0 degrees freezing, 100 boiling point Kilogram is mass of 1 liter of water at its melting point. 1. measures of length, capacity, weight and temp derive from each other 2. Measures are derived via properties of natural objects - earth, water 3. Uses base 10 4. Global use - every country uses it except 3 5. US children put at disadvantage because they need to learn 2 systems Thousand - kilo Thousandth - milli 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. 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
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.
It is important to use advanced organizers because helps the student organize new, incoming information. • An outline or a table can be considered an advanced organizer because it organized information into categories that are more easily understandable. It also gives the students a visual. • Students should preview their textbooks because it gives you an opportunity to learn about your students' interests and sets the tone for the year so students can keep topics in the back of their head. Data Analysis and probability, measurement, geometry, algebra
Explain why Postage Stamp Riddles from Riddle Math are an example of productive instruction.
It is productive instruction because it is practicing multiple outcomes. The students can each make their own riddles, or solve one made by the teacher and it helps them all to gain a better understanding of substitution and practicing computation. It has multiple objectives which include critical thinking (solving puzzles), language arts (reading and writing), algebra (substitution) and number and operations (addition and multiplication). Math and literacy.
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 If the tangram is 1, then each of the 7 tans (pieces )will be a fraction large triangles = 1/4 each medium triangle = 1/8 small triangles = 1/16 each square = 1/8 parallelogram = 1/8 If the tangram is 16, then each of the 7 tans (pieces )will be a whole number. large triangles =4 each medium triangle = 2 small triangles = 1 each square = 2 parallelogram = 2
What is the need for leap year? Give all the leap years between 1776 and 1812 in the Gregorian calendar, in the Julian calendar. How can you tell if any year is a leap year?
Leap year is needed to make up for the .25 not accounted for in 365 days. Leap years are divisible by four, not 1800 though because century years did not count unless divisible by 400
Translate 2,344 into Roman, Mayan, Babylonian and Egyptian.
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 ______ . . ______ ______ ______ .... Roman: MMCCCXLIV 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
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 is Descartes invention used to identify locations around the world? What is the point (0,0) in Chicago?
Only need to know: The coordinate plane helped create latitude and longitude; point (0,0) in Chicago is State and Madison;
Describe 3 types of transformations. Why is the importance of the point or vertex in transformations?
Rotation-spins around one vertex Translation-Every vertex is shifted equally Reflection-reflected around one point All need point or vertex to properly transform 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.
51. 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?
Saying "balance" instead of "equals" in an equation helps students see that math does work. 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
29. 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
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.
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] 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.
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? What is phi? Why is it important? How do we derive pi?
The real numbers include natural numbers or counting numbers, whole numbers, integers (need them for subtraction), rational numbers (fractions and repeating or terminating decimals, need them for division), and irrational numbers. Pi is C/d. Irrational numbers are important for square roots. Infinite number of irrational numbers?
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 (add each counting number) Square 1, 4, 9, 16, 25, 36, 49, 64 Pentagonal (1,5. 4 then add three to each) 1, 5, 12, 22, 35 ,51, 70, 92 Hexagonal (add 4) 1, 6, 15, 28, 45, 66, 91, 120 Heptagonal (ends in 6 and 1, = 6,11,26,21) 1, 7, 18, 34, 55, 81, 112, 148 Octagonal (add 6 to every difference) 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 (0,1,3,6,10,5, 28 then 21, 28)
Patterns with figurate polygonal numbers: • illustrate the first 4 terms of the following sequences (2-D); • give the next three terms (1-D); • write the rule for each; • give the formula for each; • show how the sum of square and triangular numbers are related; and • show how the sum of triangular numbers and oblong numbers are related. i. triangular (sum of consecutive counting numbers) ii. square (sum of consecutive odd numbers)
Triangular -1, 3, 6, 10, 15, 21, 28 -rule: add next counting number -formula: (n+1)n/2 Square - 1, 4, 9, 16, 25, 36, 49, 64 -rule: add the next odd number -formula: n^2 two consecutive triangular numbers equal the next square number 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...
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 sextillions, four hundred billion, seventy eight 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
51. Why do we need integers? Rational numbers? Irrational numbers? What are the Real Numbers?
We need integers for subtraction. We get negative numbers. 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)) Rational Numbers → Division Irrational Numbers → measurements (square root) Integer → subtraction (negative #s) Whole → subtraction (zero) Counting → adding and multiplying
What is the fundamental theorem of arithmetic? Give an example of it. What mathematician stated it?
every integer is a prime number or a product of prime number; factor trees; Gauss
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
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?
helps determine meaning of test scores, demonstrates variability - can see where students did relatively the same or if results were scattered A - 2% F - 2% B - 14% D - 14% C - 68% 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)
Give an example of productive computation for geometry.
measurement coincides with geometry algebra with formulas coincides with geometry
List ten math concepts that require multiplication.
measurement, place value, probability, prime factorization, slope, figurative numbers, order of operations, distributive principle, proportions, geometric sequences (exponential growth), algebra: substitution, cross multiplication, Pascal's triangle.
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 quiz answers are not dependent on the answer before it while in matching quizzes the answer is dependent on the answer before it Independent: FCP. 3x3x3 Dependent: 3x2x1 (only get that flavor once)
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.
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
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
Sketch a graph of an arithmetic sequence. How can you use the coordinate plane to teach multiplication? Why would this be considered productive teaching?
y = 2x x y 0 0 1 2 2 4 3 6 4 8 5 10 6 12 You can explain how the x when replaced by the 1 or 2 or 3 ect. being put into the equation to make it y=2(1) is the same as a normal multiplication equation, which is why 2(1)=2 and 2(4)=8 and so on. Its productive teaching because you're showing how two separate parts of math can be found in each other or how all math works together. See graph 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
