Module 1
What is Inductive Reasoning?
A type of reasoning accomplished by first observing patterns and then predicting answers for more complicated similar problems.
What is the Principles of school mathematics equity goal?
Excellence in mathematics education requires equity—high expectations and strong support for all students.
(a) Place the digits 12, 14, 15, 16, and 17 in the circles to make the sums across and vertically equal 43. (b) Is there more than one answer to part (a)?
(a) 15 on top, 14 on left, 16 on bottom, 17 on right, 12 in the middle (b) No. There are no different solutions other than interchanging the numbers on the left and right or top to bottom or rotating the figure
NCTM presents itself as?
"the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research".
Study this sequence of numbers: 3, 4, 7, 11, 18, 29, 47, 76. Note that 3+4=7, 4+7=11, 7+11=18, and so on. Use the same rule to complete the sequences in parts (a)-(e). (a) 1, 2, 3, ___, ___, ___, ___ (b) 3, ___, 11, ___, ___, ___, ___ (c) 4, ___, ___, 14, ___, ___, ___ (d) 2, ___, ___, ___, ___, 36 (e) 4, ___, ___, ___, ___, 22
(a) 1, 2, 3, 5, 8, 13, 21 (b) 3, 8, 11, 19, 30, 49, 79 (c) 4, 5, 9, 14, 23, 37, 60 (d) 2, 6, 8, 14, 22, 36 (e) 4, 2, 6, 8, 14, 22
A rectangular table seats 10 people: 1 person on each end and 4 on each of the longer sides. Thus, two tables placed end to end seat 18 people. (a) How many people can be seated if n tables are placed in a line end to end? (b) How many tables, set end to end, are required to seat 140 people?
(a) The total number of seats available is 8n + 2. (b) 18 tables are needed to seat 140 people
(a) How many red playing cards must be drawn from a single deck of cards to be sure that at least two are the same suit question mark (b) How many red playing cards must be drawn from a single deck of cards to be sure that at least eleven are diamonds or at least eleven are hearts?
(a) There must be at least 3 red playing cards drawn from a single deck of cards to ensure there are at least 2 of the same suit. (b) 21 red playing cards drawn from a single deck of cards would result in 11 being in one pigeonhole.
Use counting to determine the whole number that corresponds to the cardinality of these sets. (a) A={x|x is an element of N and 26<x< or =34} (b) B={x|x is an element of N and x + 5=x}
(a) n(A)=8 (b) n(B)=0
What is Conditional Statement?
- the statement "if p then q" symbolized by pImpliesq.
The equations given below illustrate a suspected number pattern. Determine the next equation. 1 = 1^2 1 + 2 + 1 = 2^2 1 + 2 + 3 plus 2 plus 1 = 3^2 1 + 2 + 3 + 4 + 3 + 2 + 1 = 4^2
1 + 2 +3 +4 +5 +4 +3 +2 +1 = 5^2
How many terms are there in the following arithmetic progression? 1, 9, 17, ..., 89
12 terms in the given progression.
Lola is thinking of a number. If you multiply her number by 92, add 9, and divide by 3, you obtain 463. What is Lola's number? Solve this problem by working backwards.
15
A survey of 1100 households revealed that 300 had only a TV, 200 had only a computer, and 100 had neither a TV nor a computer. How many households have both a TV and a computer?
500 households have both a TV and a computer
The spring concert at a certain high school sold 181 tickets. Students were charged $3 each and adults $5 each. The income from the sale of tickets was $735. How many students and how many adults bought tickets? Use algebra to solve this problem.
85 students bought tickets, and 96 adults bought tickets.
The spring concert at a certain high school sold 165 tickets. Students were charged $3 each and adults $5 each. The income from the sale of tickets was $639. How many students and how many adults bought tickets?
93 students and 72 adults bought tickets.
What is Set?
A collection of particular things.
What is Venn Diagram?
A diagram used to illustrate relationships among sets.
What is Deductive Reasoning?
A formal structure based on a set of axioms and a set of undefined terms.
What is Algebra?
A generalization of arithmetic. Letters called variables are used to denote numbers, which are related by laws that hold, or are assumed for any members in the set.
What is Negation?
A logical connective that changes the truth value of a given statement.
What is Associative Property?
A property of grouping that applies to addition and multiplication. If a, b and c are real numbers, then (a + b) + c = a + (b + c) and (ab)c = a(bc). If you change the grouping, not the order, it does not change the outcome.
What is Commutative Property?
A property of order that applies to addition and multiplication. If a and b are real numbers, then a + b = b + a and ab = ba. If you change the order it does not change the outcome.
What is Distributive Property?
A property which uses multiplication over addition. If a, b, and c are real numbers, then a(b + c) = ab + ac. It is the only operation which uses multiplication and addition together.
What is Set-Builder Notation?
A technical notation for defining a set. For example, {a|a is an integer, 5 < a < 100}, means "the set of all elements a such that a is an integer between 5 and 100."
What is Division?
A/b = x is a divided by b equals x and means a = bx.
The Curriculum Focal Points helped create what in 2010?
Common Core State Standards
What is the Principles of school mathematics teaching goal?
Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well.
Name each of the six Principles for school mathematics
Equity, curriculum, teaching, learning, assessment, and technology.
Five runners, Andy, Beth, Caleb, Darnell, and Ella are in a one-mile race. Andy finished 7 seconds before Caleb. Caleb finished 2 seconds before Beth. Beth finished 6 seconds after Darnell. Ella finished 8 seconds after Darnell. In which order did the runners finish the race?
First Place Andy Second Place Darnell Third Place Caleb Fourth Place Beth Fifth Place Ella
What is Multiplication?
For bNot Equal0, a x b means b + b + b +...+b (a addends).
What is Contrapositive Statement?
For the implication pImpliesq, the contrapositive is ~q Implies~p.
What is Converse Statement?
For the implication pImpliesq, the converse is qImpliesp.
What is Inverse Statement?
For the implication pImpliesq, the inverse is ~pImplies~q.
Bob can cut through a log in four minutes. How long will it take Bob to cut a 30-foot log into 6-foot sections?
It will take Bob 16 minutes to cut a 30-foot log into 6-foot sections.
Peter and Jill each worked a different number of days, but earned the same amount of money. Use these clues to determine how many days each worked. Peter earned $40 a day. Jill earned $50 a day. Peter worked six more days than Jill.
Jill worked 24 days and Peter worked 30 days.
To get an A in math class, John Curtis must average at least 93 on 6 tests. John's scores on the first five tests were 95, 100, 86, 93, and 92. What is the lowest score that he can get on the last test and still get an A?
John must score at least 90 on the last test to get an Upper A.
Moe, Joe, and Hiram are brothers. One day, in some haste, they left home with each one wearing the watch and shirt of one of the others. Joe was wearing Hiram's shirt and Moe's watch. Whose watch and shirt was each one wearing?
Moe was wearing Joe's shirt and Hiram's watch and Hiram was wearing Moe's shirt and Joe's watch.
What is the Principles of school mathematics curriculum goal?
More than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades
Mr. Akika has 34 21-cent and 34-cent stamps all told. The stamps are worth $10.00. How many of each kind of stamp does he have?
Mr. Akika has 12 21-cent stamps and 22 34 dash cent stamps in his possession.
Mr. Akika has 46 24-cent and 28-cent stamps all told. The stamps are worth $12.12. How many of each kind of stamp does he have?
Mr. Akika has 19 24-cent stamps and 27 28 dash cent stamps in his possession.
Joni has two U.S. coins in her pocket. One of the coins is not a penny. If the total value of the coins is 6 cents, what are the two coins?
Nickel and penny
Are the sets {d, t, w} and {d, t, 1, m} equivalent?
No
Are the sets {0} and empty set equivalent?
No
What is Factor?
Number or letters which are being multiplied.
What is Addition?
One of the fundamental undefined operations applied to the set of counting numbers.
What is the Principles of school mathematics learning goal?
Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.
List the five content standards.
number and operations, algebra, geometry, measurement, and data analysis and probability.
What is Divisor?
The dividing number in a division problem.
How many posts does it take to support a straight fence 150 feet long if a post is placed every 15 feet?
The fence needs 11 posts.
What is Principles & Standards for School Mathematics?
The mathematical understanding, knowledge, and skills that students should acquire from Pre-K through grade 12.
What is Identity of addition?
The number 0 is the identity of addition
What is Identity of multiplication?
The number 1 is the identity of multiplication
What is Minuend?
The number before the minus sign or the top number in a subtraction problem.
What is Dividend?
The number being divided in a division problem.
What is Subtrahend?
The number following the minus sign or the bottom number in a subtraction problem.
A number is tripled and then increased by five. The result is seventy dash seven. What is the original number?
The number is 24.
Alice is thinking of a number. The number is odd. It is more than 1 but less than 100. It is greater than 40. It is less than 5 x 11. The sum of its digits is 13. It is evenly divisible by 7. What is the number? Was all the information needed?
The number is 49 and not all the information was needed.
Seven people come to a party and shake hands with each other. If each person shakes the hand of every other person, how many handshakes occur?
The number of handshakes is 21.
What is Quotient?
The number which is the result of a division problem.
What is Product?
The number which results in a multiplication problem.
Ms. Gardner is thinking of a number. The number is not odd. The sum of its digits is divisible by 5. The number is a multiple of 13. It is greater than 4 x 5. It is a multiple of 6. It is less than 7 x 8 + 23. What is the number? Is more than one answer possible?
The only possible number is 78.
What is Subtraction?
The operation of it is defined by: a - b = x, means b + x = a
What is NCTM (National Council for Teachers of Mathematics)?
The organization for the curriculum standards for teaching mathematics.
What is Difference?
The result of a subtraction problem.
A rectangle is twice as long as it is wide and has the same perimeter as a square whose area is 1 square foot larger than that of the rectangle. What are the dimensions of both the rectangle and the square?
The width is 2 feet, and the length is 4 feet; the length of the sides is 3 feet.
A rectangle is twice as long as it is wide and has the same perimeter as a square whose area is 9 square feet larger than that of the rectangle. What are the dimensions of both the rectangle and the square?
The width is 6 feet, and the length is 12 feet, and the length of the sides is 9 feet.
How much dirt is there in a hole 13 meters long, 5 meters wide, and 3 meters deep?
There are 0 cubic meters of dirt in the hole.
Old MacDonald has 100 chickens and goats in the barnyard. Altogether, there are 258 feet. How many chickens and how many goats are in the barnyard?
There are 71 chickens and 29 goats in the barnyard.
What is Addends?
Two or more numbers to be added.
Are the sets {1, 2, 3, 4, 5} and {x|x is a letter in the phrase "I LIKE MIKE"} equivalent?
Yes
{z, e, r, o} and {f, o, u, r} equivalent?
Yes
Consider this mathematical machine. input--> add 3 --> multiply by 9 --> subtracted 31 --> divided by 2 --> output a) What number would you have to use as input if you wanted 61 as the output? b) What would you have to input to obtain an output of 70?
a) 14 b) 16
The given diagram should hold the same pattern when completed. Here, only the sums are given. {19} c a {16} b {23}
c=6 a=13 b=10
The controversial 1989 NCTM Standards called for more emphasis on
conceptual understanding and problem solving informed by a constructionist understanding of how children learn.
What is the Principles of school mathematics technology goal?
essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning.
List the five process standards.
problem solving, reasoning and proof, connections, and representation.
What is the Principles of school mathematics assessment goal?
should support the learning of important mathematics and furnish useful information to both teachers and students.
A child has nine coins with a total value of 97 cents. What coins does the child have?
the child has 2 pennies, 1 nickel, 4 dimes, and 2 quarters.
Lisa is thinking of a number. If you multiply it by 7 and subtract 12, you get 30. Find this number.
the number Lisa is thinking of is 6.
If you multiply me by 9 and subtract 47, the result is 511. Who am I?
the number is 62.
Who am I? If you multiply me by -7 and add 25, the result is -143.
the unknown number is 24
In 2006 Curriculum Focal identified
what it believes to be the most important mathematical topics for each grade level, including the related ideas, concepts, skills, and procedures that form the foundation for understanding and lasting learning.
Predict the next three numbers in the pattern. 4, 6, 8, 10, 12, __, __, __
4, 6, 8, 10, 12, 14, 16, 18
A rectangle is four times as long as it is wide and has the same perimeter as a square whose area is 9 square feet larger than that of the rectangle. What are the dimensions of both the rectangle and the square?
The width of the rectangle is 2 feet, and the length of the rectangle is 8 feet; the length of the sides of the square is 5 feet.
