MAE Midterm

What are Polya's four principle to problem solving?

1. Understand the Problem 2. Devise a Plan 3. Carry out the Plan 4. Look Back - Check your work

Set

A collection of objects

Closed Figure

A figure that begins and ends at the same point

Ray

A line with a start point but no end point (it goes to infinity)

zero pair

A pair of numbers whose sum is zero; used to illustrate addition and subtraction problems with positive and negative integers

Vertex

A point where two or more straight lines meet. A corner; the endpoint

Trapezoid

A quadrilateral that has exactly one pair of opposites sides that are parallel

Infinite Set

A set that has an unlimited number of elements

Open Figure

A shape made of line segments, but there is at least one line segment that isn't connected to anything at one of its endpoints.

Compensation

A strategy used in mental math in which you change one addend to a multiple of ten and then adjust the other addend to keep the balance

Right Trapezoid

A trapezoid that has two adjacent right angles

Obtuse Triangle

A triangle in which one of its angles is an obtuse angle

Right Triangle

A triangle in which one of the angles is a right angle

Equiangular Triangle

A triangle with three congruent angles; these are also equilateral triangles

Equilateral Triangle

A triangle with three congruent sides; these are also equiangular triangles

Reflective Symmetry

A type of symmetry where one half is the reflection of the other half; both halves would match exactly if the image was folded in half.

Reflexive Property

A whole number equals itself EX: a = a

Acute Triangle

All three of the angles are acute angles

Obtuse Angle

Angle greater than 90° but less than 180°

Acute Angle

Angle less than 90°

Vertical Angles

Angles opposite each other when two lines cross. They are always congruent.

Concave vs. Convex

Concave: Curved inwards Convex: Curved outwards.

"Borrowing" and "carrying" are terms currently used when explaining regrouping. TRUE/FALSE

FALSE

The complete set of digits - {1, 2, 3, 4, 5, 6, 7, 8, 9} TRUE/FALSE

FALSE

The correct expression for the representation below is 4 x 3 XXXX XXXX XXXX TRUE/FALSE

FALSE

The position of the unknown in a simple equation does not effect the cognitive complexity (difficulty). TRUE/FALSE

FALSE

You can divide by zero. TRUE/FALSE

FALSE

TRUE/FALSE If a = b and b = c, then c = b and b = a best represents the transitive property.

FALSE. Transitive Property: If a = b and b = c, then a = c

SSS Congruence Postulate

If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent

ASA Congruence Postulate

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

AA Similarity Postulate

If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

SAS Similarity Postulate

If two side lengths of one triangle are proportional to two side lengths of another triangle, and the included angles are congruent, then the two triangles are similar.

SAS Congruence Postulate

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent

Line

In geometry a line is straight (no curves), has no thickness, and extends in both directions without end (infinitely).

Solve a Similar, but Simpler, Problem

Look for easier numbers, patterns, and relationships. Then use previously learned concepts to solve the original problem.

Element

Members of a set

Fractal

Patterns of geometric shapes which have continually smaller similar iterations

Corresponding Sides

Sides that have the same relative positions in geometric figures

An algorithm is a step-by-step procedure used to solve a mathematical problem. TRUE/FALSE

TRUE

Any number over itself (e.g. 6/6) except 0 x a = a would best correctly identified as the Multiplication Identity Element. TRUE/FALSE

TRUE

If the remainder in the division problem you just finished is equal to or greater than the divisor then you should go back and check your work. TRUE/FALSE

TRUE

In a simple equation (EX: 4 x 9 = 36), the product can be determined with repeated addition. TRUE/FALSE

TRUE

Instant recognition of pattern to determine cardinality is called subitizing. TRUE/FALSE

TRUE

a x (b + c) = (a x b) + (a x c) is a representation of the distributive property of multiplication over addition TRUE/FALSE

TRUE

Line Segment

The part of a line that connects two points. It has definite end points. Adding the word "segment" is important, because a line normally extends in both directions without end.

Zero Multiplication Property

The product of 0 and any whole number is 0.

Multiplicative Identity Property

The product of any whole number and the multiplicative identity, 1, is the whole number.

Commutative Property of Multiplication

The product of two whole numbers does not depend on their order.

Associative Property of Multiplication

The product of whole numbers does not depend on how the factors are grouped.

Hypotenuse

The side opposite the right angle in a right-angled triangle; it is also the longest side of the right-angled triangle

Additive Identity Property

The sum of any whole number and the additive identity, 0, is the whole number.

Commutative Property of Addition

The sum of two whole numbers does not depend on how the addends are ordered.

Associative Property of Addition

The sum of whole numbers does not depend on how the addends are grouped.

Isosceles Triangle

Triangle with at least two sides that are congruent

Scalene Triangle

Triangle with three sides of different lengths; no congruent sides

An array is a square or rectangular arrangement of like objects TRUE/FALSE

True

Equal set

Two sets are equal if an only if they have the same elements

Transitive Property

Two whole numbers that are equal to the same whole number are equal to each other EX: if a = b and b = c, then a = c.

Rotational (Point) Symmetry

When a rotation of 180° (a half-turn) or less around a point in the figure produces an image that fits exactly on the original figure

Corresponding Angles

When two lines are crossed by another line (the transversal), the angles in matching corners are called corresponding angles

Expanded Notation

Writing a number to show the value of each digit; it is shown as a sum of each digit multiplied by its matching place value. EX: 4,265 = 4x1,000 + 2x100 + 6x10 + 5x1

Transversal line

a line that crosses at least two other lines

Rhombus

a parallelogram that has four congruent sides

Guess, Check, and Revise

a problem-solving strategy in which students begin by making a reasonable guess about the solution. After checking the guess in the statement of the problem, students adjust the guess by making a new, refined guess. T

Looking for a Pattern

a problem-solving strategy in which students look for patterns in the given information. Students look for items or numbers that are repeated, or for a series of events that repeats.

Drawing a Diagram

a problem-solving strategy in which students make a visual representation of a problem. By representing units of measurement and other objects visually, students can begin to think about the problem mathematically.

Working Backwards

a problem-solving strategy involving a sequence of events in which students are given a final event and are asked to find an original event. Students begin at the end, with the final event, and work through a process in reverse order to establish what happened in the original situation.

Making a Table or List

a problem-solving strategy that students use to solve mathematical word problems by writing the given information in an organized format. This strategy allows students to discover relationships and patterns. It encourages students to organize information in a logical way and to look critically at the information to find patterns and develop a solution.

Rectangle

a quadrilateral that has four right angles; opposite sides of a rectangle are congruent

Parallelogram

a quadrilateral that has two pairs of opposite sides that are parallel; opposite angles of parallelograms are congruent

Square

a rectangle that has four congruent sides

Finite Set

a set that has a limited number of elements

Numeral

a symbol used to represent a number EX: "3," "three," and "III"

regrouping

a term used to describe the process of changing groups of ones into tens to make adding and subtracting easier

Reflection

a transformation in which a figure flips in a line; a reflection creates a mirror image of the original figure

Dilations

a transformation in which a figure is made larger or smaller with respect to a fixed point; the original and the image are similar

Translation

a transformation in which a figure slides but does not turn; every point of the figure moves the same distance and in the same direction

Rotation

a transformation in which the figure turns about a point. The original figure and its image are congruent.

Glide Reflection

a transformation that is composed of a translation (the glide) followed by a reflection in a line that is parallel to the direction of the translation

Isosceles Trapezoid

a trapezoid whose nonparallel sides are congruent

What is the Least Common Multiple (LCM) of 63 and 42? a. 126 b. 2646 c. 252 d. 9

a. 126

Which of the below would be an accurate use of a Zero Pair Strategy (compensation) to avoid having to do regrouping? a. 42-19=42+(+1)-19(-1) b. 50+13=50+(-1)+13+(+1) c. 64-39=64(-1)-39+(+1) d. 49+17=49+(+1)+17+(+1)

a. 42-19=42+(+1)-19(-1)

(a x b) x c = a x (b x c) is correctly identified as: a. Associative Property of Multiplication b. Transitive Property of Multiplication c. Commutative Property of Multiplication d. Distributive Property of Multiplication

a. Associative Property of Multiplication

Which of the following is not true about using Number Bonds for Addition/Subtraction? a. They make use of the Associative Property b. They are good for generating fact families c. They are a part/part/whole structure d. They make use of the Commutative Property

a. They make use of the Associative Property

Whole numbers plus their negative counterparts are best identified as: a. integers b. nominal numbers c. counting numbers d. natural numbers

a. integers

Straight Angle

angle equaling 180°

Right Angle

angle equaling 90°

Which of the following best represents the Multiplication Identity Property? a. (a x 1) = (1 x a) b. (a x 1) = a c. (a + 0) = a d. a x (b + c) = (a x b) + (a x c)

b. (a x 1) = a

(3) + (-3) = 0 would be described as _______________. a. A Missing Addend b. An Inverse Operation c. The Additive Identity Property d. The Associative Property

b. An Inverse Operation

One of the goals of learning your basic facts for all operations is to reduce the number of potential facts students have to practice. Which of the below would be the least helpful to do this? a. Additive Identity Property b. Associative Property c. Multiplication Identity Property d. Commutative Property of both Addition and Multiplication

b. Associative Property

4 + 7 = ________ could best be solved by using which strategy? a. Counting Up b. Counting On c. Counting Around d. Counting Back

b. Counting On

Anne has 21 pies. She wants to give them out in equal groups of 3 pies. How many friends will get pies? This is an example of which type of problem structure? a. Comparison Division b. Measurement Division (Number of Sets Unknown) c. Partition Division (Set Size Unknown) d. Equal Groups Multiplication

b. Measurement Division (Number of Sets Unknown)

John has 18 marbles. He wants to distribute them evenly to 3 friends. How many marbles will each friend get? This is an example of which type of problem structure? a. Equal Groups Multiplication b. Partition Division (Set Size Unknown) c. Measurement Division (Number of Sets Unknown) d. Comparison Division

b. Partition Division (Set Size Unknown)

Which growing pattern does this sequence represent? {2, 7, 12, 17, 22, 27} a. geometric b. arithmetic c. special or other d. repeating

b. arithmetic

The number you divide in a division problem is known as the ________________. a. remainder b. dividend c. divisor d. quotient

b. dividend

There are 54mg of caffeine in a 12oz. can of Mountain Dew. There are 29mg of caffeine in a 12oz can of Coke. How many more milligrams of caffeine does Mountain Dew have than Coke? Which information would you need to find out to answer the question? a. the Multiplier b. the Difference c. the Smaller Set d. the Larger Set

b. the Difference

Decomposition

breaking a number apart by its place value; for example, 349 = 300+40+9

What is the greatest common factor of 77 and 91? a. 13 b. 91 is prime so 1 is the greatest common factor c. 7 d. 1001

c. 7

Which of the following is not one of Polya's four steps to problem solving? a. Understand the Problem b. Solve the Problem c. Analyze the Problem d. Make a plan for solving the Problem

c. Analyze the Problem

In a concept building lesson using Base Ten Blocks to explain subtraction and regrouping, what is the first step one should do with this example: 52-18? a. Break out your calculator b. Borrow from the 5 c. Decompose (or trade) one of the tens rods into 10 unit cubes to make 4 tens and 12 ones d. Cross out the 5 and make it a 4

c. Decompose (or trade) one of the tens rods into 10 unit cubes to make 4 tens and 12 ones

14x - 8 would be most correctly identified as a(n): a. coefficient b. variable c. expression d. equation

c. expression

Which growing pattern does this sequence represent {2, 8, 32, 128, 512, 2048...}? a. repeating b. arithmetic c. geometric d. special or other

c. geometric

Set 1 = {a, c, d, e, b, f} Set 2 = {b, c, a, e, d, f} Set 3 = {a, b, z, x, y, q} Set 4 = {a, g, z, b, r} Which of the below statements is true about these sets? a. sets 2 and 3 are equal; sets 1 and 3 are equivalent b. sets 1 and 4 are not equivalent; sets 2 and 3 are equal c. sets 1 and 2 are both equal and equivalent; sets 3 and 4 are neither equal nor equivalent d. sets 1 and 3 are both equal and equivalent; sets 1 and 4 are neither equal nor equivalent

c. sets 1 and 2 are both equal and equivalent; sets 3 and 4 are neither equal nor equivalent

Choose the most accurate and complete answer. 62,523,576 could be divided evenly by: a. 2, 3, 4, 9, but not 8 or 6 b. 0, 2, 3, 4, 6, and 9 c. 2, 3, and 9, but not 4, 6, or 8 d. 2, 3, 4, 6, 8, and 9

d. 2, 3, 4, 6, 8, and 9

The absolute value of |-8| is ________? a. Both 8 and -8 b. 16 c. -8 d. 8

d. 8

Partial Products is MOST directly related to which of the following properties? a. Associative Property b. Multiplication Identity Property c. Commutative Property d. Distributive Property

d. Distributive Property

Null set

denoted by { } or ∅, is empty and has no elements.

Interior Angles

formed by the vertex and the two sides that have the vertex as an endpoint; also called a vertex angle

Congruent

identical in form; coinciding exactly when superimposed

SSS Similarity Postulate

if all three sides of one triangle are proportional to the corresponding side lengths of another triangle, then the two triangles are similar

Making a Graph

problem-solving strategy in which students use a graph to represent information visually. Representing information using pictures makes it easier to visualize results, identify relationships, describe trends, make comparisons, and draw conclusions.

Equivalent Set

sets that have different elements but the same amount of elements.

Equal Set

sets that have the exact same elements in them, even though they could be out of order

Cardinality of a Set

the number of unique elements of the set

Symmetric Property

the order of the equality of whole numbers does not matter EX: if a = b, then b = a

Whole Numbers

the set of all cardinalities of finite sets {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,....}

Natural Numbers

the set of all cardinalities of nonempty, finite sets; also called the set of counting numbers {1, 2, 3, 4, 5, 6, 7, 8, 9....}

Complementary Angles

two angles are complementary when the sum of their measures is 90°

Supplementary Angles

two angles are supplementary when the sum of their measures is 180°

Similar

two shapes are similar when the only difference is size (and possibly the need to move, turn, or flip one around).