Math for Elementary Teachers Midterm

What is the Additive Identity?

When a number is added to 0, the result is that number: 5+0 = 0; 0+2 = 2

What is the Additive Inverse?

When a number's inverse is added to it, it yields 0: -a + a = 0.

Define "opposites" as a necessary conceptual model for operations.

When pairs of numbers whose absolute values are equal, they are opposites or negatives of each other: the opposite of +6 is -6; the negative of -6 is +6. Adding any integer and its opposite, the result is zero.

What are the Whole Number Properties?

Whole Number - any finite set that defines a cardinality and does not include fractions, decimals, place values: W = {0,1,2,3...}

Define "integer" without using the word in the definition.

Whole values, including 0, negatives and positives, that are not fractions, or decimals.

What is meant by a set's Cardinality?

"Cardinality" is the number of elements within a set; expressed in terms of a whole number: n (A) = a whole #

What is the relationship between rational numbers and fractions?

***A fraction is a rational # sometimes; a rational # is a fraction always. A rational number's value can be expressed as the quotient or ratio of two integers (Integers are natural whole numbers including 0 & negatives); a fraction is a number whose value can be expressed as the quotient or ratio of any two *numbers*.

What are the five meanings of multiplication?

*Groups of (so many "groups of" a quantity); *Rate problems that involve finding a total given a number of items and a rate affecting them; *Comparison situations involving the determination of the size of a set and how many ties larger the unknown is; *A combination problem that involves finding all the ways of combining the items of one type with those of another (Cartesian Example - the wardrobe question) *Area situations that entail finding the area of a rectangle given its length and width.

What are the five NCTM content standards?

*Number & Operations, *Algebra, *Geometry, Measurement, *Data Analysis and Probability

What are the five NCTM process standards?

*Problem Solving, * Reasoning and Proof, *Communication, *Connections, *Representation

What are Polya's four steps to problem solving

*Understand the Problem, *Devise a Plan, *Carry Out and Monitor the Plan, *Look Back

What digits are allowed in a base?

0 - (b-1), so if the base is 8, digits 0-7 are allowed in a base.

What are the Five Strands of Mathematical Proficiency?

1. Conceptual Understanding; 2. Procedural Fluency; 3. Strategic Competence; 4. Adaptive Reasoning, 5. Productive Disposition

How do you convert repeating decimals to fractions?

1. Let r = the number with repeating decimal; 2. Multiply r by 10 - 10,000, etc., until you find two results whose decimal places match; 3. Subtract those four respective and equal equations from each other for the answer, reducing if possible; 4. Divide out from the r to get the r by itself to check your work: Steps 1 & 2 (Represent & Find equivalent decimal places). (1)r = 3.987... 10r = 39.87... 100r = 398.7... 1000r = 3987.7... Step 3 (Subtract). 1000r = 3987.7... -100r = 398.7... ---------------- 900r = 3589.0 >>>>>>>> your answer from repeating decimal to fraction form is: 3589/900 Step 4 (Check). 900r = 3589.0 ÷900 ÷900 ----------------- r = 3.987...

In Number Theory, what are the Divisibility Rules?

2 if the digit in the ones place is even; 3 if the sum of the digits is divisible by 3; 4 add the ones digit to twice the 10s digit and the result is divisible by 4: (72-> 2+ (7•2) = 16), if the last two digits as a whole is divisible by 4 (i.e.: 512); If the tens place is even and the ones place is 0, 4, or 8, if the tens is odd and the ones is 2 or 6 5 if the ones place is 0 or 5 6 if it is divisible by 2 and 3 under those rules 8 if the last three digits is divisible by 8 (1,024 -> 024 is divisible by 8, so 1,024 is also divisible by 8) 9 if the sum of the digits is divisible by 9: 1269 -> 1+2+6+9=18 ->1+8=9 and 9 is divisible by 9, so 1269 is divisible by 9, too. 10 if the last digit is 0

How do you find the LCM?

2 ways: 1.Using the chart created above, represent every number from each column once, even those without 1:1 match. Multiply all of these numbers together and that's your LCM. 2. write your given numbers in a row and draw an upside down long division sign underneath. Divide out common multiples from each of the numbers and record the answers beneath those numbers, bringing any dividend straight down that was not divisible by your divisor. Repeat this step until all prime numbers are achieved. Now multiply all of the numbers you used as divisors by all of the numbers that resulted in your prime quotients. That's your LCM

Show multiplication as repeated addition.

3 • 2 = (3+3) = 6; 8 • 3 = (8+8+8) = 24; 4 • 5 = (4+4+4+4+4) = 20

What is meant by the "Union" of sets?

A "Union" contains all elements of all sets.

What is an Equivalence Set?

A set is "Equivalent" iff they have the same number of elements; aka 1:1 correspondence. The set of even numbers is equivalent to the set of Whole numbers.

What is the Closure Property?

A set of numbers is said to be closed for a specific mathematical operation if the result obtained when the operation is performed on any two numbers in the set is itself a member of the set: + Adding two whole numbers will always result in another whole number, therefore, addition process has the closure property. - Subtracting two whole numbers doesn't always result in another whole number (14-21 = - 7); therefore, subtraction does not possess the closure property. x Multiplying two whole numbers will always yield another whole number: Multiplication of whole numbers process has the closure property. ÷ Dividing two whole numbers does not always yield a whole number, therefore, division of whole numbers does not have the closure property: 2/8 = .25

What is the "Intersection" of set?

A set's "Intersection" contains only those elements that are shared between sets.

Explain the various Computational-Estimation techniques.

Addition Estimations: 75,145 + 34,135 + 55,124 = 16404 or Round: 80,000 + 30,000 + 60,000 = 170,000 Compatible: 75+35+55=110+55=165; 32,425+31,456+34,234 = 32+31+34=97,000 = 98,115 Redefined rounding: if the digit to the left of the 5 is odd, round up, if the digit to the right of the 5 is even, round down. Subtraction Estimation: Multiplication Estimation: (1) 78m/wk 52 wks/yr -> 70•50=3500; 80•60=4800; answer is somewhere between 3500 & 4800 miles this year (rounding up = upper bound; rounding down = lower bound) (2) Round one number up and the other (3) Use expanded form and estimate the sum of the four partial products: 47,752•6 -> 50,000•6 = 300,000 - or - double & halve: 48•6 = 96•3=288 or 288,000 Division Estimation: missing-factor model -> 12 times what is closest to 91? 7 1/2 Round up 489/19 ≈ 500/20 = 50/2 ≈ 25 mpg Rounding both numbers up cancels effect (in multiplication, round one number up, and the other down)

Why can't you divide by zero?

Any non-number divided by zero is undefined because it results in a zero product property with no solution, or results in infinite solutions. You can't divvy 8 into 0 subsets. Proof: Iff x÷0=k, then k•0=x, which is untrue because k•0=0.

What is Deductive Reasoning?

Deductive Reasoning is the act of reaching a conclusion based on hypothesis.

What are the two types of division?

Divvy up and partition or measure out....*How many are in each subset *How many subsets are there

What are Polya's problem-solving strategies?

Draw Pictures, Use Variables with Helpful Names, Be Systematic, Solve a simpler version of the problem, Guess and Check/Trial and Error/Guess and Test, Look for Patterns, Make a List

3 Standards for assessing alorithms

Efficient, mathematically valid, generalizable

Which type of knowledge is emphasized in traditional mathematics classrooms? Why is this problematic?

Emphasis on passive learning through rote memorization by way of teacher lecture with a few word problems at the end of a chapter to apply to real-world is the traditional math classroom. Student's aren't taught "why" they are performing certain functions, and can't apply them to real-world issues, and each student's comprehension isn't the same from student to student. Today students work in groups to problem-solve, using multiple resources, real-life application discussions, communicating ideas and understanding.

How do you convert from other bases to base 10?

Example: convert 52301 base 8 to base 10: Place values: 8ˆ4 8ˆ3 8ˆ2 8ˆ1 8ˆ0 4096 512 64 8 1 Face values: 5 2 3 0 1(v8 - or b8) now multiply the digit by the place value (5•4096) + (2•512) + (3•64) + (0•8) + (1•1) = 20480 + 1024 + 192 + 0 + 1 = 21697, therefore 5230 b8 = 21697 b10

What is the Identity Property?

Identity Properties vary from operation to operation: the Identity property of adding 0 means that the number added to 0 doesn't change; but 0 multiplied by another number will always equal 0. The identity/multiplicative property of multiplying any other number by 1 means that the number multiplied by 1 doesn't change, it's always going to be that number.

What is meant by a 1-1 set?

In a 1:1 set, every element of one set corresponds with exactly one element of another set, and vice versa. All whole and even infinite sets are always 1:1 equivalent.

What is Inductive Reasoning?

Inductive reasoning is the at of looking for regularity in found pattern.

Explain the various Mental-Computation Techniques.

Mental Addition: adding left to right, increasing by place as necessary: 39+57 = (30+50) + (9+7)=80+16=96; using compensation: 39+57=40+56=96; Break and Bridge: 68+35 = (68+30) +5 = 98+5=103 Compatible numbers: 186+125= (180+120) + (6+5) = 300+11=311 Mental Subtraction: adding up: 65-28=28+30 =58, 58+7=65, 30+7=37 -or- 28+2=30, 30+35=65, 35+2=37 compensation: 62-39 -> 43-40=23; compatible numbers: 132-36 -> 36+64=100 -> 100+32=132, so the answer is 64+32=96 Break the second number apart: 54-27 -> 64-7=57 -> 57-20=37 Subtract each place from l-r & combine the differences: 64-27 -> 60-20=40 -> 4-7=-3 ->40-3=37 Adding up 64-27 -> 27+30=57 -> 57+7=64, the answer is 30+7 = 37 Mental Multiplication: Mental Division: Canceling: 6000/20 = 600/2 ÷ 10/10 Equivalent Fractions: 6000/20->(60•100)/20 = 3•100 (after reducing/canceling) = 300 Multiply both numerator & denominator by 2: 690/10=69 Missing factor: 252/4 -> 4•?=252, then add as needed Scaffolding Algorithm: 4356÷6

Model addition and subtraction with the counters.

Red = positive; Blue = negative -3 - (-4) Start with -3: Blue Blue Blue, but then you're supposed to take away 4 blue ones, but you don't have four blue ones, yet, so: since a negative and a positive of equal amounts will cancel out to zero, so for every negative blue you need to use, you have to use that same amount in positive reds. add one blue and one red to the three blue. now you have 4 blues and one red, meaning you now have -4 blues that you can remove and have the positive 1 red remaining. 3 - (-5) Red Red Red, add 5 sets of red and blue, remove 5 blue and you're left with 8 red positive -3 + 5 blue blue blue and red red red red red, cancel out using 1:1 red:blue, and the remaining 2 red positives is the answer. -6 + 4 blue blue blue blue blue blue and red red red red, cancel out blues and red 1:1, negative 2 blue remain.

How do you find rational numbers between two given rational numbers without converting to decimals?

Take the average of the given rational numbers: 4/7 & 1 1/4 1. Add all the elements and then divide by the number of elements that have been added together 1/2 (4/7 + 1 1/4) 2. Convert Mixed fractions into Improper fractions and Find the LCD in order to add the given rational number elements: 1/2 (4/7 + 5/4) -> 1/2 (4/7•4/4 + 5/4 • 7/7) -> 1/2 (16/28 + 35/28) -> 1/2( 51/28) = 51/56 is a rational number between the two given numbers.

What is the Commutative Property?

The "Commutative" Property is the ability to perform operations within any set, or sets, of numbers in any order and come out with the same answer. Addition and multiplication are commutative and associative.

Define "absolute value" and explain why it can never be a positive number.

The "absolute value" of a number is the number of places that number resides from zero on the number line. It can never be a negative number because the "absolute value" is not a place holder, but a representative number indicating space across the number line in either direction.

What is the Multiplicative Inverse?

The Multiplicative Inverse is the reciprocal of a number which when multiplied yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a

What is the Associative Property?

The associative property means that it doesn't matter how you group or regroup your sets in order to operate on them, the answer will always be the same.

What is the Distributive Property?

The distributive property is the process of distributing a number across the sets of numbers in order to perform some operation on them: Distributive Property over addition a(b+c); Distributive Property over subtraction a(b-c)

What are the different place values for a base?

The place values of a numeral are the values of the powers of the base. Extreme right is 1 or nˆ0, then from right to left, nˆ1, nˆ2, nˆ3, etc.

How do you find the GCF?

Using prime factorization - a factor tree, breaking down each number by it's multiplicative equivalents and continuing to break down each of those numbers until you reach the prime of each extension. Repeat for each given number set. Once all of the primes are identified, list them for each given number set in a column, lining up the matching numbers. For each number that has a 1:1 representation, record that number once. For each number only represented once, don't include it. Multiply the recorded numbers together and that is your GCF.

What is the Multiplicative Identity?

any number • 1 = that same number: a•1=1•a=a

Give an example and explain each: +, -, x, ÷ for whole numbers, without using the standard algorithm.

||| combined with ||| makes |||||| ( joining two amounts or values together to create a whole.) |||||| give away ||| leaves ||| remsining (separating a whole to create separate parts.) 2+2+2+2 is 2 counted 4 times, combined altogether becomes a total 8. 27-9-9-9=0, you have to remove 9 from 27 three times.