MEGA Math Terms

Piaget: Pre-operational

Stage 2 (two - seven years): In the Stage, children begin to understand and develop the concept of one-to-one correspondence. They understand how to count accurately and can associate the number five, for example, to five objects. Children at this stage can begin to solve one-step logic problems and can begin to use their understanding of one-to-one correspondence to develop a concrete (hands-on/manipulative based) understanding of addition and subtraction. They can also recognize patterns and give the next two or three members of a simple sequence.

Bloom's: Analyzing/analysis

ANALYZING / ANALYSIS - Here is where outcomes are thought through, inferences made, elements and relationships analyzed, and alternatives considered. Keywords in questions and activities might include: how, reason, why, what are the causes, what are the consequences, categorize, compare, contrast, separate into parts, what are the steps in the process, how did you start, what are some examples, list all solutions possible, and what problems might arise? A sample problem students might encounter at this level could be: Jake now has more than $12. He had $5 at the beginning of the day and completed several jobs in his neighborhood for which he was paid. What is the minimum amount Jake could have earned working today? How did you arrive at your answer? Are other answers possible?

closure

A set is considered closed for an operation if the operation on any two elements in the set always yields a value that is a member of the set. For example, the set of whole numbers is closed under the operation of addition because the sum of any two whole numbers is a whole number. The set of whole numbers is also closed under the operation of multiplication for a similar reason. However, the set of whole numbers is not closed under either subtraction or division, because the difference - answer to a subtraction problem - of two whole numbers does not always result in a whole number (for example: 9 - 12 = -3 and -3 is an integer, but not a whole number)

multiplicative inverse

Again, inverses, in general, are opposites. When a number and its multiplicative inverse are multiplied together, they "cancel each other out"/it yields the Multiplicative Identity/they multiply to equal 1. A multiplicative inverse is also known as a reciprocal, and more formally stated, is a number that when multiplied by x yields 1. The multiplicative inverse is commonly denoted as 1/x or x−1. In fraction form, the multiplicative inverse of a/b would be b/a because their product will always be 1 (the form of 1 written !"). Only the number zero has no multiplicative inverse, no reciprocal (because division by !" zero is undefined/impossible).

commutative property

An operation if changing the order of the values involved does not change the outcome. Both addition and multiplication are commutative operations.Forexample: 2+3+7=3+7+2,and2×3=3×2.

Bloom's Taxonomy: Applying/Application

Application questions require students to use their knowledge. They should be solving problems in new situations so that they can select, transfer, and use information, rules, and processes that they have learned. This is where most problem-solving will occur in the elementary classroom. Keywords in questions and activities at this level include: compute, demonstrate, solve, what are the next steps, how can this be used, use, change, apply, estimate, or determine. Sample questions for this level include: If Jake had $7 and his father gave him an additional $3, how much money does he have now?

bloom's: evaluating/evaluation

At this level, students are asked to make judgments based on set criteria and to justify their thinking. This stage is rarely reached at the elementary level simply because of the nature of the students' cognitive development. However, students can be involved in many activities where they are asked to make choices and defend or support their choices. Activities and questions where students are asked to explain their thinking give students the opportunity to justify choices they have made. Keywords for questions prompting evaluation include: which is the best choice and why, what do you think about, what do you recommend, rate from good to poor, what is the problem, are all of the solutions the same, will all the solutions work, decide which, and justify your choice.

Blooms Taxonomy: Understanding/Comprehension

Comprehension questions have students represent information in their own words or in a different way in order to show their understanding of information. Comprehension questions require that students interpret or translate information based on prior knowledge. Keywords to aid in determining whether questions and activities are at this level include: tell in your own words, summarize, describe, interpret, compare and contrast, explain, or what does it mean. For example: What are some ways to express a sum of 10? and What other problems could you write that show how 7, 3, and 10 are related to each other?

exponents

Exponents indicate repeated multiplication. They make it easier to express something like 16 is 2×2×2×2, as 16 can be written more simply as 24 even while being written in its fully factored form. any number raised to the zero power always equals 1. (With one exception - 0 itself cannot be raised to the 0 power at all, that operation, much like division by 0, is undefined.)

greatest common factor

In the case of 12 and 15, the greatest common factor (GCF) is 3. That is because an examination of the prime factorization of 12 (2×2×3) and 15 (3×5) shows that the only value in common for the two lists is 3. (And by virtue of being the only common factor, 3 is also the greatest common factor.) In the case of 12 and 16, as explored above, the GCF was 4, the number reduced from each value in order to rewrite 12/16 in its simplest form as 3⁄4. [12 is 2×2×3, and 16 is 2×2×2×2. They each have two factors of 2, so their greatest common factor is the product of the common factors: 2×2 = 4.]

additive inverse

Inverses, in general, are opposites. When an additive inverse is added to a number, it "cancels out"/it yields the Additive Identity/it has a sum of zero. The additive inverse of a number always has the same absolute value (number's distance from zero) as the original value, but with the opposite sign, like 5 and -5. Note that 5 + (-5) equals 0. The additive inverse of -2 is 2 because -2 + 2 = 0. The additive inverse can be found by subtracting the original number from zero.

Bloom's Taxonomy: Remembering/Knowledge

Questions and activities at this level are simple memory and recall questions of information presented to children and recalled by them in a similar form, such as: How much is 2 + 3? and How many sides does a triangle have? Activities and questions at this level will ask students to write/tell, label, name, list, describe, state, define, identify, or will ask who, what, when, where.

Piaget: Sensorimotor

Stage 1 (birth - two years): children experience their world through their own senses: touch, taste, smell, sight, and hearing. As such, during this stage, teaching should be geared to the sensorimotor system. It is also important to understand that children in this stage of development are egocentric and can only experience the world from their own perspective. With respect to mathematics, then, children begin to develop the idea that one finger corresponds to one object, but they lack a full understanding of one-to-one correspondence. While most two-year-olds can count to ten, most cannot count ten objects. Children who cannot count to ten and match that number to the counting of ten objects, are not yet ready to move to the next stage of development and will need additional practice in the development of counting and one-to-one correspondence.

Piaget: Concrete-Operational

Stage 3 (seven - eleven years): brings the ability to think logically and classify based on multiple attributes other than simply visual ones. Children at this stage are able to think and reason in two and three dimensions, and so would recognize that the volume of water in the third glass in the example above is the same as the volume in the original glasses. Children at this stage are able to think about and articulate the attributes that define shapes and can classify figures according to those attributes. For example, they can classify trapezoids, parallelograms, rectangles, squares, and rhombuses as all having at least one pair of parallel sides. Children in this stage can solve problems requiring multiple steps, and can see and understand different points of view and perspectives. Accordingly, they can understand problems and problem solving approaches in ways other than their own, and can begin to understand that there is often more than one way to approach and solve a problem

Piaget: Formal Operational

Stage 4 (starting around ages eleven to sixteen and lasting through adulthood): Stage marks a major change in the ability to think and process abstractly. At this stage, there is no longer the reliance on concrete materials that were required for understanding concepts at the earlier stages of cognition. Students in this stage are able to make and test/prove conjectures. They are capable of abstract thinking including concepts such as infinity, sizes of the various number sets, limits, areas under a curve, a formal geometric proof, and complex numbers.

bloom's: creating/synthesis

This level of Bloom's is where divergent thinking, originality, and imagination occur. At this level, concepts that have been learned are taken apart and put together to form a whole that is new for that individual student. Information is compiled or combined in a novel way. Although synthesis can occur at the elementary level, it is not frequent. Keywords and phrases to prompt creation include: create, design, invent, compose, develop, suppose, how many hypotheses can you suggest, how many different ways, how else, what would happen if, or explain several possible ways.

relatively prime

Two numbers are relatively prime if they have no factors (besides 1) in common. For example, 34 and 15 are relatively prime because 34 = 2 × 17 and 15 = 3 × 5, and there are no factors in common. Note that the least common multiple of two numbers that are relatively prime can be found by multiplying the two numbers together: the LCM of 34 and 15 is 34 × 15 or 510.

multiplicative identity

identity is a number that when multiplied by any element x in a set, always yields x. The idea of the multiplicative identity is that an initial value is kept identical to what it began as, even while a quantity is multiplied with it. That quantity multiplied by, of course, must be the number 1, as in, 5 × 1 = 5. The number 1 is sometimes referred to as The Multiplicative Identity. The utility of the multiplicative identity comes in realizing that 1 comes in many forms, like 2/2 or 8/8 (as long as the numerator and denominator of a fraction are the same as each other, the quantities divide with a quotient of 1).

additive identity

identity is a number that, when added to any element x in a set, always yields x. The idea of an additive identity is that an initial value is kept identical to what it began as, even while a quantity is added to it. That quantity added to it, of course, must be zero, as in, 5 + 0 = 5. Zero itself is sometimes referred to as The Additive Identity

reducing fractions

just a disciplinary norm; while it is true that 12/16 is a valid number, since it is the equivalent of 3⁄4, mathematicians prefer to see it written with the smallest values possible, "fully reduced" to 3/4. A fraction like 12/16 is reduced by thinking of the values in its numerator and denominator in terms of their factors. Recognizing 12 as 3×4 or as 2×2×3, and 16 as 4×4, or as 2×2×2×2 allows one to divide out the common factors (4 and 4, or else two 2s), leaving behind just the 3⁄4.

denseness

means that between any two numbers, there is always at least one additional number. Some number sets are dense, while others are not. The set of rational numbers is dense because, for example, between 1⁄4 and 1/3, there is the rational number of 7/24. The set of whole numbers is not dense because there may not be a whole number between two whole numbers. For example, while it's true that 6 is a whole number between the whole numbers 4 and 10, not every pair of whole numbers has another whole number between them, like 7 and 8 - there is no whole number between them.

complex numbers

numbers written in the form a + bi where i = −1, and where a and b are real numbers. The value of i is called the imaginary unit. The number a is the real part of the complex number. The value bi is the imaginary part. If b = 0, then a complex number has no imaginary part and is simply a real number. If a = 0 and b ≠ 0, then the complex number is classified as a pure imaginary number.

integers

positive and negative counting numbers and zero; the infinite set ...-3, -2, -1, 0, 1, 2, 3,...

associative property

property concerns grouping. Specifically, if the values in an expression can be regrouped but will yield the same outcome as with the original grouping, that is an example of the associative property. For example, (2 + 3) + 7 = 2 + (3 + 7). The left side of the statement becomes 5 + 7, which equals 12. The right side becomes 2 + 10, which also equals 12. Because the answer stayed the same despite the regrouping/change of parentheses placement, this problem illustrates the associative property. (a + b) + c = a + (b + c).

natural numbers

quantity concepts that can be pointed to in nature - one tree, two people, three dogs; this set is also known as the counting numbers; it starts at 1 and goes up by 1s in the same way that children first learn to count, yielding the infinitely large set of values 1, 2, 3, 4, 5,...

irrational numbers

real numbers that cannot be represented as a ratio of two integers. In order to be expressed exactly (without rounding), irrational numbers are often written with symbols (like π, e, or 2). When irrational numbers are written using only ordinary numerals, they must be rounded, as irrational numbers, by definition, cannot be written in fraction form, and when expressed as decimals, they have infinitely many, non-repeating digits. Irrational numbers can be approximated (π ≈ 3.14 or 22/7; e ≈ 2.72, and 2 ≈ 1.41) and positioned on a number line.

conjugate

the conjugate for any complex number of the form a + bi is simply a - bi (same numeric quantities, but with the opposite operation/sign in between them).

prime factorization

the process of writing a number as a product of prime factors. All integers can be factored into a product of prime numbers or a product of prime factors and -1. The ability to use prime factorization is useful in many applications. There are multiple ways to go about doing this, but a common procedure used to find the prime factorization of a value is to use a factor tree. When creating a factor tree, one selects any pair of factors of the original value. For instance, to make a factor tree for 70, one could choose either 7 and 10, or the pair 2 and 35, or the pair consisting of 5 with 14. All primes will be circled; all composites further deconstructed. In the case of the number 70, however, no further steps are needed. Each pair found in the step just completed (2 × 5, 5 × 7 and 2 × 7) is formed of only prime numbers. Following each path explored above, it should be clear that, though the list of factors was compiled in slightly different orders (7, 2, 5 and then 2, 5, 7 and finally 5, 2, 7), they all boil down to the same list when reordered from least to greatest (as is standard practice in prime factorization): 2, 5, 7. And so the prime factorization of 70 is 2 × 5 × 7.

whole numbers

the set of natural numbers and zero; it starts at 0 and goes up by 1s to create the infinite set 0, 1, 2, 3, 4,...

least common multiple (LCM) / LCD (lowest common denominator)

the smallest number two values will both divide into evenly. For example, the least common multiple of 12 and 15 is 60 because 60 is the smallest number that can be divided by both 12 and 15 without a remainder. How is the LCM determined? One of two main methods can be applied. Either one could generate a list of multiples of 12 (12, 24, 36, 48, 60, 72, 84...) and a list of the multiples of 15 (15, 30, 45, 60, 75, 90...) and look for the smallest value on both lists (60). Or, one could use the prime factorizations of 12 as 2×2×3 and 15 as 3×5, realize that they share the factor of 3, and then bring in all remaining factors (2×2×5) to get 3×2×2×5 = 60.

quotient

answer to a division problem

prime number (and composite number included)

are natural numbers greater than 1 that have no numbers that will divide into them without a remainder, aside from 1 and themselves. The set of known prime numbers begins at 2 and also includes 3, 5, 7, 11, 13, 17, 19, 23, 29, 31... To test whether a given value is prime or composite (that is, composed of other numbers, like 6, which is made up of the factors of 2 × 3 and so is not prime), try dividing each prime number that is smaller than the approximate square root of the value in question. If any of them divide into that number evenly, the value in question is a composite number. If no prime number smaller than the value's approximate square root goes into the original value without a remainder, the original value is a prime number.

rational numbers

are numbers that can be expressed as a ratio (fraction or comparison)of two integers, !" where b ≠ 0 (simply because zero could never serve as the denominator of a fraction). Rational numbers include all number categories already mentioned (natural numbers, whole numbers, and integers), as each value from those sets can be expressed in fraction form with a denominator (the bottom of the fraction) of 1. Other values that are also rational numbers are numbers written as fractions (including mixed numbers, quantities like 41⁄2 which have both a whole number part, like the "4," and a fractional part, like the "1⁄2"), and two types of decimals. The kinds of decimals that are part of the set of rational numbers are decimals that terminate (end), like 0.25, which is also 1⁄4, and decimals that do not ever end, but do repeat a pattern, like 0.727272... (also written 0. 72), which is 8/11 when written in fraction form. All rational numbers can be written exactly in both fraction and decimal forms, and can be graphed on a number line. [Note that exact is a formal term in mathematics meaning a value that has not been rounded or approximated.]

distributive property

creates the option to multiply a factor by a sum (or difference) by multiplying the factor with each term of that sum (or difference) separately, and then finding the sum (or difference) afterward. That is, a(b + c) = ab + ac, or 3(40 + 8) = 120 + 24 = 144. This form of the distributive property can be helpful for mental math [that is, it is easier to multiply 3 × 48 in one's head if 48 is thought of 40 + 8 so that 3(48) becomes 3(40 + 8)]. form ab + ac can be rewritten as a(b + c). That is, the common factor of a in both of the terms being added can be factored out, or "undistributed," from ab + ac to a(b + c).