1
Q
Manipulatives
A
Play a valuable role in effective math instruction
2
Q
Pre operational stage
A
Manipulatives are used frequently with very young children, they NEED to see how things work and use hands on activities to learn
3
Q
Beads, buttons, small objects
A
Variety of colors, used with young children for sorting, counting, patterns
4
Q
Unfix Cubes
A
Snapped together or connected
5
Q
Counters
A
2 sided circles, red and yellow, indicate positive or negative
6
Q
Pattern Blocks
A
Teach patterns and attributes
7
Q
Dominoes dice
A
Ability to quickly recognize how many objects are in a set without having to count
8
Q
Base 10 block
A
Represent place value amounts in our base 10 number system
9
Q
Informal
A
Using reading available objects
10
Q
Formal
A
Using traditional measurement tools. Ex: paper clip, book length, rulers, stopwatch
11
Q
Graphic/pictoral
A
Provide visual models of mathematical equations
12
Q
Proportions***
A
Can be used to find a missing quantity in a new world problem that compares two units. One fraction is a known conversion and the other fraction is comprised of a given from the problem and what the problems asks for. An important aspect of setting up a proportion from a word problem is making usre the units for both ratios match. One technique to help make sure the problem the is set up correctly is to make a labeled box.
13
Q
Steps to solve Proportional Reasoning Word Problems
A
- Draw the outline of a proportion (two equal fractions) inside of a rectangle. Label the left side of the rectangle with the units from the conversion provided in the problem. On the top of the rectnagle, label the first fraction “conversion” and the second fraction “actual”.
- In the forst fraction, place the provided conversion, making sure the units to the left correspnd appropriately.
- In the second fraction, write the known amount on the situation. Place an x for the unknown amount.
- Cross mulitply and divide to solve for x.
- Check your answer.
14
Q
Probability of 1
A
When graphed, every proportional relationship is a straight line that intersects the origin, because if on x value of 0 gets multiplied by K, the result will always be 0.
[X|Y
0 |0
2 |6
4 |12
6 |18
15
Q
Simple event
A
Where one experiment happens at a time with a single outcome. An example would be flipping a coin once.
16
Q
Independent
A
Rolling of a dice twice (one dice is rolling and then another)
17
Q
Dependent
A
Drawing 2 marbles from a bag( when the first is kept after drawing)
18
Q
Compound event
A
Two or more simple events are performed together so that it can show that as one event happens so does another
19
Q
Simulations
A
Allow students to investigate probability concepts by generating large amounts of data in a short amount of time
20
Q
Experimental probability
A
Is what actually occurs during a simulation or trial. For example, When flipping a coin a few times, more are Heads than Tails.
21
Q
Theoretical Probability
A
Is what we expect to happen in theory.
For example =, if a coin is flipped 10 times, theoretically, one would expect to get 5 heads and 5 tails because there are two possible and equally likely outcomes for each coin flip. There’s a 50/50 chance of getting either heads or tails.
22
Q
Sample space
A
List or set of all possible outcomes. Ex: Head/tails with dice
23
Q
Geometric Probability
A
Offers an opportunity to use geometric shapes and their properties, especially area, to determine outcomes based on the functional relationships of areas.
The ratio of two areas, Geometric probability involves interpreting one “area of successful outcome” part of area “successful or favorable” outcome, compared with the entire area.
G.P = (Area of successful outcome/ Total area)
Probability= (Number of successful outcomes possible/ Total number of outcomes possible)
Look back at notes
24
Q
Placebo ***
A
Something that looks like the actual treatment but is designed to have no effect on the subject
25
Observational study
Researchers OBSERVE and record data for the subjects in the samples (they do not assign treatments). Can only show correlation. Ex: Survey's, Gather opinions.
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Experimental study
Researcher's assign subjects in the sample to certain treatments, then observe the effects of the treatment can show causation (cause & effect).
Typically, researchers will have two treatment groups. 1. A group that receives the actual treatment, can the treatment group (Experimental Group). 2. A group that serves as the .
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Population
Is all members of a defined group that you are studying.
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Sample
A sample is a subset of the population. Used because populations is ofton so large that is not feasible to gather data from everyone in the population.
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Random sample
Is a sample in which each member of the subset has an equal probability of being chosen. A random sample helps avoid bias, which occurs when parts of the population are over represented in the sample and some parts are under represented.
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Representative sample
A sample that accurately represents the characteristcs of a population, is also important. The random sample should have the same properties of the general population, such as gender, age, and socioeconomic distributions.
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Margin of error
Statistic expressing the amount of random sampling error in the results of a survey. The larger the margin of error, the less confidence one should have that a poll result would reflect the result of a simultaneous census of the entire population.
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Progress monitoring
Periodic assessments are given to keep track of student growth toward a specific goal or objective.
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Performing based assessment
Students apply knowledge or skills to compare a process or create a product
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Norm- referenced Test
Tests that compare an individuals performance/ achievement to a group called the “norm group”
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Control Group
Is a group assigned to no treatment or to a (something that looks like the actual treatment but is designed to have no effect on the subject). A common placebo in a medical trial... come back n finish
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Culturally Responsive Teaching
Is an approach that recognizes the importance of including students' cultures in all aspects of learning. Ex: Teach concepts in context, Draw connections to the real world, and Emphasize the diversity of contributions to math.
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Measurement
Number that shows the size or amount of something
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Invariance
Between types of units. Ex: 12 inches =1 foot
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Error
Numerical value that tells us how far “off” a measurement is compared to the correct or accepted value
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Approximation
Occurs when a students uses numbers that are similar, but not exactly equal to, the actual dimensions
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Unit/Dimensional Analysis
Process of converting within or between systems by multiplying by unit factors. Since the items on top equal the items on bottom; each of these fractions is actually equal to one.
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Linear relationships
A relationship with constant rate or change that creates a straight line when graphed
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Sequence
List of numbers, shapes, or symbols that go in a specific order. Most sequences also follow some sort of pattern
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Shape Pattern
A sequence of shapes that follow a pattern
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Arithmetic Sequence
Sequences that have a common difference
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Common Difference
Sequence is the amount of change that is added or subtracted to get one term to the next in the sequence
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Geometrical sequence
sequences that have a common ratio, or multiplier
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Common ratio
In the sequence is the amount you multiply by to get from one term to the next
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Function table
Describes a function by displaying inputs and corresponding outputs in a table
50
Linear ***
There will be a constant rate of change and the (change in y/ change in x) will be the same throughout.
Slope= y=mx+b - y intercept
51
English/ Metric
English
•Length - inches, feet, yards miles
• Weight - ounces, pounds, tons
• Volume- Ounces, pints, gallons
Metric
•Length-centimeters, meters, kilometers
•Weight-milligrams, grams, kilograms
•Volume-millimeters, liters, kilometers
52
Area Model
Can help students visualize multiplication and division with both whole numbers and decimals. Ex: For example, 12 x 25:
1. To use the area model for multiplication, write the multiplicand and multiplier in expanded form by decomposing the number based on place value. 12= 10+ 2, 25= 20+5.
2. Write the expanded forms on the top and left side of the model, making each tern it's own row or column header.
3. Multiply the row and column header of each box to fill in the rectangles.
4. Add the products to calculate the final result.
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Area Model Part 2
When used for division, area models are also known as the . For example, 825/ 4.
1. Write the dividend in the expanded form. The expanded form can be decomposed by place value or by values that are multiples of the divisor: 825 -> 800+20+5|or 400+400+20+4+1
2. Set up the area model by writing the components of the expanded form you choose in each section of the model. Write the divisor to the left of the model.
3. Divide the value in the first rectangle by the divisor, write times quotient at the top of the rectangle. In this example, 800/ 4= 200, which written at the top of the first rectangle.
4. Multiply the quotient by the divisor and subtract the result from the value in the first column: 4x 200= 800
5. Carry the remainder from the first column to the second and add it to the second value: 20+0=20.
54
Fundamental Counting Principle
Also called the counting rule, is used to calculate the infinite number of outcomes in a probability problem. The number of outcomes for each event is multiplied to find the total number of outcomes for the overall problem. -Unique outcomes= outcomes for event A x outcomes for event B x outcomes for event C...
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*Important Note
Come back
If a relationship is linear, and equation in slope-intercept form or (y=mx+b) can be used to represent the relationship, y=mx+b. (Change in y/ Change in x) [Y-intercept]
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Linear
There will be a constant rate of change and will be the same throughout.
57
Unit Rates
Are similar to ratios in that they show the relationship between two values. A common example of this is the speed we drive, 60 miles per hour, or the gas our cars use, 13 miles per gallon. UNIT RATES ALWAYS HAVE/ AS A PART OF THE RADIO. In the example given, speed is the number of miles driven in one hour and gas usage is rated at miles down in one hour. Ex: 60miles/1hour and 1.3miles/1gallon.
58
Unit Rates
*Any relationship can be expressed using part to part or part to whole comparisons.
Unit rates are also helpful when determining which item to purchase at a store. If Steve wants to get the best deal on toilet paper, he will need to compare unit rate and not overall price. The store has 12 rolls for $4.80 or 20 rolls for $7.00. Assuming the rolls are all the same size he needs to determine the cost per roll. The ratio for the first package is 12:480 which simplifies to 1:0:35. This means each roll costs $0.35 making it a better deal even though the overall cost is greater.
59
Ratio
*Creating proportional ratios only works when multiplying or dividing both numbers by the same amount.
Is a comparison that shows the relative size of two or more values. A ratio can be a part to part comparison or a part to whole comparison. Suppose there are 2 yellow blocks and 5 blue blocks. We can write the part to part ratio in several ways. All of these representations can be read as "for every 2 yellow blocks, there are 5 blue blocks". By using a colon: 2:5, By using "to' 2 to 5, by using a fraction.
60
Proportions
(Extra knowledge)
These fractions are proportional because they all reduce to 1/2. -Note that creating proportional
Is a pair of equal fractions. It is a statement that two ratios are equivalent. 2 numbers are proportional to another pair of numbers if their ratios are the same. Two ratios are the same if they can simplify to the same ratio. In other words, if you can multiply or divide both numbers in a ratio by the same value, the resulting ratio is equivalent and the numbers are proportional.
61
Probability Model
^Note that each probability is between 0&1 that 0.25+0.50+0.25/ 1.
Is a math representation of a random phenomenon. If there are a finite number of outcomes.
1. The probability of each possible value will be between 0 and 1.
2. All of the probabilities will add to 1(100%). Ex: Create a probability model for the number of heads showing when flipping 2 coins.
*Probability Model: Number of heads showing 0,1,2 /(Divided by) Probability 0.25, 0.50, 0.25
62
Number Lines
Can be used after students have masked skip counting as a way to show multiplication and division as repeated as addition and subtraction.
For multiplication, students will use the factors to determine how many times to hop forward and what number to skip count by for each hop.
For division, students will use the same strategy except they will start on the dividend number and hop backward by the divisor until they get to zero. The number of hops needed to get to zero will be the quotient.
63
Repeated Addition and Subtraction
Can help students see what is happening when you multiply and divide numbers. When used to multiply this strategy breaks down the factors to represent that addition problem to solve. One factor will take the place of the addend and the other factor will represent how many times you add that addend. Once added together, the total will be the product of the factors.
-Repeated subtraction can be used to help students divide. Students will start with the dividend and then subtract the divisor until they get a remainder of 0. The qoutient will be the number of times needed to subtract until you get a remainder of 0. 15/ 5= 3 (15-5=10-5=5-5=0)
64
Equal Groups
Is a great strategy to help young learners visualize how to solve multiplication and add division problems with pictures. Students will use the numbers of the first factor to create groups and will use the second factor to draw dots in each group. The total group of dots drawn will be the product. It is best to teach students to draw larger circles to create their own groups and add smaller dots inside each circle.
65
Strip Diagrams
Help show students how factors are used to find the product in a multiplication problem. They can also be a good way for students to connect multiplication to repeated addition, since they are similar to part-part-whole models.
66
Common Denominator
When two fractions have the same denominator, they are said to have ; means that the object (s) used to represent the fractions and divided into equally -sized pieces.
-Adding and subtracting fractions using common denominators. Continuing the previous pizza example, the pizza slice from pizza 1 is 1/4 of a pizza, and the slice from pizza is 1/8. If a student wants to know how much larger one slice is than the other, they would need to subtract (1/4)-(1/8)
67
Least Common Multiple
Multiples of 4: 4,8,12,16, etc.
Multiples of 8: 8,16, 24, etc. LCM=8
68
Decimals
Are numbers less than 1 displayed using place values and the powers of 10. They are usually first introduced as "decimal fractions", which are fractions with denominators that are powers of 10.
Ex: 1/10 is shown to be equivalent to 0.1%. It is important to recognize the appropriate format for the solution to a problem.
Ex: A money problem will use decimals and a pizza problem will use fractions.
69
Percentages
Usually represent part of a whole but can represent larger value.
' 40% means 4 out of 10 parts of one unit or 40 out of 100 units.
' 200% means twice as much as 1 unit.
70
Fractions
Usually represent partial numbers but can represent whole numbers.
'1/2 means one half of one unt. 4/1 means 4 out of one unit. The top part of a fraction is referred to as the fractions numberator. The bottom of a fraction is its denomination. You can remember this when the mnemonic "denominator is down". The fraction bar itself should be understood to be a division symbol.
71
Real Numbers
Can be represented in a variety of ways, with the most common being fractions, percents, and decimals.
|Fraction | Percent| Decimals|
1/4 25% 0.25
1/3 33.3% 0.33
1/2 50% 0.5
2/3 66.6% 0.66
3/4 75% 0.75
1 100% 1.00
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Fraction Composition
Add together fractions with like denominators to get one fraction. The fraction will keep its denominator the same and the new numerator is the sum of its orignal numbers.
For example: 1/4+1/4+1/4= 3/4 and 3/4 is the fraction being composed.
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Fraction Decomposition
Which means to break down a faction. The fraction will keep its denominator the same and the new numerators should add up to the original fraction. Ex: 5/8 can be decomposed in a variety of ways. 1/8+1/8+1/8+1/8+1/8 or 2/8 or 3/8.
74
Statistical Experiment
Are not experiments in the sense of science. They are trials of an event that has more than one probability based outcome. An example of a statistical experiment is a coin toss. A coin toss is a statistical experiment because it has predictable outcomes that are determined by chance.
-All statistic experiments have the following characteristics.
-Have more than one possible outcome, all of the possible outcomes can be determined beforehand.
-The outcomes are determined by chance (at random).
-Rolling a six-sided die is also a statistical experiment.
75
Normal distribution
Is a graph with a bell-shaped curve, where the graph is symmetric and has no SKEW.
-The mean & median are approximently equal to each other.
-The line down the center represents the mean (upside down h).
-Each vertical line separates one standard deviation (o') of data from the mean.
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Mean, Median, and Mode
Are all measures of central tendency.
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Mean
The mean is found by adding all of the numbers in a data set and dividing that sum by the number of numbers (the n) in the data set. The mean is commonly known as the average and the greek symbol (upside down h) is often used to represent the mean.
78
Median
Is found by first putting all the numbers in order from LEAST TO GREATEST. The median is either the middle number of a set that has an odd quantity of values, or, when there is an even number of values. The median is found by calculating the mean of the middle two numbers.
79
Mode
Is the most frequent number in a data set. Some sets of data have no mode (if all data values appear in the set the same number of times). Some sets of data have more than one mode (of multiple pieces of data in a set appear the same number of times & appear more than any other data values.
80
Range
The difference between the highest data value and the lowest data value. In general, the greater the range, the more spread out the data is. A small range indicates data that is more clustered together.
-Guess and check involves a reasonable guess about a solution that can be checked, or a finite number of solutions that can be checked. This is NOT random guessing.
-This strategy can be useful on multiple choice problems, particularly for problems involving equation solving as each potential solution can simply be tested in the original question to see whether or not it works.
*If x=0, then 0(0-1)>0 | 0(-1)>0 | 0>0 which is false. Therefore, 0 is not a solution.
*If x= 2, then 2(2-1)>2 2(1)>2, 2>2 which is false. Therefore, 2 is not a solution.
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Formal Reasoning
(Problems have a single right answer)
Is used to answer questions and solve problems that have a single solution (a right answer) by using rules of logic and algorithms (systematic methods that always produce a correct solution to a problem) to reach a conclusion.
Ex: How much paint should we buy to paint the bathroom?
82
Inductive Reasoning
(Broad statements are made from some data to create conjectures)
Generalize knowledge from one area to another.
An example of faulty inductive reasoning is a snickers is a candy bar and snickers have peanuts, therefore all candy bars have peanuts.
83
Informal Reasoning
(Questions have more open-ended conclusions)
Is used to answer questions and solve problems that are complex and open-ended(without a definitive solution) by using everyday knowledge to synthesize in got and reach a conclusion.
84
Deductive Reasoning
Uses two or more known premises to draw a conclusion.
Ex: All cats say meow (premise #1) Jackie is cat (Premise). Therefore, we can deduce that Jackie says meow. {Conclusion}
85
Estimating
Involves rounding or approximating numbers to quickly perform math operations. Values can be rounded to the nearest whole number, tens, hundreds, or thousands depending on the problem. For example, the sum: 24+56+77 -> 20+ 60+80=100
86
Benchmark Numbers
Are numbers that are easily added, subtracted, multiply, or divided. Typical benchmark numbers include: 10,25, and 100, but can also be multiples of other numbers in the problem.
Ex: Divide 35 by 11 using benchmark numbers. Because 33 is a multiple if 11, the qoutient can be estimated by completing the problem 33/11, which is equal to 3.
87
Partitioning
Involves breaking up numbers by place value before adding, subtracting, multiplying, or dividing and can be a helpful mental math strategy. The process of breaking numbers up by place value is known as decomposing. When this process is reversed at the end of the problem, this is called composing.
- Start by decomposing the numbers by place value.
-Conjectures= Guesses/ an inference created without proof.
88
Compensation
Is another mental math strategy in which one number is changed to a value that simplifies the computation, such as a multiple of 10. The result is adjsuted at the end of compensate for the change. For example, find the sum of 52 and 49 using compensation start by changing either value to a multiple of 10: 52-2=10 Then, completed the addition: 50+49=99. Adjust for changing 52 to 50 by adding 2 (99+2=101).
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Addition Property of Equality
Same amount added to both sides. If a=b, then a+c= b+c
90
Subtraction Property of Equality
Same amount subtracted from both sides. If a=b, then a-c= b-c
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Multiplication Property of Equality
Both sides multiplied by same amount. If a=b, then ac=bc
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Division Property of Equality
Both sides divided by same amount. If a=b, then a/c = b/c
93
Real Numbers
Are numbers that have a specific value. Real numbers can be rational or irrational. Within rational numbers, there-are integer, whole, and natural number.
-Have a specific value
Real Number ex: -1,-2,-3,-4, 0,1,2,3,4, 0.111, (-8/2)
Irrational Number ex: (√ 2 (√ 3 3.14)
94
Rational Numbers
Are numbers that can be expressed as a ratio (fraction or comparison) of two integers, (a/b) where b+0 (because zero can't be the denominator of a fraction). 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 math meaning a value that has not been rounded or approximated.
95
Rational Numbers part 2
Many numbers are rational numbers:
-Mixed numbers - like (1 1/2) which have both a whole number part, like the "1/2"
-Decimal numbers that end such as 0.25
-Decimal numbers that repeat the same pattern forever such as 0.727272.... (also written 0.72, which is 8/11 written in fraction form)
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Natural Numbers
Real numbers that can be pointed to in nature - one tree, two people, 3 dogs. This set is also known as the counting numbers; it starts at 1 and goes up by 1's in the same way that children first learn to count, yielding the infinitely large set of values 1,2,3,4,5,...
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Whole Numbers
The set of natural numbers and "zero". It starts at 0 and goes up by is to create the infinite set 0,1,2,3,4.
98
Integers
Positive and negative counting numbers and zero; the infinite set -3,-2,-1,0,1,2,3...
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Irrational Numbers
Real numbers that CANNOT be represented exactly as a ratio of two integers.
-In order to be expressed exactly (w/o rounding) irrational numbers are often written with symbols (like (pi), e, or (√ 2)
-When irrational numbers are written using only ordinary numerals, they must be rounded.
100
Absolute Value
Distance a number is from zero; always a positive number. Ex 1-91=9 and 191=9; -123<-122.
-When comparing the magnitude of numbers in different forms, convert them all of the same form.
101
Number Line
Is a representation used in math. It is a straight line and each whole number is equal distance from the next one. It can start and end with any number and can include negatives, fractions, and decimals. The numbers always increase left to right, *left is lesser, right is greater*.
102
Unit Fractions
Are fractions that have 'i' in the numerator. For example, 1/2,1/7,1/8 are all unit fraction.
-For example, 1/4 is smaller than 1/2
-If the denominators are the same, the larger the numerator, the larger the fraction. For ex, 1/4 is smaller than 3/4. 4/4>3/4>2/4>1/4
-If the numerators are the same, the smaller the denominator, the larger fraction.
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*Probability
Is the likelihood of an event occuring. Math equals the number of possible successful outcomes divided by the total number of possible outcomes. Consider trying to roll a 5 on a standard six sided die. There is 1 way to have a successful outcome, and 6 possible total outcomes, so the probability is (1/6). The notation used for probability is a "P followed by parenthesis that are used to indictate what probability you are finding. For example, P(S)= (1/6) means "the probability of rolling a 5 on a die is one sixth".
-All possibilites fall from 0 to 1, that is 0 less or equal to (any event) less or equal to 1
- *Probability= (#of successful outcomes possible/ Total number of outcomes possible)
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Probability of 1
Means that an outcome is certain, or guaranteed. For example, P(n<10)=1 because any roll on a standard six sided die will guarantee number smaller than 10.
Ex: What is the probability of drawing an eight from a standard deck of cards?
There are four numbers eight cards in a standard deck of 52 total cards. 50, 4 over 52, which simplifies to 1 over 13.
Probability= (numbers of successful outcomes possible/ total number of outcomes possible)
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Probability of 0
Means that an outcome is IMPOSSIBLE. For example, P(7)=0 because it is impossible to roll a 7 using one standard six-sised die.
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