EC-6 Practice Questions

regular polygon

regular polygon

Identify the equation below that models xª • xᵇ = xª⁺ᵇ? A. 5³ • 5⁴ = 25⁷ B. 5 • 5 • 5 + 5 • 5 • 5 • 5 = 5⁷ C. 5 + 5 + 5 • 5 + 5 + 5 + 5 = 5⁷ D. 5³ • 5⁴ = 5⁷

D. 5³ • 5⁴ = 5⁷ - Would NOT BE 5³ • 5⁴ = 25⁷ - does not fit the pattern and is incorrect because 25⁷ would be equivalent to (5²)⁷ or 514. - 5 • 5 • 5 + 5 • 5 • 5 • 5 = 5⁷ would be the model for the equation x³ + x⁴ = x⁷ and would mean that 125 + 625 = 78,125: NOT a true statement.

acute triangles

angle smaller than 90 degrees - HAS ALL acute angles

Order for geometry learning

visualization--> analysis --> abstract

horizontal lines have a slope of ___

ZERO /0

Factor tree

factor tree

A parabola

parabola y = x²

What is the value of the "3" in the number 17,436,825? A. 30,000 B. 300,000 C. 3,000 D. 300

A. 30,000

A school fundraiser is selling candy bars to raise money for a new gymnasium. If Billy sells a total of $650 worth of candy bars for $x per candy bar, which equation would be used to find out how many candy bars Billy sold? A. 650 / x B. 650 + 3x C. 650 * x D. 650 - x

A. 650 / x - To find out how many candy bars were sold, simply divide the total amount of money by the price per candy bar to find the total number of candy bars sold. This would be 650 / x. - Another way to write this is x*N = 650, where x is the price per candy bar and "N" is the number of candy bars sold.

A runner is running a 10k race. The runner completes 30% of the race in 20 minutes. If the runner continues at the same pace, how long will it take to complete the race? A. 67 minutes B. 60 minutes C. 85 minutes D. 62 minutes

A. 67 minutes - To find the answer to this question, set up a ratio; remember that 30% = .3 and 100% = 1. (20 / .3) = (x / 1) - When you cross multiply to solve for x, you get the equation .3x = 20. - Divide each side by .3 to isolate x and the answer is 66.66666. The best answer choice is 67 minutes.

A teacher is monitoring her class while the students are involved in a group activity exploring the size of angles in a set of triangles. She moves from group to group, pausing and watching the group dynamics. What the teacher is doing can best be described as: A. informal formative assessment. B. informal summative assessment. C. formal summative assessment. D. formal formative assessment.

A. informal formative assessment. - The teacher is listening and watching as she informally assesses student understanding of the concept: angle measures in triangles.

Which of the following lists ALL the equations that are linear? (Select all that apply) A. y = -3x - 5 B. y = -1/3 x - 5/3 C. y = x - 2 D. y = x

A. y = -3x - 5 B. y = -1/3 x - 5/3 C. y = x - 2 D. y = x - A linear equation is an equation whose graph is a line. - The x and y variables contained in a linear equation are always first degree, meaning that they have no exponents - or they have exponents of 1 which are not usually written. - So, x and x¹ are equivalent expressions. All quadratic equations contain one second degree variable: a variable with an exponent of 2. - When quadratic equations are plotted, their graphs are parabolas.

Which of the following correctly combines like terms for the following equation? 3x - 4 + 6x + 5 - 4x - 2 = 16 A. (3x - 4) + (6x + 5) - (4x + 2) = 16 B. 5x - 1 = 16 C. 14x - 10 = 16 D. 13x - 11 = 16

B. 5x - 1 = 16 - This is the only answer option that combines the terms correctly. - Another way to write the problem is 3x + 6x - 4x - 4 - 2 + 5 = 16; this can then be written 5x - 1 = 16.

What is the digit in the hundreds place in the product of 63 * 31? A. 1 B. 9 C. 5 D. 3

B. 9

Mrs. Adamson's student asks her how much space a cube takes up. Mrs. Adamson said to answer this question, the student would need to calculate the volume of the cube. Which of the following measurable attributes is the formula for a cube based upon? A. Mass B. Intensity C. Length D. Capacity

C. Length - Before the volume of a cube can be calculated, the length, width, and height must be measured. Length is the best answer.

What is the place value of the "3" in the number 15,436,129? A.Thousands B. Hundred Thousands C. Ten Thousands D. Millions

C. Ten Thousands

What is the place value of the "5" in the number 15,436,129? A. Billions B. Hundred Thousands C. Trillions D. Millions

D. Millions

all irregular polygon

all irregular polygon

visualization

in this stage of development learn to classify shapes holistically and often with reference to visual prototypes EX: Students recognize a circle b/c it looks like a clock or the sun, or they identify a rectangle b/c it looks like a door or a box. - Though learners can identify shapes with some accuracy at this developmental stage, they cannot articulate properties of the shapes and still have difficulty identifying atypical shapes.

vertical lines have _______

no slope

Given A(15) = 20 and a₁ = -8, what is d? A. d = 130 B. d = 2 C. d = 1.5 D. Cannot be solved due to insufficient information given.

B. d = 2

What is the value of the "7" in the number 432.0769? A. 7/1,000 B. 7/10 C. 7/100 D. 7/10,000

C. 7/100

Which of the following is not equivalent to ½? A. 2 ÷ ¼ B. 50% C. ⅓ ÷ ⅔ D. 1 ÷ √4

A. 2 ÷ ¼ 2 ÷ 1/4 is asking the question: how many "groups" of size 1/4 are there in 2? - This is a good time to, perhaps, think about money. 1/4 of a dollar is $0.25. - If $2 were separated into quarters, how many quarters would we have? 8 quarters.

A pancake recipe requires 1 tablespoon of baking powder per 2 cups of flour. If 2 cups of flour make 4 pancakes, how many tablespoons of baking powder are needed to make 12 pancakes? A. 3 B. 6 C. 1 D. 9

A. 3 - 1 tablespoon of baking powder per 2 cups of flour makes 4 pancakes -> 1 tablespoon of baking powder is used to make 4 pancakes. - To make 12 pancakes, 3 tablespoons of baking powder are needed. - To find this, divide 12 by 4 to the proportional increase in pancakes. 12 / 4 = 3 -> because the number of pancakes is 3 times as many as the recipe size of 4; multiply 1 tablespoon by 3 to get 3 tablespoons of baking powder.

How many different outcomes are possible when a pair of standard dice are rolled? A. 36 B. 10 C. 12 D. 24

A. 36 - There are 36 possible outcomes. Each combination takes into account that a 1 on the first die is different from a 1 rolled on the second. - The roll of each die is an independent event. There are 36 possible outcomes or 36 different ways to roll the sums of 2 through 12.

Which of the following civilizations is most closely associated with the development of algebra? A. Arabian B. Egyptian C. Roman D. Mayan

A. Arabian - Al-Khwarizmi (770 - 840 C.E.) is generally accepted as the Father of Algebra even though there is evidence that many concepts from algebra were known many thousands of years before Al-Khwarizmi. - The word, algebra, comes from the Arabic al-jabr, meaning to unite or combine.

Tymon is building a bookshelf. He has the entire outside completed. He must purchase wood to make the 5 shelves. Each shelf needs to be 26 3/4 inches long. He can purchase wood up to 16 feet long. He needs to allow 1/8 in for each saw cut, how many feet of wood does he need? A.11 1/8 B. 11 3/16 C. 11 5/16 D. 133 19/24

B. 11 3/16

Congruent triangles can be used to explore geometric relationships. Which of the following is NOT representative of a position of congruent triangles? A. Reflection B. Mirror C. Rotation D. Translation

B. Mirror

Which of the following requires the most advanced understanding of relationships between arithmetic operations? A. 53 - 47 + 4 = 57 - 47 =10 B. 100 - 36 = 64 C. 47 + 53 = 50 + 50 = 100 D. (4 * 3) + (4 * 3 - 2) = (12) + (12 - 2) = 22

D. (4 * 3) + (4 * 3 - 2) = (12) + (12 - 2) = 22 - This requires the knowledge of the Order of Operations, as well as multiplication. -This is the most advanced answer option.

Which method below does not define a function? A. Passes the vertical line test. B. A correspondence assigning only one range value to each domain value. C. A set of ordered pairs where the first component is never repeated. D. Passes the horizontal line test.

D. Passes the horizontal line test.

Mrs. Johnson lets her students choose between two word problems: Problem A: If you are digging for dinosaurs and need to fence off your dig site, what's the biggest site you can fence off with 40 ft. of fence? Problem B: What is the largest area you can create with 20 inches of rope? Mrs. Johnsons finds a significant majority of her students chose to work Problem A. Which of the following is the most likely reason more students chose Problem A instead of Problem B? A. Problem A requires a lower mathematical knowledge B. Problem B is harder than Problem A C. Students who work Problem A get a greater reward from Mrs. Johnson than students who work Problem B D. Problem B is less interesting than Problem A

D. Problem B is less interesting than Problem A - Students are more likely engaged when presented with a problem about digging for dinosaurs than a simple mathematical word problem. - The way problems are presented can impact students' engagement in the learning process.

Scale 1- Triangle and square left and 2 cylinders on right Scale 2- triangle on right and 1 square and 1 cylinder on right In both figures shown in the model, the scales are balanced. Which of the following scales would also be balanced? (View attachment for larger image) ANSWER: 1 triangle on the left and 3 squares on right

- The scale on the left is scale 1 and the scale on the right is scale 2. - Step 1: In scale 1 substitute a cylinder and a cube in place of the triangle, as per the equality in scale 2. - Step 2: Using the properties of equality, cancel a cylinder from both sides, obtaining a balanced scale, which will be called scale 3, with two cubes on a side and a cylinder on the other. - Step 3: In scale 2 substitute two cubes in place of the cylinder, as per equality in scale 3, obtaining a balanced scale with a triangle on one side and three cubes on the other.

Vertical angles

- Vertical angles are made with 2 intersecting lines that cause two opposite pairs of congruent angles. - Congruent means equal, therefore, making angle 3 also measure 70°.

Formal assessment

- When students are graded on explicit criteria such as the written work of a worksheet or quiz, they are being given a formal assessment - occur whenever students know they are being assessed for a grade - are contrasted with informal assessments, such as when a teacher observes students' responses to questions posed during a class discussion

criterion-referenced assessment

- assessments where student achievement is measured against a particular set of (predetermined) standards, goals, or criteria. -the primary purpose of criterion-refernced tests is to determine whether or not students have acquired a specific set of knowledge of skills - when student achievement is measured against the desired outcome (and not viewed in comparison to other students' achievement) , it is a criterion-based assessment - An appropriate goal for all teachers, then, would be high scores for all students on all criterion-referenced assessments.

Summative assessment

- give information about what students have learned by the end of a chapter or unit, such as a midterm or final exam. - allow teachers, parents, students, administrators, or other stakeholders to measure and report the level of mastery that students have attained once the teaching and learning process in a topic have been competed. - typically use a grade or score that does not provide detailed information about what students do or do not know, but allows for comparisons among students within a particular course, graduating class, school district, county or even across the globe. - happen infrequently within academic year, and they can be high stakes, like the SAT or ACT.

Histograms

- represents quantatative data - used to summarize information from large sets of data - data is organized into intervals containing a range of numeric data values called bins or classes - BINS are labels given to the horizontal axis - FREQUENCY, or count of the data, within that bin is of the BIN is located on the vertical axis (y axis) - Bars have the same width, the area of each rectangle is proportional to the amount of data in that interval - BARS in histogram TOUCH EACH OTHER b/c the data they contain are quantitative - should be NO GAPS between bars

Analysis

- sometimes referred to as a "descriptive stage" - students in this stage of development begin to analyze shapes as having properties. - They can identify characteristics or attributes of shapes, and so atypical versions of shapes or unusual orientations of shapes should no longer confuse the learner. - Though students at this stage can learn and discuss properties of geometric figures, such as rectangles having 2 pairs of congruent sides that are parallel-to each other; their understanding is compartmentalized such that they do not recognize overlap between shapes - for EX: (so have a hard time recognizing that a square is a rectangle)

geoboard

- used to explore concepts of perimeter, area, angles, fractions, congruence, and more - use this type of manipulative to form line segments and polygons by stretching rubber bands around pegs on a board

Mrs. Granato would like to introduce her students to careers and professions which use math in the workplace. She has made arrangements for a banker to come in and discuss with her class the many roles a banker has within a bank. Which of the following TEKS might Mrs. Granato be able to apply to this lesson? Click each TEK below which could apply to this lesson. A. Develop a system for keeping and using financial records B. Represent categorical data with bar graphs or frequency tables and numerical data C. Balance a simple budget D. Define income tax, payroll tax, sales tax, property tax E. Identify the advantages and disadvantages of different methods of payment F. Describe actions that might be taken to balance a budget when expenses exceed income

A, B, C, E, and F

Simplify: 30 - 2 * 50 + 70 A. 0 B. 570 C. -210 D. -70

A. 0 - To simplify the equation, follow the order of operations: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction (PEMDAS) and work left to right. The steps to simplify the expression would be: 30 - 2 * 50 + 70 = 30 -100 + 70 = -70 + 70 = 0. - PEMDAS can be slightly deceiving. The order of operations is: Parentheses, then exponents, then multiplication and division, then addition and subtraction. - When performing multiplication/division, you perform the actions as they appear left to right; multiplication does not necessarily come before division - they have equal priority. The same goes for addition and subtraction.

Two angles are complementary. If the measure of one of the angles is 68°, what is the measure of the other angle? A. 22° because the sum of the measures of complementary angles is 90° B. 112° because the sum of the measures of complementary angles is 180° C. 68° because complementary angles are congruent to each other D. 292° because the sum of the measures of complementary angles is 360°

A. 22° because the sum of the measures of complementary angles is 90° - NOT 112º b/c it - names the supplement of 68 degrees. Two angles are SUPPLEMENTARY if their sum is 180 degrees. - COMPLEMENTARY angles are not congruent; congruent angles have the same angle in degrees. Meanwhile, being COMPLEMENTARY does not mean they are congruent; complementary angles are two angles whose sum is 90°. - 292º would be the solution if they asked for the EXPLEMENTARY angle. Angles are EXPLEMENTARY when they add together to equal 360°.

Simplify: 15 - 3(8 - 6)² A. 3 B. 540 C. 51 D. -21

A. 3 - To get the correct final value of 3, the Order of Operations must be followed. - In this problem, there are several operations occurring: subtraction, multiplication, and use of exponents. - Because the order of operations, commonly abbreviated by the acronym PEMDAS, requires that what is inside parentheses be addressed first, the first step in this problem is to perform the subtraction of 8 - 6 = 2. - From there, the problem becomes 15 - 3(2)². - Once the work inside of the parentheses has been completed, exponents should be handled next. In this case, 2 is raised to the second power, resulting in 15 - 3(4). - The subtraction and multiplication remain, but as "M" precedes "S" in PEMDAS, multiplication of the 3 with 4 must be completed before subtraction can occur, leaving 15 - 12 = 3. NOT- Note that a common error in this problem may include erroneously distributing the 3 from the front of the parentheses into the quantity "8 - 6" inside the parentheses. - Because distributing 3 is a form of multiplication, and because exponents must be used before multiplication can occur, the 3 cannot be distributed to the quantity in the parentheses but must be multiplied only after the power is raised.

What is the degree of the monomial 3x²y³? A. 5 B. 2 C. 6 D. 3

A. 5 - To find the degree of a monomial, add the number of variables in the monomial. - The monomial 3(x²)(y³) can be written: 3*x*x*y*y*y. There are five variables, so the degree of the term 3(x²)(y³) is five.

There are 5 children in the Drake family. The oldest children are 12-year-old twins, and the youngest child is 4 years old. If the sum of all the children's ages is 47, what could be the ages of the other 2 Drake children? A. 8 and 11 years old B. 9 years old C. 3 and 6 years old D. 7 and 11 years old

A. 8 and 11 years old - The sum of the ages of the five children is 47. - The twins are 12 (2 x 12 = 24) and the youngest child is 4 years old. So, for these three children we have used 24 + 4 = 28 of the sum of 47, leaving 19 as a sum for the ages of the two remaining children when each of the two children must be between the ages of 4 and 12. The only choice that sums to 19 is 8 and 11.

A tennis ball has a diameter of about 3 inches. What is the approximate volume of the cylindrical container if it holds three tennis balls? A. About 64 in³ B. About 27 in³ C. 108 in³ D. 82 in³

A. About 64 in³ - To find the volume of a cylinder, the B (area of the base) is multiplied by the height. The tennis ball can is three tennis balls high or about 9 inches. B, the area of the base, would be the area of the circle with the diameter of the tennis ball, or 3 inches. - If the diameter is 3 inches, the radius would be 1.5 inches and the area would be: B = A of circular base = πr² = π(1.5)² = π(2.25) ≈ 7.07 in². So, the volume of the cylinder would be: V = Bh ≈ 7.07(9) = 63.63 in³. 64 in³ is the best approximate answer to this question.

(Select all that apply) Which of the manipulative materials would be most suitable for teaching decimal notation to the hundredths place? A. Decimal squares B. Pattern blocks C. Base ten blocks D. Tangrams E. Color Tiles F. Geoboards

A. Decimal squares & C. Base ten blocks - Decimal squares are tag-board pictures of 10 x10 grids that have portions of the 100 smaller squares shaded. Students are asked to name the decimal represented by the shaded or unshaded area. They see that the sum of the shaded and unshaded areas always equals 100 hundredths or 1. - Base ten blocks are hands-on manipulatives consisting of a large cube (made up of 1000 smaller cubes), a flat (10 x 10 grid or a 100 square), a long (1 x 10), and a unit cube (1 x 1). Base ten blocks allow the representation of decimals from 0.001 (the smallest cube) to 1 whole (the largest cube).

The students in homeroom 232 are exploring equivalencies when an addend or minuend is missing. The teacher has given the students the following equation: 18 +__?__= 25 The teacher's goal is for the students to be able to explain how to solve this problem. Which set of questions could the teacher use to have the students think at a higher level on the Bloom's Taxonomy chart? A. How might we solve for this problem? Can you explain what would make this equation true? B. What number represents the question mark? Why? C. What is the solution? Why is that the solution? D. Why didn't you just subtract?

A. How might we solve for this problem? Can you explain what would make this equation true?

Mrs. Azul had each of her first graders separate a small bag of M&Ms into groups by color, arrange the groups into side by side bars, and determine which color they had the most of. What concept is Mrs. Azul introducing to her first grade class? A. Mode B. Mean C. Range D. Median

A. Mode - Mrs. Azul is having her students arrange their data (M&Ms) into bars in order to compare the heights or lengths of the bars. - This is a visual representation of the bar that is the longest or has the most M&Ms - the mode. - A good way for students to remember mode is to think about the word most. The mode is the piece of data (M&Ms in this case) that occurs the most frequently.

(Select all that apply) Using money in mathematical examples is a good strategy to promote student engagement in activities. A teacher decides to begin teaching about place value by using money, specifically with the example of 10 pennies = 1 dime and 10 dimes = $1. Why is this strategy probably not a good beginning strategy? A. The relationships above are too abstract for young learners. B. The coins are not proportional with respect to shape and size. C. Most young learners would rather have 8 pennies rather than 1 dime.

ALL- A. The relationships above are too abstract for young learners. B. The coins are not proportional with respect to shape and size. C. Most young learners would rather have 8 pennies rather than 1 dime. - Money can be a good motivator, but very young children do not have a grasp on the value. - They tend to be more interested in "how many" and not "how much." - There is also a problem because there is no proportional relationship. - Why are ten pennies worth less than ten dimes, especially when a penny is physically larger than a dime?

James has saved $35.25. He wants to save his money to buy a bicycle that costs $85.00. His brother's bike cost $92.00. If sales tax is 8%, about how much more must he save to purchase his bike, including tax? A. $55 B. $60 C. $50 D. $70

B. $60 - This is the correct answer. The math used: 8% is close to 10% sales tax on $85.00, or about $8.50 tax. So $85.00 + $8.50 = $93.50. - Notice this is an overestimate so James' target will be a bit more than he actually needs. James needs to save about $93. If he has saved about $35, he will need an additional $58. ($93 - $35 = $58) - Therefore, if rounded up this would be the best choice: $60. When dealing with money, generally an overestimate is more reasonable.

There is a 15% increase in tuition at UT for next fall. If the current tuition is $3,500 per semester, which equation could be used to find x, the new tuition for the fall? A. 0.15 • 3500 = x B. 1.15 • 3500 = x C. 0.85 • 3500 = x D. (15/100) = (x/3500)

B. 1.15 • 3500 = x 0.15 • 3500 = x tells us how much the increase alone is but not the total tuition. - The answer from this equation would have to be added to $3,500 in order to determine the new tuition. - NOT- 0.85 • 3500 = x would actually result in a decrease since 0.85 < 1, and 0.85 • 3500 < 3500. - NOT- When (15/100) = (x/3500) is simplified by cross multiplication, it becomes: 100x = 52500 and x, the new tuition, would be only $525 and definitely not a tuition increase.

A student tosses a six-sided die, with each side numbered 1 though 6, and flips a coin. What is the probability that the die will land on the face numbered 1 and the coin will land showing tails? A. 1/3 B. 1/12 C. 1/6 D. 1/4

B. 1/12 - The probability of getting any number on a six-sided die is one out of six (1/6), because there are six equally possible sides. - The probability of a coin landing on tails is one in two (1/2) because there are two equally possible outcomes. - To find the probability of a 1/6 instance happening at the same time of a 1/2 instance, simply multiply the probabilities together. This yields (1/6) * (1/2) = (1/12).

Simplify: 200 - 3(6 - 2)³ + 10 A. 174 B. 18 C. 246 D. -2

B. 18 - To get the correct final value of 18, the Order of Operations must be followed. - In this problem, there are several operations occurring: subtraction, multiplication, use of exponents, and addition. - Because the order of operations, commonly abbreviated by the acronym PEMDAS, requires that what is inside parentheses be addressed first, the first step in this problem is to perform the subtraction of 6 - 2 = 4. - From there, the problem becomes 200 - 3(4)³ + 10. Once the work inside of the parentheses has been completed, exponents should be handled next. In this case, 4 is raised to the third power, resulting in 200 - 3(64) + 10. - The subtraction, multiplication, and addition remain. As "M" precedes both "A" and "S" in PEMDAS, multiplication of the 3 with 64 must be completed before subtraction or addition can occur, leaving 200 - 192 + 10. - In this next stage, the fact that Subtraction is a form of Addition (subtraction is addition of opposites), and so Addition and Subtraction form a pair of equally ranked operations, means that the subtraction and addition in this problem are to be simplified in the order in which they appear when reading the problem from left to right. Accordingly, 200 - 192 + 10 = 8 + 10 = 18.

12π ÷ 9 is approximately equivalent to: A. 4/3 B. 4 C. 4π D. π

B. 4 - 4 is the correct answer. - Recall that a good estimation for π is 3. So, 12π ≈ 3. 12π ÷ 9 12 X 3 = 36 36 π ÷ 9 ---- ---- 1 1 36 π X 1 ---- ---- = 1 9 π 36 --- = 4 9

A small company employs 19 hourly wage workers. The hourly wage range is from $10 to $25 per hour. If three workers earn the median wage of $13.50 per hour, how many workers earn more than $13.50 per hour? A. 6 B. 8 C. 9 D. 11

B. 8 - Median is the number that is halfway into a number set. - This means that if 3 workers earn the median wage, then the remaining 16 workers are split, with half of the workers earning more than $13.50 an hour and the other half earning less than $13.50 an hour. - 8 of the 19 workers earn more than the median wage, if 3 workers earn the median wage.

What is the equation for the nth term of the arithmetic sequence -8, -6, -4, -2, 0,...? A. A(n) = -2n - 6 B. A(n) = 2n - 10 C. A(n) = -6n + 6 D. A(n) = 2n - 6

B. A(n) = 2n - 10 A(n) = 2n - 10 comes from seeing that a₁ = -8 and d = -6 - (-8) = +2, and then plugging those components into the formula A(n) = a₁ + d(n - 1) as A(n) = -8 + 2(n - 1), and then simplifying the statement as far as possible. From A(n) = -8 + 2(n - 1), distributing the +2 gives: A(n) = -8 + 2n - 2. Upon combining like terms (-8 and -2) and putting the variable portion first, the final answer is A(n) = 2n - 10.

What is the range and mode of the data set? 10, 8, 5, 3, 7, 4, 5, 9, 2, 3, 7, 3, 8, 6, 4, 1, 2, 1, 10, 3 A. Range: 10; Mode: None B. Range: 9; Mode: 3 C. Range: 10; Mode: 3 and 4 D. Range: 9; Mode: 3 and 4

B. Range: 9; Mode: 3 - The largest piece of data in this set is 10, and 1 is the smallest. - Therefore, the range = 10 - 1 = 9. The mode is 3 because there are more 3's (there are 4 of them) than any other number in the data.

A teacher tells the students the mode of the scores on the most recent exam was 85. Which of the following is the most accurate interpretation of the result? A. Half the students scored below 85 percent B. The most common score on the test is 85 percent C. The highest score on the test was 85 percent D. The average score on the test is 85 percent

B. The most common score on the test is 85 percent - This would be correct if the mode score was 85. Mode is the number that appears the most in a data set.

In a fifth grade math class, the teacher will be introducing representing real numbers as percentages. What prior knowledge of the students could the teacher draw on to relate percentages to real life situations? A. The teacher should help the students recall knowledge of fractions. B. The teacher should implement an activity relating fractions and decimals to percentages. C. The teacher does not need to access prior knowledge because students should be able to recall what they have already learned. D. The teacher should show a video of how to find percentages.

B. The teacher should implement an activity relating fractions and decimals to percentages.

A mathematics teacher noticed the students' performance on the recent assessment of the multiplication introductory unit was much lower than expected. Which of the following strategies could the teacher utilize to help the students understand the concept of multiplication? A. Require students to memorize the multiplication table for numbers 1 through 9 B. Use manipulatives to demonstrate that the product of 3 and 3 is the same as the sum of adding 3 together three times C. Have students complete a multiplication table at home with their parents D. Have students complete the multiplication table in groups

B. Use manipulatives to demonstrate that the product of 3 and 3 is the same as the sum of adding 3 together three times - This is a great activity that uses concrete manipulatives to demonstrate the concept of multiplication. - Students are able to scaffold their knowledge of addition into the concept of multiplication.

Each student has been given a small package of colored chocolate candies. The students will sort and classify their candies by color and the number of candies of each color. Students will then make a chart and a graph to show the number of each color in their packages. This activity supports which of the following TEKs? A. use data to create picture and bar-type graphs B. collect, sort, and organize data in up to three categories using models/representations such as tally marks or T-charts C. identify examples and non-examples of charting and graphing data D. draw conclusions and generate and answer questions using information from picture and bar-type graphs

B. collect, sort, and organize data in up to three categories using models/representations such as tally marks or T-charts

First grade students learning math fluency have been using several methods to support their learning. Which of the following would be the least effective method for first graders to learn math fluency? A. practice addition fluency with a hundreds chart B. write the fluency vocabulary C. practice adding with skip counting D. use concrete objects to add numbers

B. write the fluency vocabulary

Mr. Habib bought 8 gifts. If he spent between $2 and $5 on each gift, which is a reasonable total amount that Mr. Habib spent on all of the gifts? A. Under $10 B. $45 C. $32 D. More than $50

C. $32 - Mr. Habib spent at least $16 and at most $40: $16 if every gift cost exactly $2 and $40 if every gift cost exactly $5. - So, only amounts within this range are reasonable. - The first choice is too small while the second and fourth choices are too large. - Only this option falls within the designated range.

Mrs. Harris writes all the numbers from 4 to 24 on slips of paper and places them in a hat. She then asks a student to pick a number from the hat. What is the probability that the number chosen by the student will be a prime number? A. 1/24 B. 3/10 C. 1/3 D. 9/20

C. 1/3 - Mrs. Harris writes down 21 numbers (4-24). This number set has 7 prime numbers (5, 7, 11, 13, 17, 19, and 23). - That leaves 7 numbers out of 21 being prime. A student has 7 chances in 21 to choose a prime number. - This can be simplified to 1 out of 3, or 1/3.

Express 0.7083 as a fraction. A.708 1/3 B. 39 5/4 C. 17/24 D. It is a repeating decimal; impossible to write as a fraction.

C. 17/24

Which two numbers have a product of 56 and a quotient of 14? A. 7 and 8 B. 14 and 4 C. 28 and 2 D. 56 and 1

C. 28 and 2 - 7 and 8, 14 and 4, and - 56 and 1 all have a product of 56, - but only 28 and 2 has a quotient of 14.

Antone- Arm span= 36in / Height = X Antones's brother- Arm span = 56 in/ Height = 60in The ratio of Antone's height to arm span is the same as the ratio of his brother's height to arm span. About how tall is Antone? (View attachment for larger image) A. 42" B. 46" C. 39" D. 32"

C. 39" - The correct choice, 39", is found by solving the proportion: (56/60) = (36/x). - In this proportion, x is Antone's height. In the first ratio, 56 is Antone's brother's arm span and 60 is his height. - In the second ratio, 36 is Antone's arm span, and x, his height. - Proportions are solved by cross multiplying. When cross multiplication is performed on this proportion, we get: 56 x = 60 36. - When simplified, we get: 56x = 2160; then you divide each side by 56 resulting in x = 2160/56 = 38.57, but because the question says "about", you would use 38.57 39".

Students are working to solve the following question: ½ - x = ¼. The teacher then gives the following as an example: "If you are sharing a pizza with somebody and there is half a pizza left, how much must the other person eat so that you only have one-quarter of the pizza left?" As the teacher engages with several students, the teacher observes students are still having difficulty understanding the concept of fractions. The teacher then uses a pie chart to help explain the concept. Which of the following types of assessments has the teacher used? A. Summative B. Criterion C. Formative D. Formal

C. Formative - This is correct because a formative assessment involves teachers adjusting their instruction based on the assessment of students; a formative assessment helps form a teacher's instruction. - The teacher engages with the students, observes student difficulty with fractions, and then adjusts instruction.

Which answer above is the most appropriate for the following situation? Sixteen teachers placed a book order for books to be used in their classrooms. The bill, totaling $350, is to be shared equally. How much will each teacher pay? A. I B. II C. III D. IV

C. III - This situation involves money. We are looking for an answer that can be "translated" into money. - You can use this option by taking the exact answer of 21.875 and rounding it to $21.88.

The teacher provides a word problem for her students: Quacky Donald's Donuts sells glazed donuts in packages of six and donut holes in packages of 10. If Quacky Donald's Donuts sold the same number of glazed donuts and donut holes yesterday, what is the minimum amount of donut holes that Quacky Donald's Donuts sold? Based on the world problem above, which of the following concepts is the teacher most likely to cover in the lesson? A. Least common factor B. Greatest common factor C. Least common multiple D. Greatest common multiple

C. Least common multiple - The least common multiple will find the lowest number that will equally divide two integers. - By finding the LCM, students will be able to find the correct answer.

Jose bought lunch for himself and his brother and sister. His lunch cost $4.75, his brother's was $3.70, and his little sister's was $2.25. How much change should Jose receive from his $20 bill? A. One 5 dollar bill, four 1 dollar bills, two quarters, and two dimes B. Two 5 dollar bills, two quarters, one dime, and one nickel C. One 5 dollar bill, four 1 dollar bills, one quarter, and one nickel D. One 5 dollar bill, four 1 dollar bills, one quarter, three dimes, and three nickels

C. One 5 dollar bill, four 1 dollar bills, one quarter, and one nickel - Jose spent a total of: $4.75 + 3.70 + 2.25 = $10.70. So, he should have $20 - 10.70 = $9.30 in change. - Choice A adds up to $9.70, Choice B adds up to $10.70, Choice C adds up to $9.30, and Choice D adds up to $9.70.

In the pair of dice that Tim rolled 25 times, he recorded a sum of 4 on three of those rolls. What is the difference between the theoretical probability and the experimental probability of rolling a pair of dice and getting a sum of 4 based on Tim's experiment? A. The experimental probability and the theoretical probability are the same B. The theoretical probability is about 8% less than the experimental probability C. The experimental probability is about 4% greater than the theoretical probability D. The experimental probability is 3/25, but we don't know the theoretical probability for Tim's simulation

C. The experimental probability is about 4% greater than the theoretical probability - Theoretical probability is determined by analyzing the situation, in this case the rolling of a pair of dice, and determining the number of successful outcomes divided by the number of possible outcomes. - The experimental probability is the probability that occurred in this specific experiment. The theoretical probability of rolling a sum of 4 is PT(4) = 3/36 = 1/12, about 8.3%. - When Tim rolled his dice 25 times, he got a 4 on three of those rolls. While the theoretical probability of rolling a pair of dice and getting a sum of 4 is always about 8.3%, the probability that occurred in Tim's experiment, the experimental probability, is PE(4) = 3/25 = 0.12 = 12%. - This means his experimental probability was about 4% greater than the theoretical probability (12% - 8.3% = 3.7%, about 4%).

A teacher wants to help her students understand the rule of the order of operations. The student simplified the following expression: 3 * (8 - 2) + 6 (24 - 6) + 6 18 + 6 24 Which of the following best describes the student's work? A. The student performed the order of operations correctly B. The student subtracted at the inappropriate stage C. The student multiplied before simplifying the operation within the parenthesis D. The student simplified the operation from right to left

C. The student multiplied before simplifying the operation within the parenthesis - The student should have performed the operations in the parenthesis prior to multiplying. The order of operations is: Parentheses (left to right), Exponents (left to right), Multiplication and Division (left to right), Addition and Subtraction (left to right). This can be remembered with the acronym PEMDAS.

Susan surveyed the people leaving Mama Mia's Pizza Palace to determine America's favorite food when eating out. What is the best explanation for why the results of Susan's survey might NOT be valid? A. The survey is biased because it did not include everyone in America B. The survey should have been conducted in front of a different restaurant C. The survey is biased because it only interviewed people who had just eaten at a specific restaurant D. The survey should have been conducted online

C. The survey is biased because it only interviewed people who had just eaten at a specific restaurant - This gives the best explanation of why this survey is invalid: it only interviewed people who had just eaten at a specific restaurant, so the sample surveyed was definitely not random.

Two standard pieces of 8.5 x 11 paper are rolled into cylinders - open at both ends. One is rolled so that the height is 8.5 inches and the circumference is 11 inches and the other is rolled so that the height is 11 inches and the circumference is 8.5 inches. Which of the following statements is true about these two cylinders and their LA (lateral surface area)? (View attachment for larger image) A. Their LA is equal and the volumes are equal for A and B B. The LA for A is greater than B and the volume for A is greater than B C. Their LA is equal, but the volume for A is greater than B D. Their LA is equal, but the volume for B is greater than A

C. Their LA is equal, but the volume for A is greater than B - The lateral surface area is the surface area of the cylinder: the surfaces of the figure excluding the bases. The LA of figure A is: LA = Ch = 11π(8.5). - The LA of figure B is: LA = Ch = 8.5π(11). Using the associative and commutative properties for multiplication, both LAs are equivalent to: (11)(8.5)(π) and so they are equal. - The volumes are as follows: Figure A, V = Bh, where B is the area of the base. In this case the base, a circle, has a radius of 11/2 = 5.5, h = 8.5, and the area of the base, B, is equal to: B = π(5.5)² . - So, the volume of figure A becomes: V = π(5.5)2(8.5) ≈ 251.4π u³. For figure B, h = 11, the radius of the base is 8.5/2 = 4.25, and B = π(4.25)². - So, the volume for figure B becomes: V = πr²h = π(4.5)²(11) ≈ 198.7π u³. Since 251.4 π > 198.7 π, this means that the volume of cylinder A > volume of cylinder B.

The teacher is introducing ways to chart and graph the measures of central tendency. Which of the following are the most comprehensive options for the students to apply the central measures of tendency to creating charts and graphs? A. bar graphs, picture graphs, data tables, T charts B. circle graphs, dot plots C. frequency tables, dot plots, stem-and-leaf plots D. pie charts, frequency tables, stem-and-leaf plots

C. frequency tables, dot plots, stem-and-leaf plots

Mr. Andrews is in a city whose street grid can be represented by a coordinate plane. Mr. Andrews wants to walk from his current location at a local barbershop, represented on the coordinate plane as (1, 2), to his apartment. Mr. Andrews walks 5 blocks east, or to the right, and 1 block south, or down. Which of the following ordered pairs correctly depicts the location of Mr. Andrews' apartment? A. (6, 0) B. (0, 7) C. (2, 7) D. (6, 1)

D. (6, 1) - The first number in the coordinates represents the X-axis, the second number represents the Y-axis. - The X-axis runs horizontally and the Y-axis runs vertically. - If Mr. Andrews walks 5 blocks east, or right, then it would be correct to add 5 to 1, which would be 6. - If Mr. Andrews walks 1 block south, or down, then it would be correct to subtract 1 from 2, which would be 1. - The correct answer is (6, 1).

Which of the following lists all the factors of a composite number? A. 1, 7, 21 B. 1, 11 C. 2, 3, 6 D. 1, 5, 25

D. 1, 5, 25 - When listing all of the factors of a composite number, ALL factors means that the number itself and 1 must always be included. - Remember also that a composite number is a positive integer that has a positive divisor other than one or itself. - Students often find it helpful to list the factor pairs of the composite number. For example, 6 and 1; 2 and 3 - would be the factor pairs for 6. This means that all the factors for the composite number 6 would be: 1, 2, 3, and 6. - The factors are generally listed in ascending order and if a factor repeats, it is only listed once. The factor pairs for the composite number 25 are 1 and 25; 5 and 5. - Notice that 5 x 5 causes the 5 to be a repeated factor. So, when listing the factors of the composite number 25, we have 1, 5, and 25.

Tim rolls a pair of dice 25 times and records the sum of the numbers shown. How many different sums are possible? A. 12 B. 10 C. 36 D. 11

D. 11 - The possible sums when rolling a pair of dice are: 2, 3, 4, 5, 6, 7, 8, 9 10, 11, 12. There are 11 possible sums.

In the expression 3x² + 6x +3, how many terms are in the equation? A. 2 B. 1 C. 4 D. 3

D. 3 - A term is either a single number, a variable, or numbers and variables multiplied together. - The equation 3x² + 6x +3 has three terms: 3x², 6x, and 3.

Which of the following does not represent an explicit rule? A. # of Triangle # of Sides 1 3 3 9 4 12 7 21 B. In order to find the number of sides, multiply the number of triangles by 3 C. 3x = y D. 3 + 6 = 9

D. 3 + 6 = 9 Answer D is correct. The number sentence is not an explicit rule. It is an implied rule that each time 3 + 6 is added the sum will be 9. The correct answer is D.

The varsity basketball team has 3 freshmen, 5 sophomores, 3 juniors, and 4 seniors. Approximately what percentage of the basketball team is comprised of sophomores? A. 30% B. 25% C. 20% D. 33%

D. 33% - A total of 15 students are on the basketball team: 3 + 5 + 3 + 4 = 15. - There are 5 sophomores on the team. - The ratio of sophomores to the whole team can be represented by 5:15 = 1:3. 1/3 = .33 or 33%.

Jamie baked cookies to give to her friends. She gave 3 cookies to Anna and gave Elle 5 more than twice what was given to Anna. She gave half of what she had left to her best friend Grace. She now has 10 cookies. How many cookies did Jamie have to begin with? A. 18 B. 24 C. 30 D. 34

D. 34 A good strategy for working this problem is to work backward. Beginning at the end, Jamie ends up with 10 cookies after she gave ½ to Grace. - Before she gave any to Grace, she had 20. She gave Elle 5 more than twice what was given to Anna (3 were given to Anna - so twice 3 plus 5 more would be 6 + 5 = 11). - So, before any cookies were given to Elle, Jamie had 20 + 11 = 31 cookies. Before Jamie gave any cookies to Anna, she had 31 + 3 = 34 cookies. Jamie had 34 cookies to begin with.

How many terms are in the equation 4x - 4 = 6 * 9? A. 3 B. 2 C. 1 D. 4

D. 4 - A term is either a single number, a variable, or numbers and variables multiplied together. - The equation 4x - 4 = 6 * 9 has four terms: 4x, 4, 6, and 9. The terms 6 * 9 are counted as two separate terms because they are not combined with a variable; terms that multiply a variable AND a number are counted as one term.

A teacher provides students a table on the historical populations of the United States during the 19th century, divided by decade. Which of the following would be the most appropriate display for the information? A. A histogram B. A pie chart C. A Venn diagram D. A line graph

D. A line graph - A line graph is best used to identify the change of a variable over time, such as the change in population over the length of a century.

A fifth-grade student was asked to multiply 15 and 35. His answer was: 35 X 15 -------- 1525 35 ------ 1560 As his teacher, how would you begin remediation for this student? A. Flash cards to practice multiplication facts B. A quick remedial lesson on two-digit multiplication C. A remedial lesson on estimation and reasonableness D. A remedial lesson on place value

D. A remedial lesson on place value - This student has absolutely no problem with multiplication. All multiplication has been done correctly, therefore, flashcards to practice multiplication facts would not be productive or corrective. »»» - Two-digit multiplication might seem to be the primary problem, but if the teacher looks more closely, it seems there is more than just proper placement of digits in a long multiplication at play. - While a helpful and necessary tool, a remedial lesson would not address the fundamental misunderstanding in this problem.

Maria baked 6 dozen cookies for her classmates. There are 28 students in her class, each child received 2 cookies and Maria gave 6 cookies each to her teacher and her principal. Which equation could be used to find C, the number of cookies she had left over? A. C = 6 • 12 - (2 • 28 - 2 • 6) B. C = 6 • 12 - 2 • 28 - 6 C. C = 6 • 12 - 2 • 28 + 2 • 6 D. C = 6 • 12 - (2 • 28 + 2 • 6)

D. C = 6 • 12 - (2 • 28 + 2 • 6) - C = 6 • 12 - 2 • 28 + 2 • 6 is incorrect because it ADDS the number of cookies given to the teacher and the principal back into the equation "+ 2 * 6" instead of subtracting "- 2 * 6", therefore, the answer would be incorrect. - NOT C = 6 • 12 - (2 • 28 - 2 • 6) is algebraically the same equation as the first choice. The parentheses are needed to express the word problem, but because they include all of the numbers to be subtracted, the numbers within the parentheses are added to each other before their sum is subtracted in the overall equation from the original number of cookies. NOT- C = 6 • 12 - 2 • 28 - 6 is almost correct except that there are 6 cookies for the teacher and not for the principal - or maybe vice-versa.

Bobby is buying gumballs for 7 of his friends. There are 51 gumballs before Bobby makes his purchase at the store. Bobby wants to give each of his friends the same amount of gumballs and not have any gumballs left. Which of the following approaches can Bobby use to find the greatest number of gumballs he can purchase to give his friends? A. Divide 51 by 7 B. On a piece of paper draw 51 gumballs and then circle groups of 7 gumballs and then count how many gumballs are left not circled C. Create a table where one side of the table represents the number of gumballs and the other side represents the number of friends D. Make a list of the multiples of 7 and then purchase the highest multiple of 7 that is less than 51

D. Make a list of the multiples of 7 and then purchase the highest multiple of 7 that is less than 51 - This is the best answer as Bobby will know the number of gumballs to purchase.

A first grade teacher is planning a lesson to engage algebraic reasoning for the students. The teacher will first model two sides of an equation using base ten blocks then write the equation for the students to see the relationship between the model and the expression. What other model would be beneficial for students in making connections between the concrete and the abstract in algebraic reasoning? A. Figure model B. Equation model C. Math model D. Pictorial model

D. Pictorial model

Mrs. Johnson lets her students choose between two word problems: Problem A: If you are digging for dinosaurs and need to fence off your dig site, what's the biggest site you can fence off with 40 ft. of fence? Problem B: What is the largest area you can create with 20 inches of rope? Mrs. Johnsons finds a significant majority of her students chose to work Problem A. Which of the following is the most likely reason more students chose Problem A instead of Problem B? A. Problem A requires a lower mathematical knowledge B. Problem B is harder than Problem A C. Students who work Problem A get a greater reward from Mrs. Johnson than students who work Problem B D. Problem B is less interesting than Problem A

D. Problem B is less interesting than Problem A - This is the best answer. Students are more likely engaged when presented with a problem about digging for dinosaurs than a simple mathematical word problem. - The way problems are presented can impact students' engagement in the learning process.

Third grade students are exploring the area of rectangles. To provide a hands-on investigation, the teacher has provided rectangle templates in various shapes and sizes and square crackers for students to place the crackers inside the rectangles in a way that will fill each rectangle. Students will work in partners to fill each rectangle and record the amount of crackers it takes to fill a rectangle. The number of crackers will indicate the area of the rectangle. By exploring in this way, the teacher is not only providing a means for exploration but also a way for students to enrich their learning experience. To further the investigation and justify their solutions, what next step must students take? A. Move on to the next problem and solve for its area B. Record their solutions in their math journals C. Prove their solutions by comparing them with the findings of the teacher D. Prove their solutions by comparing them with other classmates and with the findings of the teacher

D. Prove their solutions by comparing them with other classmates and with the findings of the teacher

Mr. Barns wants to demonstrate socioeconomic diversity in his mathematics class. Which of the following activities would best allow Mr. Barns to demonstrate socioeconomic diversity for his students? A. Encouraging students to reduce their lunch budget so they can understand how lower socioeconomic students handle purchasing food B.Graphing the weekly allowance of students in the class with the cost of the clothes they wear C. Providing students the salary schedules of teachers at the school and comparing the average car driven by different teachers at the school D. Requiring students to create household budgets based on various income levels

D. Requiring students to create household budgets based on various income levels - Creating household budgets is the best activity because students will be able to alter different variables (i.e., housing, food, clothes, utilities) to understand how a lower income requires a lower standard of living and the tough choices families with lower incomes must make.

Mr. Williams teaches a 3rd grade math class specifically for English Language Learners. His students are representative of four different cultural and country backgrounds. Some of the students do not currently speak English fluently. While this does present a challenge for Mr. Williams when communicating with the class as a whole, he designs his instruction in such a way that each student feels valued and understood. Which approach may be most beneficial for Mr. Williams' students as he assesses them over a current concept? A. Assess the students then determine if the assessment was an accurate reflection of the material which had been taught. B. Administer the assessment to small groups of students who all speak the same language. C. Give every student the same assessment and plan to reteach the most commonly missed concepts to the whole class. D. Scaffold the assessment to the level of conceptual comprehension- concrete, semi-abstract, abstract- to assess each student's ability and knowledge of the concept.

D. Scaffold the assessment to the level of conceptual comprehension- concrete, semi-abstract, abstract- to assess each student's ability and knowledge of the concept.

Mr. Harris is planning to teach a unit on division to students for the first time. Which of the following would be the best first instructional lesson for Mr. Harris to present to his students? A. Having students memorize their multiplication tables B. Asking students their understanding of division prior to presenting the concept C. Presenting and solving a division problem using an abstract form or concept D. Solving a division problem using a concrete manipulative

D. Solving a division problem using a concrete manipulative - This is the best starting point for the instructional unit so that students have a concrete example of division before discussing the more abstract concept of division and the process of division.

Mr. Isaka wants to buy a new car for his wife and needs to borrow $16,500. The bank will loan him $16,500 that he must pay back in 48 equal monthly payments. The amount to be paid back will include the amount he borrowed plus interest. What other information is necessary to determine the amount of Mr. Isaka's monthly payment? A. The amount of Mr. Isaka's down payment B. The amount of Mr. Isaka's monthly salary C. The kind of car Mr. Isaka is purchasing D. The interest rate that the bank charges

D. The interest rate that the bank charges - What Mr. Isaka's down payment was is not relevant at this point. - We already know that Mr. Isaka needs to borrow $16,500 from the bank, so the down payment has already been considered. - Mr. Isaka's salary is not relevant at this point, although it probably was a factor in whether or not the bank would loan him the money to begin with. - The kind of car has nothing to do with the monthly payment amount. The interest rate will definitely affect the amount of money Mr. Isaka will have to repay the bank and, therefore, will impact the monthly payment.

Which of the following questions is written at the highest level of Bloom's Taxonomy? A. What is 7 - 3? B. What number must 3 be subtracted from in order to have 4 left? C. What is the difference between 7 and 3? D. What question could you ask that could be answered by subtracting 3 from 7?

D. What question could you ask that could be answered by subtracting 3 from 7? (EVALUATION) - This choice is written at the Application Level of Bloom's. - In this choice, the students must not only understand how to work the problem 7 - 3, but they must be able to demonstrate this knowledge by applying the problem to a situation where that problem might occur. (is concerned with the ability to judge the value of material (statement, novel, poem, research report) for a given purpose. The judgements are to be based on definite criteria. These may be internal criteria (organization) or external criteria (relevance to the purpose) and the student may determine the criteria or be given them. Learning outcomes in this area are highest in the cognitive hierarchy because they contain elements of all the other categories, plus conscious value judgements based on clearly defined criteria.) - "What is 7 - 3?" is a memory or recall question; this is the lowest level of Bloom's, the Knowledge Level. (KNOWLEDGE) - "What is the difference between 7 and 3?" is written at the Comprehension Level. The question asks in words rather than just symbols, giving a slightly different interpretation of the problem 7 - 3 = 4 by supplying the minuend and subtrahend and having the student still search for the difference. (COPREHENSION) - "What number must 3 be subtracted from in order to have 4 left?" is written at the Comprehension Level. The question asks in words rather than just symbols, giving a slightly different interpretation of the problem 7 - 3 = 4 by supplying the difference and subtrahend and having the student determine the minuend. (COMPREHENSION)

A class of second grade students are classifying two-dimensional polygons. Students are given a rectangle and asked to decompose it to form new two-dimensional polygons. Which of the following identifies the geometric parts that could NOT be formed from decomposing the rectangle? A. two right triangles, two trapezoids B. two congruent rectangles, 1 circle C. two congruent trapezoids that can be decomposed into two right triangles and two equilateral triangles D. two right congruent triangles and one parallelogram

D. two right congruent triangles and one parallelogram

If 15 ml is equivalent to ½ oz., which equation could be used to find x, the number of ml in 1 cup? A. x = 15 ÷ ½ + 8 B. x = 15 • ½ • 8 C. x = 15 ÷ 8 • ½ D. x = 15 • 8 ÷ ½

D. x = 15 • 8 ÷ ½ - There are 8 oz. in one cup, but the problem references ½ oz. - So, how many ½ ounces are in 8 oz.? 8 ÷ ½ = 16. If each ½ ounce is equivalent to 15 ml, then 15 • 16 would give us the number of ml in 8 ounces or 1 cup = 240 ml. x = 15 • 8 ÷ ½ = 120 ÷ ½ = 240 ml. - x = 15 ÷ ½ + 8 option gives us 38 ml, but the correct answer is 240 ml. - This leads to the wrong answer because the conversion was not done properly. - The correct equation representing the conversion is 15 * 8½. - x = 15 • ½ • 8 = 7.5 • 8 = 60 ml. - x = 15 ÷ 8 • ½ = 1.875 • ½ = .9375 or 15/16.

Which of these equations does NOT pass through quadrants III or IV? A. y = x - 2 B. y = -2x² + 3 C. y = x D. y = x²

D. y = x² - This is a problem that can be solved by graphing the four answer options. - Then note that the four quadrants are top right I, top left II, bottom left III, and bottom right is IV. - This also means that the (x,y) in each quadrant are as follows: I (+,+), II (-,+), III (-,-), and IV (+,-). - Therefore, quadrants III and IV are the bottom two quadrants which are below the x-axis; answer option II is the only option given that does not pass through these two quadrants.

Operations have inverses , other operations that "undo" them

EX: addition and subtraction EX: multiplication and division EX: squaring and taking a square root of pairs of inverse operations

Estimate

Estimates generally involve basic calculations and/or mathematical reasoning with rounded numbers EX: One could estimate the salary of someone earning $15 /hr by multiplying $15/hr by 40hours/week by 50 weeks/year 15 X 40 X 50 = 600 X 50 = 30,000 so someone making $15/hr probably makes about $30,000 per year This account doesn't take all 52 weeks into account, or unpaid holidays, taxes, possible union fees, etc

Exponent to 0

The value of any non-zero raised to the power of 0 power is always 1. EX; 9^0 = 1 EX: (-3.6)^0 = 1 EX: X ^ 0 = 1 The value of 0 is undefined

Vertical lines. or points that lie on the same vertical line-

Veritcal lines. or points that lie on the same vertical line- in not a FUNCTION!!

Rational #

a number that can be written in ration (fraction) format where both the numerator adn denominator of the fraction are integers (and the denominator is not 0) EX: 4/7 EX: - 3 2/3 EX: 192/1 Rational numbers can appear as integers, such as 192 as terminating decimals (decimals that stop like 0.47 or -6.125) non-terminating decimals repeating decimals (decimals that don't stop, but do have a pattern that goes on forever)

Natural numbers

are the set of positive integers 1, 2, 3, ....etc.

Mean

average

congruent angles

congruent angles - They don't have to point in the same direction. - They don't have to be on similar sized lines. - Just the same angle.

Probability

defined as the likelihood of an event occurring -Procedure to determine the probability of an event is to divide the number of ways that the particular event could occur by the total number of possible outcomes - this fraction is usually reduced and can be written in its fraction, decimal, or percentage formats - Formula: EX: Given a hat containing the names of each of the 22 students in a class, the probability of selecting a male students name from the hat, if there are 12 male students is: -Either 6/11 OR 0.545454...or...-54.5% can be said to be the probability of selecting one name from this bag and it being a male student. - answer comes from that there are 12 male students (12 successful outcomes) and 22 names in the bag b/c each of the 22 members of the class has his/her name in the bag (22 potential outcomes, one for each being drawn)

Median

middle number - if odd average 2 middle numbers

Irrational numbers

numbers that cannot be written as the ratio of 2 integers -square root of numbers that are not perfect squares EX: square root of 5 - and other famous irrational values are written with a letter to represent them such as PIE

Informal Deduction

sometimes referred to as "abstraction" - students at this stage not only know properties of objects, but they understand relationship between properties - b/c of this deeper grasp of the connections between shapes, students can apply what they know to begin to for definitions and also to explain why s square is a rectangle but not all rectangles are squares EX: why they reason in basic ways about geometric figures, students at this stage cannot follow a complex argument or truly reason deductively

Distributive property

the distributive property of multiplication OVER addition states that the result will be the same to multiply a quantity by a sum as it is to multiply that quantity by each value added to create the sum. EX: 4(2x+11) is the same as 4(2x) + 4(11) OR 8x + 44

Area

the measurement of how much flat surface is contianed within a 2-dimensional figure - such as - rectangle - triangle -circle AREA is measured in square units

Volume

the measurement of how much space is taken up by a 3-dimensional figure - such as - prism - pyramid - sphere VOLUME is measured in cubic units

Mode

value that appears the most - some sets of data have no mode (if all data values appear the same amount of times) - Some sets have more than one mode (if multiple pieces of data in a set appear the same number of times and appear more than any other data values) - would say both are modes of the data then

The square root function

y = √x y = square root of x

Multiplicative inverse

- a numbers multiplicative inverse is also known as its "RECIPROCAL" - it is the number that when multiplied with an original value makes a product of 1 EX: the multiplicative inverse of 3 is 1/3 EX: the multiplicative inverse of -5 is -1/5 EX: the multiplicative inverse of 2 1/2 is 2/5 (b/c the mixed number 2 1/4 is written as 5/2 it's an improper fraction or format) or 0.4

Which answer above is the most appropriate for the following situation? Sixteen teachers placed a book order for books to be used in their classrooms. The bill, totaling $350, is to be shared equally. How much will each teacher pay? 350/16 = ?

- This situation involves money. We are looking for an answer that can be "translated" into money. - You can use this option by taking the exact answer of 21.875 and rounding it to $21.88.

Tesselation

- A Tessellation (or Tiling) is when we cover a surface with a pattern of flat shapes so that there are no overlaps or gaps.

Additive inverse

- A numbers ADDITIVE INVERSE is also known as its "opposite the additive inverse is the number that, when added to an original value, makes a sum of 0 EX: the additive inverse of 2 is -2 EX: the additve inverse of -17.4 is 17.4

Inverse (slope intercept)

- An inverse occurs algebraically when the x's and y's are interchanged and the resulting equation is solved for y. - In this case, the inverse of the equation y = x + 2 would be found by swapping the x and y and solving for y. - So, we would have: x = y + 2 and solving for y we would have: y = x - 2. C is the correct answer.

What is the degree of the monomial 66? A. 3 B. 0 C. 2 D. 1

B. 0 - To find the degree of a monomial, simply add the number of variables in the monomial. - The monomial 66 has no variable, so the degree of the monomial is 0.

A baseball mitt is on sale for 30% off. If the regular price is $78, what is the sale price, excluding tax? A. $48 B. $54.60 C. $101.40 D. $23.40

B. $54.60 - There are a couple of different ways to work this problem. - You can simply take 30% of $78 (0.3 • 78 = 23.4 or $23.40) and find out how much you will save, then subtract the savings from the original price: $78 - 23.40 = $54.60. - Or, you can also obtain the answer by: 100% - 30% = 70% and 0.70 • 78 = $54.60.

What are the prime factors of 36? A. (2³) * (3²) B. (2²) * (3²) C. 2 * 3 D. (2²) * (3³)

B. (2²) * (3²)

ABC ISD is improving its curriculum needs through technology and hands-on experiences. Which of the following would provide a hands-on experience of being able to manipulate, rotate, transfer, and decompose three-dimensional shapes? A. cut-out images of the geometric shapes B. visual media C. a worksheet for students to complete about three-dimensional shapes D. two dimensional polygons

B. visual media

Which of the following is equivalent to the inequality? 5 > x -1 A. x < 7 B. x > 4 C. x < 6 C. x < 4

C. x < 6 - To simplify the expression 5 > x -1, add +1 to both sides of the equation: 5 + 1 > x -1 + 1 = 6 > x.

Composite numbers

Composite Number- Can be written as the product of at least one integer aside from 1 and itself. EX: 6 b/c not only can it be considered the product of 1 X 6, it is also composed of other numbers, the prime factors 2 x 3 EX: 27 is composite b/c , not only can it be considered the product of 1 x 27, but it's also composed of other numbers , such as 3 x 9. EX: 9 can be divided exactly by 3 (as well as 1 and 9), so 9 is a composite number.

convex and concave polygon

Convex polygon: - all interior angles measure less than 180 degrees -vertices appear to point out away from the figure - All triangles are CONVEX Concave polygon: - A polygon that has at least one interior angle that measures more than 180 degrees - As a result at least on vertex appears to be pointing inward or looks "caved" in

perpendicular lines

lines that form a 90 degree angle where they meet -

Range

subtract the smallest data value from largest data value

obtuse triangle

- Has an angle greater than 90 degrees - triangle is a 3 sided POLYGON - equilateral/equiangular triangles are always also ACUTE b/c their 3 interior angles each measure 60 degrees -

Bar graph

- used to represent data organized by categories such as countries, languages, colors, food groups, activities, etc. - the total number of pieces of data found in each category must be determined and then a bar is constructed with a height represetative of the amount of data in that category - vertical axis shows frequency - bars don't touch each other b/c the the data that they contain are not part of a continuous set (like the numeric data represented in a histogram)

The Booster Club at Martin MS is selling spirit buttons for homecoming. The buttons cost $0.75 to make and will be sold for $2 each. How many buttons, b, must be sold to make a profit of $500? A. $500 = $2b - $0.75b B. $500 = $2b + $0.75b C. $500 + $2b = $0.75b D. $500 - $0.75b = $2b

A. $500 = $2b - $0.75b - The profit is equal to the selling price minus whatever costs are applicable. - So, if the profit is to be $500, then enough buttons must be sold to reach that profit. - If we are selling the buttons for $2 each, but it costs $0.75 to make each one, then there is a profit of $2 - .75 or $1.25 on each button. - How many buttons will we have to sell to reach $500 profit: $500 = $1.25b. This is: $500 = $2b - 0.75 c = 1.25b. - So, the Booster Club will have to sell 400 buttons.

Sally and Susie place 30 white marbles and 9 black marbles in a sack. They play a game where the object of the game is to accumulate more white marbles than the other person. On each turn a person can take as many marbles as they would like, but if they grab a black marble then they must return all their marbles taken on that turn. On Sally's turn she picks up 3 white marbles and no black marbles. If Susie grabs one more marble, what are the odds that she will grab a black marble? (Note: Sally does not return her 3 marbles to the sack.) A. 1 in 4 B. 1 in 5 C. 1 in 3 D. Not enough information

A. 1 in 4 - To find the odds of Susie grabbing a black marble, add all the remaining marbles together and divide the black marbles by the remaining marbles. - Sally grabbed 3 white marbles, which leaves 27 white marbles in the sack. 27 white marbles plus 9 black marbles equal 36 total marbles. - Choosing a black marble out of all marbles can be represented as 9/36 = ¼.

Simplify: 100 ÷ 4 × 5 A. 125 B. 1,250 C. 5 D. 2,000

A. 125 - To get the correct final value of 125, the Order of Operations must be followed. - In this problem, there are two operations occurring: division and multiplication. - The order of operations is commonly associated with the acronym PEMDAS (Parentheses Exponents Multiplication Division Addition Subtraction). - A common misperception of the acronym is that it is to be taken completely literally with Multiplication always preceding Division and Addition always preceding Subtraction. However, Division is a form of Multiplication (division is multiplication by a reciprocal) and Subtraction is a form of Addition (subtraction is addition of opposites). - Therefore, both pairs of operations (Multiplication & Division and also Addition & Subtraction) are pairs of equally ranked operations that happened to have been named in an arbitrary order to make a pronounceable and memorable acronym for the Order of Operations. "PEMDAS" could have been "PEDMAS" or "PEDMSA" or "PEMDSA" and have the exact same meaning. - Accordingly, when both of two equal ranked operations are present in one problem, they are to be simplified in the order in which they appear when reading the problem from left to right. - In this case, division is encountered before multiplication and so the problem 100 ÷ 4 × 5 becomes 25 × 5 = 125.

Which of the following word problems is the correct question to be written and solved with the given equation? 8 - 5 + 3 = 6 A. 8 birds were sitting on a fence. 5 birds flew away and 3 more birds landed on the fence. How many birds are on the fence now? B. 6 children were sitting at a table. 5 got up and left and 3 more sat down. How many children are now at the table? C. Molly's mom made 8 hot dogs for the BBQ. 5 were eaten by her cousins and 3 by her and her two brothers. How many hot dogs are left? D. A dog named Daisy had a litter of 8 puppies, then a litter of 5 puppies. 3 of the puppies were adopted and taken away. How many puppies does Daisy have left?

A. 8 birds were sitting on a fence. 5 birds flew away and 3 more birds landed on the fence. How many birds are on the fence now? - A really good strategy for helping children understand what the numbers represent is for them to explain what their equations actually mean, term by term. - Learning how to "translate" the mathematics in a situation into a meaningful and correct equation is very abstract and children need to be able to explain what the numbers and symbols mean in the context of the problem being posed. - In first grade when students are just beginning to write number representations for a situation, they need to be able to explain, in words, what the numbers mean in the context of the problem they are solving. - You ask them to explain what each of the numbers mean and why there is an + or - between them. - Students need to understand that the numbers and symbols in an equation or an expression are not arbitrary; the numbers mean something specific and so does the order in which the numbers appear in the equation/expression and the mathematical symbols that are used. This is the correct word problem for the equation.

A third-grade class begins working on a mathematics project at 9:50 a.m. and stops working on the project at 11:10 a.m. How many minutes did the class work on the project? A. 80 minutes B. 20 minutes C. 60 minutes D. 100 minutes

A. 80 minutes

What is the value of the "8" in the number 17,436,825? A. 800 B. 80 C. 8 D. 8,000

A. 800

What is the 18th term of the arithmetic sequence -13, -9, -5, -1, 3,...? A. A(18) = 55 B. A(18) = 59 C. A(18) = -81 D. A(18) = -153

A. A(18) = 55 - A(18) = 59 comes from forgetting the "- 1" part of the "n - 1" component of the formula. - A(18) = -81 comes from mistaking d = +4 for d = -4. - A(18) = -153 comes from inappropriately combining the -13 with the +4 after seeing the statement A(18) = -13 + 4(18 - 1). - Because the -13 represents an amount of ones and the 4 represents an amount of 17s (from 18 - 1), they are not like terms and so cannot be added. - Also, multiplication comes before addition in PEMDAS, so the 4, which is involved in multiplication with the "18 - 1" and also addition with the -13 must be used in the multiplication step first.

Mallory has $2.16 in her pocket to buy an apple and a bag of chips. What information is needed to determine how much money Mallory will have left after she makes her purchase? A. The cost of the apple and the chips B. How many coins she has remaining C. How many apples Mallory has in all D. How many coins she used to make her purchase

A. The cost of the apple and the chips -Nowhere in the problem is the cost of the apple and chips mentioned. - Until we know how much she spent, we cannot figure out how much money she has left.

Mrs. Brooks is a first-grade mathematics teacher. She wants to incorporate workstations into her lesson. She sets up the following stations: Station 1: Students toss two dice and record the numbers on each die plus the sum of the two dice. They repeat the process ten times. Station 2: Students build a tower consisting of nine cubes and each cube must have either a red or blue color on a side. Students then count the number of red sides and blue sides on each side of the tower. Station 3: Two students place 13 marbles on the table. The students take turns removing from 1-12 marbles from the table and the other student has to figure out how many marbles the other student removed. The students then record the two numbers. Which of the following concepts is Mrs. Brooks most likely trying to explore with the workstations for her students? A. Part-part-whole B. Spatial concepts C. Benchmarking numbers D. One more and one less

A. Part-part-whole - This is correct because in this activity the students are conceptualizing that a number is made up of two or more parts.

Mrs. Peters teaches a class with native English speakers and English-language-learners (ELL). She needs to introduce new mathematics terminology for the upcoming instructional unit. Which of the following would be the best strategy for implementing the new terminology? A. Mrs. Peters explains each term to the class and then uses the term in a variety of sentences B. Have each student write down the new word with the corresponding definition C. Have students practice using the terminology with each other using English in their conversation D. Because mathematics terminology can be difficult, have students create their own spelling for the new terminology

A. Mrs. Peters explains each term to the class and then uses the term in a variety of sentences - Explaining each term and using them in a sentence is the best strategy because Mrs. Peters will present the terms in context and then provide various examples. - This will provide both the native English speakers and ELL students an opportunity to be exposed to the new terms in a variety of contexts. - Explaining each term and using them in a sentence is the best strategy because Mrs. Peters will present the terms in context and then provide various examples. - This will provide both the native English speakers and ELL students an opportunity to be exposed to the new terms in a variety of contexts.

Which of the following learning goals is most appropriate when teaching a class of third-grade students about money? A. Students will be able to identify and determine the value of a specific coin or bill B. Students will understand the progressive system to collect taxes in the United States C. Students will be able to divide units of money to the penny D. Students will be able to calculate the basic interest payment on a mortgage

A. Students will be able to identify and determine the value of a specific coin or bill - This is the most appropriate learning level for a third-grade classroom.

A student asks the teacher who invented the number system. Which of the following answers would be most appropriate? A. The base-ten number system was developed by the Hindu-Arabic civilizations B. The current number system was developed by the Greek and Roman empires C. The base-ten number system was invented by Isaac Newton in the late 17th century D. The current number system has evolved over a period of thousands of years and each culture contributed to its development

A. The base-ten number system was developed by the Hindu-Arabic civilizations

Using the Present Value Formula, students have calculated a house payment for a loan set up on an annuity. The principal is $127,000 and the monthly payments are $583.56. The students want to know how much interest over the period of the note (30 years) will accrue. They decide the first step is to calculate the total amount to be invested in the house. For this, they determine the following equation should be used. Payment amount x Number of payments = Total amount invested What should the next step of the algorithm be? A. Total amount invested - Principal B. Total amount invested (Payment amount * Number of payments) C. Principal - Total amount invested D. Impossible to determine the next step without knowing the interest rate

A. Total amount invested - Principal

A mathematics teacher noticed the students' performance on the recent assessment of the multiplication introductory unit was much lower than expected. Which of the following strategies could the teacher utilize to help the students understand the concept of multiplication? A. Use manipulatives to demonstrate that the product of 3 and 3 is the same as the sum of adding 3 together three times B. Require students to memorize the multiplication table for numbers 1 through 9 C. Have students complete a multiplication table at home with their parents D. Have students complete the multiplication table in groups

A. Use manipulatives to demonstrate that the product of 3 and 3 is the same as the sum of adding 3 together three times This is a great activity that uses concrete manipulatives to demonstrate the concept of multiplication. Students are able to scaffold their knowledge of addition into the concept of multiplication.

Mrs. Brown would like for her students to use information they have collected in the form of a frequency table on population of cultures within the school to create a scatter plot. From the scatter plot, she plans for the students to show the normal distribution of the population of cultures and draw inferences about the populations of the school. What information would the students need to collect in order to be able to show the normal distribution Mrs. Brown is requesting? A. the number of students indicating their association with each culture B. number of total students in the school and the number of cultures within the school C. types of cultures within the school D. types of students within the school

A. the number of students indicating their association with each culture

Which of the equations, when graphed, will contain the ordered pair (1,1)? (Select all that apply) A. y = -2x² + 3 B. y = x - 2 C. y = x D. y = x²

A. y = -2x² + 3 C. y = x D. y = x² When you substitute (1,1) for (x,y), this is the only option that is NOT true (B)

A baseball mitt is on sale for 30% off. If the regular price is $78, what is the sale price, excluding tax? A. $23.40 B. $54.60 C. $101.40 D. $48

B. $54.60 - $23.40 is the amount saved, not the sale price. By multiplying 78 by .3, one can arrive at this number. - However, this number should be subtracted from 78 to find the sale price. »»» ««« $48 results from subtracting: 78 - 30 = 48. - If this choice is selected as being correct, there is a basic misunderstanding about exactly what percent is AND a misunderstanding about the adding/subtracting only of LIKE units. - Here, a percentage (%) is being subtracted from dollars ($): % and $ are unlike units. »»» ««« $101.40 results from adding the savings to the regular price. - This error might indicate a misunderstanding of exactly what the 23.4 represented (amount saved), or, quite possibly, the student was working too quickly and simply added rather than subtracted.

James has saved $35.25. He wants to save his money to buy a bicycle that costs $85.00. His brother's bike cost $92.00. If sales tax is 8%, about how much more must he save to purchase his bike, including tax? A. $55 B. $60 C. $50 D. $70

B. $60 - The math used: 8% is close to 10% sales tax on $85.00, or about $8.50 tax. - So $85.00 + $8.50 = $93.50. Notice this is an overestimate so James' target will be a bit more than he actually needs. James needs to save about $93. If he has saved about $35, he will need an additional $58. ($93 - $35 = $58) - Therefore, if rounded up this would be the best choice: $60. When dealing with money, generally an overestimate is more reasonable.

The state sales tax is 7.5%. Which number could also represent 7.5%? A. 0.0075 B. 3/40 C. 3/4 D. 0.75

B. 3/40 - 7.5% means 7.5 parts of 100 or 7.5/100. 7.5/100 = 75/1000 or 0.075. - Unfortunately, 0.075 is not a choice, but it does rule out 0.0075. - 3/4 is equivalent to .75 or 75%; 75% is not the same as 7.5%. - .5% means 7.5 parts of 100 or 7.5/100. 7.5/100 = 75/1000 or 0.075. Unfortunately, 0.075 is not a choice, but it does rule out 0.75.

(picture of 100 squares with 48 squares shaded) If each square in the decimal square has a value of 0.1, then which of the following is the decimal numeric representation of the shaded area? A. 48 B. 4.8 C. 0.48 D. 0.048

B. 4.8 - 48 is the total number of squares shaded, but this is not the decimal that represents the value of the shaded area. - 0.48 does not multiply by the value per square of 0.1 but instead views the decimal square with its whole being 1 - which assigns the value of 0.01 to each small square because each small square is 1/100 of the whole. -0.048 does not display that the value per square is .1 but instead .001. - Therefore, this is not the correct answer.

Order the following numbers from greatest to least: -2, ½, 0.76, 5, √2, π. A. 5, π, √2, 0.76, -2, ½ B. 5, π, √2, 0.76, ½, -2 C. -2, 0.76, ½, √2, π, 5 D. -2, ½, 0.76, √2, π, 5

B. 5, π, √2, 0.76, ½, -2 Pie = 3.15 1/2= 0.5 square toot of 2 = 1.14

Below is an example of a student's work: 4/16 - 1/8 = 3/8 10/13 - 3/8 = 7/5 3/5 - 2/3 = 1/2 If the student continues making the same error, the student's most likely answer to the problem 9/16 - 3/4 would be? A. 12/20 B. 6/12 C. 3/5 D. 0

B. 6/12 - In each example the student is subtracting both the numerators and denominators. - When subtracting fractions, the student should find a common denominator and then subtract only the numerators. - The student is not finding a common denominator but subtracting both the numerators from each other and the denominators from each other. - Thus the most likely answer if the student performed (9/16) - (3/4) would be (6/12).

What is meant by the term "fact family"? A. A set of math facts that all have one common addend B. An addition fact plus the additional addition fact and two subtraction facts that relate to it C. A set of math facts that have the same sum D. A subtraction fact and the two addition facts that are related to it

B. An addition fact plus the additional addition fact and two subtraction facts that relate to it - A set of facts that have the same largest number (sum) is called a number family. 0 + 5 = 5; 5 + 0 = 5; 5 - 0 = 5; 1 + 4 = 5; 4 + 1 = 5; 5 - 1 = 4; 5 - 4 = 1; 2 + 3 = 5; 3 + 2 = 5; 5 - 3 = 2; and 5 - 2 = 3 make up the number family for the number 5. - A fact family is a subset of a number family. - A set of math facts that all have one common addend is not the correct definition of a fact family. - It is important to note that a subtraction fact has only one addition fact to which it is related, not two, meaning that this choice is not an accurate statement.

Which of the following equations is written in slope-intercept form? A. 3y - 5x = 10 B. y = 3x + 5 C. y + 5 = 3x D. x = 3y + 5

B. y = 3x + 5 - Slope-intercept form is y = mx + b where M can be a numerical value and B is a constant. - Or described another way, M is the slope of x and y is the point where the line intercepts with the y-axis. - This is why it is called slope-intercept form.

Mrs. Summer's students are having difficulty with the concept of multiplication. She wants to use calculators to help students better understand the concept of multiplication. Which of the following would be the most appropriate activity for Mrs. Summer to provide her class? A. Have students add together the ages of each of their classmates to find the average age of the class B. Have students type 8 + 8 + 8 + 8 + 8 and then 8 x 5 and then write down why the values of the two equations are equal C. Have students race against the other classmates to answer equations that Mrs. Summer writes on the board D. Calculators cannot be used to enhance the students' conceptual understanding of multiplication

B. Have students type 8 + 8 + 8 + 8 + 8 and then 8 x 5 and then write down why the values of the two equations are equal - Having students are able to visually see and manipulate the same mathematical problem using addition and multiplication methods is the best way.

Which situation could best be represented by the equation: 12x = 54? A. Marty earns $12 for typing a paper. If her rate is $54 per hour, what is x, the number of hours it actually took to type the paper? B. Marty made car payments on her car for 54 months until it was paid off. What is x, the number of years it took Marty to pay off her car? C. Marty had 54 minutes left on his cell phone plan. If he uses 12 minutes, what is x, the number of minutes remaining on his cell phone plan? D. Marty collected 12 dozen eggs every day for 54 days. What is x, the total number of dozens of eggs she collected?

B. Marty made car payments on her car for 54 months until it was paid off. What is x, the number of years it took Marty to pay off her car? - This is the correct situation for the given equation. - 54 months divided by 12 months per year gives us 4½ years.

Mrs. Green is introducing subtraction to her kindergarten students. Mrs. Green will provide blocks for her students to use to indicate separating to represent subtraction. The lesson sequence will follow a whole group lesson, small group lessons, partner work and independent student work. During the initial small group lessons, which of the following will be important for Mrs. Green to link to the concrete representation of the blocks as she checks for understanding with the students? A. Blocks and number sentences B. Pictorial models and blocks C. Number sentences and spoken words D. Counters and number sentences

B. Pictorial models and blocks

Students are learning about probability. The teacher explains the theoretical probability of a two-sided coin landing on any one side. Students then work in groups and flip a two-sided coin several times. The first group of students flips the coin 20 times and finds the coin lands on side A 13 times and side B 7 times. Which of the following observations, based on the scenario, about probability is accurate? A. The experimental probability of a coin landing on side A was less than the theoretical probability B. The theoretical probability of the coin landing on side A is less than the experimental results recorded C. The experimental would have produced results closer to the theoretical probability if the number of trials was decreased D. The theoretical probability of the coin landing on side B is .5, which is equal to the results of the experiment

B. The theoretical probability of the coin landing on side A is less than the experimental results recorded - The students' coin landing on Side A was greater than the theoretical probability, which would be 10 times.

Given two even integers, a and b, determine what could be the least common multiple (LCM)? A. ab B. ab⁄2 C. Same as the least common multiple for two odd integers. D. greatest common factor

B. ab⁄2 Answer B is correct. Of the choices given, ab⁄2 is the LCM. Since both numbers are even, both have factors of 2. So, the product can be reduced by 2. If the numbers have only one factor of 2 in either of them such as in 6 and 16, then 96 is the product and a multiple. However, if 96 is reduced by 2, then 48 is the LCM.

Determine the choice that illustrates the commutative property for (a + b) + c = A. a + (b + c) B. c + (a + b) C. a + b + c D. c (a + b)

B. c + (a + b)

Simplify: 10 - 3 + 4 A. 3 B. -9 C. 11 D. 9

C. 11 - To get the correct final value of 11, the Order of Operations must be followed. In this problem, there are two operations occurring: subtraction and addition. - The order of operations is commonly associated with the acronym PEMDAS (Parentheses Exponents Multiplication Division Addition Subtraction). - A common misperception of the acronym is that it is to be taken completely literally with Multiplication always preceding Division and Addition always preceding Subtraction. However, Division is a form of Multiplication (division is multiplication by a reciprocal) and Subtraction is a form of Addition (subtraction is addition of opposites). - Therefore, both of these pairs of operations (Multiplication & Division and also Addition & Subtraction) are pairs of equally ranked operations that happened to have been named in an arbitrary order to make a pronounceable and memorable acronym for the Order of Operations. - "PEMDAS" could have been "PEDMAS" or "PEDMSA" or "PEMDSA" and have the exact same meaning. - Accordingly, when both of two equal ranked operations are present in one problem, they are to be simplified in the order in which they appear when reading the problem from left to right. In this case, subtraction is encountered before addition and so the problem 10 - 3 + 4 becomes 7 + 4 = 11.

Larry started the following number pattern: 900, 888, 876, 864, ... Which number could not be a part of Larry's pattern? A. 756 B. 816 C. 736 D. 624

C. 736 - Subtracting by 12, 816 is the 8th number in the sequence. - Subtracting by 12, 756 is the 13th number in the sequence. - Subtracting by 12, 624 is the 24th number in the sequence.

In what context is it important that students be exposed early on to diversity in acquisition of mathematical concepts? A. All students need to be exposed to mathematical concepts across cultures in order to apply the knowledge within cultural settings. B. All students need to be exposed so STEM classes will take off and flourish within all schools. C. All students need to be exposed so that more women and minorities will pursue mathematical careers and broaden the field of experts across cultures. D. All students need to be exposed so as to limit white males from leading mathematical ideas into new frontiers.

C. All students need to be exposed so that more women and minorities will pursue mathematical careers and broaden the field of experts across cultures.

Mrs. Campbell is teaching a lesson on slope-intercept form. She requires each student to create a formula that represents a graph they find visually interesting. Once each student creates a formula, she has the student present and explain his equation and graph to the class. Which of the following learning theories best matches the activity Mrs. Campbell uses with her students? A. Social learning theory B. Behaviorism learning theory C. Constructivist learning theory D. Sociocultural learning theory

C. Constructivist learning theory - Constructivist learning theory proposes that students learn by combining knowledge and meaning through interactions with experiences and ideas. - Having the students research, create a presentation, and then present that idea to the class is constructivist because the student is provided many opportunities to interact with his idea in multiple contexts.

Some friends had a contest to guess the number of M&M's in a jar without going over. Tina guessed 270, Terry guessed 50% more than Tina, Twila's guess was twice Tina's, and Toby guessed 500. There were actually 450 M&Ms in the jar. Which friend guessed closest to the correct answer? A.Tina B. Twila C. Terry D. Toby

C. Terry - Tina guessed 270, Terry guessed 50% more than Tina - 270 + 135 = 405, Twila guessed 2 x 270 = 540, and Toby guessed 500. - Terry's guess of 405 is 45 M&Ms under the actual number while Toby is 50 M&Ms over. So, Terry is the closest.

Mrs. McCoy writes two equations on the board for her class. +5 - 5 = 0 -20 + 20 = 0 Which of the following properties of integers is Mrs. McCoy most likely trying to illustrate? A. The associative property of addition B. The commutative property of addition C. The additive inverse property D. The additive identify property

C. The additive inverse property The additive inverse property is that each number has an inverse that, when added to the number, equals 0: -a + a = 0; a - a = 0. Combining a number with its additive inverse yields the answer "0".

John made a circular garden in his back yard. The garden has a diameter of 20 feet. He used ⅓ of the garden for tomatoes, his favorite vegetable. He enclosed the entire garden with a picket fence that was 12 inches high. Which of the following questions could NOT be answered with the information provided? A. How many feet of fence does John need? B. What is the area of John's garden? C. What is the volume of the dirt in the garden? D. What is the area of the tomato patch?

C. What is the volume of the dirt in the garden? - The number of feet needed to enclose the circular garden would be the circumference of the circle with diameter 20 ft. C = 20π ≈ 63 feet of fence. - The area of the garden would be A = πr² = π10² ≈ 314 ft². - A = πr² = π10² ≈ 314 ft², therefore, the area of the tomato patch would be ⅓(A) = ⅓(314) ≈ 104.7 ft².

The greatest benefit of providing elementary students with mathematical tools such as geoboards is that these tools — A. increase student's awareness of the practical applications of mathematics. B. facilitate student's activation of their own prior knowledge with regard to mathematics activities. C. provide students with visual representations that promote their conceptual understanding. D. prompt students to engage in mathematical games and activities outside the classroom.

C. provide students with visual representations that promote their conceptual understanding.

Personal financial literacy was added to the TEKS in 2012 so that students might learn to apply "mathematical process standards to manage one's financial resources effectively for lifetime financial security." (TEKS, 2012) Which of the following would NOT relate to one's financial security? A. fixed and variable expenses B. balanced budget C. volunteer jobs D. distinguish between wants and needs

C. volunteer jobs

Which of the following points on a number line is the greatest distance from .5? A. 0 B. 1.5 C.-1.5 D. 1

C.-1.5 -1.5 is 2 units away from .5 on the number line.

Sheila has a large collection of stickers. She gives ½ of her collection to Sue, ½ of what is remaining to Sandra, and then gave ⅓ of what was left over to Sarah. If she has 30 stickers remaining, how many stickers did she begin with? A. 90 stickers B. 270 stickers C. 120 stickers D. 180 stickers

D. 180 stickers - This is another problem that can be worked backwards. - Sheila is left with 30 stickers after she gave ⅓ of what she had to Sarah. - That means that 30 stickers represent ⅔ of what Sheila had before she gave any stickers to Sarah. 30 is ⅔ of 45; so, Sheila had 45 stickers before she gave any to Sarah. - 45 is half of what was left when half of the collection was given to Sandra. - This means that Sandra received 45 stickers and that Sheila had 90 stickers before she gave any to Sandra. - 90 stickers is how many Sheila had after she gave ½ of what she had to Sue; this means that Sue received 90 stickers and that Sheila had 180 stickers before she gave any away to Sue.

In the expression 3x² + 6x +3, what is the degree of 3x²? A. 3 B. 4 C. 1 D. 2

D. 2 - To find the degree of a monomial [(3x²) is a monomial - simply a single term in an equation], add the number of variables in the monomial. - The monomial 3x² can be written: 3*x*x. There are two variables, so the degree of the term 3x² is two.

What are the prime factors of 18? A. (2²) * (3²) B.(2²) * 3 C. 2 * 9 D. 2 * (3²)

D. 2 * (3²)

All of the following equations, when simplified, equal the same value except one. Which of the following equations, when simplified, is NOT equal to the other equations? A. (20 / (1/5) ) - (5 / (1/23) ) B. .25 * 100 - 40 C. 200 * (1/8) - 2(40-20) D. 2(40 * .2) + 5(.4 * (1/4) )

D. 2(40 * .2) + 5(.4 * (1/4) ) NOT D b/c- 2(40 * .2) + 5(.4 * (1/4) ) = 2(8) + 5(.1) = 16 + .5 = 16.5

Students learning to solve for missing values through input/output tables which increase the output (or y value) by the same value are laying a foundation to be able to work with which kind of function as they mature in their mathematical knowledge? A. Rational functions B. Quadratic functions C. Nonlinear functions D. Linear functions

D. Linear functions

model of 4 hexagons - 8 squares = 1 hexagon +m 1 square A model of an equation is above. What is the value of M? A. M = 2 B. M = 4 C. M = 5 D. M = 3

D. M = 3 - The equation modeled is: 4M - 8 = M + 1. - Solving this equation, we get: 3M = 9 or M = 3. - Replacing the values given in each choice can also solve the problem. If M = 3, the model becomes 4(3) - 8 = 3 + 1 or 4 = 4, which is a TRUE statement.

A student walks into class and tells the teacher that a local candy store had a "buy one candy bar, get one candy bar free" promotion this weekend. The teacher asks the student how many candy bars she purchased, but the student was only able to remember how many candy bars she received. The teacher uses this opportunity to create a word problem for the class. She asks the class to create an equation that would help the student find out how many candy bars a person would receive (r) if for any candy bar they purchased (p), they received one candy bar for free. One student, Sally, says the equation would be 2p + 1 = r. Which of the following activities would best help Sally recognize the error in her answer? A. Sally's equation is the correct equation B. Create a model of the equation C. Graph the equation D. Create a function table

D. Create a function table - This is the correct answer because it will clearly show the relationship between different pairs and allow the students to see the pattern of the equation.

Fifth graders at Smith Elementary were introduced to number bonds as first graders. Which of the following, when applied to fifth grade concepts, would relate back to number bonds? A. Shaded regions B. Patterns C. Fraction strips D. Diagrams

D. Diagrams

A mathematics teacher determines that the median score for the most recent test was 80 percent. Which of the following is the most accurate interpretation of the result? A. The most common score on the test is 80 percent B. The highest score on the test was 80 percent C. The average score on the test is 80 percent D. Half the students scored below 80 percent

D. Half the students scored below 80 percent - Median is the middle number in a data set. It is calculated by arranging the numbers from least to greatest and finding the number in the middle; if there is not one middle number, then the median is the average of the two numbers. - For example, in the data set 1, 2, 3, 4 the median is 2.5 because 2.5 is the average of 2 and 3. If the median score is 80, then half the students scored below 80 and half the students scored above 80.

Bobby is buying gumballs for 7 of his friends. There are 51 gumballs before Bobby makes his purchase at the store. Bobby wants to give each of his friends the same amount of gumballs and not have any gumballs left. Which of the following approaches can Bobby use to find the greatest number of gumballs he can purchase to give his friends? A. Divide 51 by 7 B. On a piece of paper draw 51 gum-balls and then circle groups of 7 gum-balls and then count how many gum-balls are left not circled C. Create a table where one side of the table represents the number of gum-balls and the other side represents the number of friends D. Make a list of the multiples of 7 and then purchase the highest multiple of 7 that is less than 51

D. Make a list of the multiples of 7 and then purchase the highest multiple of 7 that is less than 51 - This is the best answer as Bobby will know the number of gum-balls to purchase.

A sixth-grade student asked his teacher the value for 0 ÷ 0. What is the answer to this student's question? A. It is infinity B. It is equal to 0 C. It is equal to 1 D. It is indeterminate

D. It is indeterminate - In division, 6 ÷ 3 = 2 because 3 • 2 = 6; and any number divided by itself is equal to 1. - In general, a ÷ b is asking the question, "How many groups of size b are there in a?" So, it is pretty well understood that 0 ÷ n = 0, where n is any number. - There are 0 groups of size n in 0, and n • 0 = 0. It is also pretty well understood that division by 0 is not allowed; the mathematical term used is "undefined". - But the problem of 0 ÷ 0 is rather different. It is a number divided by itself, so it seems that we could say that 0 ÷ 0 = 1. It also follows that 0 • 1 = 0. So, what is the problem? - The problem is that one could say that 0 ÷ 0 = 99 because 99 • 0 = 0! In fact, we could say that 0 ÷ 0 = z, where z is ANY NUMBER you want it to be. So, because there is not a specific "answer" for 0 ÷ 0, it is said that 0/0 is indeterminate - unable to be determined.

Mathematics revision and reform is an ongoing debate to reach a conclusion as to what is "best practices" to improve mathematics education. "It is clear that the work of reform requires large investments of time and energy in order to enact critical change in mathematics education" (Ellis, Berry, 2005). What is one way teachers can work towards reform in mathematics education? A. Teachers can read journal articles and perform research to determine what is "best practices". B. Teachers can return to school for a higher degree in mathematics while continuing to teach. C. Teachers can join a union and support what the union leaders are advocating. D. Teachers can design learning environments to allow for mathematical discussion and the connection of mathematical ideas.

D. Teachers can design learning environments to allow for mathematical discussion and the connection of mathematical ideas.

In a sequence which begins 25, 23, 21, 19, 17,..., what is the term number for the term with a value of -11? A. n = -17 B. n = 1.5 C. n = 17 D. n = 19

D. n = 19 n = -17 comes from mistaking the common difference as +2 (perhaps by doing the subtraction of terms in the wrong order, such as 25 - 23). - However, the presence of a negative sign on an answer that is supposed to be the term number of the value "-11" in this sequence should be a warning that something has gone wrong. The solutions for "n" can ONLY be natural/counting numbers greater than or equal to 1. Any fractions/decimals or negative signs indicate an error in the setup or solving process of such a question. NOT- n = 1.5 comes from inappropriately combining the 25 with the -2 after seeing the statement -11 = 25 - 2(n - 1). Because the 25 represents an amount of ones and the -2 represents an amount of "n - 1"s, they are not like terms and so cannot be added. However, the presence of a decimal on an answer that is supposed to be the term number of the value "-11" in this sequence should be a warning that something has gone wrong. The solutions for "n" can ONLY be natural/counting numbers greater than or equal to 1. Any fractions/decimals or negative signs indicate an error in the setup or solving process of such a question. NOT- n = 17 comes from a failure to distribute the negative sign with the 2 when the equation -11 = 25 - 2(n - 1) is simplified to -11 = 25 - 2n - 2.

Prime numbers

numbers greater than 1 that have no numbers that will divide into them without a remainder, aside from 1 and themselves - Prime numbers are contrasted with composite numbers

Which of the following numbers is neither prime nor composite? A. 2 B. 1 C. 3 D. 4

B. 1 - One is neither a prime number nor a composite number.

Greatest common factor is

-GCF of a set of numbers is the largest value that divides evenly into each number in the set - The GCF of a pair of numbers is the biggest value that can "go into" all of the other numbers without a remainder EX: The GCF of 8, 12, and 20 is 4 b/c 4 is the largest number to divide into 8, 12, and 20 with a remainder of 0. - Sometimes a set of numbers may have GCF of just 1. - For EX: 6, 7, and 8 have no number that divides evenly into each of them except the number 1.

After a lesson on rounding and estimation, a teacher tells students that the football concession stand has purchased 590 candy bars to sell for the 6 football home games this year. The teacher asks the students to estimate the average number of candy bars that will be sold at each home game. Which of the following would be the correct estimation? A. 98.3 B. 100 C. 105 D. 90

B. 100 - An estimate is finding an approximation of a value. - Estimates are used to quickly find an answer that is close, but probably not precise. - Estimates are good for quickly answering an equation with a useful number. - While the exact number of candy bars sold per game is 98.3, the correct estimate - or approximate answer - would be 100.

A fifth-grade teacher is beginning a unit on equivalent fractions with her students. If this is an introductory lesson, which of the following activities would be the most effective in helping the students understand the concept of equivalent fractions? A. Begin with the concept that 50¢ is ½ of $1; 25¢ is ½ of 50¢; 5¢ is ½ of 10¢ B. Compare pictures showing ½ of a variety of different objects C. Use pattern blocks to model fractions equivalent to ½ of the hexagon D. Find as many fractions as possible equivalent to ½ in one minute

B. Compare pictures showing ½ of a variety of different objects - Since this is an introductory activity, concrete, proportional manipulative materials like this should be used for concept development. - It is important not to rush past this step and to use a variety of different materials to develop and reinforce understanding of this concept.

What is the place value of the "9" in the number 6,587.9213? A. Thousands B. Tenths C. Ones D. Hundredths

B. Tenths

In a kindergarten class, two students have discovered that four butter tubs full of sand will fill a plastic pitcher. This learning is best described as: A. informal standard measurement. B. formal non-standard measurement. C. formal standard measurement. D. informal non-standard measurement.

D. informal non-standard measurement. - Formal activities are generally teacher-developed and completed by all students. Informal activities are developed or discovered by the student, and with younger students this discovery often occurs during play. - This "play" activity is informal and results in a discovery about the relationship between butter tubs and a pitcher; both are non-standard measuring tools. Therefore, this is the correct answer.

In a sequence which begins 25, 23, 21, 19, 17,..., what is the term number for the term with a value of -11? A. n = -17 B. n = 1.5 C. n = 17 D. n = 19

D. n = 19 - n = -17 comes from mistaking the common difference as +2 (perhaps by doing the subtraction of terms in the wrong order, such as 25 - 23). - However, the presence of a negative sign on an answer that is supposed to be the term number of the value "-11" in this sequence should be a warning that something has gone wrong. The solutions for "n" can ONLY be natural/counting numbers greater than or equal to 1. - Any fractions/decimals or negative signs indicate an error in the setup or solving process of such a question. - n = 1.5 comes from inappropriately combining the 25 with the -2 after seeing the statement -11 = 25 - 2(n - 1). - Because the 25 represents an amount of ones and the -2 represents an amount of "n - 1"s, they are not like terms and so cannot be added. - However, the presence of a decimal on an answer that is supposed to be the term number of the value "-11" in this sequence should be a warning that something has gone wrong. - The solutions for "n" can ONLY be natural/counting numbers greater than or equal to 1. - Any fractions/decimals or negative signs indicate an error in the setup or solving process of such a question. - n = 17 comes from a failure to distribute the negative sign with the 2 when the equation -11 = 25 - 2(n - 1) is simplified to -11 = 25 - 2n - 2.

The following is a word problem: When Bob eats a sandwich, Bob always eats only half his sandwich at lunch and saves the other half for a snack later in the day. If Bob eats a sandwich for 15 days in a row, what is the number of full sandwiches Bob ate during lunch in those 15 days? Which of the following equations should a student use to solve the problem? A. 15 / (½) B. 15 * ½ C. 15 * 2 D. 15 - ½

B. 15 * ½ - Bob eats ½ of a sandwich during lunch. To find how many sandwiches Bob ate during lunch during a given time period (x), the question is ½(x). - Since the question is defining x as 15, the equation is ½(15) or 15 * ½.

If the number 180 is written as the product of its prime factors in the form a²b²c, what is the numerical value of a + b + c, where c = 5 and a and b do not equal 1? A. 10 B. 16 C. 17 D. 22

A. 10 - The prime factors are the numbers that, when multiplied together, equal a number. - In this problem there are three prime factors, where two are squared. We know that c = 5 so the equation (a²)(b²)(c) can be written (a²)(b²)(5) = 180. - We can then divide by 5 and simplify this to (a²)(b²) = 36. - Take the square root from each side simplifies the equation further to (a)(b) = 6. - We know that a and b do not equal 1, so they must equal 2 and 3. So (2²)(3²)(5) = 180 and a + b + c is 2 + 3 +5 = 10.

The only prime factors of a number are 2, 3, and 5. Which of the following could be that number? A. 25 * 36 B. 20 * 21 C. 12 * 24 D. 18 * 24

A. 25 * 36 - The prime factors are the prime number(s) that divide the integer exactly. - The prime numbers then can be multiplied together to equal that number. - Most numbers must use exponents in the prime factorization to find the prime factors. The prime factors of 25 are 5² and the prime factors of 36 are (2²) * (3²).

A gas pump can pump a quarter gallon of gas every five seconds. If a person is filling up an empty gas tank that can hold 18 gallons of gas, how long will it take the gas pump to fill the empty gas tank? A. 6 minutes B. 8 minutes and 30 seconds C. 4 minutes and 30 seconds D. 3 minutes

A. 6 minutes - If a gas pump can pump a quarter of a gallon every five seconds, then the pump can deliver a gallon of gas every 20 seconds, and 3 gallons of gas every minute. If the tank is 18 gallons, then it will take 6 minutes (3 gallons per minute * 6 minutes = 18 gallons).

A student asks a teacher when would knowing the likelihood of a six being rolled on a dice be useful in real life. Which of the following examples would be the most appropriate response for the student? A. A health insurance company estimating the expected number of payouts over the next fiscal year B. A pharmacist measuring the correct amount of medication C. A builder cutting materials for a house D. A teacher averaging a student's grade for the semester

A. A health insurance company estimating the expected number of payouts over the next fiscal year - Insurance companies estimate the number of claims they will have to pay over the course of a given time period. - This is a great example of probability and statistical analysis.

A school cafeteria offers 5 different meals and serves each meal on a set day of the week. A first-grade teacher takes a survey among her students of which of the 5 meals is their favorite. Which of the following should the teacher use to display the results of the survey? A. A pie chart B. A line graph C. A histogram D. A table with the values

A. A pie chart - A pie chart is best to show how a whole data set is divided into parts. - A pie chart is a great way to visually depict how many students named each of the 5 meals as their favorite because it visually divides a whole by percentages.

Which of the following situations might require the use of a common denominator? (Select all that apply) A. Addition of fractions B. Multiplication of fractions C. Subtraction of fractions D. Division of fractions

A. Addition of fractions & C. Subtraction of fractions - Unless fractions have like denominators, you must always find a common denominator before you can add or subtract. - Finding a common denominator requires finding a common multiple of the two (or more) denominators. - It is important to note that you do not have to find the LCM or lowest common multiple; any common multiple will work. - However, if the LCM is found and used, there will be considerably less simplifying to do in order to reduce to the lowest terms. - Multiplication and division never require finding a common denominator.

Tom wants to mentally calculate a twenty percent tip on his bill of $40. Which of the following is best for Tom to use in the mental calculation of $40? A. 40 * (20/100) B. 40 * (200/1000) C. 40 * .1 * 2 D. 40 * .02

C. 40 * .1 * 2 - Tom can quickly find 10% of 40 and then double it. - In this case the answer is $8 because 10% of 40 is 4 and 4 * 2 is 8

(Select all that apply) Which of the manipulative materials would be most suitable for teaching decimal notation to the hundredths place? A. Decimal squares B. Pattern blocks C. Base ten blocks D. Tangrams E. Color Tiles F. Geoboards

A. Decimal squares & C. Base ten blocks - Decimal squares are tag-board pictures of 10 x10 grids that have portions of the 100 smaller squares shaded. - Students are asked to name the decimal represented by the shaded or unshaded area. - They see that the sum of the shaded and unshaded areas always equals 100 hundredths or 1. - Base ten blocks are hands-on manipulatives consisting of a large cube (made up of 1000 smaller cubes), a flat (10 x 10 grid or a 100 square), a long (1 x 10), and a unit cube (1 x 1). - Base ten blocks allow the representation of decimals from 0.001 (the smallest cube) to 1 whole (the largest cube).

A kindergarten class is beginning a unit on data collection. Which of the following would be the best first activity? A. Give each student a collection of colored tiles to sort by color B. Have each student bring or draw a picture of their favorite pet and arrange them into a class graph C. Show the class a bar graph representing different favorite fruits and have them tell you which fruit is the most favorite, least favorite, etc. D. Any of the above would be an equally good first activity for this unit

A. Give each student a collection of colored tiles to sort by color - Having each student bring a picture of their favorite pet would be a good follow-up activity to the correct answer. - Showing the class a bar graph is the most abstract because students do not have any direct input or relationship to the graph. - This activity might be a good first grade activity, but it might not be appropriate for most kindergarten classes.

When asked to answer the question what is ½ ÷ 2; Jon answered 1. How could this problem be presented to Jon so that he would better understand the problem and how to answer it? A. Have Jon draw a picture to represent ½ a pizza, then divide it into two equal parts. Ask him how much of the whole pizza each part represents. B. Tell Jon that he has answered the question, "What is ½ of 2?" Then have him rework the correct problem. C. Have Jon draw a picture to illustrate his thought process to you. D. Tell Jon to think about having ½ a pizza. Ask him, "if he were to share the pizza equally with a friend, how much of the pizza would each of you get?"

A. Have Jon draw a picture to represent ½ a pizza, then divide it into two equal parts. Ask him how much of the whole pizza each part represents. - This choice allows Jon to go to a pictorial representation of the problem; this will aid him in understanding what the problem is asking, as well as reinforce the strategy of drawing a picture or diagram as a valid problem solving strategy.

In a first-grade class, the students have been working with manipulative materials and pictures as they investigate the concept of addition. Through both formative and summative assessments, the teacher has determined that the students are ready to move to more abstract (pencil and paper) ways to represent addition. How should she begin this process? A. Have the children model pictorial representations of problems like 7 + 2 = 9 that include the numbers that represent each step B. Model one of the problems, 7 + 2 for example, for the children by writing: 7 + 2 = 9. Then have the students repeat the process with a different problem C. Relate the symbolic representation of addition facts to models the children have created, modeled, or drawn in their math lessons D. By giving the students a page of one digit addition problems with sums of 10 or less and having them draw a picture to match the sum

A. Have the children model pictorial representations of problems like 7 + 2 = 9 that include the numbers that represent each step - This choice builds on pictorial models by adding spaces below the model to record the symbolic representation for each step. - This is the next logical step in learning to write addition problems symbolically. - Student readiness to move to the symbolic will vary from student to student.

A teacher discovers that each student in his class receives an allowance from their parents. Which of the following examples would best demonstrate to the students the power of saving their allowance instead of spending all of their allowance? A. Show students the expected return of 5% allowance savings over a 10-year period B. Have students calculate the amount of federal tax owed on their allowance if it was taxed C. Have students set aside 10% of their allowance each week D. Have students research a charity and ask how their allowance money could impact those whom the charity serves

A. Show students the expected return of 5% allowance savings over a 10-year period - Demonstrating how much money students could make by saving their allowance would best demonstrate the power of saving money.

A student asks the teacher who invented the number system. Which of the following answers would be most appropriate? A. The base-ten number system was developed by the Hindu-Arabic civilizations B. The current number system was developed by the Greek and Roman empires C. The base-ten number system was invented by Isaac Newton in the late 17th century D. The current number system has evolved over a period of thousands of years and each culture contributed to its development

A. The base-ten number system was developed by the Hindu-Arabic civilizations - The base-ten number system, which is the foundation of the modern number system, was developed by Arabic and Hindu civilizations.

Which of the following scenarios would be the most appropriate situation to estimate? A. The number of pizzas needed for a birthday party B. A pharmacist filling a prescription C. An accountant filling out a tax form for a client D. The number of votes cast in an election

A. The number of pizzas needed for a birthday party -This is an appropriate situation to estimate. Some people might eat more, some people might eat less. - It is a common practice to estimate the number of pizzas needed at a birthday party.

Bill went to the store to purchase new clothes for the upcoming school year. Bill purchased 8 shirts, 4 pairs of shorts, and 2 pairs of pants. If a single outfit consists of one shirt and either one pair of shorts or one pair of pants, how many outfits can Bill create with the clothes he purchased? A. 42 B. 48 C. 32 D. 36

B. 48 -CANNOT BE 36 b/c- the answer can be found by multiplying the number of shirts by the number of pairs of shorts and pairs of pants. This would create the equation (8 shirts) * (4 pairs of shorts + 2 pair of pants) = 8 * (4+2) = 8 * 6 = 48.

What is the value of the "9" in the number 432.0569? A. 9/1,000 B. 9/10,000 C. 9/10 D. 9/100

B. 9/10,000

In a sixth-grade class, students are beginning a unit on algebraic reasoning. They are asked to measure the sides of several different squares and then determine the perimeter of the square. They are to record the information in a table like this: What would be the best question to assess student understanding about this concept? A. What is the relationship between the length of side of a square and its perimeter? B. What is a good definition of perimeter? C. If a square has a perimeter of 18 cm, what is the length of one side? D. If a square has a perimeter of "P", what is the length of a side?

A. What is the relationship between the length of side of a square and its perimeter? - "If a square has a perimeter of 18 cm..." is specific to a particular perimeter and not a reflection of understanding of the concept. This is the Application Level of Bloom's Taxonomy. - "What is a good definition of perimeter?" is a recall question and does not measure student understanding of the concept of perimeter. This is in the Knowledge Level of Bloom's Taxonomy. - "If a square has a perimeter of 'P'..." requires the students to work backwards. The current activity prepares the students to answer how to find length of a side and then find the perimeter, working backwards would be the next logical progression of the algebraic concept. This is the Application Level of Bloom's Taxonomy.

A class of sixth-grade students is given the following problem: ¹/₂ + ³/₄ = Many of the students arrive at the answer: ⁴/₆ = ²/₃ What should the teacher consider with respect to remediation? (Select all that apply) A. Students need more practice working with equivalent fractions at the concrete level. B. Students need more practice finding equivalent factors using scale factors. C. Students need more work finding common denominators.

ALL- A. Students need more practice working with equivalent fractions at the concrete level. B. Students need more practice finding equivalent factors using scale factors. C. Students need more work finding common denominators. -All represent prerequisite skills that must be mastered before adding or subtracting fractions with unlike denominators. - Students must understand that fractions with unlike denominators must be rewritten as equivalent fractions with the same denominators before addition can happen.

Mrs. Keller writes down the following numbers on the board: 4, 2, 6, 8, 9, 1, 3 She instructs her students to write down the smallest number possible with the 8 in the thousands place and the 1 in the tens place. Which of the following would be the correct answer? A. 2,486,913 B. 1,628,943 C. 2,968,314 D. 4,698,312

C. 2,968,314

The mathematics teacher and art teacher work together to create an interdisciplinary lesson using tessellations, which are basic geometric shapes set to a repeating pattern. The students cover a large piece of poster board with the patterns they create. Which of the following mathematical concepts is most closely reflected in this activity? A. Perimeter B. Infinity C. Conservation D. Number Sense

B. Infinity - The tessellations will continue in infinity. - The teacher is introducing a mathematical concept that does not end, but repeats continually. - This is the concept of infinity.

15 X 19 ----- 135 25 ------ 385 A student is working through a double-digit multiplication problem and turns in the work pictured above. Which of the following best describes the student's error? (View attachment for larger image) A. The student erred when multiplying 9 and 15 B. The student carried over the hundreds value from the 9 and 15 multiplication C. The student added the products incorrectly D. The student's understanding of the base ten numerical system needs remediation

B. The student carried over the hundreds value from the 9 and 15 multiplication - The student placed a two in the hundreds place on the second row of the addition instead of a one. - The student carried the additional hundreds value from the (9 * 15) calculation when calculating the tens product (1 * 15) or (10 * 15). - The student should add 135 and 15 (or 150) not 25 or (250). The correct answer would be 285.

Given A(22) = 188 and d = 19, what is the value of a₁? A. a₁ = 587 B. a₁ = -211 C. a₁ = 606 D. a₁ = 9.9

B. a₁ = -211 - a₁ = 587 comes from mistaking the A(22) value for a₁ and solving the resulting (incorrect) equation: a₁ = 188 + 19(22 - 1). - a₁ = 606 comes from forgetting the "- 1" part of the "n - 1" component of the formula. - a₁ = 9.9 comes from not using the formula at all and simply dividing the given term value 188 by the difference value 19.

A class is learning about ratios and percentages. The teacher tells the class that at last Friday night's football game there were between 800 and 1000 people. Of those at the football game, about 13-17 percent of the people had blonde hair. Which of the following is the most reasonable estimate of the number of people at the football game with blonde hair? A. 100 B. 200 C. 135 D. 170

C. 135 - The best estimate of people with blonde hair at the football game would be 135. - Using 15% (the average number of 13% and 17%) as the percentage estimate, 15% of 800 people (.15 * 800) would be 120 and 15% (.15 * 1000) would be 150 people. - It is a reasonable estimate that between 120 and 150 people at the football game had blonde hair. 135 is the only number that fits between this range.

A sixth-grade class is asked to estimate the answer to the following question: 75.8 + 326.79 + 488.92 ÷ 11 = _____. Which of the following would be the best answer? A. 900 B. 80 C. 450 D. 82

C. 450 - For 900, the sums of the first three numbers were estimated, but no division by 11 happened. - For 80, this reflects that the total sum was divided by 11 (incorrect since only the last addend 488.92 is to be divided by 11) and rounded to the nearest 10. - For 82, this reflects that the total sum was divided by 11 (incorrect since only the last addend 488.92 is to be divided by 11) and rounded to the nearest whole number.

The west wall of a square room has a length of 13 feet. What is the perimeter of the room? A. There is not enough information B. 169 C. 52 D. 48

C. 52 - A square has four sides of equal length. To find the perimeter of a square, simply multiply one side by 4. Thus: 13 * 4 = 52.

What are the prime factors of 25? A. 5 B. (5²) * 2 C. 5² D. 5 * 2

C. 5²

Which of the following is an example of the commutative property of multiplication? A. 123 + 345 = 345 + 123 B. 2(2+2) = 2(2-2) C. 6 x 5 = 5 x 6 D. 4(2-1) = 2(4-1)

C. 6 x 5 = 5 x 6 - The commutative property of multiplication states that it does not matter the order of the multiplication sequence, the answer will be the same. This can be represented by the equation a * b * c = c * b * a.

Given a₁ = 4, d = 3.5, n = 14, what is the value of A(14)? A. A(14) = 97.5 B. A(14) = 53 C. A(14) = 49.5 D. A(14) = 55.5

C. A(14) = 49.5 A(14) = 49.5 comes from seeing that a₁ = 4, d = 3.5, and n = 14, and then plugging those components into the formula A(n) = a₁ + d(n - 1) as A(14) = 4 + 3.5(14 - 1), and then simplifying the statement as far as possible. If following the Order of Operations, 14 - 1 will be subtracted to produce 13 inside the parentheses: A(14) = 4 + 3.5(13). - Next, multiplication is performed (before addition) for: A(14) = 4 + 45.5. Finally, 4 + 45.5 is added to yield the final answer of A(14) = 49.5.

In a unit on personal finance, a teacher wants students to be able to identify the difference between fixed and variable costs. Which of the following examples would best highlight this difference? A. Analyzing the money spent on gas each month of an average American and looking at how much a person drives impacting the price they will pay in gas B. Looking at the differences between a tax deduction and a tax credit C. Categorizing the expenses of a local restaurant into expenses that depend on the number of customers and expenses that do no not depend on the number of customers D. Having students ask their parents what fixed costs they pay each month

C. Categorizing the expenses of a local restaurant into expenses that depend on the number of customers and expenses that do no not depend on the number of customers - This will help students highlight the differences between variable and fixed costs because the student is actively having to categorize an expense into one of the two categories.

The teacher provides a word problem for her students: Quacky Donald's Donuts sells glazed donuts in packages of six and donut holes in packages of 10. If Quacky Donald's Donuts sold the same number of glazed donuts and donut holes yesterday, what is the minimum amount of donut holes that Quacky Donald's Donuts sold? Based on the world problem above, which of the following concepts is the teacher most likely to cover in the lesson? A. Least common factor B. Greatest common factor C. Least common multiple D. Greatest common multiple

C. Least common multiple - The least common multiple will find the lowest number that will equally divide two integers. - By finding the LCM, students will be able to find the correct answer.

In a kindergarten class, the students are lining up for lunch. The teacher begins calling the first four or five students to line up in boy-girl order. She stops after the fifth student and asks the question, "Class, who do you think I might pick to go next in line? Why?" What math concept is the teacher most likely teaching? A. Ordinal number of a set member B. Cardinal number of a set C. Patterns to make prediction D. Fairness in making choices

C. Patterns to make prediction - The teacher is teaching or rehearsing an ABABA pattern sequence. In this case, it is boy, girl, boy, girl, boy. - Based on the teacher's choices thus far, the next person should be a girl.

Which of the following activities is most effective in helping kindergarten students understand measurement of the lengths of small items, such as juice boxes or lunchboxes? A. Watching the teacher estimate the length of the item using a student's arm or leg B. Tracing the items on construction paper and cutting the construction paper to have a two-dimensional replica of the item C. Placing same-size objects, such as Legos or cubes, next to the object and counting the number of objects D. Listening to a teacher explain how to use a ruler to measure the objects

C. Placing same-size objects, such as Legos or cubes, next to the object and counting the number of objects - This is correct because having the student count the number of cubes and placing the cubes next to each item provide a visual for the student to use in determining the length.

Any town ISD wants all elementary students to be able to use computational strategies fluently and estimate appropriately. Which of the following learning objects best reflects this goal? A. Students will memorize multiplication tables B. Students will use calculators to perform their computations and check for accuracy C. Students evaluate the reasonableness of their answers D. Students understand the theoretical reasoning of basic mathematical rules

C. Students evaluate the reasonableness of their answers - If a student is able to use computational strategies, strategies for computing an answer, as well as estimate properly, then the student should be able to evaluate the reasonableness of his final answer. - A student who is fluent in computational strategies and is good at estimating will know if his answer is about what he expected the answer to be, or if he should review his answer because it does not match what he expected.

(Picture of cube flat) Students in a sixth-grade class are asked to find all possible ways to arrange five squares so that the resulting net could be folded into an open cube, as in the picture above. Which level of Bloom's Taxonomy best describes this activity? A. Application Level Question B. Analysis Level Question C. Synthesis Level Question D. Evaluation Level Question

C. Synthesis Level Question - There are eight possible nets that can be formed into an open cube. - Sometimes it is difficult to tell the difference between an Analysis and a Synthesis Level question. - One way to tell the difference is to think about whether the student is being asked to create something new and different or think about why something works. - If students are being asked to think about how or why something works, they are analyzing - Analysis Level. - If they are being asked to CREATE SOMETHING NEW (to them) or to use what they have learned in a new and different way, they are operating at the Synthesis Level. - It is also important to understand that Bloom's levels may differ as students become more proficient in their understanding of mathematics. - This problem given to a group of high school geometry students might be an Application Level problem because the students have likely seen and done problems like this before. - It is not new and different. Evaluation Level is not correct because the student is not asked to make a judgment using a set of criteria.

What is the place value of the "3" in the number 6,587.9213? A. Thousandths B. Tenths C. Ten Thousandths D. Hundredths

C. Ten Thousandths

Given A(54) = 299 and d = -4, what is the value of the first term? A. a₁ = 87 B. a₁ = -74.75 C. a₁ = 511 D. a₁ = 836

C. a₁ = 511 - a₁ = 511 comes from seeing that A(54) = 299 implies that n = 54, from using d = -4, and from keeping a variable expression in the place of a₁ so that the formula A(n) = a₁ + d(n - 1) becomes 299 = a₁ - 4(54 - 1), and then simplifying the statement before isolating the unknown a₁. - If following the Order of Operations, 54 - 1 will be subtracted to produce 53 inside the parentheses: 299 = a₁ - 4(53). - Next, multiplication is performed for: 299 = a₁ - 212. Finally, 212 is added to each side of the equation in order to isolate the unknown, a₁. Because 299 + 212 = 511, a₁ = 511.

What is the common difference in an arithmetic sequence with a first term of 17 and A(6) = 4½? A. d = 0.2 B. d = 4.3 C. d = -2.5 D. Cannot be solved due to insufficient information given.

C. d = -2.5

In a sequence which begins -7, 4, 15, 26, 37,..., what is the term number for the term with a value of 268? A. Cannot be solved due to insufficient information given. B. n = 68 C. n = 26 D.n = 24.4

C. n = 26 - n = 68 comes from inappropriately combining the -7 with the +11 after writing the equation 268 = -7 + 11(n - 1). Because the -7 represents an amount of ones and the 11 represents an amount of "n - 1"s, they are not like terms and so cannot be added. - n = 24.4 comes from not using the formula at all and simply dividing the given term value 268 by the difference value 11. - Someone selecting the option "Cannot be solved due to insufficient information given" is likely to draw this false conclusion due to not understanding the meaning of "the term number" as requested in the problem really meaning "solve for n in the formula A(n) = a1 + d(n - 1)." - As long as it is understood that A(n) means the value of the term in the nth position, the 268 can be substituted into the A(n) position and the "n" can be left as a variable.

(Table) Sarah - 12 Anne- 8 Ross- 10 Sarah wants to compare the ages of the children in her family. The table above shows the children's ages. Which of the following compares the children's ages correctly? A. 10 > 8 >12 B. 12 > 8 < 10 C. 8 > 12 < 10 D. 12 > 10 > 8

D. 12 > 10 > 8 - Remember that when using inequality signs in a math expression, the signs must face the same direction. 1 < 2 < 5 or 5 > 2 > 1; but not 5 > 1 < 2.

Anytown School District provides a 50 multiple-choice question mathematics assessment to all 4th grade students. The students complete the assessment, the tests are scored, and the scores are compared throughout the school district. Which of the following mathematics component is most likely the goal of this type of assessment? A. Rate B. Flexibility C. Automaticity D. Accuracy

D. Accuracy The assessment most likely is designed to measure the students' accuracy of answering questions.

A fifth-grade teacher writes the problem 5(10-5) on the board. She asks students to solve the problem mentally and raise their hands when they find a solution. When all students have raised their hands, the teacher begins a dialogue with a student: Teacher: "John, what solution did you come up with?" John: "25" Teacher: "That is the correct answer! (to the class) Raise your hands if you thought 25 was the correct answer." Every student then raises their hands and the teacher writes a new problem on the board. Which of the following adjustments could the teacher make to best assess all the students' understanding of the concept? A. Require John to demonstrate how he found the answer on the board in front of the entire class B. Require students to write their answers on a sheet of paper and then collect the sheets of paper at the end of class C. Ask John to explain how he found the answer and then move on to another problem D. Ask multiple students for their answers and allow them to explain how they found the answers prior to identifying the correct answer

D. Ask multiple students for their answers and allow them to explain how they found the answers prior to identifying the correct answer - This is the best answer as it gives the teacher and students opportunities to explore multiple answers and multiple ways to solve the problem. - It is best to ask students to share before revealing the correct answer so that students are not intimidated if their answers were incorrect.

A sixth-grade teacher is beginning a unit on probability. She utilizes the following steps in planning her unit: I. Determine the necessary prerequisite skills. II. Begin planning probability activities that involve the collection of data. III. Determine what the students already know by using a KWL chart. IV. Plan the final assessment for the unit. What is the best order for the teacher to organize these steps? A. I, III, II, IV B. I, II, III, IV C. I, III, IV, II D. IV, I, III, II

D. IV, I, III, II - The teacher should begin by planning the final assessment. - This serves as her destination goal, where she wants her students to be at the end of the unit. - Once the assessment is developed, the teacher needs to determine what prerequisite skills are necessary for successful mastery. - This step should rely heavily on what the students mastered in prior grades as a beginning point. The third step would be to assess what the students already know about probability. - One way to accomplish this is with a KWL chart. The "K" stands for know. Begin with a brainstorming session by asking students what they already know about probability. - Prod for additional information or clarification. When finished, ask the students what they are wondering about when they think about probability and list student responses in the column under "W". - This is the most difficult part of this procedure, and the teacher might need to provide some direct guidance here. Make sure that your learning goals are addressed. - At the end of the unit, you will return to this chart and complete the "L" column - learned. In this column, students will list what they have learned making sure that all wonderings or questions listed in the "W" column are answered. - Finally, the teacher will begin planning a variety of probability activities designed to help students discover the answers to questions they are seeking. - The teacher should follow the steps: assess, determine prerequisite skills, determine what students already know, and plan activities to get to the destination.

Which of the following would be the best concept to introduce to students in a second-grade class? A. The concept of infinity B. The proportions of the Earth to the sun and to the solar system C. Pouring water from a wide, short glass into a tall, thin glass does not mean there is more water in the second cup D. That a square is a rectangle and a rectangle can be square

D. That a square is a rectangle and a rectangle can be square - Children at this age should recognize the conservation of matter principle. - This concept would be appropriate for younger children, but children ages 7 and up should already know this concept. - The proportions of the Earth to the sun is abstract and too advanced for students at the "concrete operations" stage. - This concept would be appropriate for students ages 12 and up. - The concept of infinity is abstract and too advanced for students at the "concrete operations" stage. - This concept would be appropriate for students ages 12 and up.