Math

Which of the following numbers cannot be used to express probability? A. -0.004 B. 0.2 C. 0 D. 0.99

(A) A probability can be expressed as a number from 0 to 1, or with percentage or decimal numbers. A percent is a representation of a decimal number relative to 100. Therefore, the only number shown which does not fit is the negative number. (802-005 Probability and Statistics)

What is the mode of the following data set? 4, 12, 3, 15, 8, 3, 9, 9, 8, 10, 12, 14, 3, 2. A. 3 B. 9 C. 8 D. 14

(A) Choice (A) is the correct answer. The mode is the number that occurs most often. From looking at the data we see that the number 3 occurs 3 times, the most of all the numbers. (802-005 Probability and Statistics)

In Ms. Leal's class, there are 18 blondes, 17 brunettes, and 5 redheads. Rodrigo is asked to determine the probability of Ms. Leal randomly selecting a redhead from her class to answer a question. His answer is 8. Which of the following best describes how Rodrigo went wrong in finding the solution? A. He divided 40 by 5. B. He divided 5 by 40. C. He added 7 and 1. D. He subtracted 1 from 9.

(A) The correct answer is (A). Choices (C) and (D), while matching Rodrigo's answer, would seemingly come from nowhere which is highly unlikely. Choice (B) would produce a number less than 1, the correct answer for the problem asked of Rodrigo, and therefore would not match Rodrigo's answer. It is most likely that Rodrigo divided 40, the number of students in the class, by 5, the number of redheads, to obtain his answer of 8. (802-005 Probability and Statistics)

Divide 6.2 by 0.05. A. 124 B. 1.24 C. 12.4 D. 0.124

(A) The correct answer is (A). The traditional whole number division algorithm (method) is helpful when dividing decimals longhand.

A card is drawn from a standard deck of cards. What is the probability that the card is a queen or black four? A. 6/52 B. 8/25 C. 25/52 D. 12/52

(A) The correct answer is (A): 6/52. In a standard deck of cards there are 52 total cards. Of the 52 cards, there are four suits and 13 cards per suit. Therefore, there are four queens in the deck. Additionally, of the four suits, two are black and two are red. So, there are two black fours giving a grand total of six cards of 52 that meet the criteria. (802-005 Probability and Statistics)

Carlos wants to bring cookies for each student in his entire elementary school. He knows there are 800 students in the school. His teacher asks him to make sure he buys enough bags to carry all of the cookies, assuming each bag can hold 38 cookies. Carlos goes to the store and buys 20 bags, then brings the cookies to school the next day. He finds out that he doesn't have enough cookies for all the students at the school. Which of the following is Carlos most likely to have done to obtain the incorrect number of bags? A. Rounded 38 to 40 B. Neglected to count 2 classrooms of students C. Baked at 375 degrees for too long D. Used plastic bags

(A) The correct response is (A). It is most likely that Carlos rounded 38 to 40 in order to make his division process easier. This would then explain how he came to purchase 20 bags because 800 students divided by 40 yields 20. (802-006 Mathematical Processes)

There are 16 more apples than oranges in a basket of 62 apples and oranges. How many oranges are in the basket? A. 23 B. 39 C. 32 D. 30

(A) This problem is easily solved by using some basic algebraic reasoning. Since we are interested in determining the number of oranges that are in the basket, we will set this as a variable called r. The number of apples, a, and the number of oranges, r, sum to a total of 62. We also know that there are 16 more apples than oranges (a = r + 16). This gives r + (r + 16) = 2r + 16 = 62. Solving for r yields 23 oranges in the basket. (802-003 Patterns and Algebra)

What measurement principle do children in Pre-K through kindergarten sometimes have difficulty with? A. Conversation B. Conservation C. Condensation D. Conversion

(B) Children who are four and five years old may not understand that changes in the appearance do not necessarily change the characteristics of an object. For example, if an apple is cut in half the children may not understand that there are two pieces of one apple. Instead, many students may say that there are now two apples. The difficulty these students face is with the concept of conservation (B). (802-001 Mathematics Instruction)

Which of the following sets of numbers is not an integer followed by its square? A. -8, 64 B. 8, 64 C. 6, 32 D. -9, 81

(C) An integer number is a whole number that is either positive or negative. Remembering that a negative times a negative gives a positive means that any of the answers could be possible and we cannot rule any out by process of elimination. Knowing that 8 times 8 gives 64 and -8 times -8 gives 64 means both choices (A) and (B) are true. Similarly, -9 times -9 results in 81 (D), meaning 81 is true. However, 6 times 6 gives 36, not 32. This means that choice (C) is not a true statement and is the solution to the problem. (802-003 Patterns and Algebra)

Given that the radius of the circle shown is 6 units and that the circle is inscribed in the square, which of the following is the approximate area of the shaded region? A. 106 square units B. 31 square units C. 77 square units D. 125 square units

(B) First, it is helpful to view the shaded area as the area of the square minus the area of the circle. With that in mind, you simply need to find the area of each simple figure, and then subtract one from the other. You know that the radius of the circle is 6 units in length. That tells you that the diameter of the circle is 12 units. Because the circle is inscribed in the square (meaning that the circle fits inside of the square touching in as many places as possible), you see that the sides of the square are each 12 units in length. Knowing that, you compute that the area of the square is 144 square units (12 x 12). Using the formula for finding the area of a circle (πr2), and using 3.14 for π, you get approximately 113 square units. (3.14 x 6 x 6). Then, you subtract 113 (the area of the circle) from 144 (the area of the square) for the answer of 31. Based on the explanation given, choices (A), (C), and (D) are incorrect. (802-004 Geometry and Measurement)

If a can weighs 14 oz., how many cans would you need to have a ton? (Round your answer to the nearest ones place and pick the best answer.) A. 2285 B. 2286 C. 2287 D. 2300

(B) The correct answer is (B), 2286. An easy way to solve this problem is to use basic algebra. Knowing that there are 16 oz. in a pound and that there are 2,000 lbs. in a ton helps ease the difficulty of the problem. We want to find out the number of cans x it will take to obtain a ton. Therefore, we have \(\frac{14}{16}x = 2000 \). If both sides of the equation are multiplied by 16 and then we divide both sides by 14, we will obtain the approximate number of cans it will take to obtain one ton. \(x = \frac{16 \times 2000}{14} = \frac{32000}{14} \approx 2285.7\) We see that many of the answers are close to this value. When we round this number, we will obtain 2286. (802-003 Patterns and Algebra)

While students in a sixth-grade class work in small groups on an ungraded discovery activity involving linear functions, the teacher circulates through the class asking individual students questions such as, "Why did you choose to assign the height variable to the vertical axis?" and "What happens to the graph if you use different-sized cups for the experiment?" Based on the scenario, which of the following is a primary purpose of these questions? A. To provide a formal assessment that can be used to encourage students to work harder. B. To gauge student progress in developing conceptual understanding in a way that will not penalize the student. C. To show the students that the teacher cares about the work they are doing. D. To provide an informal summative assessment that can be used to determine each student's grade for the assignment.

(B) The correct answer is (B). Because no grades are being assigned, the students will not be penalized if they cannot answer the question. The questions themselves are focused on key concepts in linear functions such as the meaning of the y-intercept (question No. 1) and the linear function as a model of the physical world (question No. 2). Choice (A) can be ruled out because the fact that the teacher is asking the questions in a conversational tone without mention of grades being assigned makes this an informal assessment. Choice (D) can be ruled out because the teacher is assessing the students as they work on the content. This means the teacher is using a formative, not summative, assessment. Though choice (C) may initially seem appealing, it is not correct because a teacher who maintains a neutral relationship with his or her students can still perform these tasks effectively; thus, the notion of caring in itself is not integral to this test item. (802-001 Mathematics Instruction)

The concept of the zero evolved in India but was also developed, disconnectedly, by the A. Babylonians. B. Mayans. C. Arabs. D. Romans.

(B) The correct answer is (B). The Mayans developed the concept of the zero around 700 ce. However, there is no evidence to suggest that its discovery has any connection with the development of the same concept in India. Choice (A) is incorrect because there is no evidence to suggest that the Babylonians developed the concept of the zero as part of their numeric system based on 60. Choice (C) is incorrect because most historians agree that the Arabs used information from the Hindus to develop our modern numeric system, which includes the zero. Choice (D) is incorrect because there is no evidence to suggest that the Romans developed the concept of the zero as part of their numeric system. (802-003 Patterns and Algebra)

Which of the following activities is most effective in helping students understand how length, width, and height contribute to the volume of a rectangular solid? A. Have students cut out squares of various sizes and determine the area of the resulting piece of paper. B. Have students cut out squares of various sizes, bend up the sides, and determine the volume of the resulting shape. C. Have students plug in various values of x into the given formula. D. Have students watch a YouTube video of the problem.

(B) The correct answer is option (B) because having students cut out the squares gets them to see that the larger the square, the smaller the length and width. Being able to have a physical model of these solids allows students to see readily which ones are obviously larger in volume than others. Option (A) is incorrect as cutting squares out and examining the area tells students nothing about the volume. Option (C) is incorrect because although plugging different values of x will give different volumes of V, students do not see how the size of x affects the length and the width. Option (D) is incorrect because unless this video shows someone performing option (B), it cannot be as effective a learning tool. (802-001 Mathematics Instruction)

Word problem: Tripti has 10 apples and 20 oranges. Tripti's mother doubles the total number of apples and oranges she has. Meanwhile, Gahn's mother gives Gahn twice as many apples and twice as many oranges as Tripti started with. A teacher creates the word problem shown above for a math lesson. Based on the word problem, the lesson will most likely cover which of the following mathematical properties? A. Associative property B. Distributive property C. Commutative property D. Dissociative property

(B) The correct response is (B). The distributive property best describes the statement. Tripti initially had (10 + 20) apples and oranges and then the amount is doubled: 2 (10 + 20). Gahn was given twice as many apples and oranges from the beginning: (2 x 10 + 2 x 20). Choice (A) is a play on the associative property and therefore is incorrect. Answers (C) and (D) do not apply. Therefore, the best answer is choice (B). (802-006 Mathematical Processes)

Which of the following were the first mathematicians to impact the development of modern-day mathematics? A. Greeks and the Aztecs B. Egyptians and Babylonians C. Hindus and Aztecs D. Arabs and the Mayans

(B) While other groups made significant contributions to mathematics, choice (B), the Egyptians and Babylonians (third millennium BCE) were the first groups to make an impact on the development of modern-day mathematics. (802-006 Mathematical Processes)

Which of the following is not a critical step to take when solving a problem? A. Understand the problem. B. Choose a strategy and/or making a plan. C. Make a diagram of the problem situation. D. Think critically about the solution.

(C) In order to solve a problem we must understand the problem, choose a strategy and/or make a plan, carry out the plan, and think about whether the answer makes sense in the context of the problem. The only response option that is not a critical component of the process is to make a diagram of the problem situation (C). This is because not all problems lend themselves to diagrams. (802-006 Mathematical Processes)

Solve the following equation for x 44−7x=3x+5+7x−4x A. 39 B. 93 C. 3 D. 9

(C) In this problem we are asked to find the value of x. In order to aid in solving this problem we want to group like terms and simplify. Therefore, the first step is to group all of the terms associated with x on one side of the equals sign and all of the constant whole numbers on the other. Once this is done we are left with 13x=39. To solve for x we divide both the left and right hand sides of the equation by 13 to isolate the variable. When this is done we are left with x=3. (802-003 Patterns and Algebra)

Two fair two-sided coins are tossed at the same time. What is the probability that only one head is obtained in each of the tosses? A. 0.25 B. 0.75 C. 0.5 D. 0.1

(C) Since tossing two separate coins does not affect the outcome of the other coin, we know that the probability of getting heads on either coin is 1/2. The probability of the outcome is thus 2/4 or 0.5. Alternatively, the possibilities are HH, HT, TH, and TT. Since there are 4 possibilities and 2 of them have only one H, the probability is 2/4=1/2=0.5. (802-005 Probability and Statistics)

The main advantage of using hands-on activities in mathematics is to A. enhance students' ability to think abstractly. B. make the lesson more enjoyable. C. lead the students to active learning and guide them to construct their own knowledge. D. promote equity, equality, and freedom for the diverse ethnic groups in the nation.

(C) The correct answer is (C). Hands-on activities can make the curriculum more relevant and guide children to construct their own knowledge. Hands-on activities can probably lead children to think abstractly (A), but the activity is not designed exclusively to accomplish this goal. Hands-on activities can make the class more interesting and enjoyable (B), but they are not the reasons for the activities. (D) is incorrect because there is no evidence to suggest that hands-on activities can promote equity, equality, or freedom among students. (802-001 Mathematics Instruction)

For what value of x is the expression 5/x undefined? A. 9 B. 1,000,000,000.11 C. 0 D. 1E - 14

(C) Until more advanced mathematics are incurred, the standard definition of division states that an expression is undefined if it contains division by 0. Option (D) is a number very close to zero. Although option (D) will make the expression a huge number, it is still not undefined. Therefore, the response that best corresponds to the question is choice (C). (802-003 Patterns and Algebra)

Perform the indicated operation: (-36) - 11. A. 47 B. 25 C. -47 D. -25

(C) When subtraction involves any negative numbers, a good rule to use is, "Don't subtract the second number. Instead, add its opposite." Using that rule, the original expression, (-36) - 11, becomes (-36) + -11. To be "in debt" by 36, then to be further "in debt" by 11, puts one "in debt" by 47, shown as -47. (802-002 Number Concepts and Operations)

Which of the following would be the best set of units to use when measuring a football field? A. Centimeters B. Inches C. Meters D. Miles

(C) While all of these units will accurately express the length of the football field, centimeters and inches would not be a convenient scale to use, as the resolution of the measurement would be too great resulting in an extremely large number (or a long time to measure). Miles are another inconvenient method to measure a football field, as a mile is much larger than a single football field. (802-002 Number Concepts and Operations)

An example of a prime number is A. 9 B. 682 C. 49 D. 67

(D) A prime number is a number whose only factors are one and itself. Even numbers greater than 2 can always be factored by 2, eliminating (B). (A) can be factored as 3 times 3 and (C) can be factored as 7 times 7. Therefore, (D) must be the right answer as it only has factors of 1 and 67. (802-002 Number Concepts and Operations)

An isosceles triangle is a polygon with two equal sides. What else does this imply? A. It is equilateral. B. Its angles sum to greater than 180°. C. It is also scalene. D. It has two equal angles.

(D) In any triangle, the sum of the angles must equal 180°. If a triangle has two equal sides, the third side cannot be unique and must be a set length. The lines associated with the equal sides will intersect the third side at the same angle. Therefore, the only answer that can be determined to be true from the information is option (D). (802-004 Geometry and Measurement)

Mr. Anderson tells his class that in a two-week period (including weekends and holidays), Max spends $71.47 on lunch. Two students in Mr. Anderson's class, Dave and John, are asked to state the approximate average value Max spends on lunch over this time. Dave says Max spent $4.50 on average per day. John says Max spent $5 on average. What concept needs to be addressed further in Mr. Anderson's class? A. Multiplication B. Integration C. Differentiation D. Rounding

(D) The correct answer is (D), rounding. Choices (B) and (C) are related to calculus, and are not involved in the problem. Multiplication (A) is not needed as the problem asks to find the average value spent on lunch, which implies division. Based on the outcomes of Dave's and John's answers, both being relatively close to each other and less than the true average, it appears that rounding should be addressed further. (802-006 Mathematical Processes)

Which of the following statements best represents the value of manipulatives as part of an instructional strategy? A. Manipulatives are more appropriate for students at the preoperational stage of cognitive development. B. The use of manipulatives should be restricted to children in kindergarten to fourth grade. C. The use of manipulatives is more appropriate for teaching computation skills and geometry. D. Manipulatives can be used to teach mathematics in grades Pre-K to high school.

(D) The correct answer is (D). Manipulatives can be used to simplify the teaching of mathematics in all grade levels -- Pre-K to high school. Choices (A) and (B) are incorrect because the use of manipulatives does not have to be restricted to early childhood (pre-kindergarten through fourth grade). Choice (C) is incorrect because manipulatives can be used to teach concepts including computation skills and geometry but its use is not limited to these two components. (802-001 Mathematics Instruction)

A local Brownie troop uses the model y=2x−25 to calculate the money earned in a bake sale, where x is the number of cookies sold. If the troop sold 75 cookies, how much money did the troop earn? A. $75 B. $150 C. $175 D. $125

(D) The correct answer is (D). The Brownie troop sold $125 worth of cookies. The amount of money, y, is found by substituting the number of cookies sold, 75, for x in the given equation: y = 2x - 25 = 2(75) - 25 = 150 - 25 = 125. (802-006 Mathematical Processes)

Simplify: 6⋅2+3÷3. A. 18 B. 5 C. 10 D. 13

(D) The correct answer is (D). The order of operations must be obeyed here. Remembering the saying "Please Excuse My Dear Aunt Sally (PEMDAS)" allows us to remember the order in which mathematical operations must be carried out: Parentheses Exponent Multiply Divide Add Subtract. Following this, multiply 6 by 2 to obtain 12. Then, divide 3 by 3 obtaining 1. Finally, add the two results together to obtain 12 + 1 = 13. (802-001 Mathematics Instruction)

What is the formula for the relationship between the number of faces, vertices, and edges of a cube? A. F+E=V+2 B. E+V=F+22 C. F+V=E−2 D. F+V=E+2

(D) The correct answer is (D): F+V=E+2. Recall that the Face (F) of a cube is a plain region of a geometric body, the Edge (E) is a line segment where two faces of a three-dimensional figure meet, and a Vertex (V) is the union of two segments or the point of intersection of two sides of a polygon. Knowing this we can see that for a cube there are 6 faces, 8 vertices, and 12 edges. Substituting these numbers into the appropriate spot in each formula allows us to determine option (D) as the only correct answer. (802-004 Geometry and Measurement)

As a formative assessment, Mr. Williams asked his students to create and draw a map of the town. Johana and Giovani drew complementary angles for their roads. Which of the following statements should the students include in their angle explanations? A. Complementary angles are each 90°. B. Complementary angles are made of two angles whose sum is 90°. C. Complementary angles are made of two angles whose sum is 180°. D. None of the above.

B. (B) The sum of complementary angles must equal 90°. (802-004 Geometry and Measurement)

What is an example of a rational number?

15/4 When represented in decimal format is 3.75. Recalling that an irrational number is one which cannot be represented by a fraction and whose decimal form never repeats or ends (it cannot be represented by a ratio of integers.

Formula for area of a triangle

A = 1/2 bh

Formula for area of a circle

A = pi r^2

How many unique factors are there for the number 9? A. 3 B. 4 C. 2 D. None of the above.

A. (A) In order to make 9 using multiplication, one can multiply 1 and 9 or one can multiply 3 and 3. The problem asks for the unique factors are 1, 3, and 9. (802-002 Number Concepts and Operations)

On Halloween the local grocery store offered a sale on bags of candy. The 72-ounce bags were regularly $8.95 each and were discounted to $5.99 each on purchases of 2 bags. The 36-ounce bags of candy were regularly priced $5.49 but were $3.99 each on purchases of 4 bags. The 18-ounce bags of candy were regularly priced $4.49 per bag and were discounted to $2.89 per bag on purchases of 8 bags. For a purchase of 288 ounces of candy, which option provides the best value? (The sale price on each option extends for multiples of the required amount of bags only.) A. 72-ounce bag B. 36-ounce bag C. 18-ounce bag D. All are the same

A. (A) The correct answer is (A). One way to solve this problem is to figure out how many bags of each size you would need to buy in order to obtain 288 ounces. Then, multiply the number of bags by the price per bag. This will then tell the total price to be paid for each option. The 72-, 36-, and 18-ounce bags are all multiples of the required 288 ounces. One needs four 72-ounce bags, eight 36 ounce bags, and sixteen 18-ounce bags to obtain 288 ounces. For the sale price to be applied to each option, we need a multiple of 2 for the 72-ounce bags, a multiple of 4 for the 36-ounce bags, and a multiple of 8 for the 18-ounce bags. Therefore, the sale price applies to all bags. After multiplying the required number of bags by the sale price, we see that the best option to pay the least is by buying the four 72-ounce bags of candy. (802-006 Mathematical Processes)

What is the probability of rolling a 7 with a pair of dice? A. 1/6 B. 1/36 C. 1/12 D. 5/36

A. (A) With a pair of dice there are 36 different combinations of outcomes. Of the 36 outcomes there are 6 different ways to roll a 7: 1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2, 6 + 1. Note that 1 + 6 is not the same as 6 + 1. If one die is green and the other red, 1 red, 6 green is not the same as 6 red, 1 green. Therefore there is a 6/36 chance of rolling a 7. This may be simplified to 1/6. (802-005 Probability and Statistics)

In the Texas curriculum, the teaching of place value is required in which of the following? A. First grade B. Third grade C. Kindergarten D. Fourth grade

A. A) One of the primary focal areas in grade 1 is the teaching of place value as a way to teach problems involving addition and subtraction. Teaching place values represent a concrete way to teach basic arithmetic in a tangible fashion. In second grade this continues being a focal point to aid in building a foundation to teach multiplication. In grade 6, place values continue being used to teach number and operation, parts-to-whole relationships and equivalence. Based on this explanation, the rest of the options are ruled out. (802-001 Mathematics Instruction)

In a bag there are 12 green marbles, 11 red marbles, and 7 blue marbles. What is the probability of drawing a blue marble? A. 7.4 B. 7/30 C. 11/30 D. 12/30

B. (B) Since there are 7 blue marbles and a total of 30 marbles in the bag, the odds of drawing a blue marble are 7/30. (802-005 Probability and Statistics)

The use of digital cameras, balanced metric weight sets, and virtual manipulatives available on the Internet is most appropriate for children in which of the following grade levels? A. Grades 1 to 6 B. Upper elementary C. Kindergarten to grade 2 D. Grade 6

B. (B) The use of electronic equipment, virtual manipulatives, and devices to measure using the metric system are generally used in upper elementary. Teaching the metric system begins in third or fourth grade. Allowing young learners (K-2) the use of digital cameras can be risky. Students in early childhood enjoy working with objects that they can touch and manage. The use of virtual manipulatives is less concrete than physical ones, and for that reason they are more effective with children in upper elementary. (802-001 Mathematics Instruction)

Dana uses 2 gallons of milk in 3 weeks. How many gallons of milk would she use in 9 weeks? A. 5 B. 6 C. 13.5 D. 7

B. (B) This problem can be solved multiple ways. One method is to set up a proportional problem. This can be done by placing the number of gallons she uses over the time period in which they are used. (2 gallons)/(3 weeks)= (x gallons)/(9 weeks). Multiplying both sides of the equation by 9 weeks to find out how many will be used in this amount of time we obtain gallons = (9 weeks x 2 gallons)/(3 weeks)=6 gallons. (802-003 Patterns and Algebra)

Suzie has 5/7 of a pie and Liz has 3/5 of a pie. The full size of each pie is identical. Which of the following concepts would be utilized to determine who has more pie? A. Greatest common factor B. Lowest common denominator C. Matching numerator D. Greatest common factor

B. (B) To compare fractions, one should obtain the lowest common denominator to compare. (802-002 Number Concepts and Operations)

What is the result when 7x^5 and 5x^4 are multiplied together? A. 12x^9 B. 35x^9 C. 35x^20 D. 12x^20

B. (B) When multiplying exponential numbers, we multiply the constant terms and then add the exponents of the terms with the same base. So, we multiply the constants 7 and 5 to obtain 35 and then add the 5 and 4 exponents associated with the x base. (802-003 Patterns and Algebra)

Formula for circumference (perimeter) of a circle

C = 2 pi r OR C = pi d

Solve the following equation for x. 44−7x=3x+5+7x−4x A. 39 B. 93 C. 3 D. 9

C. (C) In this problem we are asked to find the value of x. In order to aid in solving this problem we want to group like terms and simplify. Therefore, the first step is to group all of the terms associated with x on one side of the equals sign and all of the constant whole numbers on the other. Once this is done we are left with 13x=39. To solve for x we divide both the left and right hand sides of the equation by 13 to isolate the variable. When this is done we are left with x=3. (802-003 Patterns and Algebra)

A newscaster commenting on President Obama's interest in opening American markets to more Asian trade indicated that at the beginning of his presidency, Obama had opposed this idea, but now he was promoting it. To conceptualize this change of heart, he suggested that President Obama had undergone a 360-degree change in his stance. From a mathematical point of view, what is the problem with this characterization of the Obama policy? A. The newscaster is presenting a subjective view of the president's policy. B. Having a 360-degree change is an exaggeration. C. The use of the 360-degree allusion implies that the president has not changed his policy at all. D. He misled the audience because most Americans are not familiar with how the mathematical concept of degree can be used in politics.

C. (C) Mathematically, if you make a 360-degree change, you end up in the position that you started from. If the newscaster wanted to imply that President Obama changed his position to the opposite, he should have said that the president had made a 180-degree shift. Based on this explanation, the rest of the options are ruled out. (802-001 Mathematics Instruction)

Anu has a pocket full of change. She has a total of 3 quarters, 7 dimes, and 10 nickels. She goes to the vending machine to purchase a soda. What is the probability that Anu will pull out 60 cents if she reaches into her pocket twice, removes a single coin on each draw, and does not place the coin back in her pocket before drawing the second coin? A. 0.16% B. 43% C. 0% D. 6%

C. (C) The correct answer is (C). The problem asks you to analyze the probability of Anu removing $0.60 from her pocket while only removing a single coin from her pocket on 2 consecutive opportunities. Knowing that with the specific coins in Anu's pockets (3 quarters, 7 dimes, and 10 nickels) there is no way for any 2 coins to sum to $0.60 reveals a probability of 0 that this can happen. (802-005 Probability and Statistics)

What is the mean of the following data set? 4, 12, 3, 15, 8, 3, 9, 9, 8, 10, 12, 14, 3, 2. A. 3 B. 9 C. 8 D. 14

C. (C) The mean is the average of all the numbers in the data set. To find the mean, we first sum all of the numbers and then divide by the number of data points in the set. The sum of the numbers is 112 and there are 14 total numbers. 112/14 = 8. (802-005 Probability and Statistics)

What is one of the greatest accomplishments achieved by the Babylonians? A. Concept of 0 B. Rational numbers cannot accurately express all mathematical values C. Base-60 number system D. Angles at the base of an isosceles triangle are equal

C. (C) Typically the Arabs are given credit for incorporating the concept of 0 in our current number system. The Greeks are given credit for the equal angles at the base of an isosceles triangle and for discovering that rational numbers cannot accurately express all mathematical values. The Babylonians implemented a base-60 number system. (802-006 Mathematical Processes)

Jennifer Gray, a third grade teacher, took her class grocery shopping at the local supermarket. She organized the class in groups of five and guided them to study and purchase products based on best value and caloric content of the products. The goal is to purchase the least expensive products with the lowest caloric content. What type of information should the teacher provide students prior to taking them to the market? A. Model how to find nutritional-value information, and how it is measured. B. Explain the difference between the standard American and the metric systems. C. Review mathematical processes the students might need to solve problems. D. All of the above.

D. (D) All the components listed will help students in this project. Finding and reading labels is vital to making informed decisions (A). Determining how the product is measured (B) will facilitate the students' decision. Most foreign-made products will use the metric system, which can create confusion among students. This activity represents a meaningful mathematics/science application that requires specific mathematic process (C) to find solution to problems. (802-001 Mathematics Instruction)

What is 7.5−4 1/4 represented as a decimal? A. 1.75 B. 2.25 C. 3.35 D. 3.25

D. (D) There are numerous methods that can be used to solve this problem. Since the problem asks for the answer as a decimal, we will convert all numbers into decimal form before carrying out any other operation. The fraction represented as a decimal is 4.25. Subtracting this from 7.5 yields a difference of 3.25. (802-002 Number Concepts and Operations)

Carlos wanted to bring cookies for the entire school. He is going to bring in 400 cookies in bags. If each bag can hold 38 cookies, how many bags will he need? A. 10.52 B. 10 C. 12 D. 11

D. (D) To find the number of bags required for Carlos to package the cookies for his schoolmates, divide the number of cookies by the number of cookies that are able to fit into a single bag. Using a technique called dimensional analysis, we can see that the answer will then have the units of bag (or bags). # cookies divided by # cookies/bag, and using our properties of division with fractions we can change this to be # cookies x # bag/cookies . Since cookies are both in the numerator and the denominator, these cancel and our answer when using the actual numbers will be shown in units of bags. Doing this yields 10.52 bags. Since Carlos cannot have a fraction of a bag, we round up to 11 because were counting the number of bags. (802-006 Mathematical Processes)

What is 5 less than three times a number represented as? A. (5−x)3 B. (x−5)3 C. (x−5)/3 D. 3x−5

D. (D) To find the proper representation of the question it is easiest to work backwards. 5 less than three times a number means that we are subtracting 5 from a number multiplied by 3. Therefore, we multiply a number x by 3 and then subtract 5. (802-002 Number Concepts and Operations)

Students are working independently to solve the equation -4 + ? = -10 . The teacher says the following to help them understand the problem. "If you owe somebody $4, you have a negative $4 balance with that person. If you borrow more money from the person, you will owe more and have a more negative balance with that person." After speaking with several students, the teacher finds that some of them are still having trouble with the concept of negative numbers. As a result, the teacher then reteaches the concept using a number line. Which of the following types of assessments has the teacher used? A. Formative B. Summative C. Formal D. Criterion

Option A is correct because formative assessment involves teachers' adjusting their instruction based on their assessment of students. The teacher in the scenario has conversations with the students and teaches again based on what is observed. Option B is incorrect because a summative assessment involves a teacher evaluating a student's learning, often at the end of a unit. It is usually a high-stakes test. Option C is incorrect because a formal assessment will give a teacher data. It is usually a test that has been used before, and data can be compared between students. Option D is incorrect because most criterion-referenced tests are used to simply tell if a student has learned the material, not to adjust instruction as described in the scenario.

Which of the following activities is most effective in helping kindergarten students understand measurement of the lengths of small objects, such as pencils or cups? A. Placing interlocking cubes next to the objects and counting the cubes. B. Cutting sheets of construction paper so that they are the same dimensions as the objects. C. Listening to the teacher explain how to line up a ruler next to the objects and mark their lengths. D. Watching the teacher demonstrate how to estimate the lengths of the objects using a child's hand or shoe.

Option A is correct because having the students count the number of interlocking cubes and placing the cubes next to each item provides a visual for the student to use in determining length. Option B is incorrect because cutting the construction paper would help in developing the concept of area and not length. Option C is incorrect because effectively measuring with a ruler involves concepts that have not been introduced yet in kindergarten. Option D is incorrect because using a hand or a shoe to estimate a length is more appropriate when the order of magnitude of the objects to measure is greater.

Which of the following learning goals is most appropriate for a third-grade unit on money? A. Students will be able to determine the value of a collection of coins and bills. B. Students will be able to represent the value of a collection of coins as a fraction of a dollar. C. Students will be able to differentiate between money received as income and money received as gifts. D. Students will be able to solve problems involving money by performing operations on decimals to the hundredths place.

Option A is correct because this option correctly describes a Texas Essential Knowledge and Skill (TEK) for third grade math. A learning goal identifies what students will learn or be able to do as a result of instruction, not what they will be asked to do to demonstrate such learning. Options B, C and D are incorrect because they do not describe a learning goal that is appropriate for third grade.

Word problem: A paint store is having a sale, and for every gallon of paint a customer purchases, the customer will receive one additional gallon for free. Write an equation for p, the number of gallons of paint received, in terms of x, the number of gallons of paint purchased. A teacher asks students to solve the word problem shown. One student, John, says the answer is 2 + x = p. Which of the following activities will best help John recognize his misconception? A. Generating a model B. Creating a function table C. Using mental math D. Graphing the numbers

Option B is correct because a function table is a table of ordered pairs that follow a rule. The table will help the student identify the pattern. Option A is incorrect because generating a model is more appropriate for a geometric figure. Option C is incorrect because using mental math is less effective than teaching a student how to accurately use a strategy. Option D is incorrect because graphing the numbers is a skill that follows understanding and seeing the pattern and relating it to an expression.

Which of the following geometric solids has five faces, eight edges and five vertices? A. A pentagonal pyramid B. A rectangular pyramid C. An octagonal prism D. A triangular prism

Option B is correct because a rectangular pyramid has one rectangular base and four triangular faces, eight edges and five vertices. Option A is incorrect because pentagons have five sides, so a pentagonal pyramid has six faces — one pentagonal base and five triangular faces. Option C is incorrect because an octagonal prism has ten faces — two octagonal faces and eight rectangular faces. Option D is incorrect because a triangular prism has five faces, nine edges and six vertices.

A fifth-grade teacher writes the problem 56 12 × on the board. Students begin to solve the problem mentally, and as each student finds a solution, he or she signals the teacher with a thumbs-up signal. When almost every student has given a thumbs-up signal, the teacher has the following dialogue with a student. Teacher: "Billy, what answer did you come up with?" Billy: "792." Teacher: "Great job, Billy! That is the correct answer. Raise your hand if you found 792 to be the product, like Billy." Almost every student in the class raised a hand. The teacher writes the next problem on the board. Which of the following instructional adjustments can the teacher make to best assess all of the students' understanding of multiplying two-digit numbers? A. Allowing students to write their answers on paper, then collecting the papers at the end of the lesson B. Asking multiple students to share and defend their solutions before acknowledging the correct answer C. Asking students who did not hold up their thumbs to share their answer and explain D. Having Billy work the problem out on the board in front of the class

Option B is correct because the discussion gives the teacher an opportunity to hear how students are solving the problem. It will also give students an opportunity to share before they know their answer is wrong. Option A is incorrect because the students are only turning in their answers, not their work or their thinking. It will not show the teacher the level of their understanding. Option C is incorrect because it is only focusing on the students who did not raise their hand. More students may have missed the problem but were too shy to admit it. Option D is incorrect because the teacher is learning only about Billy's level of understanding, not that of the rest of the class.

After learning the theoretical probability of a two-sided coin landing on any one side, students work in groups to flip the coin several times and get the following results: 9 heads and 6 tails. Based on the scenario, which of the following observations made by students about probability is accurate? A. The theoretical probability of a coin's landing on heads, 0.5, is equal to the experimental probability obtained. B. The theoretical probability of a coin's landing on heads is less than the experimental probability obtained. C. The experimental probability of a coin's landing on heads, 0.6, is lower than expected. D. The experimental probability would have been more accurate if the students had decreased the number of trials.

Option B is correct because the experimental probability and the theoretical probability of the coin's landing on heads are respectively 0.6 and 0.5. The theoretical probability is thus less than the experimental probability. Option A is incorrect because the experimental probability and the theoretical probability of the coin's landing on heads are not equal. Option C is incorrect because the experimental probability of the coin's landing on heads is greater than the theoretical probability. Option D is incorrect because as the number of trials (in this case a trial is flipping a coin) increases, the experimental probability gets closer to the theoretical probability.

Which of the following statements best explains why the algebraic formula for the area of a triangle is 1/2 bh? A. A parallelogram can be transformed into a rectangle if a triangular piece is moved from one side to the other. B. The height of a triangle is not equal to the length of one of its sides, and the length must be divided by 2 to be used to find the area of the triangle. C. A parallelogram is composed of two congruent triangles, so the area of a parallelogram with the same base and height as the triangle can be divided by 2 to find the area of the triangle. D. This formula is only true for scalene triangles because all of their sides are different lengths, so one has to use the base and height to find the area.

Option C is correct because every parallelogram is made up of two congruent triangles. The formula for the area of a parallelogram is bxh, so the area of each of the two congruent triangles is half the area of the parallelogram. Two congruent triangles can always be arranged to form a parallelogram with the same base and the same height as the triangles. The area of the triangle will therefore be one-half as much as that of the parallelogram. Option A is incorrect because while the statement is true, it does not explain the formula for the area of a triangle. Option B is incorrect because it is logically faulty. Option D is incorrect because the area of every triangle can be found using the formula shown.

Word problem: Samantha's Bakery sells cupcakes in packages of 12 and cookies in packages of 20. The bakery sold the same number of cupcakes and cookies yesterday. What is the minimum number of cupcakes that the bakery could have sold? A teacher creates the word problem shown for a math lesson. Based on the word problem, the lesson will most likely cover which of the following mathematics concepts? A. Least common factor B. Greatest common factor C. Least common multiple D. Greatest common multiple

Option C is correct because finding the least common multiple will identify the smallest number of cupcakes sold. In fact, let x be the number of packages of cupcakes that were sold and y be the number of packages of cookies that were sold. The number of sold cupcakes will be 12x, and the number of sold cookies will be 20y. Clearly 12 divides 12x and 20 divides 20y. Since 12x = 20y, 20 must also divide 12x. Since 12 and 20 both divide 12x, 12x is a common multiple of 12 and 20. The problem asks for the minimum (or least) number of cupcakes that could have been sold, so the least common multiple must be found to answer the question. Option A is incorrect because the least common factor is always 1. Option B is incorrect because the greatest common divisor is 4. Option D is incorrect because there is no greatest common multiple.

3 x (2+6)^2 - 4 [3(2) + 3(6)]^2 - 4 (24)^2 - 4 576 - 4 572 A student simplifies the initial expression by applying the rule of order of operations. Which of the following best describes the student's error when simplifying the expression? A. The student added before evaluating the power of the exponents. B. The student evaluated the exponent before subtracting. C. The student multiplied before simplifying within grouping symbols. D. The student simplified the expression from left to right.

Option C is correct because the student multiplied before evaluating within the grouping symbol, violating the rule for order of operations: parentheses left to right, exponents left to right, multiplications or divisions left to right, additions or subtractions left to right. Option A is incorrect because the distributive property is not a rule in order of operations. Option B is incorrect because if the student had simplified the expression from left to right, the student would have gotten 3x2=6; 6+6=12; 12^2=144; 144-4=140.Option D is incorrect because the student was supposed to add before evaluating the power of the exponent, since addition fell within evaluating the expression in parentheses.

A first-grade teacher has set up the following math workstations for students to work in pairs. Station 1: Students toss 7 two-color counters from a cup and record the addition equation represented. They repeat the process ten times. Station 2: Each student builds a tower of 8 cubes using two different colors, and then records the addition equation that the colors represent. Students then exchange towers and record the addition equation for the new towers. Station 3: Students are provided with 9 counters each. One student hides some of the counters. The other student looks at how many counters remain present and determines how many are hidden. The students then record the equation that the missing counters represent. Which of the following relationships are the students most likely exploring in the stations? A. Spatial concepts B. One more and one less C. Benchmarking numbers D. Part-part-whole

Option D is correct because in the activity, the students are conceptualizing that a number is made up of two or more parts. Option A is incorrect because in doing addition based on spatial relationships, students recognize the number of objects based on the way they are arranged. Option B is incorrect because in one-more-one-less relationships, students add and subtract 1 to a number. Option C is incorrect because when students use benchmarks, they relate a number to 5 or 10.

Celeste is buying erasers for 8 of her friends. There are 76 erasers left at the store. Which of the following approaches can Celeste use to determine the greatest number of erasers she can buy to give each of her friends the same number and have none remaining? A. Drawing a picture of 76 erasers and circling groups of 8 before counting the number of groups created B. Creating a table in which one column represents the number of erasers at the store and the other represents the number of erasers each friend receives C. Using a standard algorithm learned previously in class to solve 76 divided by 8 D. Making an organized list of the multiples of 8 to see which one is closest to 76

Option D is correct because the approach described provides the correct answer of how many erasers Celeste should buy at the store, 72. Options A, B and C are incorrect because the approaches described provide the number of erasers each friend will get, i.e., 7, not the total number of erasers Celeste should buy.

Formula for volume of a cylinder

V = pi r^2h

When can shapes be congruent?

When they have the same shape AND size. The other shape can be rotated.