EC-6 Practice Questions

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

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.

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

B. (2²) * (3²)

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

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

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

Factor tree

factor tree

Median

middle number - if odd average 2 middle numbers

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

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

Range

subtract the smallest data value from largest data value

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

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

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.

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

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.

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.

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.

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).

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 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 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

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.

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.

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.

(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.

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 mo

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.

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.

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 - 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.

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 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.

(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?

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.

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.

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.

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.

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.

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

(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.

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.

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

B. 9

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

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.

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.

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.

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

B. Tenths

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.

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

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.

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

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

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.

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

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.

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.

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

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.

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.

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.

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.

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.

(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.

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

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.

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 a

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.

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.

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

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.

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.

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.

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