FTCE Middle Grades Mathematics 5-9 Ultimate Guide2019-03-02T18:52:08+00:00

FTCE Middle Grades Mathematics 5-9: Ultimate Guide and Practice Test

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FTCE Middle Grades Mathematics 5-9

FTCE Middle Grades Mathematics 5-9 Quick Facts

Florida teacher certificate candidates are required to pass this test which covers mathematics for grades 5-9.

Cost:

First attempt: $200

Retake: $220

Scoring:

A scaled score of at least 200 is needed to pass the exam.

Pass rate:

The pass rate is 71%.

Study time:

In order to pass, you should break the test topics down into your strengths and weaknesses. Give yourself enough time to work on each topic until you feel confident with the majority of the subjects. Create a study plan with the time you need to study each topic and the days you plan to do that.

What test takers wish they would’ve known:

  • A Texas Instruments TI-30X IIS scientific calculator is provided at the test site. Examinees may not bring their own calculator.
  • Unofficial pass/non-pass status is provided immediately after testing.
  • You will not be penalized for incorrect responses so make an educated guess if you are not sure.

Information and screenshots obtained from the National Evaluation Series website: http://www.fl.nesinc.com/testPage.asp?test=025

Exam Content

Overview

The exam has ten competencies:

  • Problem-Solving and Reasoning (13%)
  • Manipulatives, Models, and Instructional Technology (6%)
  • Assessment (9%)
  • Connections Among Mathematical Concepts (7%)
  • Number Sense, Operations, and Proportionality (9%)
  • Foundations of Algebra (14%)
  • Algebraic Thinking (11%)
  • Data Analysis, Statistics, and Probability (7%)
  • Two-Dimensional Geometry (15%)
  • Measurement and Spatial Sense (9%)

So, let’s talk about Problem-Solving and Reasoning first.

Problem-Solving and Reasoning

This competency includes about 10 multiple-choice questions which makes up about 13% of the entire exam.

These questions test your knowledge of problem-solving strategies and mathematical expressions, as well as your ability to use logic and reasoning.

Let’s talk about some concepts that you will more than likely see on the test.

Teaching Strategies: Problem-Solving

One strategy for helping middle schoolers problem solve, particularly with word problems, is to have them break up the problem into the following steps:

  1. Write down the information that you know.
  2. Write down what you need.
  3. Draw any relevant diagrams.
  4. Use what you have to work toward what you need.
  5. Circle the solution. Make sure that you answered the question that was asked and that you have the correct units.

Another problem-solving strategy that helps students find key information in the problem is called C.U.B.E.S:

Circle the important numbers

Underline the question statement

Box the keywords

Eliminate extra information

Show work

Deductive versus Inductive Reasoning

Deductive reasoning, sometimes called top-down reasoning, starts with premises that are known to be true and continues towards a logical conclusion.

For example, humans breathe air. David is a human; therefore, David breathes air.

Inductive reasoning, sometimes called bottom-up reasoning, starts with specific observations and combines them to make broad generalizations.

For example, you notice that Julia shows up late to work. Julia is a teenager; therefore, you conclude that teenagers are irresponsible.

Conclusions reached using inductive reasoning can be wrong sometimes, but this type of reasoning is still important. Inductive reasoning is most often used to form hypotheses, which can then be tested using the scientific method.

Manipulatives, Models, and Instructional Technology

This competency includes about 5 multiple-choice questions which makes up about 6% of the entire exam.

These questions test your knowledge of manipulatives, technology, and multiple representations of mathematical concepts.

Here are some concepts that you may see on the test.

Using Models

Models provide visual pictures of mathematical concepts, which can help to remember concepts, as well as understand them better.

For example, below is a model for squaring the binomial expression 5x – 3:

Graphing Calculators

In the middle grades, graphing calculators are useful for graphing linear and quadratic equations. They are particularly helpful for students to check their work, once they understand how to graph these kinds of equations by hand.

A graphing calculator can also be used to quickly find statistical measurements from a list of data using the list feature, including minimum, maximum, mean, median, and upper and lower quartiles. This can be helpful in analyzing data from a probability simulation.

Graphing calculators are most helpful when they are not relied on as a crutch for not understanding mathematical concepts. As such, they should be used on a limited basis.

One of the easiest ways to teach students how to use a graphing calculator is to use an online calculator simulator so that they can follow along as you press buttons in sequence on an overhead screen or projector.

Assessment

This competency includes about 7 multiple-choice questions which makes up about 9% of the entire exam.

These questions test your knowledge of diagnosing students’ needs, interpreting performance, and determining appropriate assessments.

Let’s talk about some specific concepts you need to know.

Formative versus Summative Assessments

Formative assessment is given regularly throughout the school year to provide ongoing feedback and help teachers determine concepts that need to be covered in more detail. Formative assessments also help students identify where their own understanding can be improved. Formative assessment is usually lower stakes and can include very informal tasks such as games, projects, and group work, as well as writing a summary about the main point of the lesson that day or submitting an outline of a paper before writing it.

Summative assessments are usually given at the end of a chapter, unit, or course to determine how much the student has learned and retained. These types of assessments are usually higher stakes and include tests, quizzes, final papers, or cumulative projects.

Audience-Response Systems

Audience-response systems are technology that allows the teacher to poll a large group of students very quickly. They usually involve some kind of handheld voting instrument that is given to each student. Then the students vote on a topic and the results are summarized and sent to the teacher.

For example, the teacher can put a multiple-choice question on the board and ask the students to select a, b, c, or d. After the students make their choices, the teacher can then determine within seconds whether the concept is understood by the majority of students or if the concept needs to be covered more thoroughly.

Connections Among Mathematical Concepts

This competency includes about 5 multiple-choice questions which makes up about 7% of the entire exam.

These questions test your knowledge of mathematical errors, relationships between skills, and common misconceptions.

Let’s take a look at a concept that will more than likely appear on the test.

Common Misconceptions in Mathematics

Students often confuse area and perimeter, especially when given a shape other than a rectangle. They try to find the area of the figure by adding the lengths of the sides together.

When working with box plots, one common misconception is that the median shown is actually the average value or mean of the data.

Another common misconception is to think that when taking the square root of both sides of an equation, you only get the positive root. It is very easy for students to forget to include both the positive and negative root even after they have been explicitly taught to.

Number Sense, Operations, and Proportionality

This competency includes about 7 multiple-choice questions which makes up about 9% of the entire exam.

These questions test your knowledge of estimation, factorization, ratios, and relative size.

Here are some concepts that you may see on the test.

Cube Roots

The cube root of a number is the value that when multiplied by itself 3 times, gives you that number.

For example, the cube root of 64 is 4 since 4*4*4 = 64.

The cube root of -729 is -9 since -9*-9*-9 = -729.

Cube roots are not always integers. For example, the cube root of 21 is not an integer since it can only be factored as 1*21 or 3*7. However, we can say that the cube root of 21 is approximately 2.7589…

GCF and LCM

The GCF (Greatest Common Factor) is the largest factor in common for two numbers.

For example, let’s find the GCF of 18 and 108:

First, find all the factors of each number.

18 has factors 1, 2, 3, 6, 9, 18.

108 has factors 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108.

The factors in common between these numbers are 1, 2, 3, 6, 9, and 18.

The largest of these factors is 18; therefore, the GCF of 18 and 108 is 18.

The LCM (Least Common Multiple) is the smallest value that is a common multiple of two numbers.

For example, let’s find the LCM of 6 and 15:

First, list the multiples of each number.

*NOTE: As a rule of thumb, if you are not sure when to stop listing, keep going until you reach the product of the two numbers (in this case 6*15 = 90). This number is always a common multiple of the two; however, it may not be the least common multiple.*

The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, …

The multiples of 15 are 15, 30, 45, 60, 75, 90, …

30, 60, and 90 are all common multiples of 6 and 15; however, since 30 is the smallest, it is the LCM.

Foundations of Algebra

This competency includes about 11 multiple-choice questions which makes up about 14% of the entire exam.

These questions test your knowledge of patterns, inequalities, linear equations, and square roots, as well as your ability to simplify expressions.

Let’s talk about some concepts that you will more than likely see on the test.

Solving Inequalities

Example 1: Solve the inequality 4x < 2x + 8.

Isolate x by collecting all of the terms with a variable on one side of the inequality. To do this, subtract 2x from both sides:

4x – 2x < 8

2x < 8

x < 4

Example 2: Solve the inequality below:

-7x + 10 ≤ 31

To solve this inequality, again we need to isolate the variable x. Subtract 10 from both sides:

-7x ≤ 31 – 10

-7x ≤ 21

Divide both sides by -7:

x ≥ -3

*NOTE: When you multiply or divide an inequality by a negative number, you must reverse the inequality (flip the sign).*

We can represent the solution x ≥ -3 on a number line. Since this inequality uses the symbol “≥,” the solution includes the value -3 and everything greater than -3. So, we shade to the right of -3 on the number line:

Determining the Equation of a Line

The equation of a line can be determined using two points on the line or with the y-intercept and slope of the line.

This information can be plugged into various forms of linear equations:

Point-Slope Form: y-y1=m(x-x1) where m is the slope and (x1, y1) is a point on the line.

Slope-Intercept Form: y = mx + b where m is the slope and b is the y-coordinate of the y-intercept.

The slope of a line is calculated using the formula m = (y2y1)/(x2x1).

Example 1: You are given the graph below.

The slope can be calculated using any two points on the line. We will choose (2, 0) and (0 , -4):

Using the point-slope form of an equation of a line, y – 0 = 2(x – 2).

Example 2: Given that a line has a slope of -8 and a y-intercept at (0, 4), find the equation of the line.

In this case, it is easier to find the equation of the line in slope-intercept form. We know that m = -8 and b = 4. Therefore, y = mx + b = -8x + 4. The equation of the line is y = -8x + 4.

Algebraic Thinking

This competency includes about 8 multiple-choice questions which makes up about 11% of the entire exam.

These questions test your knowledge of solving systems of equations and quadratic functions, as well as finding zeros of polynomials. You also need to know the laws of exponents.

Here are some concepts you need to know.

Factoring Polynomials

Factoring is writing a polynomial as the product of two other polynomials. This is usually applied to simplify an expression involving polynomials or to find the roots of a polynomial. There are many strategies for factoring polynomials including factoring out the greatest common multiple, factoring perfect square trinomials, factoring by grouping, and several other strategies.

Let’s take a look at some examples.

Laws of Exponents

For a (any real number), b (any non-zero real number), and n and m (natural numbers), the following properties are true:

Data Analysis, Statistics, and Probability

This competency includes about 5 multiple-choice questions which makes up about 7% of the entire exam.

These questions test your knowledge of statistical measures, graphical representations of data sets, population samples, and probabilities.

Let’s talk about a concept that you will more than likely see on the test.

Central Tendency

Measures of central tendency help us to determine how data is distributed by identifying a single value at roughly the center of the data. We will consider several measures of central tendency below.

The mean of a data set is average value of a data set. This can be found by adding together all of the values and dividing by the total number of values.

The mode of a data set is the value that occurs most frequently in the data set. It is possible to have more than one mode in a data set if several values occur the most.

The median of a data set is the middle value in the set. If there is an even number of data points, there will not be an exact middle. In this case, the median is found by taking the average of the two data points closest to the middle.

For example, suppose that the ages for a group of ten students were collected and are listed below:

9, 11, 13, 11, 8, 7, 13, 9, 9, 12

The mean of this data set can be found by adding all of these ages together and dividing by 10, since there are 10 students:

Mean = (9+11+13+11+8+7+13+9+9+12) / 10=10.2

To find the mode and median of a data set, it is helpful to reorder the set from lowest to highest.

7, 8, 9, 9, 9, 11, 11, 12, 13, 13

Now we can see that the mode of the data set is 9, since 9 occurs 3 times, which is more than any other data point.

The middle of the data set is between the two 11’s in the middle of the set; therefore, the median of the set is 11.

Two-Dimensional Geometry

This competency includes about 11 multiple-choice questions which makes up about 15% of the entire exam.

These questions test your knowledge of geometric figures such as triangles, quadrilaterals, angles, and circles, including relevant theorems, as well as the coordinate plane.

The following concept is likely to pop up on the test.

Trigonometric Ratios

There are 6 trigonometric ratios that you should know:

When given x, an angle measure in a right triangle, and two of the sides, opposite and adjacent to the angle or the hypotenuse, this provides the information needed to find the length of the missing side.

Example 1: Find the hypotenuse of the triangle below.

First, consider which trigonometric identity is appropriate to use. Here we are given an angle and the opposite side, and we would like to know the hypotenuse. The appropriate trig identity is therefore the sine function, since it involves the opposite side and the hypotenuse:

Example 2: Find the missing angle x, of the right triangle, if you know that cotangent x = 6.

Now, to solve for the angle x, we need to use the inverse tangent function:

*NOTE: If you are working in degrees, make sure your calculator is in degree mode when using trigonometric functions.*

Measurement and Spatial Sense

This competency includes about 7 multiple-choice questions which makes up about 9% of the entire exam.

These questions test your knowledge of three-dimensional figures, area, and volume, as well as your ability to convert between units.

Let’s talk about some concepts that you will more than likely see on the test.

Characteristics of Three-Dimensional Figures

When looking at three-dimensional figures, it can be helpful to identify the different parts of the figure. A face of a three-dimensional figure is a flat surface. An edge is the line where two faces meet. A vertex is the point where three faces meet.

Below is a table of shapes and the number of faces, edges, and vertices each shape has.

Identifying Nets

A net of a three-dimensional figure is a two-dimensional representation of the shape that can be folded into the three-dimensional figure by the faces along the edges.

Here are some tips for identifying which net matches with a given 3D shape:

  • Make sure that the net has the correct number of faces.
  • If you are having trouble visualizing what the shape looks like folded, cut out the net and actually try folding it up.
  • Consider what shapes the faces of the 3D shapes are. For example, in a cylinder, we know that two of the faces are circles and one is a rectangle. This means that the net has to contain two circles and a rectangle.

Below are several examples of 3D shapes and their nets.

And that’s some basic info about the FTCE Middle Grades Mathematics 5-9 exam.

Exam Content Practice Test

Question 1

Order the following from least to greatest:

-3, √17, 8, ¾, π, -2.6.

  1. -3, -2.6, ¾, √17, π, 8
  2. ¾, -2.6, -3, π, √17, 8
  3. -2.6, -3, ¾, √17, π, 8
  4. -3, -2.6, ¾, π, √17, 8

Correct answer: 4. The smallest number in the set is -3; -2.6 would be next smallest; 8, √17, and π are all greater than 1 which makes them larger than ¾ so ¾ is next in line. √17 is bigger than 4, or √16, and smaller than 5, or √25, so √17 is just a little bigger than 4. This makes π smaller than 4 and makes 8 the largest number listed. So, the correct order from least to greatest would be -3, -2.6, ¾, π, √17, 8.

Question 2

John worked the following division problem:

7/8 ÷ 3/4 = (7 ÷ 3)/(8 ÷ 4) = (7 ÷ 3)/2 = 3.5/3 = 35/30 = 1(5/30) = 1 (1/6)

What is the error in John’s work?

  1. John does not understand how to divide fractions
  2. John’s work is mathematically correct
  3. When John reached the step (7÷3)/2, he should have distributed the 2 over the 7 and the 3 to get 3.5 / 1.5 = 2(1/3)
  4. John does not understand how to divide fractions and when he reached the step, (7 ÷ 3)/2 he should have distributed the 2 over the 7 and the 3 to get 3.5 ÷ 1.5 = 2(1/3).

Correct answer: 2. John may not divide fractions by the method traditionally taught, but he knows how to divide fractions. Division is not distributive over multiplication. Another way of looking at this step would be (7 ÷ 3)/2 = (1/2)(7÷3) = (1/2)(7/3) = (7/6) = 1(1/6). It is correct to think of the divisor, 2, as a factor of 1/2.

Question 3

Consider the following problem:

3x² + 7 – 12 + 3 + 5 = -12

3x² + 10 + 5 – 12 = -12

3x² + 15 – 12 = -12

3x² + 15 = 0

3x² = -15

x² = -12

x = ±√12

What property could be used to justify step 2 in the problem above?

  1. The distributive property
  2. The reflexive property
  3. The associative property
  4. The addition property of equality

Correct answer: 3. In this problem, 3x² + 7 – 12 + 3 + 5 = -12, the terms were rearranged so that addition could be done at one time followed by the subtraction. The new arrangement became 3x² + 7 + 3 + 5 –12 = -12, and the 7 and 3 were added giving us step 2: 3x² + 10 + 5 – 12 = -12. The associative and commutative properties justify this step.

Question 4

The array above models which of the following?

  1. 2/3 > 3/4
  2. 3/4 – 2/3 = 1/12
  3. 2/3 • 3/4 = 6/12
  4. 3/4 > 2/3

Correct answer: 3. 2/3 > 3/4 could certainly be used to show how 2/3 and 3/4 are related to each other by comparing the number of small rectangles that 2/3 contains, 8, with the number of small rectangles contained in 3/4, 9. Since 9 > 8, this would mean that 3/4 > ⅔, because it is 1 rectangle larger. But that is not what is being modeled here, so this option is ruled out. 3/4 – 2/3 = 1/12 is not what is being shown in the picture given. The 3/4 > 2/3 model could certainly be used to show how 2/3 and 3/4 are related to each other by comparing the number of small rectangles that 2/3 contains, 8, with the number of small rectangles contained in 3/4, 9. Since 9 > 8, this would mean that 3/4 > ⅔, because it is 1 rectangle larger. But, that is not what is being modeled here, so this option is ruled out.

Question 5

Which set of manipulatives would be most appropriate for a fifth-grade class?

  1. Small objects to sort, counting blocks, geometric shapes
  2. A graphing calculator, base-ten blocks, attribute shapes
  3. A standard calculator, geoboards, pattern blocks
  4. Measuring tools, beans, popsicle sticks

Correct answer: 3. This is the best option for fifth-grade students.

Question 6

What would be the least appropriate use for handheld calculators in the classroom?

  1. To justify and explain your thinking
  2. To convert between decimals and fractions when ordering sets of rational numbers consisting of fractions, decimals, and percents.
  3. To answer questions on a computation test
  4. To explore linear relationships

Correct answer: 3. If students are taking a computation test, they are being tested on their ability to add, subtract, multiply, or divide without the aid of a calculator. All the other options could be very appropriate uses for calculators.

Question 7

In solving the following equation for x, John justified his work as follows:

Solve:  -3(x – 5) = -18.

  1. -3x + 15 = -18 distributive property
  2. -3x + 15 – 15 = -18 – 15 subtraction property of equality
  3. -3x + 15 – 15 = -33 combining terms
  4. -3x + 0 = -33 identity property for addition
  5. -3x = -33 identity property for addition
  6. x = 11 division property for division

At which point is John’s justification incorrect?

  1. Step 3 to Step 4
  2. Step 2 to Step 3
  3. Step 4 to Step 5
  4. Step 1 to Step 2

Correct answer: 1. Step 1 to 2 – This would mean that the justification for moving from step 1 to step 2 would be subtraction. You subtract 15 from each side of the equation. This is correct. Step 2 to 3 – When doing a proof and supplying justification for a step in the proof, it is important to remember that the reason is the justification for moving from one statement to the next: the reason is placed opposite the resulting statement. Step 4 to 5 – The correct reason for moving from step 3 to step 4 is the additive inverse property for addition: a number plus its additive inverse = 0. A correct application of the identity property for addition is the reason for moving from step 4 to step 5.

Question 8

If y varies directly as x and y = -6 when x = 4, what is the constant of proportionality?

  1. -10
  2. -3/2
  3. -2
  4. -2/3

Correct answer: 2. If y varies directly as x, then y = kx. Substituting the values given for x and y, -6 = k(4). And then, k = (-6/4) = (-3/2). The constant of proportionality is k = (-3/2).

Question 9

Which of the following is the equation of a line that is perpendicular to the line in this graph?

  1. y = -3x – 1
  2. y = ⅓x + 2
  3. y = 3x + 2
  4. y = -⅓x + 2

Correct answer: 1. Lines that are perpendicular have slopes that are negative reciprocals of each other. In this problem, the points (-6,0) and (0,2) are points on the line. So, the slope can be found by the following: M = (y₁ – y₂)/(x₁ – x₂) = (0-2)/(-6-0) = (-2/-6) = ⅓; this means that the slope of the perpendicular line would be -3 since the negative reciprocal of ⅓ would be -3/1 or -3. Only this option has -3 for its slope, so this is the only possible correct answer. y = ⅓x + 2 is actually the equation of the line that is graphed. It is important to understand that the graph pictured has a positive slope, because it is slanting upwards as you move from left to right. A line that is perpendicular would have to have a slope that is negative.

Question 10

What is the slope, the y-intercept, and the x-intercept of: 4x – 2y = 3?

  1. Slope: 4; y-int.: 3; x-int.: ¾
  2. Slope: 2; y-int.: 3; x-int. -¾
  3. Slope: 4; y-int.: 2; x-int.: 3
  4. Slope: 2; y-int.: -3/2; x-int.: ¾

Correct answer: 4. The equation 4x – 2y = 3 is a linear equation in standard form. Let’s first change this equation to slope-intercept form: y = 2x – (3/2). From this form, it is easy to see that the slope is 2, and the y-intercept is (-3/2), the x-intercept would be: 0 = 2x – (3/2), and then 0 = 4x – 3, so x = (¾) So, the slope is 2, the y-intercept is (-3/2), and the x-intercept is ¾.

Question 11

Which equation is graphed above?

  1. x = y²
  2. y = x²
  3. y = 3x + 1
  4. x = 3y – 1

Correct answer: 4. Since the graph is a line, the choices are immediately reduced to only y = 3x + 1 and x = 3y – 1, the only two linear equations. y = 3x + 1 has a y-intercept of 1 and a slope of 3. This means that the equation would cross the y-axis at 1. From that point, (0,1), for every move of 1 to the right, a vertical change of + 3, or 3 up, is required. The graph pictured does not go through the point (0,1) nor does it have a slope of 3. By the process of elimination, x = 3y – 1 would be the correct answer, but it is important to understand why. x = 3y – 1 in slope intercept form would be y = ⅓x + ⅓. This means the slope is ⅓ and the y-intercept is also ⅓. So, locate the y-intercept, (0, ⅓), and from that point for every move to the right, there is a vertical shift of ⅓ up that is required. This means that 2 moves to the right would require a move of ⅔ of a unit upwards, ending with the ordered pair (2, 1). Both the y-intercept (0, ⅓) and the ordered pair (2,1) are points on the graph pictured. This means the slope is ⅓ and the y-intercept is also ⅓. So, locate the y-intercept, (0, ⅓), and from that point for every move to the right, there is a vertical shift of ⅓ up that is required. This means that 2 moves to the right would require a move of ⅔ of a unit upwards, ending with the ordered pair (2, 1). Both the y-intercept (0, ⅓) and the ordered pair (2,1) are points on the graph pictured.

Question 12

Which of the following is not a proportional relationship?

  1. The cost of several bunches of bananas that sell for $.50 a pound
  2. The amount of gasoline used on a trip and the length, in miles, of the trip
  3. The cost of a gym membership that includes an initial membership fee followed by a monthly fee based on the frequency of use
  4. The ratio of the perimeter of a square to the length of one of its sides

Correct answer: 3. A proportional relationship occurs when there is a constant ratio of proportionality: common ratio. In P = 4s, the common ratio is 4, so the perimeter of a square to the length of one of its sides is a proportion. The common ratio is whatever the mpg is for the specific car. Say that ratio is x. Then, each mile you drive 1/x gallons of gas will be used. So, the constant ratio is 1/x: amount of gas needed = 1/x(distance). The cost of bananas will be $.50 lb. The common ratio is $.50, so this is also a proportion.

Question 13

Which could be the graph of the inequality: -2y > 4x – 6?

  1. Graph D
  2. Graph C
  3. Graph A
  4. Graph B

Correct answer: 3. This is the correct answer. To graph an inequality, first graph the linear equation with which it is related. So, for the inequality, -2y > 4x – 6, first graph the linear equation: -2y = 4x − 6 -> y = -2x + 3. The inequality tells us that y is less than -2x + 3, and this means that the area below the graph of the line should be shaded. One other important point about graphing inequalities to remember is that if the inequality is < or >, then the linear equation is graphed using a dotted line, because the actual points on the line are not in the solution set; if the inequality is ≤ or ≥, the linear equation is graphed using a dark solid line, because the points on the line are included in the solution set.

Question 14

Which ordered pair is the solution of the system:

x + 2y = -7

2x − 3y = 0

  1. (-3,-2)
  2. (1,-4)
  3. (-6,-40)
  4. (-1,-3)

Correct answer: 1. To arrive at the solution to this problem, you could do a “guess and check” and substitute the values given for x and y in each ordered pair into BOTH equations in the system to see if the ordered pair is a solution to the system: x + 2y = -7 and 2x – 3y = 0. Using point (-3, -2), we would get -3 + 2(-2) = -3 + -4 = -7, a true statement. For the second equation, 2(-3) – 3(-2) = -6 + 6 = 0, this is also a true statement. So, (-3, -2) is a correct solution. None of the other choices are true for BOTH equations in the system.

Question 15

An addition is to be added on to an existing home. The area of the addition is 425 ft². If the length is 8 feet longer than the width, what will be the length and the width of the new room?

  1. w = 15 ft, l = 23 ft
  2. w = 17 ft, l = 25 ft
  3. w = 25 ft, l = 17 ft
  4. w = 25 ft, l = 33 ft

Correct answer: 2. This is an area of a rectangular problem. The dimensions of the rectangle are w and w + 8 and the area is 425 ft².

Question 16

Solve x² – 6x − 7 = 0 by factoring.

  1. (x + 7)(x – 1) = 0
  2. (x – 7)(x + 1) = 0
  3. (x – 7)(x − 1) = 0
  4. (x + 7)(x + 1) = 0

Correct answer: 2. The only factors of 7 are 7 and 1; 7 is a prime number. Since 7 is negative, the factors for -7 are +1 and -7 or -1 and +7. So the binomial factors for this equation would be either (x − 1)(x + 7) = 0 or (x + 1)(x − 7) = 0. When we multiply the binomial factors to see which is correct we get the following: (x − 1)(x + 7) = x² + 7x − 1x − 7 = x² + 6x − 7 (x + 1)(x − 7) = x² − 7x − 1x + 7 = x² − 6x − 7 Since the equation to be factored is x² − 6x − 7 = 0, the second set of factors would be the correct factors for this equation.

Question 17

A rectangle has a length that is 8 cm longer than its width. If it is inscribed within a circle with a radius of 7 cm, which expression below represents the area of the un-shaded region?

  1. -w(w + 8)
  2. w(w + 8) – 49π
  3. w(w + 8) + 49π
  4. 49π – w(w + 8)

Correct answer: 4. w(w + 8) – 49π would yield a negative area and this is not reasonable. -w(w + 8) has the incorrect expression for the area of the circle: the radius, 7, should be radius squared, 7². w(w + 8) + 49π adds the areas of the circle and the rectangle together and does not correctly find the area of the unshaded area.

Question 18

What is the mean, median, mode and range for the data set?

75, 82, 68, 95, 74, 72, 91, 60, 72, 80

  1. Mean: 76.9; Median: 74.5; Mode: 72; Range: 31
  2. Mean: 80; Median: 73; Mode: 72; Range: 35
  3. Mean: 77; Median: 73.5; Mode: 72; Range: 31
  4. Mean: 76.9; Median: 74.5; Mode: 72; Range: 35

Correct answer: 4. The mean of the data is sum of all scores / total # of scores = 76.9. The median is the middle score when the scores are ordered: 60, 68, 72, 72, 74, 75, 80, 82, 91, 95 so 74 and 75 are the middle scores. The median is 74.5, the average of the two middle scores. The mode is the score that occurs the most frequently: 72. The range is the maximum value – the minimum value: 95 – 60 = 35.

Question 19

Which of the following could the dot frequency diagram actually represent?

  1. The ages of children in a third-grade classroom
  2. The temperatures over an 11-month period for a town (Feb.-Dec.)
  3. The possible outcomes when a pair of dice are rolled and their faces are summed
  4. All of the above

Correct answer: 3. The ages of children in the third grade would be between 8 and 9, not between 2 and 12, inclusive. The average temperatures for a town over a 11-month period would not be a count (frequency).

Question 20

Students who are 6 feet tall or taller are 182.88 cm or greater on the scatterplot.  Which of the following statements about this group is true?

  1. In general, their arm spans are less than their heights
  2. In general, their arm spans are greater than their heights
  3. In general, their arm spans are greater than or equal to their heights
  4. In general, only tall students have arm spans and heights that are the same

Correct answer: 3. Draw a horizontal line at 183 cm on the vertical axis and a vertical line at 183 cm on the horizontal axis. These lines will divide the graph into four parts; the upper left portion is the part we are interested in. In this portion, most of the points are either on the blue line or above it. This means that their arm spans are greater than or equal to their heights.

Question 21

Which of the nets pictured above could be folded into a cube?

  1. 2, 3, and 5
  2. 1, 2, and 4
  3. 1, 4, and 5
  4. 3, 4, and 5

Correct answer: 4. 1, 2, and 4–If you attempt to fold 1 or 2 into a cube, two of the faces will overlap. 2, 3, and 5–If you attempt to fold 2 into a cube, two of the faces will overlap. 1, 4, and 5–If you attempt to fold 1 into a cube, two of the faces will overlap.

Question 22

What is the area of figure B?

  1. 20 u²
  2. 12.5√2 u²
  3. 20√2 u²
  4. 40 u²

Correct answer: 1. This triangle has a height on the grid of 5 units and a base of 8 units. Since A =1/2(bh), the area would 20 u².

Question 23

What is the reasoning behind step 4?

  1. AAS
  2. SSS
  3. SAS
  4. ASA

Correct answer: 3. SAS because the vertical angle is between the two pairs of sides that are corresponding and congruent.

Question 24

What is the measure of angle a?

  1. 115°
  2. 25°
  3. It cannot be determined
  4. 65°

Correct answer: 4. Since lines x and y are parallel, line m is a transversal, and then the corresponding angles must be congruent. Since angle, a and angle g are corresponding angles and angle g is 65° (given information), then angle a must also be 65°.

Question 25

What is the volume of the figure above?

  1. 75 in³
  2. 120 in³
  3. 132 in³
  4. 60 in³

Correct answer: 4. The volume is B(h). The height is the distance between the two bases: 10 in for this prism. The area of the base, B, is ½(3)(4) = 6 in². V = 6(10) = 60 in³.

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