GACE Middle Grades Mathematics: Ultimate Guide and Practice Test
Preparing to take the GACE Middle Grades Mathematics exam?
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GACE Middle Grades Mathematics Quick Facts
The GACE Middle Grades Mathematics exam is designed to test the knowledge and skills of beginning middle school math teachers in Georgia.
Tests are scored from a range of 100-300. There are two passing score levels:
- 220–249: induction level
- 250: professional level
So, you need to score at least a 220 to pass.
The amount of time you will need to spend preparing for the GACE Middle Grades Mathematics exam depends upon your existing content knowledge.
One way to determine your aptitude for the GACE Middle Grades Mathematics exam is to use 240Tutoring materials and practice questions to gage your understanding of the contents of the exam. Which concepts do you struggle with the most?
After identifying your areas of need, you can use 240Tutoring tools to strengthen your knowledge of these concepts until you’re ready for the big day! Remember, it’s best to spend some time studying each day instead of cramming for the exam shortly before you take it. That way, you’ll retain what you learn and you’ll also have less stress during the exam.
What test takers wish they would’ve known:
- You will have access to an on-screen calculator. Check out this free tutorial to become familiar with the calculator before the exam: http://www.gace.ets.org/prepare/tutorials/calculator
- It’s a great strategy to track your time while taking the exam. You can monitor your time by periodically checking the timer in the upper right-hand corner of your screen.
- Test-takers tend to overestimate their abilities to perform well on GACE assessments. Many students regret not putting more time and effort into preparing for GACE assessments beforehand. Fortunately, it’s easy to avoid this mistake by using test preparation materials early. If you’re reading this, you’re already starting off on the right foot!
- Because time management is crucial, skip questions that you find extremely difficult and move forward to questions that you find easier to answer. Don’t worry, you can mark the questions you skip as you take the test. Try to finish the other questions with 10 to 15 minutes remaining and use that extra time to return to the more challenging questions. If you are unsure of an answer, it is better to guess than to leave a question blank.
- When answering the selected-response questions, you should read all possible answers before marking the correct one. You wouldn’t want to miss out on the best answer by not reading all of the responses!
- You’ll feel more confident if you check out GACE’s free guide to taking computerized tests.
Information and screenshots obtained from the ETS GACE website: https://gace.ets.org/prepare/materials/013
Arithmetic and Algebra
The Arithmetic and Algebra subarea has about 29 questions. These questions account for 65% of the entire exam.
This subarea can be neatly divided into 3 sections:
- Numbers and Operations
- Functions and Their Graphs
So, let’s talk about the Numbers and Operations section first.
Numbers and Operations
This section tests your ability to manipulate numerical expressions, as well as use ratios and proportions.
Let’s talk about some concepts that you will more than likely see on the test.
Prime factorization is breaking an integer down into its prime factors.
34 can be factored as 217. Since both 2 and 17 are prime numbers (they cannot be factored), 2(17) is the prime factorization of 34.
180 can be factored as 1810. Both of these factors can be factored again. For example, 18 can be factored as 92, and 10 can be factored as 52:
180 = 18(10) = 9(2)(5)(2)
Now, all of these factors are prime except for 9, which can be factored as 33. Therefore:
180 = 3(3)(2)(5)(2)
Now, let’s group the factors that are duplicates:
180 = 2²3²5
2²3²5 is the prime factorization of 180.
Absolute and Relative Error
Absolute error is the discrepancy between the actual value and the measurement you found.
For example, you have 20.2 mL of water in a vial. When you measure the volume of the water, you record the volume as 20.1 mL; therefore, the absolute error in your measurement is 0.1 mL.
However, suppose you estimate the internal temperature of a turkey at 170℉, when the actual temperature is 172℉. The absolute error in this situation is 2℉.
Now we have two different situations. In the first one, the absolute error is 0.1 mL, and in the second one, the absolute error is 2℉. How can we compare these errors since they are in different units?
This is where relative error is useful. Relative error is equal to the absolute error divided by the measurement.
In the case of the vial of water, the relative error = 0.1/20.1.≈.005. In the turkey situation, the relative error ≈ 2/170≈.0118.
This section tests your ability to solve linear equations and inequalities, as well as represent patterns using sequences.
Here are some concepts you should know.
Linear Equations and Inequalities
Take a look at this linear equation in one variable:
5x – 9 = 4
To solve this equation, we need to isolate the variable x. We can move the -9 to the other side by adding 9 to both sides:
5x – 9 + 9 = 4 + 9
5x = 13
Now we can move the 5 to the other side by dividing both sides by 5:
x = 13/5
This is the solution to the equation.
Here is a linear inequality in one variable:
-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.
Quadratic equations are equations where the variable has degree 2.
Let’s solve the quadratic equation:
Some quadratic equations can be solved by factoring; however, this equation can only be solved algebraically by using the quadratic formula or by graphing. We will solve using the quadratic formula.
To use this method, first put the equation into the form ax2+bx+c=0. To do this, subtract 2 on both sides of the equation:
Now we can see that a = 5, b = 1, and c = -2. We will plug these coefficients into the quadratic formula:
x=(-1√1²-4(5)(-2))/2(5) = (-1±√41)/10
Therefore, x =(√41)/10-(1/10) and x = -(√41)/10–1/10.
Functions and Their Graphs
This section tests your knowledge of functions and their different representations, including notation. You’ll be evaluating functions, modeling with functions, and finding various properties of functions.
Let’s talk about some concepts.
Domain and Range
The domain of a function is the set of inputs that a function can take which produce a real number output.
The range of a function is the set of outputs a function can take on.
Example 1: Consider the function f(x) = √(4-x²). For the x-values to be in the domain of this function, the expression under the square root cannot be negative, or the function will take on imaginary values. Therefore, we must solve:
2 ≥ x
Therefore, x must be less than or equal to 2. We can write the domain as the interval from 2 to infinity using the notation [2, ∞).
The range of f(x) is the set of values the function can take on. Since the output can never be negative for this function, the range is the set of all real numbers where x 0. We can write that in set notation as [0, ∞).
Example 2: Consider the function g(x) = (6x-2)/(3x+12).
This function can take on any value of x and produce a real number output, unless the denominator is zero; therefore, to find the value which is not in the domain, solve the equation 3x + 12 = 0:
3x = -12
x = -4
The domain of g(x) is therefore all real numbers where x≠-4.In set notation, this is written as (-∞, -4) U (-4, ∞).
The range of g(x) in this case is all real numbers where y ≠ 2.You can see this from the graph of the function:
To evaluate the function graphically at -1, find the y-coordinate of the point on the graph that corresponds to x = -1.
From the graph, you can see that the y-coordinate of the function at x = -1 is -3.
Therefore, f(-1) = -3.
Example 3: Consider the function represented by the table below. Find f(3).
To find f(3), we need to find the value of the function at x = 3.
By looking at the table, we can see that f(3) = -2.
And that’s some basic info about the Arithmetic and Algebra subarea.
Now, let’s look at a few practice questions in each area to see how these concepts might actually appear on the real test.
Geometry and Data
The Geometry and Data subarea has about 16 questions. These questions account for 35% of the entire exam.
This subarea can be neatly divided into 2 sections:
- Geometry and Measurement
- Probability, Statistics, and Discrete Mathematics
So, let’s talk about the Geometry and Measurement section first.
Geometry and Measurement
This section tests your knowledge of circles, triangles, quadrilaterals, area, volume, geometric transformations, congruence, properties of lines, and measurement.
Let’s take a look at some concepts that you definitely need to know for the test.
Angle relationships help us to find missing angles when we are given a diagram, often involving parallel and intersecting lines.
Some important angle relationships are supplementary, complementary, vertical angles, alternate interior angles, and alternate exterior angles.
Supplementary angles are angles whose measures add to 180 degrees. If the angles are next to each other, they will form a straight line.
Complementary angles are angles whose measures add to 90 degrees. If the angles are next to each other they will form a right angle.
For example, if ∡a=48° and ∡b=42°, then angle a and angle b are complementary since they sum to 90°.
Vertical angles are opposite each other on other sides of two intersecting lines. Vertical angles are always congruent (have equal measure).
For example, in the diagram below, angle a and angle c form a vertical pair. Angle b and angle d are also vertical to each other; therefore, we can say that ∡b=∡d and ∡a=∡c.
Also, angle a and angle d are supplementary, as are angle a and angle b. Other supplementary pairs are:
Angle b and angle c
Angle c and angle d
A transversal is a line that cuts through at least two other lines.
When we have transversals that cut through more than one line, they form alternate angles. Alternate interior angles are on opposite sides of the transversal but inside the two lines being cut by the transversal.
For example, in the diagram below, a and b are alternate interior angles, and c and d are alternate interior angles.
*NOTE: When the two lines cut by the transversal are parallel to each other, then alternate interior angles are always congruent.*
Likewise, alternate exterior angles are congruent if the two lines cut by a transversal are parallel. These are angles on the outside of the two lines, but opposite each other.
For example, in the diagram below, the two lines are parallel which we know by the arrow markings on them.
Angle e and angle h are therefore congruent, since they are alternate exterior angles. Angle g and angle f are also alternate exterior angles and therefore congruent.
The Pythagorean Theorem is used to find the missing side of a right triangle.
For the right triangle, where a and b are the shorter sides of the triangle, and c is the hypotenuse, then a2+b2=c2.
Example 1: Suppose we know one side of a right triangle is 10 cm, and the hypotenuse of the triangle is 26 cm. Find the remaining side.
Let a = 10 cm and c = 26 cm. Then we will use the Pythagorean Theorem to find b:
Therefore, the remaining side of the triangle is 24 cm.
Example 2: Suppose two people start walking from the same point. After 1 hour, Person A has walked 2.8 miles due north. Person B has walked 3.2 miles due east. Find the distance between them at this time.
Since the two people are walking at right angles to each other, we can apply the Pythagorean Theorem. In this case, the distance between Person A and Person B is the hypotenuse of a right triangle.
Let a = 2.8 and b = 3.2. Then a²+b²=c² gives:
2.8² + 3.2² = c²
18.08 = c²
c ≈ 4.25
Therefore, Person A and Person B are approximately 4.25 miles apart after 1 hour.
Probability, Statistics, and Discrete Mathematics
This section tests your knowledge of statistical measures, probability, algorithms, and charts.
Here are some concepts you need to know.
Measures of Central Tendency
Measures of central tendency help us to determine how data is distributed. 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:
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.
Common Features of a Data Set
Other common features of a data set that we consider are range and outliers.
The range of a data set is the difference between the largest value in the set and the smallest value in the set.
An outlier is a data point that is far outside of the normal range of data points. In other words, it is far away from all the other data.
For example, consider the list of speeds of cars on the highway below in miles per hour:
75, 70, 83, 42, 72, 81, 75, 80, 76, 69, 68
First reorder the set from least to greatest:
42, 68, 69, 70, 72, 75, 75, 76, 80, 81, 83
Now we can see that the smallest value in this set is 42, and the highest speed is 83. The range is the largest value minus the smallest value:
Range = 83 – 42 = 41
However, you may have noticed that the speed of 42 was much slower than the rest of the speeds recorded. Therefore, 42 is considered an outlier.
In many cases, the outliers of a data set are discarded so that the range provides a more accurate picture of the data distribution. If the speed of 42 is ignored, then the range of the data set (without that outlier) is 83 – 68 = 15.
And that’s some basic info about the Probability, Statistics, and Discrete Mathematics subarea.
Now, let’s look at a few practice questions in each area to see how these concepts might actually appear on the real test.
Arithmetic and Algebra Practice Test
Directions: Each of the questions or incomplete statements below is followed by four suggested answers or completions. Select the one that is best in each case.
Which of the following is a correct statement?
- Whole numbers are a proper subset of natural numbers
- Rational and irrational numbers are both proper subsets of real numbers
- Real numbers are a proper subset of the irrational, rational, whole, and natural number sets
- The set of irrational numbers is a proper subset of the set of integers
Correct answer: 2. A proper subset is a subset whose members are entirely contained within a larger set. For example, the set of whole numbers is a proper subset of the set of integers because every whole number is also an integer. Real numbers are made up of all rational and irrational numbers and all their subsets.
When the following numbers are ordered from greatest to least, which number will appear last?
⅔, ⅝, 0.67, 0.82, 0.72
Correct answer: 1. It is probably easiest to convert the fractions to decimals to compare them: ⅔ ≈ 0.667; ⅝ = 0.625. (The other two answer choices are already written in decimals.) Arranging the numbers in order from greatest to least, we would have 0.82, 0.67, .667 (⅔), .625 (⅝). So, ⅝ is the smallest number and would be last when the numbers are arranged from greatest to least.
Simplify: 17 – 6 + 13 + 2(25 ÷ 5)² – 50/√25
Correct answer: 3. Remember to use the correct order of operations for this problem. This is how this problem would be simplified: 17 – 6 + 13 + 2(25÷5)² – 50/√25 = 17 – 6 +13 + 2(5)² – 50/5 17 – 6 + 13 + 2(25) – 10 = 17 – 6 + 13 + 50 – 10 = 80 – 16 = 64.
In a sequence which begins 25, 23, 21, 19, 17,…, what is the term number for the term with a value of -11?
- n = 1.5
- n = -17
- n = 17
- n = 19
Correct answer: 4. 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 set up 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 set up 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.
Joni and her friends Susan, Maria, and Irsla had lunch together at their favorite restaurant. The bill, including tax, was $62.50. Susan paid $18, Maria paid ¼ of the bill, Irsla paid 20%, and Joni paid the remainder. Who paid the greatest part of the bill?
Correct answer: 1. First, we need to find out exactly how much of the bill each girl paid. Susan paid $18; Maria paid ¼ of $62.50 or $15.625 or about $15.62; Irsla paid 20% of $62.50 or $12.50; and Joni paid what was left: 62.50 – (18 + 15.62 + 12.50) = 62.50 – (46.12) = $16.38. So, Susan paid the largest portion of the bill, $18.
Which of the following is not a function?
- x = 3y -1
- y = x²
- y = 3x + 1
- x = y²
Correct answer: 4. x = y² fails the vertical line test. For all values of x except 0, there are two y-values. The first equation, y = x², is a parabola that opens upward with a vertex on (0,0). The other two options are linear equations. Only this choice is NOT a function.
Write an equation in Standard Form of the line passing through (6,1) with a slope of ½.
- x – 2y = 4
- ½x – y = 2
- y = ½x – 2
- 2y = x – 4
Correct answer: 1. Begin with the slope intercept form of a linear equation: y = mx + b. Since we are given that the slope, m, is ½ and passing through the point (x,y) = (6,1), substitute what we are given into the slope-intercept form of the equation: 1 = ½(6) + b. Simplifying, this gives us -2 = b. Our equation in slope intercept form is y = ½ x – 2. In standard form the equation becomes -1/2x + y = -2 or, multiplying through by -2, we get x – 2y = 4. In standard form, Ax + By = C, whereas A, B, and C are all real numbers, so, x + y = -2 is also a correct answer, but it is not a choice here.
The linear function, f(x) = – ½(3x -4), is graphed. Where will this graph cross the y-axis?
- At y = -2
- At y = -4
- At y = 4
- At y = 2
Correct answer: 4. When the -1/2 is distributed over (3x – 4), the equation becomes: F(x) = (-3/2)x – (4/-2) = (-3/2)x + 2 So, the y-intercept is , and the graph will cross the y-axis at (0,2): y=2.
Which of the following is linear?
- x = y²
- x = 3y -1
- y = 3x² + 1
- y = x²
Correct answer: 2. x = 3y -1 is not easily recognizable as linear, but it can be transformed into the equivalent equation: x – 3y + 1 = 0 (a linear equation in standard form). The other three options contain second degree variable terms and, therefore, cannot be linear.
Which could be the graph of the inequality: -2y > 4x – 6?
- Graph C
- Graph A
- Graph D
- Graph B
Correct answer: 2. 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.
Geometry and Data Practice Test
What is the mean, median, mode and range for the data set?
75, 82, 68, 95, 74, 72, 91, 60, 72, 80
- Mean: 76.9; Median: 74.5; Mode: 72; Range: 31
- Mean: 80; Median: 73; Mode: 72; Range: 35
- Mean: 77; Median: 73.5; Mode: 72; Range: 31
- 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.
There are 12 red marbles, 8 blue marbles, and 10 green marbles in a jar. What is the probability of drawing a green marble, replacing it, and drawing a second green marble?
- 20/30 = 2/3
- 100/900 = 1/9
- 10/30 = 1/3
- 90/870 = 3/29
Correct answer: 2. There is a total of 30 marbles in the jar; only 10 of them are green. The probability of drawing a green marble is 10/30 = 1/3. If that marble is replaced, the probability that the second marble is green is also 1/3. The probability that both events will occur is a product of these probabilities: 1/3 • 1/3 = 1/9.
There are 12 red marbles, 8 blue marbles, and 10 green marbles in a jar. What is the probability of drawing a green marble followed by a second green marble without replacing the marble previously drawn?
- 10/30 = 1/3
- 90/870 = 3/29
- 100/900 = 1/9
- 20/30 = 2/3
Correct answer: 2. There is a total of 30 marbles in the jar; only 10 of them are green. The probability of drawing a green marble is 10/30 = 1/3. Then, if we do not replace the marble, there are only 29 marbles remaining in the jar and 9 would be green. (This is sometimes referred to as the probability of A given B and is written like this: P(A/B). This means that we are finding the probability of A happening if B has already happened.) The probability of drawing a green marble and without replacing it and then drawing a second green marble is 10/30 • 9/29 = 90/870 or 3/29.
John has just taken the state standardized test and his scores showed that he was in the 80th percentile. Which of the following statements about the students who took this test would be FALSE?
- In a box plot, John’s score would be in the “box”
- In a box plot, John’s score would be on a “whisker”
- 80% of the students scored lower than John
- 20% of the students scored higher than John
Correct answer: 1. Recall that in a box plot, 25% of the data (25th percentile) is on the left whisker; the middle 50% of the data (from the 25th percentile to the 75th percentile) would be in the box; and the data in the fourth quartile, the right whisker, would be 75th percentile to 100th percentile. So, the 80th percentile would fall on the right whisker.
In a recent survey in My Town, USA, random eligible voters were asked who they favored in an upcoming runoff for mayor: candidate A, B, or C. 23% said they would vote for candidate A; 53% said they would vote for candidate B; and 24% said they would vote for candidate C. The margin of error was ±4. Keeping in mind that a winner must have at least 50% of the vote or there would be a second run-off between the top two candidates, which of the following statements is probably true?
- There will probably be a run-off between candidates A and C
- There will probably be a run-off between candidates A and B
- There will probably be a run-off between candidates B and C
- A second runoff will probably not be necessary
Correct answer: 4. With a margin of error of ±4, this means that candidate B, polling at 53%, is likely to have somewhere between 49% and 57% of the votes. According to this information, it would seem that a runoff would be unlikely because candidate B holds at least 50% of the vote.
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?
- In general, their arm spans are less than their heights
- In general, their arm spans are greater than their heights
- In general, their arm spans are greater than or equal to their heights
- 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.
What is the measure of angle f?
In the triangle above, if AB = 15 in and AC = 12 in, what is the length of BC?
- 9 in
- About 19 in
- 12 in
- 13.5 in
Correct answer: 1. “About 19 in” occurs when 12² and 15² are added, not recognizing 15 as the hypotenuse but as a leg. »»» ««« 13.5 in. is the average of 12 and 15 and has no relevance to this specific problem.
What is the exact circumference of figure C?
- 4π units
- 16π units
- 32π units
- 8π units
Correct answer: 4. Whenever an exact answer is called for, leave your answers as radicals, in terms of pi, or as a fraction if the fraction will not convert to a terminating decimal. In figure C, the radius is 4 units, so the circumference is C = 2πr = 2π4 = 8π.
What is the total surface area in the figure above?
- 132 in²
- 120 in²
- 75 in²
- 60 in²
Correct answer: 1. The total surface area is the area of both bases + the area of 3 rectangular faces. The triangle has a base of 3 and a height of 4, so its area would be ½(12) = 6 in². The faces have areas of (10)(5); (10)(3); and (10)(4). So, SA = 2(6) + 50 + 30 + 40 = 132 in². Remember that the 6 was multiplied by 2 because there are two bases, both right triangles.