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MTEL General Curriculum Mathematics Practice
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Question 1 |
Which of the lists below contains only irrational numbers?
\( \large\pi , \quad \sqrt{6},\quad \sqrt{\dfrac{1}{2}}\) | |
\( \large\pi , \quad \sqrt{9}, \quad \pi +1\) Hint: \( \sqrt{9}=3\) | |
\( \large\dfrac{1}{3},\quad \dfrac{5}{4},\quad \dfrac{2}{9}\) Hint: These are all rational. | |
\( \large-3,\quad 14,\quad 0\) Hint: These are all rational. |
Question 1 Explanation:
Topic: Identifying rational and irrational numbers (Objective 0016).
Question 2 |
The equation \( \large F=\frac{9}{5}C+32\) is used to convert a temperature measured in Celsius to the equivalent Farentheit temperature.
A patient's temperature increased by 1.5° Celcius. By how many degrees Fahrenheit did her temperature increase?
1.5°Hint: Celsius and Fahrenheit don't increase at the same rate. | |
1.8°Hint: That's how much the Fahrenheit temp increases when the Celsius temp goes up by 1 degree. | |
2.7°Hint: Each degree increase in Celsius corresponds to a \(\dfrac{9}{5}=1.8\) degree increase in Fahrenheit. Thus the increase is 1.8+0.9=2.7. | |
Not enough information.Hint: A linear equation has constant slope, which means that every increase of the same amount in one variable, gives a constant increase in the other variable. It doesn't matter what temperature the patient started out at. |
Question 2 Explanation:
Topic: Interpret the meaning of the slope and the intercepts of a linear equation
that models a real-world situation (Objective 0022).
Question 3 |
How many factors does 80 have?
\( \large8\) Hint: Don't forget 1 and 80. | |
\( \large9\) Hint: Only perfect squares have an odd number of factors -- otherwise factors come in pairs. | |
\( \large10\) Hint: 1,2,4,5,8,10,16,20,40,80 | |
\( \large12\) Hint: Did you count a number twice? Include a number that isn't a factor? |
Question 3 Explanation:
Topic: Understand and apply principles of number theory (Objective 0018).
Question 4 |
Use the table below to answer the question that follows:
Gordon wants to buy three pounds of nuts. Each of the stores above ordinarily sells the nuts for $4.99 a pound, but is offering a discount this week. At which store can he buy the nuts for the least amount of money?
Store AHint: This would save about $2.50. You can quickly see that D saves more. | |
Store BHint: This saves 15% and C saves 25%. | |
Store C | |
Store DHint: This is about 20% off, which is less of a discount than C. |
Question 4 Explanation:
Topic: Understand the meanings and models of integers, fractions, decimals,percents, and mixed numbers and apply them to the solution of word
problems (Objective 0017).
Question 5 |
The first histogram shows the average life expectancies for women in different countries in Africa in 1998; the second histogram gives similar data for Europe:
How much bigger is the range of the data for Africa than the range of the data for Europe?
0 yearsHint: Range is the maximum life expectancy minus the minimum life expectancy. | |
12 yearsHint: Are you subtracting frequencies? Range is about values of the data, not frequency. | |
18 yearsHint: It's a little hard to read the graph, but it doesn't matter if you're consistent. It looks like the range for Africa is 80-38= 42 years and for Europe is 88-64 = 24; 42-24=18. | |
42 yearsHint: Read the question more carefully. |
Question 5 Explanation:
Topic: Compare different data sets (Objective 0025).
Question 6 |
Below is a pictorial representation of \(2\dfrac{1}{2}\div \dfrac{2}{3}\):
Which of the following is the best description of how to find the quotient from the picture?
The quotient is \(3\dfrac{3}{4}\). There are 3 whole blocks each representing \(\dfrac{2}{3}\) and a partial block composed of 3 small rectangles. The 3 small rectangles represent \(\dfrac{3}{4}\) of \(\dfrac{2}{3}\). | |
The quotient is \(3\dfrac{1}{2}\). There are 3 whole blocks each representing \(\dfrac{2}{3}\) and a partial block composed of 3 small rectangles. The 3 small rectangles represent \(\dfrac{3}{6}\) of a whole, or \(\dfrac{1}{2}\).Hint: We are counting how many 2/3's are in 2 1/2: the unit becomes 2/3, not 1. | |
The quotient is \(\dfrac{4}{15}\). There are four whole blocks separated into a total of 15 small rectangles.Hint: This explanation doesn't make much sense. Probably you are doing "invert and multiply," but inverting the wrong thing. | |
This picture cannot be used to find the quotient because it does not show how to separate \(2\dfrac{1}{2}\) into equal sized groups.Hint: Study the measurement/quotative model of division. It's often very useful with fractions. |
Question 6 Explanation:
Topic: Recognize and analyze pictorial representations of number operations. (Objective 0019).
Question 7 |
In the triangle below, \(\overline{AC}\cong \overline{AD}\cong \overline{DE}\) and \(m\angle CAD=100{}^\circ \). What is \(m\angle DAE\)?
\( \large 20{}^\circ \) Hint: Angles ACD and ADC are congruent since they are base angles of an isosceles triangle. Since the angles of a triangle sum to 180, they sum to 80, and they are 40 deg each. Thus angle ADE is 140 deg, since it makes a straight line with angle ADC. Angles DAE and DEA are base angles of an isosceles triangle and thus congruent-- they sum to 40 deg, so are 20 deg each. | |
\( \large 25{}^\circ \) Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal. | |
\( \large 30{}^\circ \) Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal. | |
\( \large 40{}^\circ \) Hint: Make sure you're calculating the correct angle. |
Question 7 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles,
including real-world applications. (Objective 0024).
Question 8 |
Use the expression below to answer the question that follows.
\( \large 3\times {{10}^{4}}+2.2\times {{10}^{2}}\)
Which of the following is closest to the expression above?
Five millionHint: Pay attention to the exponents. Adding 3 and 2 doesn't work because they have different place values. | |
Fifty thousandHint: Pay attention to the exponents. Adding 3 and 2 doesn't work because they have different place values. | |
Three millionHint: Don't add the exponents. | |
Thirty thousandHint: \( 3\times {{10}^{4}} = 30,000;\) the other term is much smaller and doesn't change the estimate. |
Question 8 Explanation:
Topics: Place value, scientific notation, estimation (Objective 0016)
Question 9 |
If x is an integer, which of the following must also be an integer?
\( \large \dfrac{x}{2}\) Hint: If x is odd, then \( \dfrac{x}{2} \) is not an integer, e.g. 3/2 = 1.5. | |
\( \large \dfrac{2}{x}\) Hint: Only an integer if x = -2, -1, 1, or 2. | |
\( \large-x\) Hint: -1 times any integer is still an integer. | |
\(\large\sqrt{x}\) Hint: Usually not an integer, e.g. \( \sqrt{2} \approx 1.414 \). |
Question 9 Explanation:
Topic: Integers (Objective 0016)
Question 10 |
What fraction of the area of the picture below is shaded?
\( \large \dfrac{17}{24}\) Hint: You might try adding segments so each quadrant is divided into 6 pieces with equal area -- there will be 24 regions, not all the same shape, but all the same area, with 17 of them shaded (for the top left quarter, you could also first change the diagonal line to a horizontal or vertical line that divides the square in two equal pieces and shade one) . | |
\( \large \dfrac{3}{4}\) Hint: Be sure you're taking into account the different sizes of the pieces. | |
\( \large \dfrac{2}{3}\) Hint: The bottom half of the picture is 2/3 shaded, and the top half is more than 2/3 shaded, so this answer is too small. | |
\( \large \dfrac{17}{6} \) Hint: This answer is bigger than 1, so doesn't make any sense. Be sure you are using the whole picture, not one quadrant, as the unit. |
Question 10 Explanation:
Topic: Models of Fractions (Objective 0017)
Question 11 |
A family went on a long car trip. Below is a graph of how far they had driven at each hour.
Which of the following is closest to their average speed driving on the trip?
\( \large d=20t\) Hint: Try plugging t=7 into the equation, and see how it matches the graph. | |
\( \large d=30t\) Hint: Try plugging t=7 into the equation, and see how it matches the graph. | |
\( \large d=40t\) | |
\( \large d=50t\) Hint: Try plugging t=7 into the equation, and see how it matches the graph. |
Question 11 Explanation:
Topic: Select the linear equation that best models a real-world situation (Objective 0022).
Question 12 |
Use the samples of a student's work below to answer the question that follows:
\( \large \dfrac{2}{3}\times \dfrac{3}{4}=\dfrac{4\times 2}{3\times 3}=\dfrac{8}{9}\) \( \large \dfrac{2}{5}\times \dfrac{7}{7}=\dfrac{7\times 2}{5\times 7}=\dfrac{2}{5}\) \( \large \dfrac{7}{6}\times \dfrac{3}{4}=\dfrac{4\times 7}{6\times 3}=\dfrac{28}{18}=\dfrac{14}{9}\)Which of the following best describes the mathematical validity of the algorithm the student is using?
It is not valid. It never produces the correct answer.Hint: In the middle example,the answer is correct. | |
It is not valid. It produces the correct answer in a few special cases, but it‘s still not a valid algorithm.Hint: Note that this algorithm gives a/b divided by c/d, not a/b x c/d, but some students confuse multiplication and cross-multiplication. If a=0 or if c/d =1, division and multiplication give the same answer. | |
It is valid if the rational numbers in the multiplication problem are in lowest terms.Hint: Lowest terms is irrelevant. | |
It is valid for all rational numbers.Hint: Can't be correct as the first and last examples have the wrong answers. |
Question 12 Explanation:
Topic: Analyze Non-Standard Computational Algorithms (Objective 0019).
Question 13 |
The window glass below has the shape of a semi-circle on top of a square, where the side of the square has length x. It was cut from one piece of glass.
What is the perimeter of the window glass?
\( \large 3x+\dfrac{\pi x}{2}\) Hint: By definition, \(\pi\) is the ratio of the circumference of a circle to its diameter; thus the circumference is \(\pi d\). Since we have a semi-circle, its perimeter is \( \dfrac{1}{2} \pi x\). Only 3 sides of the square contribute to the perimeter. | |
\( \large 3x+2\pi x\) Hint: Make sure you know how to find the circumference of a circle. | |
\( \large 3x+\pi x\) Hint: Remember it's a semi-circle, not a circle. | |
\( \large 4x+2\pi x\) Hint: Only 3 sides of the square contribute to the perimeter. |
Question 13 Explanation:
Topic: Derive and use formulas for calculating the lengths, perimeters, areas,
volumes, and surface areas of geometric shapes and figures (Objective 0023).
Question 14 |
What is the mathematical name of the three-dimensional polyhedron depicted below?
TetrahedronHint: All the faces of a tetrahedron are triangles. | |
Triangular PrismHint: A prism has two congruent, parallel bases, connected by parallelograms (since this is a right prism, the parallelograms are rectangles). | |
Triangular PyramidHint: A pyramid has one base, not two. | |
TrigonHint: A trigon is a triangle (this is not a common term). |
Question 14 Explanation:
Topic: Classify and analyze three-dimensional figures using attributes of faces,
edges, and vertices (Objective 0024).
Question 15 |
The expression \( \large{{8}^{3}}\cdot {{2}^{-10}}\) is equal to which of the following?
\( \large 2\) Hint: Write \(8^3\) as a power of 2. | |
\( \large \dfrac{1}{2}\) Hint: \(8^3 \cdot {2}^{-10}={(2^3)}^3 \cdot {2}^{-10}\) =\(2^9 \cdot {2}^{-10} =2^{-1}\) | |
\( \large 16\) Hint: Write \(8^3\) as a power of 2. | |
\( \large \dfrac{1}{16}\) Hint: Write \(8^3\) as a power of 2. |
Question 15 Explanation:
Topic: Laws of Exponents (Objective 0019).
Question 16 |
Elena is going to use a calculator to check whether or not 267 is prime. She will pick certain divisors, and then find 267 divided by each, and see if she gets a whole number. If she never gets a whole number, then she's found a prime. Which numbers does Elena NEED to check before she can stop checking and be sure she has a prime?
All natural numbers from 2 to 266.Hint: She only needs to check primes -- checking the prime factors of any composite is enough to look for divisors. As a test taking strategy, the other three choices involve primes, so worth thinking about. | |
All primes from 2 to 266 .Hint: Remember, factors come in pairs (except for square root factors), so she would first find the smaller of the pair and wouldn't need to check the larger. | |
All primes from 2 to 133 .Hint: She doesn't need to check this high. Factors come in pairs, and something over 100 is going to be paired with something less than 3, so she will find that earlier. | |
All primes from \( \large 2\) to \( \large \sqrt{267}\).Hint: \(\sqrt{267} \times \sqrt{267}=267\). Any other pair of factors will have one factor less than \( \sqrt{267}\) and one greater, so she only needs to check up to \( \sqrt{267}\). |
Question 16 Explanation:
Topic: Identify prime and composite numbers (Objective 0018).
Question 17 |
What set of transformations will transform the leftmost image into the rightmost image?
A 90 degree clockwise rotation about (2,1) followed by a translation of two units to the right.Hint: Part of the figure would move below the x-axis with these transformations. | |
A translation 3 units up, followed by a reflection about the line y=x.Hint: See what happens to the point (5,1) under this set of transformations. | |
A 90 degree clockwise rotation about (5,1), followed by a translation of 2 units up. | |
A 90 degree clockwise rotation about (2,1) followed by a translation of 2 units to the right.Hint: See what happens to the point (3,3) under this set of transformations. |
Question 17 Explanation:
Topic:Analyze and apply geometric transformations (e.g., translations,
rotations, reflections, dilations) (Objective 0024).
Question 18 |
Here is a number trick:
1) Pick a whole number
2) Double your number.
3) Add 20 to the above result.
4) Multiply the above by 5
5) Subtract 100
6) Divide by 10
The result is always the number that you started with! Suppose you start by picking N. Which of the equations below best demonstrates that the result after Step 6 is also N?
\( \large N*2+20*5-100\div 10=N\) Hint: Use parentheses or else order of operations is off. | |
\( \large \left( \left( 2*N+20 \right)*5-100 \right)\div 10=N\) | |
\( \large \left( N+N+20 \right)*5-100\div 10=N\) Hint: With this answer you would subtract 10, instead of subtracting 100 and then dividing by 10. | |
\( \large \left( \left( \left( N\div 10 \right)-100 \right)*5+20 \right)*2=N\) Hint: This answer is quite backwards. |
Question 18 Explanation:
Topic: Recognize and apply the concepts of variable, function, equality, and equation to express relationships algebraically (Objective 0020).
Question 19 |
Some children explored the diagonals in 2 x 2 squares on pages of a calendar (where all four squares have numbers in them). They conjectured that the sum of the diagonals is always equal; in the example below, 8+16=9+15.
Which of the equations below could best be used to explain why the children's conjecture is correct?
\( \large 8x+16x=9x+15x\) Hint: What would x represent in this case? Make sure you can describe in words what x represents. | |
\( \large x+(x+2)=(x+1)+(x+1)\) Hint: What would x represent in this case? Make sure you can describe in words what x represents. | |
\( \large x+(x+8)=(x+1)+(x+7)\) Hint: x is the number in the top left square, x+8 is one below and to the right, x+1 is to the right of x, and x+7 is below x. | |
\( \large x+8+16=x+9+15\) Hint: What would x represent in this case? Make sure you can describe in words what x represents. |
Question 19 Explanation:
Topic: Recognize and apply the concepts of variable, equality, and equation to express relationships algebraically (Objective 0020).
Question 20 |
The Venn Diagram below gives data on the number of seniors, athletes, and vegetarians in the student body at a college:
How many students at the college are seniors who are not vegetarians?
\( \large 137\) Hint: Doesn't include the senior athletes who are not vegetarians. | |
\( \large 167\) | |
\( \large 197\) Hint: That's all seniors, including vegetarians. | |
\( \large 279\) Hint: Includes all athletes who are not vegetarians, some of whom are not seniors. |
Question 20 Explanation:
Topic: Venn Diagrams (Objective 0025)
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