Hints will display for most wrong answers; explanations for most right answers. You can attempt a question multiple times; it will only be scored correct if you get it right the first time.

I used the official objectives and sample test to construct these questions, but cannot promise that they accurately reflect what’s on the real test. Some of the sample questions were more convoluted than I could bear to write. See terms of use. See the MTEL Practice Test main page to view questions on a particular topic or to download paper practice tests.

## MTEL General Curriculum Mathematics Practice

Question 1 |

#### The prime factorization of n can be written as n=pqr, where p, q, and r are distinct prime numbers. How many factors does n have, including 1 and itself?

\( \large3\) Hint: 1, p, q, r, and pqr are already 5, so this isn't enough. You might try plugging in p=2, q=3, and r=5 to help with this problem. | |

\( \large5\) Hint: Don't forget pq, etc. You might try plugging in p=2, q=3, and r=5 to help with this problem. | |

\( \large6\) Hint: You might try plugging in p=2, q=3, and r=5 to help with this problem. | |

\( \large8\) Hint: 1, p, q, r, pq, pr, qr, pqr. |

Question 2 |

## AHint: \(\frac{34}{135} \approx \frac{1}{4}\) and \( \frac{53}{86} \approx \frac {2}{3}\). \(\frac {1}{4}\) of \(\frac {2}{3}\) is small and closest to A. | |

## BHint: Estimate with simpler fractions. | |

## CHint: Estimate with simpler fractions. | |

## DHint: Estimate with simpler fractions. |

Question 3 |

#### 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 4 |

#### 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 5 |

#### Which of the following sets of polygons can be assembled to form a pentagonal pyramid?

## 2 pentagons and 5 rectangles.Hint: These can be assembled to form a pentagonal prism, not a pentagonal pyramid. | |

## 1 square and 5 equilateral triangles.Hint: You need a pentagon for a pentagonal pyramid. | |

## 1 pentagon and 5 isosceles triangles. | |

## 1 pentagon and 10 isosceles triangles. |

Question 6 |

#### Use the expression below to answer the question that follows.

#### \( \large \dfrac{\left( 4\times {{10}^{3}} \right)\times \left( 3\times {{10}^{4}} \right)}{6\times {{10}^{6}}}\)

#### Which of the following is equivalent to the expression above?

## 2Hint: \(10^3 \times 10^4=10^7\), and note that if you're guessing when the answers are so closely related, you're generally better off guessing one of the middle numbers. | |

## 20Hint: \( \dfrac{\left( 4\times {{10}^{3}} \right)\times \left( 3\times {{10}^{4}} \right)}{6\times {{10}^{6}}}=\dfrac {12 \times {{10}^{7}}}{6\times {{10}^{6}}}=\)\(2 \times {{10}^{1}}=20 \) | |

## 200Hint: \(10^3 \times 10^4=10^7\) | |

## 2000Hint: \(10^3 \times 10^4=10^7\), and note that if you're guessing when the answers are so closely related, you're generally better off guessing one of the middle numbers. |

Question 7 |

#### Which of the following is not possible?

## An equiangular triangle that is not equilateral.Hint: The AAA property of triangles states that all triangles with corresponding angles congruent are similar. Thus all triangles with three equal angles are similar, and are equilateral. | |

## An equiangular quadrilateral that is not equilateral.Hint: A rectangle is equiangular (all angles the same measure), but if it's not a square, it's not equilateral (all sides the same length). | |

## An equilateral quadrilateral that is not equiangular.Hint: This rhombus has equal sides, but it doesn't have equal angles: | |

## An equiangular hexagon that is not equilateral.Hint: This hexagon has equal angles, but it doesn't have equal sides: |

Question 8 |

#### Use the graph below to answer the question that follows:

#### The graph above best matches which of the following scenarios:

## George left home at 10:00 and drove to work on a crooked path. He was stopped in traffic at 10:30 and 10:45. He drove 30 miles total.Hint: Just because he ended up 30 miles from home doesn't mean he drove 30 miles total. | |

## George drove to work. On the way to work there is a little hill and a big hill. He slowed down for them. He made it to work at 11:15.Hint: The graph is not a picture of the roads. | |

## George left home at 10:15. He drove 10 miles, then realized he‘d forgotten something at home. He turned back and got what he‘d forgotten. Then he drove in a straight line, at many different speeds, until he got to work around 11:15.Hint: A straight line on a distance versus time graph means constant speed. | |

## George left home at 10:15. He drove 10 miles, then realized he‘d forgotten something at home. He turned back and got what he‘d forgotten. Then he drove at a constant speed until he got to work around 11:15. |

Question 9 |

#### Which of the following nets will not fold into a cube?

Hint: If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate). | |

Hint: If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate). | |

Hint: If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate). |

Question 10 |

#### 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 11 |

#### Solve for x: \(\large 4-\dfrac{2}{3}x=2x\)

\( \large x=3\) Hint: Try plugging x=3 into the equation. | |

\( \large x=-3\) Hint: Left side is positive, right side is negative when you plug this in for x. | |

\( \large x=\dfrac{3}{2}\) Hint: One way to solve: \(4=\dfrac{2}{3}x+2x\) \(=\dfrac{8}{3}x\).\(x=\dfrac{3 \times 4}{8}=\dfrac{3}{2}\). Another way is to just plug x=3/2 into the equation and see that each side equals 3 -- on a multiple choice test, you almost never have to actually solve for x. | |

\( \large x=-\dfrac{3}{2}\) Hint: Left side is positive, right side is negative when you plug this in for x. |

Question 12 |

#### A publisher prints a series of books with covers made of identical material and using the same thickness of paper for each page. The covers of the book together are 0.4 cm thick, and 125 pieces of the paper used together are 1 cm thick.

#### The publisher uses a linear function to determine the total thickness, T(n) of a book made with n sheets of paper. What are the slope and intercept of T(n)?

## Intercept = 0.4 cm, Slope = 125 cm/pageHint: This would mean that each page of the book was 125 cm thick. | |

## Intercept =0.4 cm, Slope = \(\dfrac{1}{125}\)cm/pageHint: The intercept is how thick the book would be with no pages in it. The slope is how much 1 extra page adds to the thickness of the book. | |

## Intercept = 125 cm, Slope = 0.4 cmHint: This would mean that with no pages in the book, it would be 125 cm thick. | |

## Intercept = \(\dfrac{1}{125}\)cm, Slope = 0.4 pages/cmHint: This would mean that each new page of the book made it 0.4 cm thicker. |

Question 13 |

#### On a map the distance from Boston to Detroit is 6 cm, and these two cities are 702 miles away from each other. Assuming the scale of the map is the same throughout, which answer below is closest to the distance between Boston and San Francisco on the map, given that they are 2,708 miles away from each other?

## 21 cmHint: How many miles would correspond to 24 cm on the map? Try adjusting from there. | |

## 22 cmHint: How many miles would correspond to 24 cm on the map? Try adjusting from there. | |

## 23 cmHint: One way to solve this without a calculator is to note that 4 groups of 6 cm is 2808 miles, which is 100 miles too much. Then 100 miles would be about 1/7 th of 6 cm, or about 1 cm less than 24 cm. | |

## 24 cmHint: 4 groups of 6 cm is over 2800 miles on the map, which is too much. |

Question 14 |

#### M is a multiple of 26. Which of the following cannot be true?

## M is odd.Hint: All multiples of 26 are also multiples of 2, so they must be even. | |

## M is a multiple of 3.Hint: 3 x 26 is a multiple of both 3 and 26. | |

## M is 26.Hint: 1 x 26 is a multiple of 26. | |

## M is 0.Hint: 0 x 26 is a multiple of 26. |

Question 15 |

#### Which of the lists below is in order from least to greatest value?

\( \large -0.044,\quad -0.04,\quad 0.04,\quad 0.044\) Hint: These are easier to compare if you add trailing zeroes (this is finding a common denominator) -- all in thousandths, -0.044, -0.040,0 .040, 0.044. The middle two numbers, -0.040 and 0.040 can be modeled as owing 4 cents and having 4 cents. The outer two numbers are owing or having a bit more. | |

\( \large -0.04,\quad -0.044,\quad 0.044,\quad 0.04\) Hint: 0.04=0.040, which is less than 0.044. | |

\( \large -0.04,\quad -0.044,\quad 0.04,\quad 0.044\) Hint: -0.04=-0.040, which is greater than \(-0.044\). | |

\( \large -0.044,\quad -0.04,\quad 0.044,\quad 0.04\) Hint: 0.04=0.040, which is less than 0.044. |

Question 16 |

#### What is the length of side \(\overline{BD}\) in the triangle below, where \(\angle DBA\) is a right angle?

\( \large 1\) Hint: Use the Pythagorean Theorem. | |

\( \large \sqrt{5}\) Hint: \(2^2+e^2=3^2\) or \(4+e^2=9;e^2=5; e=\sqrt{5}\). | |

\( \large \sqrt{13}\) Hint: e is not the hypotenuse. | |

\( \large 5\) Hint: Use the Pythagorean Theorem. |

Question 17 |

#### 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 18 |

#### Use the samples of a student's work below to answer the question that follows:

#### This student divides fractions by first finding a common denominator, then dividing the numerators.

\( \large \dfrac{2}{3} \div \dfrac{3}{4} \longrightarrow \dfrac{8}{12} \div \dfrac{9}{12} \longrightarrow 8 \div 9 = \dfrac {8}{9}\) \( \large \dfrac{2}{5} \div \dfrac{7}{20} \longrightarrow \dfrac{8}{20} \div \dfrac{7}{20} \longrightarrow 8 \div 7 = \dfrac {8}{7}\) \( \large \dfrac{7}{6} \div \dfrac{3}{4} \longrightarrow \dfrac{14}{12} \div \dfrac{9}{12} \longrightarrow 14 \div 9 = \dfrac {14}{9}\)#### Which of the following best describes the mathematical validity of the algorithm the student is using?

## It is not valid. Common denominators are for adding and subtracting fractions, not for dividing them.Hint: Don't be so rigid! Usually there's more than one way to do something in math. | |

## It got the right answer in these three cases, but it isn‘t valid for all rational numbers.Hint: Did you try some other examples? What makes you say it's not valid? | |

## It is valid if the rational numbers in the division problem are in lowest terms and the divisor is not zero.Hint: Lowest terms doesn't affect this problem at all. | |

## It is valid for all rational numbers, as long as the divisor is not zero.Hint: When we have common denominators, the problem is in the form a/b divided by c/b, and the answer is a/c, as the student's algorithm predicts. |

Question 19 |

#### The following story situations model \( 12\div 3\):

#### I) Jack has 12 cookies, which he wants to share equally between himself and two friends. How many cookies does each person get?

#### II) Trent has 12 cookies, which he wants to put into bags of 3 cookies each. How many bags can he make?

#### III) Cicely has $12. Cookies cost $3 each. How many cookies can she buy?

#### Which of these questions illustrate the same model of division, either partitive (partioning) or measurement (quotative)?

## I and II | |

## I and III | |

## II and IIIHint: Problem I is partitive (or partitioning or sharing) -- we put 12 objects into 3 groups. Problems II and III are quotative (or measurement) -- we put 12 objects in groups of 3. | |

## All three problems model the same meaning of division |

Question 20 |

#### In January 2011, the national debt was about 14 trillion dollars and the US population was about 300 million people. Someone reading these figures estimated that the national debt was about $5,000 per person. Which of these statements best describes the reasonableness of this estimate?

## It is too low by a factor of 10Hint: 14 trillion \( \approx 15 \times {{10}^{12}} \) and 300 million \( \approx 3 \times {{10}^{8}}\), so the true answer is about \( 5 \times {{10}^{4}} \) or $50,000. | |

## It is too low by a factor of 100 | |

## It is too high by a factor of 10 | |

## It is too high by a factor of 100 |

List |

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