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## MTEL General Curriculum Mathematics Practice

Question 1 |

#### 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 2 |

#### Which of the following is equivalent to

#### \( \large A-B+C\div D\times E\)?

\( \large A-B-\dfrac{C}{DE}
\) Hint: In the order of operations, multiplication and division have the same priority, so do them left to right; same with addition and subtraction. | |

\( \large A-B+\dfrac{CE}{D}\) Hint: In practice, you're better off using parentheses than writing an expression like the one in the question. The PEMDAS acronym that many people memorize is misleading. Multiplication and division have equal priority and are done left to right. They have higher priority than addition and subtraction. Addition and subtraction also have equal priority and are done left to right. | |

\( \large \dfrac{AE-BE+CE}{D}\) Hint: Use order of operations, don't just compute left to right. | |

\( \large A-B+\dfrac{C}{DE}\) Hint: In the order of operations, multiplication and division have the same priority, so do them left to right |

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 4 |

#### The chairs in a large room can be arranged in rows of 18, 25, or 60 with no chairs left over. If C is the smallest possible number of chairs in the room, which of the following inequalities does C satisfy?

\( \large C\le 300\) Hint: Find the LCM. | |

\( \large 300 < C \le 500 \) Hint: Find the LCM. | |

\( \large 500 < C \le 700 \) Hint: Find the LCM. | |

\( \large C>700\) Hint: The LCM is 900, which is the smallest number of chairs. |

Question 5 |

#### The "houses" below are made of toothpicks and gum drops.

#### Which of the following does not represent the number of gumdrops in a row of h houses?

\( \large 2+3h\) Hint: Think of this as start with 2 gumdrops on the left wall, and then add 3 gumdrops for each house. | |

\( \large 5+3(h-1)\) Hint: Think of this as start with one house, and then add 3 gumdrops for each of the other h-1 houses. | |

\( \large h+(h+1)+(h+1)\) Hint: Look at the gumdrops in 3 rows: h gumdrops for the "rooftops," h+1 for the tops of the vertical walls, and h+1 for the floors. | |

\( \large 5+3h\) Hint: This one is not a correct equation (which makes it the correct answer!). Compare to choice A. One of them has to be wrong, as they differ by 3. |

Question 6 |

#### Which of the numbers below is the decimal equivalent of \( \dfrac{3}{8}?\)

## 0.38Hint: If you are just writing the numerator next to the denominator then your technique is way off, but by coincidence your answer is close; try with 2/3 and 0.23 is nowhere near correct. | |

## 0.125Hint: This is 1/8, not 3/8. | |

## 0.375 | |

## 0.83Hint: 3/8 is less than a half, and 0.83 is more than a half, so they can't be equal. |

Question 7 |

#### 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 8 |

#### 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 9 |

#### Here is a student€™s work on several multiplication problems:

#### For which of the following problems is this student most likely to get the correct solution, even though he is using an incorrect algorithm?

## 58 x 22Hint: This problem involves regrouping, which the student does not do correctly. | |

## 16 x 24Hint: This problem involves regrouping, which the student does not do correctly. | |

## 31 x 23Hint: There is no regrouping with this problem. | |

## 141 x 32Hint: This problem involves regrouping, which the student does not do correctly. |

Question 10 |

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

\( \large \dfrac{1}{2},\quad \dfrac{1}{3},\quad \dfrac{1}{4},\quad \dfrac{1}{5}\) Hint: This is ordered from greatest to least. | |

\( \large \dfrac{1}{3},\quad \dfrac{2}{7},\quad \dfrac{3}{8},\quad \dfrac{4}{11}\) Hint: 1/3 = 2/6 is bigger than 2/7. | |

\( \large \dfrac{1}{4},\quad \dfrac{2}{5},\quad \dfrac{2}{3},\quad \dfrac{4}{5}\) Hint: One way to look at this: 1/4 and 2/5 are both less than 1/2, and 2/3 and 4/5 are both greater than 1/2. 1/4 is 25% and 2/5 is 40%, so 2/5 is greater. The distance from 2/3 to 1 is 1/3 and from 4/5 to 1 is 1/5, and 1/5 is less than 1/3, so 4/5 is bigger. | |

\( \large \dfrac{7}{8},\quad \dfrac{6}{7},\quad \dfrac{5}{6},\quad \dfrac{4}{5}\) Hint: This is in order from greatest to least. |

Question 11 |

#### A biology class requires a lab fee, which is a whole number of dollars, and the same amount for all students. On Monday the instructor collected $70 in fees, on Tuesday she collected $126, and on Wednesday she collected $266. What is the largest possible amount the fee could be?

## $2Hint: A possible fee, but not the largest possible fee. Check the other choices to see which are factors of all three numbers. | |

## $7Hint: A possible fee, but not the largest possible fee. Check the other choices to see which are factors of all three numbers. | |

## $14Hint: This is the greatest common factor of 70, 126, and 266. | |

## $70Hint: Not a factor of 126 or 266, so couldn't be correct. |

Question 12 |

#### Here is a mental math strategy for computing 26 x 16:

#### Step 1: 100 x 16 = 1600

#### Step 2: 25 x 16 = 1600 ÷· 4 = 400

#### Step 3: 26 x 16 = 400 + 16 = 416

#### Which property best justifies Step 3 in this strategy?

## Commutative Property.Hint: For addition, the commutative property is \(a+b=b+a\) and for multiplication it's \( a \times b = b \times a\). | |

## Associative Property.Hint: For addition, the associative property is \((a+b)+c=a+(b+c)\) and for multiplication it's \((a \times b) \times c=a \times (b \times c)\) | |

## Identity Property.Hint: 0 is the additive identity, because \( a+0=a\) and 1 is the multiplicative identity because \(a \times 1=a\). The phrase "identity property" is not standard. | |

## Distributive Property.Hint: \( (25+1) \times 16 = 25 \times 16 + 1 \times 16 \). This is an example of the distributive property of multiplication over addition. |

Question 13 |

#### Four children randomly line up, single file. What is the probability that they are in height order, with the shortest child in front? All of the children are different heights.

\( \large \dfrac{1}{4}\) Hint: Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children. | |

\( \large \dfrac{1}{256}
\) Hint: Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children. | |

\( \large \dfrac{1}{16}\) Hint: Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children. | |

\( \large \dfrac{1}{24}\) Hint: The number of ways for the children to line up is \(4!=4 \times 3 \times 2 \times 1 =24\) -- there are 4 choices for who is first in line, then 3 for who is second, etc. Only one of these lines has the children in the order specified. |

Question 14 |

#### Which of the numbers below is a fraction equivalent to \( 0.\bar{6}\)?

\( \large \dfrac{4}{6}\) Hint: \( 0.\bar{6}=\dfrac{2}{3}=\dfrac{4}{6}\) | |

\( \large \dfrac{3}{5}\) Hint: This is equal to 0.6, without the repeating decimal. Answer is equivalent to choice c, which is another way to tell that it's wrong. | |

\( \large \dfrac{6}{10}\) Hint: This is equal to 0.6, without the repeating decimal. Answer is equivalent to choice b, which is another way to tell that it's wrong. | |

\( \large \dfrac{1}{6}\) Hint: This is less than a half, and \( 0.\bar{6}\) is greater than a half. |

Question 15 |

#### 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 |

Question 16 |

#### 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 17 |

#### The speed of sound in dry air at 68 degrees F is 343.2 meters per second. Which of the expressions below could be used to compute the number of kilometers that a sound wave travels in 10 minutes (in dry air at 68 degrees F)?

\( \large 343.2\times 60\times 10\) Hint: In kilometers, not meters. | |

\( \large 343.2\times 60\times 10\times \dfrac{1}{1000}\) Hint: Units are meters/sec \(\times\) seconds/minute \(\times\) minutes \(\times\) kilometers/meter, and the answer is in kilometers. | |

\( \large 343.2\times \dfrac{1}{60}\times 10\) Hint: Include units and make sure answer is in kilometers. | |

\( \large 343.2\times \dfrac{1}{60}\times 10\times \dfrac{1}{1000}\) Hint: Include units and make sure answer is in kilometers. |

Question 18 |

#### 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 19 |

#### The picture below shows identical circles drawn on a piece of paper. The rectangle represents an index card that is blocking your view of \( \dfrac{3}{5}\) of the circles on the paper. How many circles are covered by the rectangle?

## 4Hint: The card blocks more than half of the circles, so this number is too small. | |

## 5Hint: The card blocks more than half of the circles, so this number is too small. | |

## 8Hint: The card blocks more than half of the circles, so this number is too small. | |

## 12Hint: 2/5 of the circles or 8 circles are showing. Thus 4 circles represent 1/5 of the circles, and \(4 \times 5=20\) circles represent 5/5 or all the circles. Thus 12 circles are hidden. |

Question 20 |

#### Which of the lines depicted below is a graph of \( \large y=2x-5\)?

## aHint: The slope of line a is negative. | |

## bHint: Wrong slope and wrong intercept. | |

## cHint: The intercept of line c is positive. | |

## dHint: Slope is 2 -- for every increase of 1 in x, y increases by 2. Intercept is -5 -- the point (0,-5) is on the line. |

List |

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