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## 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?

 A $$\large3$$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. B $$\large5$$Hint: Don't forget pq, etc. You might try plugging in p=2, q=3, and r=5 to help with this problem. C $$\large6$$Hint: You might try plugging in p=2, q=3, and r=5 to help with this problem. D $$\large8$$Hint: 1, p, q, r, pq, pr, qr, pqr.
Question 1 Explanation:
Topic: Recognize uses of prime factorization of a number (Objective 0018).
 Question 2

#### A

Hint:
$$\frac{34}{135} \approx \frac{1}{4}$$ and $$\frac{53}{86} \approx \frac {2}{3}$$. $$\frac {1}{4}$$ of $$\frac {2}{3}$$ is small and closest to A.

#### B

Hint:
Estimate with simpler fractions.

#### C

Hint:
Estimate with simpler fractions.

#### D

Hint:
Estimate with simpler fractions.
Question 2 Explanation:
Topic: Understand meaning and models of operations on fractions (Objective 0019).
 Question 3

#### Five million

Hint:
Pay attention to the exponents. Adding 3 and 2 doesn't work because they have different place values.

#### Fifty thousand

Hint:
Pay attention to the exponents. Adding 3 and 2 doesn't work because they have different place values.

Hint:

#### Thirty thousand

Hint:
$$3\times {{10}^{4}} = 30,000;$$ the other term is much smaller and doesn't change the estimate.
Question 3 Explanation:
Topics: Place value, scientific notation, estimation (Objective 0016)
 Question 4

#### The expression $$\large{{8}^{3}}\cdot {{2}^{-10}}$$ is equal to which of the following?

 A $$\large 2$$Hint: Write $$8^3$$ as a power of 2. B $$\large \dfrac{1}{2}$$Hint: $$8^3 \cdot {2}^{-10}={(2^3)}^3 \cdot {2}^{-10}$$ =$$2^9 \cdot {2}^{-10} =2^{-1}$$ C $$\large 16$$Hint: Write $$8^3$$ as a power of 2. D $$\large \dfrac{1}{16}$$Hint: Write $$8^3$$ as a power of 2.
Question 4 Explanation:
Topic: Laws of Exponents (Objective 0019).
 Question 5

#### 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 10 isosceles triangles.

Question 5 Explanation:
Topic:Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices (Objective 0024).
 Question 6

#### 2

Hint:
$$10^3 \times 10^4=10^7$$, 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.

#### 20

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

#### 200

Hint:
$$10^3 \times 10^4=10^7$$

#### 2000

Hint:
$$10^3 \times 10^4=10^7$$, 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 6 Explanation:
Topics: Scientific notation, exponents, simplifying fractions (Objective 0016, although overlaps with other objectives too).
 Question 7

#### 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 7 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles (Objective 0024).
 Question 8

#### 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 8 Explanation:
Topic: Use qualitative graphs to represent functional relationships in the real world (Objective 0021).
 Question 9

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

 A Hint: If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate). B C Hint: If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate). D Hint: If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate).
Question 9 Explanation:
Topic: Match three-dimensional figures and their two-dimensional representations (e.g., nets, projections, perspective drawings) (Objective 0024).
 Question 10

#### Store B

Hint:
This saves 15% and C saves 25%.

#### Store D

Hint:
This is about 20% off, which is less of a discount than C.
Question 10 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 11

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

 A $$\large x=3$$Hint: Try plugging x=3 into the equation. B $$\large x=-3$$Hint: Left side is positive, right side is negative when you plug this in for x. C $$\large x=\dfrac{3}{2}$$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. D $$\large x=-\dfrac{3}{2}$$Hint: Left side is positive, right side is negative when you plug this in for x.
Question 11 Explanation:
Topic: Solve linear equations (Objective 0020).
 Question 12

#### Intercept = 0.4 cm, Slope = 125 cm/page

Hint:
This would mean that each page of the book was 125 cm thick.

#### Intercept =0.4 cm, Slope = $$\dfrac{1}{125}$$cm/page

Hint:
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 cm

Hint:
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/cm

Hint:
This would mean that each new page of the book made it 0.4 cm thicker.
Question 12 Explanation:
Topic: Interpret the meaning of the slope and the intercepts of a linear equation that models a real-world situation (Objective 0022).
 Question 13

#### 21 cm

Hint:
How many miles would correspond to 24 cm on the map? Try adjusting from there.

#### 22 cm

Hint:
How many miles would correspond to 24 cm on the map? Try adjusting from there.

#### 23 cm

Hint:
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 cm

Hint:
4 groups of 6 cm is over 2800 miles on the map, which is too much.
Question 13 Explanation:
Topic: Apply proportional thinking to estimate quantities in real world situations (Objective 0019).
 Question 14

#### 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 14 Explanation:
Topic: Characteristics of composite numbers (Objective 0018).
 Question 15

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

 A $$\large -0.044,\quad -0.04,\quad 0.04,\quad 0.044$$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. B $$\large -0.04,\quad -0.044,\quad 0.044,\quad 0.04$$Hint: 0.04=0.040, which is less than 0.044. C $$\large -0.04,\quad -0.044,\quad 0.04,\quad 0.044$$Hint: -0.04=-0.040, which is greater than $$-0.044$$. D $$\large -0.044,\quad -0.04,\quad 0.044,\quad 0.04$$Hint: 0.04=0.040, which is less than 0.044.
Question 15 Explanation:
Topic: Ordering decimals and integers (Objective 0017).
 Question 16

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

 A $$\large 1$$Hint: Use the Pythagorean Theorem. B $$\large \sqrt{5}$$Hint: $$2^2+e^2=3^2$$ or $$4+e^2=9;e^2=5; e=\sqrt{5}$$. C $$\large \sqrt{13}$$Hint: e is not the hypotenuse. D $$\large 5$$Hint: Use the Pythagorean Theorem.
Question 16 Explanation:
Topic: Derive and use formulas for calculating the lengths, perimeters, areas, volumes, and surface areas of geometric shapes and figures (Objective 0023), and recognize and apply connections between algebra and geometry (e.g., the use of coordinate systems, the Pythagorean theorem) (Objective 0024).
 Question 17

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

$$\large \dfrac{2}{3}\times \dfrac{3}{4}=\dfrac{4\times 2}{3\times 3}=\dfrac{8}{9}$$ $$\large \dfrac{2}{5}\times \dfrac{7}{7}=\dfrac{7\times 2}{5\times 7}=\dfrac{2}{5}$$ $$\large \dfrac{7}{6}\times \dfrac{3}{4}=\dfrac{4\times 7}{6\times 3}=\dfrac{28}{18}=\dfrac{14}{9}$$

#### 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 17 Explanation:
Topic: Analyze Non-Standard Computational Algorithms (Objective 0019).
 Question 18

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

$$\large \dfrac{2}{3} \div \dfrac{3}{4} \longrightarrow \dfrac{8}{12} \div \dfrac{9}{12} \longrightarrow 8 \div 9 = \dfrac {8}{9}$$ $$\large \dfrac{2}{5} \div \dfrac{7}{20} \longrightarrow \dfrac{8}{20} \div \dfrac{7}{20} \longrightarrow 8 \div 7 = \dfrac {8}{7}$$ $$\large \dfrac{7}{6} \div \dfrac{3}{4} \longrightarrow \dfrac{14}{12} \div \dfrac{9}{12} \longrightarrow 14 \div 9 = \dfrac {14}{9}$$

#### 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 18 Explanation:
Topic: Analyze Non-Standard Computational Algorithms (Objective 0019).
 Question 19

#### II and III

Hint:
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 19 Explanation:
Topic: Understand models of operations on numbers (Objective 0019).
 Question 20

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

Question 20 Explanation:
Topics: Estimation, Scientific Notation in the real world (Objective 0016).
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