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## MTEL General Curriculum Mathematics Practice

 Question 1

Hint:

#### 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 27 Explanation:
Topic: Understand models of operations on numbers (Objective 0019).
 Question 28

#### 0 years

Hint:
Range is the maximum life expectancy minus the minimum life expectancy.

#### 12 years

Hint:
Are you subtracting frequencies? Range is about values of the data, not frequency.

#### 18 years

Hint:
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 years

Hint:
Question 28 Explanation:
Topic: Compare different data sets (Objective 0025).
 Question 29

#### 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 29 Explanation:
Topic:Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices (Objective 0024).
 Question 30

#### A

Hint:
Rise is more than 30 inches.

#### B

Hint:
Run is almost 24 feet, so rise can be almost 2 feet.

#### C

Hint:
Run is 12 feet, so rise can be at most 1 foot.

#### D

Hint:
Slope is 1:10 -- too steep.
Question 30 Explanation:
Topic: Interpret meaning of slope in a real world situation (Objective 0022).
 Question 31

#### Which of the numbers below is not equivalent to 4%?

 A $$\large \dfrac{1}{25}$$Hint: 1/25=4/100, so this is equal to 4% (be sure you read the question correctly). B $$\large \dfrac{4}{100}$$Hint: 4/100=4% (be sure you read the question correctly). C $$\large 0.4$$Hint: 0.4=40% so this is not equal to 4% D $$\large 0.04$$Hint: 0.04=4/100, so this is equal to 4% (be sure you read the question correctly).
Question 31 Explanation:
Converting between fractions, decimals, and percents (Objective 0017).
 Question 32

#### What fraction of the area of the picture below is shaded?

 A $$\large \dfrac{17}{24}$$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) . B $$\large \dfrac{3}{4}$$Hint: Be sure you're taking into account the different sizes of the pieces. C $$\large \dfrac{2}{3}$$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. D $$\large \dfrac{17}{6}$$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 32 Explanation:
Topic: Models of Fractions (Objective 0017)
 Question 33

#### 58 x 22

Hint:
This problem involves regrouping, which the student does not do correctly.

#### 16 x 24

Hint:
This problem involves regrouping, which the student does not do correctly.

#### 31 x 23

Hint:
There is no regrouping with this problem.

#### 141 x 32

Hint:
This problem involves regrouping, which the student does not do correctly.
Question 33 Explanation:
Topic: Analyze computational algorithms (Objective 0019).
 Question 34

#### If the polygon shown above is reflected about the y axis and then rotated 90 degrees clockwise about the origin, which of the following graphs is the result?

 A Hint: Try following the point (1,4) to see where it goes after each transformation. B C Hint: Make sure you're reflecting in the correct axis. D Hint: Make sure you're rotating the correct direction.
Question 34 Explanation:
Topic: Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry, similarity, and congruence; and use these concepts to solve problems (Objective 0024).
 Question 35

#### A family has four children.  What is the probability that two children are girls and two are boys?  Assume the the probability of having a boy (or a girl) is 50%.

 A $$\large \dfrac{1}{2}$$Hint: How many different configurations are there from oldest to youngest, e.g. BGGG? How many of them have 2 boys and 2 girls? B $$\large \dfrac{1}{4}$$Hint: How many different configurations are there from oldest to youngest, e.g. BGGG? How many of them have 2 boys and 2 girls? C $$\large \dfrac{1}{5}$$Hint: Some configurations are more probable than others -- i.e. it's more likely to have two boys and two girls than all boys. Be sure you are weighting properly. D $$\large \dfrac{3}{8}$$Hint: There are two possibilities for each child, so there are $$2 \times 2 \times 2 \times 2 =16$$ different configurations, e.g. from oldest to youngest BBBG, BGGB, GBBB, etc. Of these configurations, there are 6 with two boys and two girls (this is the combination $$_{4}C_{2}$$ or "4 choose 2"): BBGG, BGBG, BGGB, GGBB, GBGB, and GBBG. Thus the probability is 6/16=3/8.
Question 35 Explanation:
Topic: Apply knowledge of combinations and permutations to the computation of probabilities (Objective 0026).
 Question 36

#### Each number in the table above represents a value W that is determined by the values of x and y.  For example, when x=3 and y=1, W=5.  What is the value of W when x=9 and y=14?  Assume that the patterns in the table continue as shown.

 A $$\large W=-5$$Hint: When y is even, W is even. B $$\large W=4$$Hint: Note that when x increases by 1, W increases by 2, and when y increases by 1, W decreases by 1. At x=y=0, W=0, so at x=9, y=14, W has increased by $$9 \times 2$$ and decreased by 14, or W=18-14=4. C $$\large W=6$$Hint: Try fixing x or y at 0, and start by finding W for x=0 y=14 or x=9, y=0. D $$\large W=32$$Hint: Try fixing x or y at 0, and start by finding W for x=0 y=14 or x=9, y=0.
Question 36 Explanation:
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021)
 Question 37

#### Which of the following is equal to one million three hundred thousand?

 A $$\large1.3\times {{10}^{6}}$$ B $$\large1.3\times {{10}^{9}}$$ Hint: That's one billion three hundred million. C $$\large1.03\times {{10}^{6}}$$ Hint: That's one million thirty thousand. D $$\large1.03\times {{10}^{9}}$$Hint: That's one billion thirty million
Question 37 Explanation:
Topic: Scientific Notation (Objective 0016)
 Question 38

#### 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 38 Explanation:
Topics: Place value, scientific notation, estimation (Objective 0016)
 Question 39

#### 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 39 Explanation:
Topic: Analyze Non-Standard Computational Algorithms (Objective 0019).
 Question 40

#### 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?

 A $$\large C\le 300$$Hint: Find the LCM. B $$\large 300 < C \le 500$$Hint: Find the LCM. C $$\large 500 < C \le 700$$Hint: Find the LCM. D $$\large C>700$$Hint: The LCM is 900, which is the smallest number of chairs.
Question 40 Explanation:
Topic: Apply LCM in "real-world" situations (according to standardized tests....) (Objective 0018).
 Question 41

#### The student used a method that worked for this problem and can be generalized to any subtraction problem.

Hint:
Note that this algorithm is taught as the "standard" algorithm in much of Europe (it's where the term "borrowing" came from -- you borrow on top and "pay back" on the bottom).

#### The student used a method that worked for this problem and that will work for any subtraction problem that only requires one regrouping; it will not work if more regrouping is required.

Hint:
Try some more examples.

#### The student used a method that worked for this problem and will work for all three-digit subtraction problems, but will not work for larger problems.

Hint:
Try some more examples.

#### The student used a method that does not work. The student made two mistakes that cancelled each other out and was lucky to get the right answer for this problem.

Hint:
Remember, there are many ways to do subtraction; there is no one "right" algorithm.
Question 41 Explanation:
Topic: Analyze and justify standard and non-standard computational techniques (Objective 0019).
 Question 42

#### In the triangle below, $$\overline{AC}\cong \overline{AD}\cong \overline{DE}$$ and $$m\angle CAD=100{}^\circ$$.  What is $$m\angle DAE$$?

 A $$\large 20{}^\circ$$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. B $$\large 25{}^\circ$$Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal. C $$\large 30{}^\circ$$Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal. D $$\large 40{}^\circ$$Hint: Make sure you're calculating the correct angle.
Question 42 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles, including real-world applications. (Objective 0024).
 Question 43

#### The least common multiple of 60 and N is 1260. Which of the following could be the prime factorization of N?

 A $$\large2\cdot 5\cdot 7$$Hint: 1260 is divisible by 9 and 60 is not, so N must be divisible by 9 for 1260 to be the LCM. B $$\large{{2}^{3}}\cdot {{3}^{2}}\cdot 5 \cdot 7$$Hint: 1260 is not divisible by 8, so it isn't a multiple of this N. C $$\large3 \cdot 5 \cdot 7$$Hint: 1260 is divisible by 9 and 60 is not, so N must be divisible by 9 for 1260 to be the LCM. D $$\large{{3}^{2}}\cdot 5\cdot 7$$Hint: $$1260=2^2 \cdot 3^2 \cdot 5 \cdot 7$$ and $$60=2^2 \cdot 3 \cdot 5$$. In order for 1260 to be the LCM, N has to be a multiple of $$3^2$$ and of 7 (because 60 is not a multiple of either of these). N also cannot introduce a factor that would require the LCM to be larger (as in choice b).
Question 43 Explanation:
Topic: Least Common Multiple (Objective 0018)
 Question 44

#### Point B is halfway between two tick marks.  What number is represented by Point B?

 A $$\large 0.645$$Hint: That point is marked on the line, to the right. B $$\large 0.6421$$Hint: That point is to the left of point B. C $$\large 0.6422$$Hint: That point is to the left of point B. D $$\large 0.6425$$
Question 44 Explanation:
Topic: Using Number Lines (Objective 0017)
 Question 45

#### Which of the following is an irrational number?

 A $$\large \sqrt[3]{8}$$Hint: This answer is the cube root of 8. Since 2 x 2 x 2 =8, this is equal to 2, which is rational because 2 = 2/1. B $$\large \sqrt{8}$$Hint: It is not trivial to prove that this is irrational, but you can get this answer by eliminating the other choices. C $$\large \dfrac{1}{8}$$Hint: 1/8 is the RATIO of two integers, so it is rational. D $$\large -8$$Hint: Negative integers are also rational, -8 = -8/1, a ratio of integers.
Question 45 Explanation:
Topic: Identifying rational and irrational numbers (Objective 0016).
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