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## MTEL General Curriculum Mathematics Practice

 Question 1

#### There are 15 students for every teacher.  Let t represent the number of teachers and let s represent the number of students.  Which of the following equations is correct?

 A $$\large t=s+15$$Hint: When there are 2 teachers, how many students should there be? Do those values satisfy this equation? B $$\large s=t+15$$Hint: When there are 2 teachers, how many students should there be? Do those values satisfy this equation? C $$\large t=15s$$Hint: This is a really easy mistake to make, which comes from transcribing directly from English, "1 teachers equals 15 students." To see that it's wrong, plug in s=2; do you really need 30 teachers for 2 students? To avoid this mistake, insert the word "number," "Number of teachers equals 15 times number of students" is more clearly problematic. D $$\large s=15t$$
Question 1 Explanation:
Topic: Select the linear equation that best models a real-world situation (Objective 0022).
 Question 2

#### 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 2 Explanation:
Topic: Analyze and justify standard and non-standard computational techniques (Objective 0019).
 Question 3

#### The median number of pairs of footware owned is between 50 and 60 pairs.

Hint:
The same number of data points are less than the median as are greater than the median -- but on this histogram, clearly more than half the students own less than 50 pairs of shoes, so the median is less than 50.

#### The mode of the number of pairs of footware owned is 20.

Hint:
The mode is the most common number of pairs of footwear owned. We can't tell it from this histogram because each bar represents 10 different numbers-- perhaps 8 students each own each number from 10 to 19, but 40 students own exactly 6 pairs of shoes.... or perhaps not....

#### The mean number of pairs of footware owned is less than the median number of pairs of footware owned.

Hint:
This is a right skewed distribution, and so the mean is bigger than the median -- the few large values on the right pull up the mean, but have little effect on the median.

#### The median number of pairs of footware owned is between 10 and 20.

Hint:
There are approximately 230 students represented in this survey, and the 41st through 120th lowest values are between 10 and 20 -- thus the middle value is in that range.
Question 3 Explanation:
Topics: Analyze and interpret various graphic and data representations, and use measures of central tendency (e.g., mean, median, mode) and spread to describe and interpret real-world data (Objective 0025).
 Question 4

#### 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 4 Explanation:
Topics: Scientific notation, exponents, simplifying fractions (Objective 0016, although overlaps with other objectives too).
 Question 5

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

 A $$\large \dfrac{4}{6}$$Hint: $$0.\bar{6}=\dfrac{2}{3}=\dfrac{4}{6}$$ B $$\large \dfrac{3}{5}$$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. C $$\large \dfrac{6}{10}$$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. D $$\large \dfrac{1}{6}$$Hint: This is less than a half, and $$0.\bar{6}$$ is greater than a half.
Question 5 Explanation:
Topic: Converting between fraction and decimal representations (Objective 0017)
 Question 6

#### 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

#### P divides 30

Hint:
2, 3, and 5 are the prime factors of 240, and all divide 30.

#### P divides 48

Hint:
P=5 doesn't work.

#### P divides 75

Hint:
P=2 doesn't work.

#### P divides 80

Hint:
P=3 doesn't work.
Question 7 Explanation:
Topic: Find the prime factorization of a number and recognize its uses (Objective 0018).
 Question 8

#### Which of the following equations could also represent A  for the values shown?

 A $$\large A(n)=n+4$$Hint: For a question like this, you don't have to find the equation yourself, you can just try plugging the function machine inputs into the equation, and see if any values come out wrong. With this equation n= -1 would output 3, not 0 as the machine does. B $$\large A(n)=n+2$$Hint: For a question like this, you don't have to find the equation yourself, you can just try plugging the function machine inputs into the equation, and see if any values come out wrong. With this equation n= 2 would output 4, not 6 as the machine does. C $$\large A(n)=2n+2$$Hint: Simply plug in each of the four function machine input values, and see that the equation produces the correct output, e.g. A(2)=6, A(-1)=0, etc. D $$\large A(n)=2\left( n+2 \right)$$Hint: For a question like this, you don't have to find the equation yourself, you can just try plugging the function machine inputs into the equation, and see if any values come out wrong. With this equation n= 2 would output 8, not 6 as the machine does.
Question 8 Explanation:
Topics: Understand various representations of functions, and translate among different representations of functional relationships (Objective 0021).
 Question 9

#### Which of the equations below could best be used to explain why the children€™s conjecture is correct?

 A $$\large 8x+16x=9x+15x$$Hint: What would x represent in this case? Make sure you can describe in words what x represents. B $$\large x+(x+2)=(x+1)+(x+1)$$Hint: What would x represent in this case? Make sure you can describe in words what x represents. C $$\large x+(x+8)=(x+1)+(x+7)$$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. D $$\large x+8+16=x+9+15$$Hint: What would x represent in this case? Make sure you can describe in words what x represents.
Question 9 Explanation:
Topic: Recognize and apply the concepts of variable, equality, and equation to express relationships algebraically (Objective 0020).
 Question 10

#### A family on vacation drove the first 200 miles in 4 hours and the second 200 miles in 5 hours.  Which expression below gives their average speed for the entire trip?

 A $$\large \dfrac{200+200}{4+5}$$Hint: Average speed is total distance divided by total time. B $$\large \left( \dfrac{200}{4}+\dfrac{200}{5} \right)\div 2$$Hint: This seems logical, but the problem is that it weights the first 4 hours and the second 5 hours equally, when each hour should get the same weight in computing the average speed. C $$\large \dfrac{200}{4}+\dfrac{200}{5}$$Hint: This would be an average of 90 miles per hour! D $$\large \dfrac{400}{4}+\dfrac{400}{5}$$Hint: This would be an average of 180 miles per hour! Even a family of race car drivers probably doesn't have that average speed on a vacation!
Question 10 Explanation:
Topic: Solve a variety of measurement problems (e.g., time, temperature, rates, average rates of change) in real-world situations (Objective 0023).
There are 10 questions to complete.

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