Math MTEL: Study Guide Framework

© 2015 Volker Ecke, Westfield State College

Objectives

Candidates should not only know

Content Areas:

Study Suggestions

Undergraduate students take mathematics coursework as part of their program at Westfield State College. The four mathematics content courses Math 153, 250, 251, 252 are aligned with the areas of the Massachusetts Curriculum Frameworks. These courses are designed to cover the various content areas tested on the mathematics subtest of the General Curriculum test. This, combined with what the student already knows from their high school experience, will help students to effectively demonstrate their knowledge of content areas.

As you start preparing for the mathematics subtest of the General Curriculum MTEL (03), here are a few points to keep in mind.

  1. Take as many of the "Foundations of Mathematics" courses as you can fit into your schedule (Math 153, 250, 251, 252). Make sure you take Foundations of Number Systems (it is now a required course in the program).
  2. If you are anxious about taking the mathematics test (or testing in general), seek support for working with testing anxiety. Make a pact with yourself not to abandon the test half-way. When you notice anxiety, put down your pen and give yourself a few minutes to check in with your body and to calm yourself.

Suggestions for Self-Guided Preparation:

  1. Create a study guide for yourself. The General Curriculum test objectives are available online. These objectives give you a great framework to build your study guide around. We encourage you to purchase a binder and divider tabs. Each divider tab should represent one objective. The contents within each section should answer to the terms outlined in an objective area. Setting up a binder like this may take some time, but with advanced planning will be well worth it!
  2. Collect and organize your resources: e.g. your class portfolio, text books, handouts, the Van de Walle book, high school textbooks, online resources etc.
  3. ALEKS or some other foundational mathematics course of study (e.g. Mary DeSouza's book). While not aligned with the MTEL, you will have a chance to gain practice and confidence with basic arithmetic and computational techniques.
  4. Practice Exams. One of the best ways to prepare for the General Curriculum test is to work with the practice exam, or at similar practice exams for other states (for details see Section [*]).
  5. Almost all of the problems on the Practice Exam are multi-step problems. A single reading is not enough to unravel what you need to do. There is no obvious formula to solve any of these problems. You will need to grapple with the material to make sense of it, and you will need to give yourself the time and focus to do that. That's normal and expected.
  6. Read carefully: it may help to underline or mark the important concepts listed in the problem. Write down whatever you know about that concept. As you prepare for the exam, write down the resources you have to fill in details about this concept.
  7. The commented solutions below are intended to show you a framework how to approach using MTEL-like problems in order (a) to identify mathematical concepts that you understand, (b) to identify mathematical concepts that you do not understand, (c) to assemble information about your resources, and (d) to observe a step-by-step process that might help you in problem-solving more generally.
  8. Below are sites that offer practice exams which may prove to be helpful study tools:
  9. Additional Resources. Please note that some of these suggestions are based on student recommendations and are not designed to ensure passage on this test, but to help provide you with supplemental information.
    1. Massachusetts Comprehensive Assessment System (MCAS). Previous test items are available online at

      http://www.doe.mass.edu/mcas/testitems.html

    2. What Your Fourth Grader Should Know by E.D. Hirsch.
    3. What Your Sixth Grader Should Know by E.D. Hirsch.

Resources

Van de Walle, Elementary and Middle School Mathematics-Teaching Developmentally is the text book required for all Mathematics Foundations courses at Westfield State College. The references given here are for the seventh edition. If you have a different edition, use the Index of your book with your key words in order to guide your search.


\begin{framed}
The ''Activities'' listed throughout the \emph{Van de Walle} book...
...em (4.) about Neptune with Activity 23.3 in Chapter 23, on p. 478.}
\end{framed}

Resources on Problem-Solving:

Improving Performance of High-Stakes Tests, Test-taking Strategies (p.89 ff).

Resources on Math Anxiety:

Metacognition, Disposition, Attitudinal Goals (p.46 ff).

Resources on Number Systems:

Chapters 8-13, 15-17, 23.

Resources on Patterns, Algebra, Functions:

Chapter 14.

Resources on Geometry and Measurement:

Chapters 19-20.

Resources on Data Analysis and Probability:

Chapters 21-22.

Tips and Suggestions for Taking this Test

Have a game plan for how you are going to approach the test.
The great thing about the MTEL is that you are not required to take the test in chronological order-how you take it is entirely up to you. We encourage you to look at the open response items first, set-up an outline about how you are going to approach answering these questions, and then go back to get started on multiple choice items. Approaching the test this way may help to alleviate some test anxiety and set a good pace for yourself with the rest of the test.
Be clear and consistent with your responses.
One of the goals of the MTEL is to determine whether or not you will be able to communicate clearly with students and parents alike. This being the case, it is essential that your handwriting on open response items is legible, your spelling and grammar are exceptional, and your answers are well thought out.
Get focused!
To calm your nerves once the test begins, it is best to give yourself a few minutes to relax and look over the test in its entirety before getting started. Given that you have a full four hours to take the test, take advantage of a restroom break. Getting up from the test for a few minutes and throwing some cold water on your face will work wonders.


Sample MTEL Problems

These problems are taken from the Practice Exam. Print a copy for complete reference (diagrams etc. are not included here).
\begin{framed}
Please note that there are many different ways to solve these pro...
....
No calculator was used and none is allowed for taking the MTEL.
\end{framed}


\begin{framed}
{\em 1. In the number 2010, the value represented by the digit 1 is what fraction
of the value represented by the digit 2?}
\end{framed}

Identify important words:

place value, digit, fraction

Identify the related mathematics concepts:

Determining the value of various digits in a decimal number. Computing a fraction of a whole.

Test-taking strategy:

Look at the answers to see whether anything stands out. Nothing obvious here.

Resources:

Van de Walle, Ch. 11, Developing Whole-Number Place-Value Concepts.

First steps:

In the number 2010, what is the value represented by the digits 1 and 2, respectively? Now, that value is 10 and 2000, respectively. So far, so good. On to the fraction part of the question: what fraction of 2000 is the value of 10? Well, it's $ \frac{10}{2000} = \frac{1}{200}$ .

Answer (B).

$ \phantom{foo}$


\begin{framed}
{\em 2. If $P$ is a positive integer, which of the following must also be a positive
integer?}
\end{framed}

Test-taking strategy:

Look at the answers to see whether anything stands out. $ 1-P$ , $ \frac{1}{P}$ , $ \sqrt{P}$ , $ P^2$ .

Identify important words:

positive integer; fraction, square root, square of a positive integer.

Identify the related mathematics concepts:

Identify subsets of the real numbers (e.g., integer, rational, irrational) and their characteristics.

Resources:

Van de Walle, Ch. 23, Developing Concepts of Exponents, Integers, and Real Numbers.

First steps:

I can think of a few approaches : (a) Try a few integers (you pick!) for $ P$ and compute these four values. The ones most likely to go wrong are $ \frac{1}{P}$ and $ \sqrt{P}$ . (Note that if you can eliminate two false answers, then even guessing between the remaining two answers gives you a pretty good chance of getting it right.). The other answer that is incorrect is $ 1-P$ . If you choose the positive integer $ P=1$ , then $ 1-P = 0$ , which is not a positive integer. Answer (D).

(b) Maybe you remember that squaring a positive integer will always give you another positive integer. (Note that for this question it's enough to find one answer you know is correct.)

(c) You might also remember the fact that the positive integers are "closed" under addition and multiplication (e.g. squaring), but not under division or taking roots ("closed": e.g. when you divide one integer by another you do not usually get an integer result).

If you consider all of the integers (positve or negative), these are "closed" under addition, multiplication and subtraction.

Answer (D).

Test-taking strategy:

If nothing else comes to mind when working with such an algebraic question, pick a few concrete values, plug them in, and see what happens.


\begin{framed}
{\em 3. According to an article in a financial journal, a certai...
...eness of
this estimate for the company's average monthly earnings?}
\end{framed}

Identify important information:

(whenever one of these sections is empty, it is your job to fill it in)

Identify the related mathematics concepts:

estimate, dollars per year versus dollars per month, average monthly earnings

Test-taking strategy:

All of the answers indicate that the estimate is incorrect. Too low or too high? By a factor of 10 or a factor of 100? So we don't need to compute precise numbers.

Objective:

Determine the reasonableness of estimates.

Resources:

Van de Walle, Ch. 13, Using Computational Estimation with Whole Numbers; Ch. 23, Developing Concepts of Exponents, Integers, and Real Numbers: Large Numbers.

First steps:

Again, look at the answers to get a sense what we need to find out? Ah, we want to see whether the estimate is too low or two high.

As the numbers are written, it's not clear how many zeros each of them has, so let's write them out fully: earnings of 3,850,000 for the year and an estimate 30,000 per month. We could try to divide 3,850,000 by twelve but that will take time (it also looks like the answer will not require those details). Remembering that there is a related multiplication problem, we can look at 30,000 times 12: as a first guess, 30,000 times 10 would be 300,000 (so we see already that the estimate is likely too low compared to 3,850,000); more precisely $ 30,000 \times 12 = 360,000$ , which is about a factor 10 too low.

Answer (B).

$ \phantom{foo}$


\begin{framed}
{\em 17.  Use the diagram below to answer the question that
fol...
... the product
5775. What value does the circled digit represent?
}
\end{framed}

Identify important information:

(your job)

Identify the related mathematics concepts:

(your job)

Resources:

Van de Walle, Developing Whole-Number Place-Value Concepts, Ch. 11.

First steps:

The circled "1" comes from multiplying 30 (in 231) times 5 (in 25), which gives $ 30 \times 5 = 150$ . In 150, the "1" is in the hundreds position.

Answer (C).

$ \phantom{foo}$


\begin{framed}
{\em 24. Use the solution procedure below to answer the question...
...h of the following is a major flaw in the procedure shown
above? }
\end{framed}

Identify important information:

Identify the related mathematics concepts:

Manipulate simple algebraic expressions and solve linear equations and inequalities. Justify algebraic manipulations by application of the properties of equality, the order of operations, the number properties, and the order properties.

Resources:

Van de Walle: Meaningful use of Symbols, Ch. 14, p. 257 ff., Student Writing, Ch. p. 44-45, 85-86.

First steps:

(B): The equal sign marked with a star is used to connect expressions that are not equal: $ 4 - 25$ is equal to $ -21$ , and not equal to $ -21 \div (-3)$ .

Answer (B).

$ \phantom{foo}$


\begin{framed}
{\em 30. Use the graph below to answer the question that follows...
...presents the equation $Wx + 4y = 12$.
What is the value of $W$? }
\end{framed}

Identify important information:

Identify the related mathematics concepts:

Objective:

Translate among different representations (e.g., tables, graphs, algebraic expressions, verbal descriptions) of functional relationships.

Test-taking strategy:

Look at the answers to see whether anything stands out. Some answers are positive, some are negative. However, the equation $ Wx + 4y = -12$ is not in the usual slope-intercept form, so it's not clear how $ W$ is related to the slope of the line.

Resources:

Van de Walle: Ch. 14 Algebraic Thinking, esp. Linear Functions (p. 274-275).

First steps:

One idea would be to transform the equation to slope-intercept form by solving for $ y$ .

Test-taking strategy:

We could also use some of the known data points from the graph to find $ W$ . Let's use the $ (0, -3)$ point (zeros are easy to compute with): $ Wx + 4y = -12$ turns into $ W \cdot 0 + 4 \cdot (-3) = -12$ , or $ -12 = -12$ . OK, the $ W$ disappeared, so that didn't help.

Test-taking strategy:

Notice that I followed a path that did not lead me any closer to the solution. That is not a problem. I take a breath and try the next idea:

Let's try the other point $ (-4,2)$ : turns into

$\displaystyle W x + 4 y$ $\displaystyle =$ $\displaystyle -12$  
$\displaystyle W \cdot (-4) + 4 \cdot 2$ $\displaystyle =$ $\displaystyle -12$  
$\displaystyle -4 W$ $\displaystyle =$ $\displaystyle -12 - 8 = -20$  
$\displaystyle W$ $\displaystyle =$ $\displaystyle 5.$  

Answer (D).

$ \phantom{foo}$


\begin{framed}
{\em 36. Use the diagram below to answer the question that follow...
...pressions represents the approximate
total length of the ribbon? }
\end{framed}

Identify important information:

Identify the related mathematics concepts:

Test-taking strategy:

Look at the answers to see whether anything stands out.

Resources:

Van de Walle: Geometric Thinking, Ch. 20, The Pythagorean Relationship (p. 418-419, 428).

What would be your first step?

Second Steps:

There are four segments of ribbon, two each on the $ x$ -$ y$ -face (top and bottom) and two each on the $ y$ -$ z$ -face (left and right). Note from the picture that there is no ribbon along the $ x$ -$ z$ -face of the box (front and back). So while the box has six faces, there are only four pieces of ribbon.

Consider the piece of the ribbon that runs diagonally up the left-hand $ y$ -$ z$ -face of the box. What is the length $ d_{yz}$ of that piece of ribbon? Use the Pythagorean Theorem.

Fill in the rest:

Answer (C).

$ \phantom{foo}$


\begin{framed}
{\em 37. Use the graph below to answer the question that follows...
... across the $x$-axis, which of the following graphs will result?
}
\end{framed}

Identify important information:

Identify the related mathematics concepts:

Test-taking strategy:

Look at the answers to see whether anything stands out. Nothing obvious. Do not jump to conclusions. This is also a multi-step problem: you apply two different operations one after the other.

Test-taking strategy:

Draw what happens after the graph above is rotated $ 180 \deg$ about the origin.

Resources:

Van de Walle: Geometric Thinking and Geometric Concepts, Ch. 20, Learning About Transformations (p. 419 ff.).

First steps:

Your solution:

Observations:

If we rotate the hand $ 180^{\circ}$ about the origin, the hand will end up in the bottom-left quadrant, with the fingers pointing down and the thumb against the negative $ y$ -axis. (Notice that for a rotation about $ 180^{\circ}$ , the clock-wise and counter-clockwise rotations give the same end result, so it's ok that the problem does not tell us which way to turn.)

If we now reflect that image across the $ x$ -axis, the hand will end up in the upper-left quadrant, with the fingers pointing up and the thumb against the positive $ y$ -axis.

Answer (C).

$ \phantom{foo}$


\begin{framed}
{\em 38. Which of the following nets can be folded to form a square pyramid?}
\end{framed}

Practice strategy:

The best way to practice visualization is to work with things. What would it look like to fold this up? Cut it out and try it!

Resources:

Paper and scissors.

Answer (B).

$ \phantom{foo}$


\begin{framed}
{\em 39. Use the figure below to answer the question that follow...
...at has $AB$ as a line of symmetry,
then the entire figure is a: }
\end{framed}

Identify important information:

Identify the related mathematics concepts:

Test-taking strategy:

Make for yourself a glossary of important mathematical terms. (Maybe you have already done this as part of your course work.)

Resources:

Van de Walle: Geometric Thinking and Geometric Concepts, Ch. 20, Learning About Transformations (p. 419 ff.).

First steps:

Draw the other congruent half onto the paper. Check that the resulting line of symmetry is in fact $ AB$ .

Second steps:

If $ AB$ is a line of symmetry, then there should be another equilateral triangle with basis $ AB$ and third point $ C'$ exactly opposite point $ C$ . If you place two equilateral triangles together along a shared side, you get a rhombus (2D polygon with four sides, all the same length): answer (D).

It's not a triangle because you get four sides. It's not a rectangle because the angle at $ C$ is equal to $ 60^{\circ}$ and not a right angle. The shape is not 3D (so not a prism).

Answer (D).

$ \phantom{foo}$


\begin{framed}
{\em 40. Use the diagram below to answer the question that
foll...
...rm a triangle, as shown
above. What is the measure of angle $x$? }
\end{framed}

Identify important information:

Identify the related mathematics concepts:

Test-taking strategy:

Annotated glossary that shows relationships between angles when various lines intersect.

Resources:

Geometry resources on "supplementary angles."

First steps:

Fill in whatever other angles you can determine.

Second steps:

There are a number of ways to think about this. One idea is to fill in other angles that we know. First, the angle opposite the $ 115^{\circ}$ angle must also be $ 115^{\circ}$ ; similarly for the $ 60^{\circ}$ angle. Next, the angles adjacent to the $ 115^{\circ}$ angle are supplementary angles, so must add to $ 180^{\circ}$ : hence, they are $ 65^{\circ}$ . In the same way, the supplementary angles of the $ 60^{\circ}$ angle are $ 120^{\circ}$ .

Now we have two of the three interior angles of the triangle, namely $ 65^{\circ}$ and $ 60^{\circ}$ , so the remaining angle must be $ 180^{\circ} - 65^{\circ} - 60^{\circ} = 55^{\circ}$ .

Finally, the angle $ x$ is supplementary to this angle of $ 55^{\circ}$ , so it's equal to $ 180^{\circ} - 55^{\circ} = 125^{\circ}$ .

Answer (C).

$ \phantom{foo}$


\begin{framed}
{\em 45. Use the spinner below to answer the question that
foll...
...f the number
of fruit baskets that the host will be giving away? }
\end{framed}

Identify important information:

Identify the related mathematics concepts:

Test-taking strategy:

Make sure all the labels inside the spinner are in fact different (e.g. not two for "Smile").

Resources:

Van de Walle: Exploring Concepts of Probability, Ch. 22, Introducing Probability (p. 456 ff.).

First steps:

Second steps:

It looks like the spinner gives equal probability to each of the five option, so the outcome "Fruit Basket" has probability $ 1/5 =20\%$ . We would expect to see the spinner land on "Fruit Basket" for about 1 in 5 guests. With 180 guests total, that's $ 180/5 = 36$ guests.

Answer (C).

$ \phantom{foo}$

About this document ...

MTEL Problems-Study Guide

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