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Dr. Tom Faulkenberry – Hi, I’m Dr. Tom Faulkenberry, director of the math and science teacher preparation program at
Texas A&M University at Commerce. I will be spending the next hour with you going over the mathematics TExES exam.
The format will be simple. I will present problems that represent the types of knowledge you will be expected to
possess. Your knowledge will be broken up into 4 domains, each containing several competencies. I will present a
written description of each domain and competency followed by an example problem that I will work in detail. You will
need a calculator for some of these problems. I will be using a TI 84 Plus Silver edition but any comparable calculator
should do the trick. Now, let’s get started.
Slide 1 - .49 seconds
Domain I: Number Concepts
This domain includes some elementary content, but it is nonetheless important for the middle school teacher to know very well.
Understanding numbers goes way beyond simple arithmetic. Ideas from number theory can be used to solve hard problems in novel ways. Also, one needs to know some history behind how the numbers we use today came to be. Why do we not use Roman numerals still?
The practice problems that follow all represent some of the different number concepts that you should be familiar with.
Slide 2 - 1 min .10 seconds
Competency 001:
The teacher understands the structure of number systems, the development of a sense of quantity, and the relationship
The teacher understands ideas of number theory and uses numbers to model and solve problems within and outside of
mathematics.
Slide 7 – 4 min .43 seconds
Dr. Faulkenberry – What is the smallest positive integer that does not evenly divide 43 factorial? Now realize that is a
factorial, I’m not just being emphatic about the statement 43!
43 factorial of course is simply the repeated product of 43 to 42 to 41 all the way down to 3 times 2 times 1. We’re
looking for the smallest positive integer that does not evenly divide 43 factorial. Now to get a handle on this problem,
first notice that 2 divides 43 factorial because 2 shows up as a factor in this factorization as does 3 and similarly 4, 5 and
up to 40, 41 42, 43. So all of the integers, 1 thru 43, do, in fact, evenly divide 43 factorial, simply by virtue of showing up
in the factorization. So that means we should probably start above 43, namely 44, so let’s write it down as a possible
candidate. Does 44 divide 43 factorial? Well, you don’t have to get a calculator out for this. All you have to do is realize
44 is the same as 4 times 11. Now, 4 does in fact show up in the factorization of 43 factorial as does 11 and hence 44
does in fact evenly divide 43 factorial. By similar reasoning, 45 will also evenly divide 43 factorial. One way to
demonstrate this would be to write 45 as 9 times 5 and note that each of these, 5 and 9 do show up in the factorization
of 43 factorial. Similar is 46. 46 is 2 times 23 and each of these factors shows up in the factorization of 43 factorial. We
finally run into a snag however at 47. 47 is prime. Since 47 is prime it has no smaller factors that are going to show up in
the factorization of 43 factorial. 47 itself does not show up in the factorization and hence, this is our smallest positive
integer that does not evenly divide 43 factorial.
Slide 8 – 7 min .03 seconds
Domain II: Patterns and Algebra
Much of algebra is developed very early by students who notice patterns. Inferring the next number in a sequence is a skill that is attainable by most young children, yet many teachers do not realize that this skill is necessary for the development of algebra.
Why? Inferring the next member of a pattern involves finding an unknown…this is the central problem in algebra. From here, children develop knowledge about functions and functional relationships.
Hence, it is important that the well-prepared teacher of 4-8 mathematics has a substantial knowledge of patterns and algebra. The problems that follow represent the types of problems that you will be expected to solve.
Slide 9 – 7 min .26 seconds
Competency 004:
The teacher understands and uses mathematical reasoning to identify, extend, and analyze patterns and understands the relationships among variables, expressions, equations, inequalities, relations, and functions.
Slide 10 – 7 min .34 seconds
Dr. Faulkenberry – Consider the following pattern. For the first entry we have 1 dot. For the second entry we have a
triangular pattern of 3 dots. For the third entry we have a triangular pattern of 6 dots and for the fourth entry we have a
triangular pattern of 10 dots. The question is how many dots would comprise the 100th figure?
Now you don’t have to go through and draw all of the figures from 5 up to a 100 to see the answer to this. You can tap
into this problem in 1 of 2 ways. The first thing that you could do is realize that these are simply the triangular numbers
and if you remember a formula for the Nth triangular number then great. Chances are you probably don’t so what I
want to look at is a classical bit of mathematics that would get us to the answer to this problem. First of all, realize that
each of the numbers of dots in these entries can be realized as a repeated sum. For example, the first dot is simply 1 but
the second pattern of dots could be looked at as 1 plus 2. The third could be looked at as 1 plus 2 plus 3 and the fourth
could be looked at as 1 plus 2 plus 3 plus 4. This is in fact the generating sequence of the triangular numbers. Using this
pattern, then the 100th figure would simply be the sum of 1 plus 2 plus 3 plus 4 plus all the way up to 98 and 99 and
finally adding 100. This is a famous sum in mathematics. You may remember a story about Gauss and how he
discovered a problem similar to this while being a very young student. The method to get this is very simple. First of all
notice you have 100 terms in this sequence. Now, the outer two sums added together, let me underline them for
emphasis, the outer two parts of the sum added together would be 101. Now move in 1 unit on both ends of the
sequence. So now we have 2 plus 99. 2 plus 99 is also 101 as is 3 plus 98. In fact, each time you move in 1 from the left
and move back 1 from the right, the sum is still going to be 101. So what we have is some number of groups of 101. Our
task now is to figure out how many groups is this? Well, now remember, we have 100 terms in this sequence. The 101
comes from a pairing of the outer most terms. So how many pairs are there in 100 terms. The answer is of course 50
pairs. So what we have here are 50 groups of 101 which we can easily figure out is 50 x 101 or 5050. So the 100th figure
would be a bit too big for us to draw but would have 5050 dots.
Geometry and measurement is an important branch of mathematics for the 4-8 mathematics teacher to have mastered. For most children, geometry is the transition from concrete ideas to abstract reasoning. This is due to the fact that we can see the things that should be true about a geometric figure; so the only thing remaining is the ability to use deductive reasoning to show that it must be true, regardless of our ability to see it.
Measurement is essential to functioning in the physical world. Most physical phenomenon are studied because they can be measured. Hence, it is essential for the well-prepared teacher of 4-8 mathematics to have a deep knowledge of the concepts of geometry and measurement.
The problems that follow are indicative of the types of problems you will be expected to solve.
Slide 18 – 22 min .41 seconds
Competency 008:
The teacher understands measurement as a process.
Slide 19 - 22 min .50 seconds
Dr. Faulenberry – A forest manager uses a clinometer to measure the angle of elevation from a point on the ground to
the top of a tree 500 feet away. The angle is measured to be 25.7 degrees. The tree is known to be 82.03 yards tall.
What is the percent error of the forest manager’s measurement? Now first of all, we need to have a talk about what
percent error really is. Percent error, which I’m going to denote as PE, is simply the difference in an experimental or trial
measurement value with an accepted or known value. So one way to look at this is you can look at it as the
experimental minus the known and this quantity, this difference, needs to be scaled by the known quantity. You want
to compare it to the known quantity. So when we find out our experimental height, so to speak for this tree, then we
can compare it to this known value of 82.3 yards and figure out what the percent error of our instrument is. So to do
this we are going to use some basic trigonometry. We have a point on the ground that we are measuring the height of
this tree from and I’m going to denote this tree just by a single line. Now this triangle is not necessarily drawn to scale
but it will give us some reasonable mental things to help us with the calculation. Now we measure this angle to be 25.7
degrees. We know that the horizontal distance from our clinometer to the base of the tree is 500 feet. What we don’t
know is this height h. But we know we can relate those quantities via the tangent function. Namely tangent of 25.7
degrees is the same as the height divided by 500, the opposite over the adjacent and hence height is 500 times tangent
of 25.7 degrees. So I’m going to pull in the calculator just real quickly. First of all, make sure that you are in degrees
Probability and statistics have historically been neglected as topics of substantial coverage in the mathematics curriculum. However, probabilistic and statistical reasoning are used every day to make common decisions in our world.
The well-prepared teacher of 4-8 mathematics must have a good working knowledge of the basic principles of descriptive and inferential statistics. The problems that follow are representative of the types of problems you’ll be expected to know how to do.
Slide 27 – 38 min .08 seconds
Competency 012:
The teacher understands how to use graphical and numerical techniques to explore data, characterize patterns, and describe departures from patterns.
Slide 28 – 38 min .16 seconds
Dr. Faulkenberry – The scores on a survey of attitudes toward mathematics and science are distributed according to the
following box and whisker plot. Would either 7 or 19 be considered as an extreme score or an outlier?
Now a box and whisker plot is a very nice way to look at the distribution of a set of data. For example, it gives us 5
important pieces of information; the lowest value, the highest value, the median, which is 12 in this case, and the 2
quartiles. The quartiles are simply the medians of the lower half and the upper half of the data. Now one of the things
you will notice about a box and whisker plot is that sometimes the whiskers are very short. For example, this side has a
much shorter whisker than this side and that’s a visual cue that 19 does not follow the distribution of the data as well as
7 does. So we have this visual indication that 19 might be considered extreme or an outlier. Now the rigorous way to do
this is to take what’s called the inner quartile range, otherwise known as the IQR. That’s simply the distance between
the two quartiles; the third quartile minus the first quartile, which in this case is 13-10 or 3. Now this inner quartile
range is used then as a metric upon which to gauge your outliers. For example, notice that 7 is exactly 1 of these boxes
away from 10. That is its 1 inner quartile range. But now 13 to 19, you could fit this box, which again we will call the
IQR, you could fit that 1, 2, 3 times between, sorry, you could fit that twice between the 13 and the 19. Since 19 is 2
times the inner quartile range away from its closest quartile, we consider 19 as an outlier and we would reject it as such.