1. New York State Teacher Certification Exam Multi-Subject Part II Mathematics

2. Make up of the exam:
Selected Response: 40 multiple choice questions – 80% of score
Topics covered - approximate percentage of problems on each content area.
(Assessment and Design Framework – will give an overview of the specific topics covered for each grade band level) NYSTCE Multi-subject (Birth - Grade 2)
NYSTCE Multi-Subject (Grade 1 - Grade 6), NYSTCE Multi-Subject (Grade 7 - Grade 12)
For Birth – 2nd grade - approximate break down in content
Number sense and operations (~20%)
Algebraic thinking (~30%)
Geometry, Measurement, and Data (~20%)
Instruction in Mathematics (~10%)
For 1st – 6th grade – approximate break down in content
Number sense and operations (~10%)
Ratio, Proportions, Percent, and Number Systems (~30%)
Algebra, Geometry, Measurement, and Data (~30%)
For 7th – 12th grade – approximate break down in content
Number sense and Ratios and proportional reasoning (~15%)
Algebra and functions (~30%)
Geometry and measurement (~15%)
Statistics (~20%)

3. http://www.ixl.com/math note you are allowed to do only about 20 problems a day without paying

4. Constructed Response: 1 constructed essay – 20% of score
Use the information in the exhibits that follow to complete the task.
Prepare a response of approximately 400 – 600 words in which you:
Identify a significant strength the student demonstrates related to the given standard, citing specific evidence from the exhibits to support your assessment;
Identify a significant need the student demonstrates related to the given standard, citing specific evidence from the exhibits to support your assessment;
Identify a learning activity or strategy that would build on the student’s strength and that would help the student improve in the area of need. Include reasons for why the strategy would be effective based on the given standard and/or Include a strategy for helping the student build a viable argument related to the given standard.
The three pieces of evidence usually consist of
1. A standard from the common core state standards for mathematics
2. Classroom discourse or teacher note of the class instruction
3. Student work either written or discourse

5. Preparing to Write the Constructed Response
There is a 3 to 4 paragraph structure for this constructed response
Paragraph 1: (~100 words)
Paragraph 2: (~150 words to 200 words)
Paragraph 3 and 4 (up to 300 words)
Give yourself one hour to write the constructed response. Suggested time to read and analyze is 20 minutes with 40 minutes to write.
Start by reading the three pieces of evidence which include but are not limited to mathematics standards from CCSSM, description of class discourse and work, description of student interview and/or student written work.
Then take 10 minutes to bullet point on the dry erase pad
 One strength of the students
 One need of the student
At least 3 instructional interventions based on the standards and the strengths and
needs

6. Common Core State Standards Mathematical Practice keep these in mind as you write the constructed response – these are the practices you want to emphasize when you write about an instructional intervention
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others.
4. Model with Mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning.

7. Strategies for Answering a Constructed Response Question - RACE steps •Reword/restate the question • Provide an Answer •Cite using evidence from text •Explain how the evidence supports your answer
Example
Other examples at end of presentation
Evidence 1:
CCSMS:
1.NBT.B.2 Number and Operations in Base Ten – Understand that the two digits of a two-digit number represent amounts of tens and ones.
2.OA.A.1 Operations & Algebraic Thinking – Represent and solve problems involving addition and subtraction - use addition within 100 to solve one-and two-step word problems involving situations of adding to take from putting together etc.
Student Work
Discussion with Student:
T: Can you explain what you are doing here (I point to the drawing)
Student: I am drawing the 29 cups of lemonade in the trains but there is 1 cup extra.
The student keeps working and I don’t interfere until she is finished
T: So you did one class for the 29 children can you explain what you did for the 28 children
Student: Well you see 10, 20, 1, 2, 3, 4, 5, 6, 7, 8 is 28 so there is 3 trays with 2 left over 9 and 10.
T: What is the question asking?
Student takes some time to read the problem.
Student: How many trays are needed?
Teacher: Did you answer the question?
Student: Yes 3 trays for one class and 3 trays for the other class.

8. Example Problems from the Selected Response
Read each question and
Try it before you read the
answer
What is the smallest positive even number that is the product of five different prime
numbers?
A. 15,015
B. 10,395
C. 2,310
D. 1,890
2. Which of the following numbers does NOT represent an integer?
A. -77
B. 0
C. 1.5
D. 13
we need to know even numbers, product and prime numbers
1. answers A and B are not possible. Why?
2. product of prime numbers - multiply prime numbers together to get an answer
3. Our choices are 2310 and if you know prime factorization then you can do
that for both numbers if not
try some prime numbers and multiply them -
trial 1 2x3x5x7x11 = 210x11= 2310 (by the way - don't multiply in order 2x5 =10 first and then 3x7
That 2x5 is going to help because both of your choices are multiples of 10)
but what if you don't get that on the first try - will the other one is 189x10 and 189 is a multiple of x63 but 63 = 7x9 and 9 is not prime.
we need to know what integers are.
Our number system is made of sets of numbers
The first and most basic is our counting numbers or natural numbers 1, 2, 3, 4, ...
Then our whole numbers - which are the counting numbers and zero 0,1,2,3,4,...
Then our integers which are whole numbers and our negative numbers - think of our number line
Then our rational numbers which are defined as a number of the form p/q where q is not zero and p and q are integers. They include all of the previous sets of numbers as well as the fractional parts of numbers.

9. 3. Which of the following is a possible solution for b in the inequality shown here 5b - 4 < -14 ?
A. -3
B. -2
C. 4
D. 10
4. An even number has two different prime factors. Which of the following could be the product of those factors?
A. 6
B. 12
C. 36
D. 48
An understanding of integers is needed here - if you are comfortable with algebraic
manipulation then you will do the following
5b - 4 < -14
5b < now divide both sides by 5 to get
b < so the only solution is -3 answer A.
The other option is to plug in all the choices and see what works.
Prime numbers - the first few are 2, 3, 5, 7, 11, 13
If we multiply 2 of them together - and only 2 of them what do we get
A. 6

10. (What do you think – there are no choices here)
5. In a standard deck of 52 cards, what is the probability of being dealt a king, a queen or a
jack? A. B. C.
D.  
6. In one high school 40% of students go to college. If two graduates of the high school are
chosen at random, what is the probability that they both went to college?
(What do you think – there are no choices here)
How many kings are there? 4
How many queens are there? 4
How many jacks are there? 4
So the chances of getting one of these out of 52 cards is 12/52 which reduces to 3/13 –> C.
lets think about the chances that one student will go to college: 40% or 40/100 or 2/5
the chances of a second student going to college for that individual is also 2/5
but the chances that two students both are going to college will have to be less then 40%
we get that probability by multiplying 2/5 by 2/5 to get 4/25 or 16%
Why do we multiply the probabilities?
The multiplication principle: The chance of two independent events both happening is the product of their individual probabilities.

11. win its first 3 games and lose the fourth? (A) (C) (B) (D)
The probability of a certain team winning is ¾, what is the probability that this team will
win its first 3 games and lose the fourth?
(A) (C)
(B) (D)
All the letters of the alphabet are written on plastic discs and placed in a bag. The vowels in the bag are A,E,I,O,U. Two letters are randomly chosen and not replaced. Both letters are consonants. What is the probability that the next letter chosen is a vowel?
A. 1/5
B. 1/3
C. 5/26
D. 5/24
A fair penny is flipped three times, and each time the penny lands heads up. What is the
probability that this same penny will land heads up on the next flip?
A. 1/16
B. 1/8
C. 1/4
D. 1/2
For the first game it is 3/4 , for the second game it is ¾, for the third game it is 3/4 , but for the fourth game it Is ¼. Because these are independent events each with their own probability to find the total probability you would multiply.
The answer would be 3/4x3/4x3/4x1/4 = 27/256 (C)
So the probability of getting a consonant is 21/26 and getting a vowel
Is 5/26. Without replacement means that the total number of discs
decreases at the next pick. So we are now down to 5/24(d).
Each flip is independent of the previous flip , so it doesn’t matter
what occurred before the current flip – the answer is (D)

12. 10. Are triangle these two triangles similar triangles?
In order for two triangles to be
congruent their corresponding sides need
To be in proportion - if this is the case then
The corresponding angles are congruent.
So lets see if the sides are in proportion in any way
Choose AB/DE = 4/6 so are any other sides in the same
ratio.? Yes BC/EC is also equal to 2/3 as it is 6/9 but what about
AC/DC… well that is 5/7.5 and if we simplify this we also get 2/3 so the triangles are similar
This triangle slid right 6 units and up 3 units. Given point
find the coordinate of M’
The grid at the right helps you with this problem
But you could be given this problem without a
Grid. In that case it is saying that the x coordinate
Is increased by 6 or x+6 and the y- coordinate is
Increased by 3 or y This is a translation that in some
Notations looks like so M’ becomes
(-2+6, -2+3) = (4, 1)

13. “All dice have six sides, and all cubes have six sides.”
Which of the following statements is true according to the statement above?
(A) All dice are cubes.
(B) Some dice are cubes.
(C) No dice are cubes.
(D) Things with four sides are not cubes.
(E) Anything with six sides must be a die or a cube
The lengths of two sides of a triangle are 7 and 11. Which inequality represents all possible values for x, the length of the third side of the triangle?
A C.
B D.
This is a logic problem and you need to go through all of the choices to see which one is definite: You know that all dice have six sides, and all cubes have six sides. You don’t know how many, if any, dice are also cubes. You also don’t know how many other six-sided objects exist besides dice and cubes, or how they overlap. All you can determine is that if all cubes have six sides, then something that does not have six sides cannot be a cube.
This problem is based on the Triangle inequality theorem which states that the sum of the lengths of any two sides
of a triangle must be greater than the third side. If these inequalities are NOT true, you do not have a triangle!
For this problem if we try to connect a segment of length 7 with a segment of length 11 you will get an angle y - if y becomes really really small or close to 0 then you get a picture like the one at right and the dashed line would need to be greater than 4 or you would get 7 on top of the 11 segment. If x gets really really large you get the closer picture and you would get two segments making one large segment of length 18 so the dashed line would need to be less than 18.

14. 14. Cubes one centimeter on a side are used to form a square pyramidal shape. The bottom
square on the pyramid measures six cubes on a side. The top of the pyramid shape has a
single centimeter cube. How many centimeter cubes are used to make the pyramid?
A. 81
B. 91
C. 100
D. 216
15. A circular pool with a radius of 10 feet is inscribed inside a square wall. What is the area of
the region outside the pool but inside the fence? (recall that the area of a circle )
You can definitely draw this shape to get an idea of what it looks like but
Lets think a 6x6 base uses 36 cubes then what would come next
A 5x5 base which is 25 cubes, …Keep going
4x4 is 16 cubes
3x3 is 9 cubes
2x2 is 4 cubes
1x1 is 1 cube which is 91 cubes. So aside from doing it this way – one thing to also notice is that
One of the choices is 216 which is how many cubes there would be in a 6x6x6 box so that
Is not possible.
This problem is given without choices to give you an opportunity to
Solve it independently.
First, we know – Area of a square is side x side but what is the side?
Second, we know – Area of a circle is πr2 - radius is 10 therefore area
Of the circle is 100π ft2.
To find the area outside of the pool but inside the fence we would
Need to subtract the pool from the entire area of the square.
Square area is 20x20 = 400 ft2. Why ? Because the diameter of the circle
Is the same length as the square wall.
Finally do the calculation: 400 – 100π. Most multiple choice will not be
Given in terms of π, so approximately this is 85.84ft2

15. connecting these two points cross the y-axis? A. (+1, 0) B. (0,0)
16. Triangle ABC is an equilateral triangle. The length of side AB is 40 centimeters. What is the
measure of angle B?
A. 30°
B. 40°
C. 60°
D. 90°
The points (-4, +3) and (+4, -3) are plotted on a coordinate grid. At what point would a line
connecting these two points cross the y-axis?
A. (+1, 0)
B. (0,0)
C. (0, -1)
D. (-1, 0)
This looks like a challenging problem – we are given a triangle with
a side measurement of 40 cm and we want to know the angle
measure – Oh No, is this trigonometry?
But wait – triangle ABC is an equilateral triangle – so we are all set
as long as we recall the definition – all angles are equal and they are all 60 ˚ - the answer is C
During the testing situation – you are given graph paper so you could plot these two points on an x-y graph and connect them and see where they meet the y – axis (although there is human error associated with drawing lines. Or you could look at the solutions: A and D are not possible – they are not points on the y axis. But what about B and C? – lets first look at the points given in relation to (0,0). From (-4, 3) to (0,0) you need to move down 3 and right 4 and and from (0,0) to (4, -3) you need to move down 3 and right 4 which is the same slope (rise/run). So the answer is B. Another option is to use the point slope equation form for a line and sub in the points to see if they work: (y-y1) = m (x-x1) is the equation and then
You need the slope m which you calculate by doing ( )/-4-4) = 6/8 or ¾
Then you sub in one of the points given points and one of the choices and see if you get an equality -4 – 0 = ¾ (3- -1) so is this an equation - -4 does not equal 9/4 – 1 ( which equals 5/4)

16. approximation of the length of the diagonal? A. 0.92 B. 1.00 C. 1.41
18. A student draws a square with an area of one square meter. Then, the student draws a
diagonal from one corner of the square to the other. Which of the following is the best
approximation of the length of the diagonal?
A. 0.92
B. 1.00
C. 1.41
D. 1.66
19. As a part of a manufacturing process, two spheres, each with a radius of 3 centimeters, are dipped into water. Engineers use the product of the water displaced as a part of their
calculations. What is the formula for the product of the water displaced by two identical
spheres? [Volume of a sphere= 4/3 πr3 ]
A. 9 π2 r6
B. 16/9 π2 r5
C. 16/9 π2 r6
D. 48 π2 r5
To start this problem it is actually best to draw the diagram they are
Talking about – a diagonal is defined for you. The hard part is to recognize
That they are asking about the Pythagorean theorem (side2 + side2 = hypotenuse2)
So = 2 which equal the hypotenuse2 therefore the hypotenuse equals the square root of 2 but they want the decimal approximation for that (for 7 – 12th grade – you have a calculator) (for 1st – 6th grade – you would either need to be aware of what the approximation is or you could multiply each of the choices by itself and see which got close to 2 or you could do a decimal approximation by following the directions on the next slide.
Don’t be discouraged by product of the water displaced - if you have
two spheres you multiply both of their volumes together
In this case V1*V2 so
which is answer C. This problem is really assessing ones ability to understand how
to multiply fractions and with exponents (laws of exponents).

17. How to approximate a square root?
Let’s work on the most common one
Now which means that has a value between 1 and 2 but where is it?
Well we could say that is 1/3 of the distance between the other perfect squares
So we could say that which is approximately 1.33 but when we square this we get
~ We are about a ¼ off from 2. So what if which is which is closer
To When you square this you get approximately very close to 2. But what is
a very close approximation for the

18. A. The surface area of the entire sphere is 18π.
A sphere with a volume of 36π is cut in half. One of the circles formed by the cut is shown. Which of the following is an accurate statement about this sphere if the formula for the volume of a sphere is and the formula for the area of a circle is ?
A. The surface area of the entire sphere is 18π.
B. The area of each circle is 9π
C. The radius of each circle is 6.
D. The surface area of half the sphere is 6π
21. Cutting through a cylinder could produce all of the figures below, except
A) C)
B) D) r
Sometimes it is just easier to work off of what we know – so we have some chooses – lets ee what they give us – if the area of the circle is 9π then the radius is 3 which would make the volume 36π - which is what we were given so the answer is B… we don’t even have to worry about the formula for surface area and I don’t even know it but lets say we did know S.A. = 4πr2 then we could use the volume formula to find the radius which would give us 3 and then we could plug into SA formula to get 36π which isn’t 18π
So this problem was given with the intent of answer D being correct but what if the height of the cylinder is the same as the diameter? Then you could get a square as opposed to a rectangle. The circle occurs when you are cutting parallel to the base and an oval or ellipse occurs when you are cutting on an angle.

19. 22. Use this diagram to answer the item that follows,
∆ABC and ∆DEF are congruent. Translate ∆ABC to ∆DEF. What are the coordinates at
point E?
A. (3,4)
B. (3,5)
C. (4,4)
D. (5,5)
Using the point which is not labeled in the lower left part of the triangle
We can see that the triangle translated from (-5,-2) to (2,1) so the
Translation would be right 7 and up 3 so point E would be from
(-4,2) so -4+7 = 3 and 2+3 = 5 so point E would be (3, 5) answer B

20. 24. Use these statements to answer the question:
23. The area of a garden is 12 square yards. How many square feet is that?
A. 4
B. 36
C. 72
D. 108
24. Use these statements to answer the question:
1. Some of the triangles are isosceles triangles.
2. Some of the triangles are equilateral triangles.
Which of the following conclusions is true?
A. Some of the triangles have three sides of equal length.
B. None of the triangles contains a right angle.
C. All of the triangles have three sides of equal length
D. None of the triangles has two sides of equal length.
I love problems like this - 1 square yard is the same as how many square feet?
Well there are 3feet on a side in a square yard so the area in square feet
Would be 9square feet.
So now we could do proportions (equivalent fractions)
1 sq. yd is to 9 sq. ft. as 12 sq yds is to ? sq. ft.
Therefore 12 sq. yds is the same as 108 sq. ft.
This is just a matter of making sense of
the statements and an understanding
Of the terms “some”, “isosceles”, “equilateral”
even what constitutes right triangles
The Answer is A because if some of the triangles
are equilateral then some of the triangles have
Three sides of equal length.

21. 25. What term is missing in this number pattern? 2 5 10 17 _______
25. What term is missing in this number pattern?
_______
26. These points are all on the same line, find the missing term?
27. The first three terms of a progression are 3, 6, 12, … What is the value of the tenth term?
(A) 1, (C) 188
(B) 2, (D) 1,536
What is happening as we go from the
First term to the second term to the
Third term? We should see something
In common. A pattern does not have to be
The same thing but something similar or common.
As you can see these are increasing by odd numbers
Similarly we are trying to see something similar
Between coordinates - this one is harder to see
You are actually looking for the slope between 2
Points – now the two bottom coordinates are harder
To deal with so lets just deal with the last three.
Slope is rise over run or change in the y coordinates over
Change in the x- coordinates
So it looks like the first 3 terms have the relationship of x2 but how do you figure out the tenth term? You could do x2 ten times or you can use a rule but what is the rule?
Option 1: 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536 done
Option 2: figure out the rule: 3+2(term # -1 ) does this work? Will for the third term it would be 3+2(2) no it doesn’t hmm
Term 1 is 3, term 2 is 6 term 3 is 12 T#x2+(T#-1) does this work? No - so don’t try to make a rule it is too hard but the formula is an= a1*rn-1 so 3*29 = 3*512 = 1536.

22. There are no choices given – what do you think?
28. Jim is asked to write down the mathematical expression for the words, “five more than three times a number,” Jim incorrectly answered 8x . What expression should Jim have written
down?
There are no choices given – what do you think?
Which one of the following functions has a domain of all real numbers except 2 and -2?
A B.
C D.
When you are trying to convert a word problem to a symbolic equation it is best to follow it
Word for symbol.
five more than three times a number
+ 3 x N –
To simplify this it looks like 5+3N
Domain - what are the possible x values that can make the function work? So this problem is asking which function can’t have 2 and -2
as a value for x.
In quotients you can’t divide by 0 so
the denominator can’t equal 0
So for each of these functions listed you see if you can sub in 2 and -2 in the denominator
The answer is B because subbing in 2 and -2 gives you 0 in the denominator.

23. When a question asks you for a value of x in terms
Of y – it is asking you to solve the equation for x. So
You need to get x = to
With 5x-1 = 6y you first add one to both sides to get
5x = 6y +1 Then you need to divide both sides by 5.
Remember what you do to one side of the equation
you must do to the other side to keep the equality.
Therefore the answer is
31. Three of the same item is for sale in a store. The original price of each item is $ What would the sale price of each item be after a 35 percent discount?
A. $161.85
B. $53.25
C. $53.95
D. $41.50
35% discount off from an $83 item. Something to think about when you do percents - what is 10% of 83? 1/10 of 83 or 1/10 x 83 = 8.3 and if you are doing 30% then this is 3x8.3 = 24.9 but we still have an additional 5% which is 4.15 or ½ of the 10% of 83 therefore 35% of $83 is $ since they are asking from sale price we need to subtract from which is $53.95
Another way to do this problem is to recognize that sale price is 65% of original price
So .65 x 83 =$

24. 32. The school received $5,300 to use for eight different activities
32. The school received $5,300 to use for eight different activities. A total of 91 percent of the
money was allocated for seven of the activities, with the remainder used for the school trip. How much money was used for the school trip?
A. $477
B. $663
C. $757
D. $4,823
33. After a discount of 25 percent, the savings for a pair of inline skates is $ What is the
sale price?
A. $48.00
B. $36.00
C. $25.00
D. $24.00
91% is used for seven of the activities which means 9% is used for the 8th activity – the school trip.
So .09 x 5300 = well we can just multiply this out or we could do 10% of 5300 and get 530 so the answer must be $477 or to be sure you would notice that 1% of 5300 is 53 so 530 – 53 = 477
Notice that Answer D is 91% of 5300
Original Price (OP) x 25% is the discount which is $12
OP x .25 = 12 or OP x ¼ = 12 or OP = 48 which is one of the choices – but be careful the problem is asking for the sale price which is $36 answer B.
But what if you aren’t sure how to do this problem you would get by doubling 12 but that is x2 and if you went the other way it would be ½ or 50%. $25 doesn’t seem to have a bearing on the problem except for 25% so you are really looking at A or B.

25. damaged by Hurricane Sandy. How many houses were on the beach? A. 294
34. The class kept track of rainy and sunny days. During the 54 days, classified rainy or sunny,
the ratio of rainy days to sunny days is 7 to 14. How many sunny days were there?
A. 4
B. 8
C. 18
D. 36
35. The disaster relief specialist found that 289, or 85%, of the houses on the beach had been
damaged by Hurricane Sandy. How many houses were on the beach?
A. 294
B. 332
C. 340
D. 400
The ratio of rainy to sunny is 7 to 14 so 1 to 2 this ratio can be rewritten to be sunny to all days or rainy to all days – lets look at these ratios
but and
therefore if there were 54 days in all the equivalent ratio we are looking for is
So 3x = 108 or x = 36 answer D
289 or 85% of the houses were damaged is 85% of some total number of houses. To think of this we need to take 85% of some number and get 289.
Therefore .85 x ____ = 289 you can try values from the choices. Or you could do
which becomes which is Answer C
Now why can’t you do 15% of 289 and add it to 289? Remember if you do 10% of 289 you get 28.9 and 5% is 14.5 so 15% is 43.4 add this to 289 to get 332 but then you are saying that 85% of 332 is 289? Which doesn’t work  33.2 x = 282.2

26. cost does it take to generate a profit of $5.46? A. $16.38 B. $42.00
36. Serena is an account executive. She receives a base pay of $18 an hour plus a 15 percent
bonus for all the sales she generates. Last week she generated $1,200 worth of sales. What is the minimum number of hours she could have worked to make $500?
A. 17
B. 18
C. 25
D. 26
37. The accountant calculates that it takes $3 in sales to generate $0.42 in profit. How much
cost does it take to generate a profit of $5.46?
A. $16.38
B. $42.00
C. $39.00
D. $0.76
18 x (# of hours) x (sales) = pay
So $1,200 worth of sales with a pay $500
Now 18 x (# of hours) x (1200) = 500  solve for the # of hours
18 x ___ x 1200 = 500
18 x ___ = 500
18 x ___ = 320
Number of hours is 320 divided by 18 which equals ~ 17 hours so she needs to work a minimum of 18 hours (B)
To double check 18x x 1200 = = 504
$3 gives $0.42 in profit
X gives $5.46 in profit - This is proportional reasoning
simplifies to .42x = so x = 39

27. 38. A car is set on cruise control to travel at a steady rate of 45 miles per hour. The scale on the map for the area is 1 inch = 10 miles. What is the most likely length of the line on the map for the distance the car will travel in 90 minutes?
A. 3 inches
B inches
C inches
D. 27 inches
39. An oil truck pumps oil at the rate of 100 gallons per hour. The truck filled an empty oil tank in three hours. How long will it take another oil truck pumping at a rate of 80 gallons per hour to fill the same empty oil tank?
A. 2 2/5 hours
B. 3 2/5 hours
C. 3 3/8 hours
D. 3 3/4 hours
90 minutes would be = because 1 hour is 45 miles and 30 minutes is half that which is 22.5
So if 1 inch = 10 miles. you are doing proportions
this simplifies to x = 6.75 inches
100 gallons per hour, so in 1 hour the tank is 100 gallons full in 2 hours, it is 200 gallons full, in 3 hours it is 300 gallons full
Which according to the problem is filled
So – this other truck would need to do 80 X what = 300 grams?
This is approximately 3 hours for 240 grams and we still need 60 gallons which would take ¾ of an hour – to get this we look at the 80 gallons and reduce it to parts of an hour. it takes 30 minutes to do 40 gallons and 15 minutes for 20 gallons therefore it takes 45 minutes for 60 gallons which is ¾ of an hour

28. 40. A satellite orbits Earth the same number of times each day
40. A satellite orbits Earth the same number of times each day. After five days, the satellite will have orbited Earth a total of 90 times. How many times will the satellite orbit Earth after one full seven-day week?
C. 108
B D. 92
41. An automobile dealer has to sell 3.5 cars for every 1 truck to achieve the optimum profit. This year, it is estimated that 3,500 cars will be sold. How many trucks must he sell to achieve the optimum profit? ( no choices here – what do you think?)
This is an interesting problem – but again it
Is just a proportions problem
to solve this you can reduce
5/90 to 1/18 and then x= 7x18 = 126
A and D as choices don’t make sense so you need to determine if B or C is the answer
No choices – this is to see what you might do to solve this problem
Here are some possible ways to think about it
3.5 cars for every 1 truck or 7 cars for every 2 trucks – we are dealing with equivalent ratios – we can keep on building 35 cars for every 10 trucks becomes 350 cars for every 100 trucks and then finally 3500 cars for every 1000 trucks – we don’t even need the intermediate step of 7 cars for every 2 trucks, we could have jumped straight to 35 cars for every 10 trucks
Using equivalent ratios becomes

29. to find about how far light travels in an hour?
42. The landscaper recommended a mix of 3 ½ pounds of rye grass seed with ¾ pound of blue
grass seed. If the lawn needs 5 ¼ pounds of rye grass seed, how many pounds of blue grass seed would that be?
A.  3/8 pounds
B. 1 1/8 pounds
C. 1 ½ pounds
D. 4 1/3 pounds
Light travels about 186,000 miles in a second. Which of the following choices shows how
to find about how far light travels in an hour?
A. Multiply 186,000 by 24
B. Multiply 186,000 by 60
C. Multiply 186,000 by 360
D. Multiply 186,000 by 3600
Mixture problems - these are just proportion problems again 3 ½ to ¾ is the same proportion as 5 ¼ to ? Blue grass. Now it may be hard to work with these as fractions so some may convert these to decimals but that may get messy too
Look at these equivalent fractions all that was done was halving or doubling but we still
don’t see the proportion for 5 ¼
Could we figure out the answer with just
These? YES 4 1/3 is too high – it is above when you have 14 lbs of rye grass, 1 ½ is for 7 lbs and 3/8 is less than ¾ so the answer is B. You could also solve the proportion.
So this is a problem that is asking how do you convert per second to per hour – it doesn’t matter that it is 186,000 miles, it could just as be
/second.
Note that there are 60 seconds in a minute and 60 minutes in an hour
so that is 3600 seconds in an hour and the answer is D. If you are not sure and need to use fraction multiplication to be sure then

30. 44. At sea level, sound travels about 34,000 centimeters per second, while light travels almost instantaneously. You see a lightning bolt, and five seconds later you hear the thunder clap associated with that lightning bolt. Which of the following is the best estimate of how far away the lightning bolt was using scientific notation?
A x 104 cm
B. 1.7 x 105 cm
C. 1.7 x 106 cm
D x 107 cm
45. Steven counted ten seconds between seeing lightning and hearing thunder, and he knows that the lightning was about 2 miles away. If he counted four seconds between the next flash of lightning and thunder, about how far away was the lightning – ( no choices)
I like these application problems – they say that if you see lightning you count 5 seconds and you know it is 1 mile away but the above problem is in centimeters per second so 34000x5 =
Which is 1.7 x choice A is correct also but it is not in correct scientific notation which says a number with 1 place to the left of the decimal point and the 10n tell you how many places after the decimal point.
If you are still not sure – say that sound travels at 1000 ft per second so in one second that’s 1000 ft, 2 seconds that’s 2000 ft and 5 seconds that 5000ft which is approximately a mile. A mile is 5, 280 ft. So we are doing the same calculation.
Using what we did above we know that for every 5 seconds the lightning was about a
Mile away. Or as it says above 10 seconds about 2 miles away. So here is a proportions
Problem.
therefore 10x = 8 or x = 4/5 of a miles, approximately 4000 ft.

31. Mathematics Learning and Instruction
46. Which of the following is the LEAST appropriate mathematics objective to teach using manipulative materials?
A. Adding single-digit numbers.
B. Solving problems using the strategy “make an organized list.”
C. Adding double digit numbers.
D. Dividing decimals.
                                                                                                                                                                                                       
47. A teacher is helping young students learn about counting. The teacher uses shapes as counters and makes sure students point to a shape each time the student says the next numeral. Why is he teacher using this approach?
The teacher wants to be sure the students are paying attention to what they are doing.
The teacher wants to be sure students are developing eye-hand coordination.
The teacher is going to ask the student questions about the shapes once they have finished counting.
D. The teacher wants to be sure the students are not just counting words.
Least appropriate is the one that doesn’t work best. If you are familiar with the new methods for computation – they can all be modeled using manipulatives be it unifix cubes, base 10 blocks or for dividing decimals - area models .25 divided by .04 which is asking how many 4 hundredths go into 25 hundredths. As you can see the blue shaded portion is .25 and the red portion is .04, so you are trying to determine how many times that size shape fits into the 25 hundredths. So the answer is B
Not mathematical more classroom management,
More operational
C may be a possibility but unlikely
D is the answer – students can just count number without understanding the cardinaility and/ or one to one tagging.

32. B. Trace each side of the shape with your finger.
After examining the diagram of the polygon below, the student states the perimeter is
23 units. Which of the following statements by the teacher is most likely to help the student?
A. Check your addition.
B. Trace each side of the shape with your finger.
C. Count the number of sides of the polygon.
Use the rule for finding perimeter.
49. These are examples of a student’s work
The student continues to make the same type of error. Which of the following is the student’s answer to 0.08 x 1.04? A B C D
In order to help student understand perimeter as opposed to just calculating it – a good
strategy is to make it more concrete. Say by showing that it is adding up ALL the sides of a
shape. So B. Helps the student see that they missed adding up some of the sides of this shape.
First you need to decipher what the error is for this student by doing the computations yourself – after doing the first one and then the second one you start to see that this student is not using the decimal point rule properly and looks like they are doing one less then the number of spaces so for .08 x the answer would be B .832 mostly because even as in problem 3 the student did place the decimal point properly on number 1 the student may not want a zero right after the decimal point.

33. B. Suggest the student get a ruler and measure.
After examining the illustration above, a student gave an answer of 2 for the area of the
larger polygon. How can the teacher best help the student understand the error?
a. Suggest the student trace the small square and compare that to the tracing of the larger square.
B. Suggest the student get a ruler and measure.
C. Suggest the student cut out the small square and find out how many small squares fit inside the large square.
d. Suggest the student estimate the increase in distance between BC and EF.
This is a good problem as it
gets us to think about what is
area. So if we are aware of area as
the measured space inside a shape.
C would make sense as you are filling in EFGH with the
unit square.

34. phone card, how much does a phone card cost?
51. Ms. Blau is making up a word problem for her students. She wants to write a word problem for (6 divided by 1/3). Which word problem would best help her students understand the concept of division?
Melissa has 6 pizzas and she wants to give a third of them to her friend. How much pizza will her friend get?
Davey has 6 cups of chocolate chips. He wants to bake cookies, and each batch requires 1/3 cup of chocolate chips. How many batches of cookies can Davey make if he uses all of the chocolate chips?
6 friends each have a third of a cookie. How many cookies would they have if they put them all together?
Jackie has 6 dollars. If he has a third of the amount needed to purchase a discount
phone card, how much does a phone card cost?
Again this is getting at what is division in the more conceptual sense, possibly as repeated subtraction or the inverse of multiplication, fair sharing or the measurement model. Upon reading through each choice – we can make a decision
6 pizzas and wants to give a third of the pizzas to her friend – that means take a part of
6 cups of CC each batch uses 1/3 cup so this problem is asking how many times can I make cookies 1/3 +1/3 +1/3 +1/3 - this is repeated addition but also could be thought of as how many times doe 1/3 go into 6 cups which is division. This can also be thought of as the measurement model or quotative approach to division.
Think about this as 1/3 6 times - which is multiplication
6 dollars is only a third of the cost of the phone card so we are really saying 6 = 1/3 x (cost of phone card) which is
multiplication but because the whole is unknown we need to solve this by division. It is not the best one for
Understanding division although we might use division to solve the problem.

35. 1) addition property of equality 2) commutative property of addition
52. When solving the equation 4(3x2 + 2) − 9 = 8x2 + 7, Emily wrote 4(3x2 + 2) = 8x as her first step. Which property justifies Emily's first step?
1) addition property of equality
2) commutative property of addition
3) multiplication property of equality
4) distributive property of multiplication over addition
53. If a student mistakenly states that it is most likely that the mistake results from a misunderstanding of which of the following:
A. Multiplication of fractions
B. Arithmetic of negative numbers
C. Associative property of Multiplication
D. Distributive property of multiplication over
addition
Emily’s process in simplifying the equation is to use addition of the 9 to both sides which is
known as the addition property of equality. So the answer is 1
As the teacher you need to be knowledgeable of these properties of numbers.
It appears that this student correctly multiplied -1/2 by -2/3 so A is out and knew that the product of two negative numbers becomes a positive number. This is not the associative property which says the order of doing multiplication with more than 2 number doesn’t matter so the answer is D. This student did not distribute the -1/2 to the ½ within the parantheses to get -1/4

36. Types of Problems on Multi-subject CST 5-9 and – can use approved calculator note that problems before this may appear on all levels of the CST – MS test
54. The length of the shortest side of a right triangle is 8 inches. The lengths of the other two sides are represented by consecutive odd integers. Which equation could be used to find the lengths of the other sides of the triangle? 1) 82 + (x + 1) = x2 2) x = (x + 1)2 3) 82 + (x + 2) = x2 4) x = (x + 2)2 55. Kelsey scored the following points in her first six basketball games: 22, 14, 19, 22, 8, and 17. What is the relationship between the measures of central tendency of these data? (1) mode > median > mean (2) mean > median > mode (3) median > mode > mean (4) mode > mean > median
This is the Pythagorean theorem but with words being changed to symbols or expressions – so the theorem says “the sum of the squares of the 2 legs of a right triangle (the shorter sides) is equal to the square of the hypotenuse (the longest side) - consecutive odd integers can be represented as x and x+2 therefore: (x)2 = (x+2)2
We need to be aware of what mean, median and mode are
Mean is the sum of all data points divided by the number of data points
Mode is the data value that occurs most often
Median is the middle value
So Mean is ( )/6 = 17
Mode is 22
And median is 18 which is the average of 17 and 19
Therefore Mode > Median > Mean

37. 57. If a sequence is defined recursively by f(0) = 2 and
56. ball is thrown into the air from the edge of a 48-foot-high cliff so that it eventually lands on the ground. The graph below shows the height, y, of the ball from the ground after x seconds. For which interval is the ball's height always decreasing?
1) 0 ≤ x ≤ 2.5
2) 0 < x < 5.5
3) 2.5 < x < 5.5
4) x ≥ 2
What is an interval for when the height is decreasing based on this graph – the y coordinate represents the height so it starts decreasing at 2.5 till it hits the ground at 5.5 so the answer is 3
57. If a sequence is defined recursively by f(0) = 2 and
f(n + 1) = −2f(n) + 3 for n ≥ 0, then f(2) is equal to
1) 1
2) −11
3) 5
4) 17
So recursively means that f(1) is based on what f(0) is and
f(2) is based on what f(1) was. Therefore if f(0) is 2
Then we plug in f(0) into the equation to get f(1)
-2(2) +3 =-1 = f(1) and then f(2) = -2(-1)+3 = 5 so the is 3)

38. This is an interesting probability problem looking at the container shown there are 24 possibilities. Now we need to know what a composite number is. It is a number that is not prime and is not the identity so we are talking about the number 4 in this problem. A number that has more than 2 factors. There are four 4’s in this container. Therefor the
Probability is 4 out of 24 or 1/6 the answer is D.
59. The third person (T) in line is taller than the first person. The first person (F) in a line is the
same height or smaller that the second person (S). Using the letters F, S, and T, which of the
following choices best represents the height order of these three people?
A. F > S > T
B. F ≥ S ≥ T
C. S > T >F
D. T > F ≤ S
This is a trichotomy property problem(because it is an ordering problem)
in which you have to determine which inequalities or equalities are
possible given the information. We know that T> F and we know that F is
less than or equal to S so S ≥ F So we need to figure out the relationship of T and S. F is les than both T and S but what is the relationship of T and S . If T > S then by the transitive property we get T>S ≥ F not a choice. If T < S then S > F and T, but T is greater then F so we get S > T > F which is choice C even though we are disregarding the same height mention above.

39. If a fourth student rolled the cube 75 times,
60. Byron has 72 coins in his piggy bank. The piggy bank contains only dimes and quarters. If he has $14.70 in his piggy bank, which equation can be used to determine q, the number of quarters he has?
(1) q = 72
(2) 0.10(q - 72) q = 14.70
(3) 0.10(72 - q) q = 14.70
(4) 0.10q (72 - q) = 14.70
Three students each rolled a wooden cube with faces painted red, white and blue. The color of the top faces is recorded each time the cube is rolled. The table below shows the results.
If a fourth student rolled the cube 75 times,
based on these experimental data,
approximately how many times can the cube
be expected to land with blue on top?
A B. 35
C D. 40
We are trying to coordinate the words in this
Problem to the symbols – so dimes are .10 d and
.25q but we know that there are 72 coins so if
We know d+q = 72 then d = 72 –q
There =.25q+.10(72-q)  the answer is 3
Looking at the students who already rolled we see the
Experimental probability for blue is 12/30 or 8/20 or 20/50 which is about 2/5 for each student so x/75 = 2/5 so x = 30 (C)

40. 62. When -3 - 2i is multiplied by its conjugate, the result is (1) -13
(2) 5
(3) -5
(4) 13
The table of values at right can be modeled by which equation?
1. f(x) = |x + 3| 3. f(y) = |y + 3|
2. f(x) = |x| f(y) = |y| + 3
-3-2i is a complex number
a conjugate of the complex number a+bi is a-bi
Therefore -3-2i is multiplied by -3+2i and when you distribute
Through on (-3-2i)(-3+2i) = 9-4i2
I2 = -1 which gives of 9+4 = 13 X y -2 5 -1 4 3 1 2
I would actually plot these and see what I got
Which looks like absolute value of x shifted up 3 units on the y axis
Which can be represented by f(x) = |x| + 3
If this doesn’t work – plug in values and see what works

41. 64. What is the product of the roots of 1. 3.
65.  The table below show the amount of decaying radioactive substance that remained for selected years after 1990.
Write an exponential regression equation for this set of data, rounding all values to the nearest thousandth.
So the calculator gives
Y = (0.786^X)
Using this equation, determine the amount of the substance that remained in 2002, to the nearest integer.
Since you have a graphing calculator – you could graph this and see what you get for the roots or you could factor the
quadratic and see what the roots are.
Let’s factor the quadratic – first set equal to zero
So 4x2 -5x – 3 = 0 Now there are only two factors to get 3  1 and 3 and there are 4 factors to get 4  2 and 2 or 1 and 4
(___+_____)(_____-_____) we know it is + and – because the c term is negative and now we try some values to see
(2x+1)(2x-3) the middle term when I do the distribution becomes -4x so that doesn’t work. Now try (x+1)(4x-3)  -3x +4x =x also doesn’t work (4x+1)(x-3) also doesn’t work hmm wait the product of the roots is a fraction and none of the whole numbers are working – I need to try the quadratic formula to get (5+/-sqrt(25-4*4*-3))/2(4) =(5 +/- sqrt(73)/8 and when you multiply those together you get -3/4
How do you write an exponential regression equation? You are
Expected to use the calculator for this. Plot the data and use ExpReg
On your calculator to get a, b, r2 and r and write the equation y= a*b^x
Years after 1990 (x) 2 5 9 14 17 19
Amount (y)
750
451
219 84 25 12 8
So now you plug in 12 for x in the equation above to get ~40

42. Constructed Response Examples
Paragraph 1
A strength that Student A demonstrates through discussion with the teacher is ________.
The student appears to ______________________based on (state the evidence from the artifacts).
A knowledge of (use the standard) helps Student A…. As a final piece, the student knows…. (use supporting evidence)
Paragraph 2
Student A’s ability to (describe the area of need).  
A significant area of need of student A is _______________ (use evidence from student work or discourse with the teacher). This is illustrated by the students work….
The student only showed (describe limited work in relation to what should have been shown).
Although the student has (state the strength), it is evident that (describe the need in relation to the standard).
Paragraph 3 and 4
The instructional intervention should begin with (here you describe a brief lesson plan). Then the teacher should ask…. Finally, the teacher would ask for …
The use of ______________________would help Student A develop an understanding of __________ therefore enabling him/her to build a viable argument regarding (the standard).
In describing these steps the writer should use supporting evidence of the mathematical content, standards, and pedagogy to support the idea for intervention.

43. Example 1 Evidence 1: Teacher Note
Derek is a student in a fourth-grade mathematics class. Ms. Stendel, Derek’s teacher, notices that Derek has trouble with some relatively simple computation problems. Derek is frequently distracted during class. Derek’s ability test scores and his insights and comments during class reveal that he is intelligent. When asked to do something he can complete it quickly or solve a problem mentally. He is great at solving logical thinking problems.
Evidence 2: Teacher Reflection
Ms. Stendel’s classroom observation of Derek in a mathematics group
The class was given a project to work on for thirty minutes that involves solving two mathematics problems. The first problem requires logical thinking, while the second problem involves complicated addition, subtraction, multiplication, and division. The students do not have their calculators. Derek is working in a group with four other students. The five students are seated around a table and each student has a copy of the problem to be solved. Derek seems disinterested. He suggests that the group talk about other things besides the problem in front of them.
Some of the students seem interested in his distraction, but the group decides to focus on the
first problem. The other students are talking about the first problem when Derek interrupts them
saying “I’ve got it.” After fifteen minutes the group has not solved the first problem, and Derek
will not share the solution he says he has. The group turns their attention to the second problem.
Students have sheets of plain paper and Derek begins to work on his calculations for the
second problem. I can see his numerals are hard to read and are not aligned well.
Evidence 3: Derek’s Discussion with the teacher
Ms. S.: Derek, you seemed to know the answer to that first problem. But you would not tell them what your answer was.
Derek: Let them get it.
Ms. S: But it could help them to know what your answer is, and help you all to see different ways of solving the problem to
determine the correct answer. What if your answer was not correct?
Derek: It was.
Ms. S.: You seem so sure, could you tell me what it was?
Derek: It’s 26.
Ms. S.: Oh Derek, that’s correct!

44. Example 2 Evidence 1: CCSSM
3.OA.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.
Evidence 2: Teacher’s note
I was teaching my students the multiples of 2, 3, and 5 and 10 using the hundred chart. I first had them color in the multiples of 2 and many children had no trouble seeing the pattern with the 2’s. Some students said that the numbers were all even, some students stated that they were all in lines. I decided to have the students do the 5’s and 10’s next. We came back to share our responses: Matthew said “these were easy, I just skip counted by 5 for example 5 then 10 then 11,12, 13, 14, 15, gets me to shade 15. Jasmine said “the 5 and 10’s are just like the 2’s - we shade in the columns.” I was pretty satisfied so I sent them off to work on the 3’s etc.
Evidence 3: Student Work

45. Example 3 Evidence 1 Evidence 2 Evidence 3
Disclaimer: Please note that some of the problems in this package were derived from Barrons, Regents, and various other resources from the past 15 years.  The most recent Barron’s prep book was used for one of the constructed response.