Praxis Core Academic Skills for Educators: Mathematics (5732)
NUMBER AND QUANTITY
RATIOS AND PROPORTIONS
Examples:
1. What is a ratio that is equivalent to the ratio 6 ?
,
8
,
Generally, this would be multiple-choice.
For the sake of our example, we will write several possible answers.
To write equivalent ratios, multiply the numerator AND denominator by the
same (non-zero) number.
2. What is the ratio of squares to stars?
4:2 (simplifies to 2:1)
Be sure to simplify.
 
 
OR
★
★
(simplifies to ) (Number of squares) : (Number of stars) OR
Number of squares
Number of stars
3. It takes Jon 6 minutes to eat 11 pancakes in a pancake-eating contest. How many
pancakes can he eat in 10 minutes?
11 pancakes
6 minutes
=
? pancakes
10 minutes
Set up a proportion.
11 x
=
6 10
Use “x” to represent the number of pancakes.
110 = 6x
Cross-multiply; solve the resulting equation.
x = 110/6 (which is approximately 18.3 pancakes)
4. Extra Practice: Set up a proportion and solve.
If there are 16 tablespoons in 1 cup, how many tablespoons are in 3 cups?
Fall 2014 PRAXIS I Workshop Notes, Ms. Meghan Carnes 48 1 Praxis Core Academic Skills for Educators: Mathematics (5732)
NUMBER AND QUANTITY
PERCENTS
Examples:
5. In a bag of 350 marbles, 20% are red and 50% are green. If the remaining
marbles are blue, how many blue marbles are there?
If 20% are red and 50% are green, that leaves 30% blue.
30% of 350 = .3 (350) =
105 blue marbles 6. 150 is 25% of what number?
150 = .25 (x)
x = 600
Call the number “x”. Set up the equation by identifying the part, the
whole, and the percent. Look for words like “is” and “of”.
Some problems will ask you to identify the “whole” (like this problem),
while others could ask for the “part” or the “percent”.
7. A meal costs $23 dollars before tax. If the tax rate is 9%, what is the total amount
owed?
23 + .09 (23) =
$25.07 8. A pair of shoes is marked down 60% from the original price.
The new price is $36. What was the original price?
(original price) – (60% of the original price) = 36
Call the original price “x”.
x – .6x = 36
.4x = 36
x = $90 Fall 2014 PRAXIS I Workshop Notes, Ms. Meghan Carnes 2 Praxis Core Academic Skills for Educators: Mathematics (5732)
NUMBER AND QUANTITY
MEAN, MEDIAN, MODE, RANGE
Example:
9. The scores on a math test for a class of 9 students were 72, 70, 63, 85, 59, 98, 81,
72 and 78.
59, 63, 70, 72, 72, 78, 85, 81, 98
What was the range of the scores? 98 – 59 =
39 Which is higher: the mean, the median, or the mode?
The mean.
Mean = (59 + 63 + 70 + 72 + 72 + 78 + 85 + 81 + 98) / 9 = 75.33
Median = 72
Mode = 72
If there were a 10th student whose test score was 80, would the range of the data
change? No. The highest value is still 98, and the lowest value is still 59. Range: highest number – lowest number
Mean: the average (Find the sum and divide by the number of data points.)
Mode: the number occurring most often in the list (Note: There can be more than one mode.)
Median: List the data in order, then find the middle number.
(If there are an even number of data points, average the two in the middle.)
10. There are six children on a team with ages 5, 13, 4, 1, 6, and x. If the average
(mean) of their ages is 6 years, find x.
Mean = (5 + 13 + 4 + 1 + 6 + x) / 6 = 6
(29 + x) / 6 = 6
29 + x = 36
x=7
Fall 2014 PRAXIS I Workshop Notes, Ms. Meghan Carnes 3 Praxis Core Academic Skills for Educators: Mathematics (5732)
NUMBER AND QUANTITY
FACTORS AND MULTIPLES
Examples:
11. Sarah goes running every 6 days, and Ansley goes running every 4 days. If they
ran together today, how many days will it be before they run together again?
To solve this, we must find the lowest number that both 6 and 4 divide into evenly. We call
this the least common multiple (LCM).
The list method:
Sarah runs on days 6, 12, 18, 24,…
Ansley runs on days 4, 8, 12, 16, 20, 24,…
So, the LCM is 12. The prime factorization method:
6=23
4=22
So, the LCM is 2  2  3 = 12.
So, Day 12 is the first day they will be running together.
12. Meg baked 30 chocolate chip cookies and 48 peanut butter cookies to package in
bags for her friends at school. She wants to divide the cookies into identical bags
so that each bag has the same number of each kind of cookie. If she wants each
bag to have the greatest number of cookies possible, how many plastic bags does
she need?
To solve this, we must find the greatest number that divides both 30 and 48 evenly. We call
this the greatest common divisor (GCD). Sometimes this is also called the greatest common factor
(GCF).
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 48: 1, 2, 3, 4, 6, 8,12, 16, 24, 48
Fall 2014 PRAXIS I Workshop Notes, Ms. Meghan Carnes So, GCD is 6. 4 Praxis Core Academic Skills for Educators: Mathematics (5732)
NUMBER AND QUANTITY
FRACTIONS
Examples:
13.
a.
3 7
! =
7 17
b.
3
2
÷
=
70 35
c.
3
2
3
4
!
=
!
= 70 35 70 70
3 35
! =
70 2
x y 5x 4y
d. + =
+
=
4 5 20 20
When adding or subtracting, first find a common denominator.
When multiplying or dividing it is not necessary to find a common denominator.
To divide by a fraction, multiply by its reciprocal.
1 3
2 8 8
14. Put these numbers in order: , , !1, ! , , ! , , 5
8 2
3 7 7
5 is between 2 and 3.
3 = 1 12
2
8 = 117
7
Rational numbers are numbers that can be written as fractions (ratios of integers). The decimal
form will either have a repeating pattern (ex. .33333…) or terminate (ex. 1.25).
Irrational numbers are numbers that cannot be written as quotients of integers. The decimal form
of will neither repeat nor terminate. (ex. 5, ! , e )
x
is between 4 and 5, what are possible values of x?
3
Generally, this would be a multiple choice so you would have options. For now, think about
POSSIBLE values.
15. Extra Practice: If
Fall 2014 PRAXIS I Workshop Notes, Ms. Meghan Carnes 5 Praxis Core Academic Skills for Educators: Mathematics (5732)
NUMBER AND QUANTITY
EXPONENTS AND RADICALS
Examples:
16. Simplify these expressions using the properties of exponents.
4
3
a. x (x )
b. (x 4 )3
10x 4 y
2x 3 y 2
c.
e. (2x)3 = (2x)(2x)(2x) = f. 210 ÷ 28 = 210!8 =
g.
5x 4!3 y1!2 = !10x ! 4 y
2x 3 y !2
!5x
!4!3 1!(!2)
y
d.
10x 2 y ! 5x 6 y 2
h.
5x 2 y
4 9
x y
x2 ! x2 ! y4 ! y4 ! y
=
10x 2 y 5x 6 y 2
!
5x 2 y
5x 2 y
=
17. Simplify.
a.
48x 3
12xy 2
b.
4x 2
=
y2
=
50x 3 y 4
= 25x 2 y 4 2x
=
Fall 2014 PRAXIS I Workshop Notes, Ms. Meghan Carnes 6 Praxis Core Academic Skills for Educators: Mathematics (5732)
NUMBER AND QUANTITY
COUNTING
Examples:
18. a. How many different 4-digit pin numbers are there if repetition is not allowed?
Using the Fundamental Counting Principle: 10  9  8  7 = 5040
5040 b. How many different 4-digit pin numbers are there if the first number is 3 and
repetition is allowed?
Using the Fundamental Counting Principle: 1  10  10  10 = 1000
c.
1000 Bob buys six shirts, three pairs of pants, and four ties. How many different
outfits consisting of a shirt, a pair of pants, and a tie can he create?
Using the Fundamental Counting Principle: 6  3  4 = 72
72 d. How many ways can the letters in the word “banana” be arranged?
Using the Fundamental Counting Principle: 6  5  4  3  2 1 = 6! = 720
Then, divide because of the letters that are repeated (3 As, 2 Ns):
60 6! / (3! 2!) = 720/12 = 60
19. Sofia is packing her bag for vacation. She has 6 dolls, but only 4 fit in her bag.
How many different groups of 4 dolls can she take?
Using combinations:
6!
6!
6 " 5" 4 "3" 2 "1 6 " 5
=
=
=
= 15
4!(6 ! 4)! 4!2! 4 "3" 2 "1" 2 "1
2
15 20. A club of 10 people wants to choose a board of 3 people to be the leaders of the
group (President, Vice President, and Secretary). How many ways are there to
choose the board from the 10 people?
Using permutations:
10!
10! 10 " 9 "8" 7 " 6 " 5" 4 "3" 2 "1
=
=
= 10 " 9 "8 = 720
(10 ! 3)! 7!
7 " 6 " 5" 4 "3" 2 "1
Fall 2014 PRAXIS I Workshop Notes, Ms. Meghan Carnes 720 7 Praxis Core Academic Skills for Educators: Mathematics (5732)
NUMBER AND QUANTITY
BASIC ARITHMETIC
Don’t waste too much time on the calculator if it is a simple calculation.
Examples:
21. a. 17 ÷ 7 = 2 R3 or b. 1798 ! 11 = c. 1982 !1900 =
19,778 82 22. Rewrite the expression using the commutative property of addition.
xy + 7 = 3
23. Rewrite the expression using the commutative property of multiplication.
xy + 7 = 3
24. Rewrite the expression using the associative property of addition.
(x + y) + 7 = 3
25. Rewrite the expression using the distributive property of multiplication over
addition.
5(x + y) = 3
Commutative Property:
Associative Property:
Distributive Property:
a+b=b+a
(a + b) + c = a + (b + c)
a (b+c) = ab + ac
ab=ba
(ab)c = a(bc)
____________________________________
A few other “number and quantity” topics to review:
Order of operations
Place value
Reading graphs and charts
Units of measurement
Fall 2014 PRAXIS I Workshop Notes, Ms. Meghan Carnes 8