Review Materials: Mathematics for Elementary Teachers

Review Materials: Mathematics for Elementary Teachers

Research on teacher effectiveness indicates that teaching subjects such as mathematics, social studies and science requires that teachers have

a sound understanding of the content they teach. These materials will help you review mathematical concepts and related terminology, skills and

problem solving related to key topics in the K-8 curriculum. The review will help you to be better prepared to teach math in your field placement

this term.

The materials are organized by content strands with subtopics listed for each strand. These strands are similar to those in New York City and

New York State Core Math curriculum and national guidelines such as Standards and Principles for School Mathematics (2000) by the National

Council of Teachers of Mathematics which have five content strands: Number and Operations, Geometry and Spatial Sense, Measurement,

Data Interpretation and Probability, and Algebra and Patterns. Review exercises are given for each subtopic. Some have examples. Answers are

given, sometimes with step-by-step solutions.

The review includes various types of questions ¨C multiple-choice, short answer, or long-answer. Some are straightforward. Others involve

several steps and/or ask you to explain your solution, as often required by questions on high-stakes exams, such as New York State Math tests

(grades 4 ¨C 8). Doing these review problems will also help you experience the types of questions which students are expected to answer.

Whole Number

W1

compute with whole numbers, order of operations

W2

apply operations on whole numbers to solve problems

W3

find place value of digits in whole numbers to 10-places

W4

find multiples of a number, common multiples and least common multiple

W5

find divisors of a number, common divisors, and greatest common divisor

W6

write a number as an expanded numeral and in exponential format

W7

determine whether a number is odd/even, prime/composite

W8

estimate amounts and answers to computations, round to the nearest ten, hundred, thousand

Fractions

F1

represent a fraction by a diagram, real-world situation, on the number line

F2

find equivalent fractions, simplify fractions (reduce to lowest terms)

F3

compare two fractions, order a set of fractions

F4

express an improper fraction as a mixed numeral and vice versa, represent mixed numerals

F5

compute with fractions, estimate answers for computations with fractions

F6

test if a fraction is a solution to an equation or inequality

F7

apply operations on fractions to solve problems

Decimals

D1

represent a decimal by a diagram, on a number line, or real-world situation

D2

represent a fraction as a decimal and vice versa

D3

find place value of digits in a decimal, compare decimals, order a set of decimals

D4

compute with decimals, estimate answers for computations with decimals

D5

test if a decimal is a solution to an equation or inequality

D6

apply operations on decimals to solve problems

D7

approximate the square root of a while number by a whole number or decimal with and without a calculator

Ratio and Percent, and Proportions

P1

represent a percent by a diagram, fraction, decimal

P2

compute percentages and apply them to solve problems, compute percentages using a calculator

P3

represent a ratio by a diagram or fraction, solve ratio problems

P4

similar figures and proportions

Measurement

M1

estimate and measure with a ruler length in metric system and English system units

M2

find the perimeter and area of plane figures, and volume of 3-dimensional figures

M3

solve problems involving perimeter, area, and volume

M4

estimate and measure angles in degrees, identify right, acute, and obtuse angles

Geometry

G1

identify parallel and perpendicular lines, congruent figures, line of symmetry of a figure

G2

identify shapes by their properties, draw shapes given certain properties

G3

identify 3-dimensional shapes and tell their properties

G4

Similar Triangle, Properties of Proportions, and Indirect measurement

Data Interpretation and Probability

S1

interpret graphs (bar, line, pictograph, circle, double bar), construct graphs given data

S2

find an average and solve problems involving averages

S3

find range, median, mean of a set of data

S4

determine combinations, and represent them by a list or a tree diagram

S5

find the probability of simple events, estimate empirical probability given data

Patterns and Algebra

A1

use variables to describe quantitative situations and diagrams

A2

compute with positive and negative integers

A3

evaluate expressions or formula involving variables

A4

check if a number is a solution to an equation or inequality, solve equations/inequalities

A5

find rule for patterns or IN-OUT functions

A6

use equations to represent real-world situations

A7

graph equations for lines and inequalities in the x-y coordinate plane

Page 1 of 16

ProficiencyReview.doc

Review Exercises

WHOLE NUMBERS

W1 A) Compute:

a) the product of 31 and 13

b) the sum of 3672 and 18

c) the difference of 31 and 13

e) the product of two numbers is 24. Their sum is 10. What is the difference of the two numbers?

d) the quotient of 3672 and 18

B) Compute using the rules for order of operations

A=3x4+5

B = 3 x (4+5)

C = 3 + 4 x 10

¡Â5

D=3x4+3x5

E = 44

¡Â 4 + (4 x 4 ¨C 4)

F = 5 x 3 ¨C 12 + 23

C) Which methods for ¡°thinking out computation¡± (i.e., mental arithmetic) are correct?

a) 7 + 8 Think: 7 plus 7 is 14, plus 1 more is 15

b) 38 - 17

Think: 38 minus 10 is 28, minus 7 more is 21

c) 999 x 3

Think: 1000 threes is 3000 minus 1 three is 2997

d) 150

¡Â3

Think: what equals 150 x 3, that¡¯s 450

W2 A) Jim drives 384 miles in his Subaru and uses 12 gallons of gas. How many miles per gallon did his car get for this trip?

?

B) Manny bought six box seat and four upper grandstand tickets for the Mets game against the San Francisco Giants.

If box seats cost $29 and grandstand cost $17, how much did the 10 tickets cost?

C) A box contains 60 oranges. Oranges cost 10 cents each. How many oranges can be shipped in a railroad car that holds 2500 boxes?

D) NJ Nets guard Jason Kidd, scored 34 points more than the NY Knicks rookie guard. The rookie scored 8 points fewer than his

teammate, Stephan Marbury, who scored 15 points. How many points did Jason Kidd score?

E) Charlie Brown and Lucy see 10 animals at the Bronx Zoo. Some are ostriches, others elephants. They count 14 legs on the ostriches.

How many elephants did they see?

F) The sum of two numbers is 12. Their product is 32. What are they?

G) June buys six pieces of fruit. She spends 95 cents.

She buys some of each type.

What could she have bought?

Find two different solutions.

FRUIT FOR SALE

Oranges

20?

Apple

10?

Banana

15?

W3 A) State the place value of the digits in $9,078,365,432.

a) digit 8

b) digit 4

c) digit 0

d) digit 9

e) The sum of the digits of a 3-digit number is 16. It¡¯s less than 300, and the ten¡¯s digit is 2 more than the one¡¯s digit. What is it?

B) State the value of digit ¡°4¡± in this number $4,756,890

If you add 2 and one half million dollars to this number, what is the digit in the millions place?

W4 A) Which are multiples of 6: 24, 42, 62, 600, 1,234?

B) List multiples of 3 and multiples of 4.

C) List the first three common multiples of 3 and 4.

D) What is the LCM (Least Common Multiple) of 3 and 4?

E) Find the least common multiple of 12 and 18.

F) The LCM of 12 and a number X is 60. What is X?

W5 A) List the six divisors of 12.

B) What is the least common divisor of 12 and 18? What is the greatest common divisor?

Page 2 of 16

ProficiencyReview.doc

3 x 103 + 2 x 102 + 0 x 101 + 4 x 100

W6 A) Write as a number and also in words:

B) Express 12,345,678 in exponential form as in A)

W7 A) Which are NOT primes? Explain. 2, 6, 13, 31, 15, 51, 151

B) Sometimes true, always true, never true. Explain for each.

i) The sum of any two odd numbers is odd

ii) The product of two odds is odd

iii) The sum of two primes is a prime

iv) The product of two primes is a prime

W8 A) Round to the nearest hundred: $1,534 and $48,390.

Round each to the nearest thousand.

B) Round first and then mentally estimate the closest answer.

396 X $9.95 =

about¡­¡­..

(a) $4000

(b) $390

(c) $40,000

(d) $39,000

(e) $3,600

C) Round first and then estimate this sum: 815 + 201 + 338 + 490

(a) 1700

(b) 1800

(c) 1900

(d) 2000

D) Estimate this quotient:

197

801

(a) 800

(b) 40

(c) 4

(d) 2

(e) 8

E) Andrew was just learning how to print the letter A. He liked to print it over and over again.

About how many letter A¡¯s the size of the one to the right could he print on this page? Select

one of these estimates, then figure out which estimate is the best. Explain your method.

(a) 100

(b) 400

(c) 600

(d) 1,000

(e) 2,000

F) Guess how many seconds there are in one day. Pick one of these choices.

(a) 5,000

(b) 10,000

(c) 50,000

(d) 100,000

(e) 1 million

Then figure out about how many seconds there are. Was your guess close?

AAA

FRACTIONS

F1 A) What fraction of the rectangle is shaded?

What fraction of the circle is shaded?

B) Shade in 4/6 of the bar.

Which is larger: 4/6 or 1/2?

C) Approximately what fraction of the bar is shaded?

(a)1/4

(b) 1/3

(c) 1/2

(d) 2/3

(e) 4/5

D) If

is 2/5, draw a bar to show the whole, 1.

F2 A) Circle all fractions that are equivalent to 4/5.

10/12

8/10

40/50

5/6

B) Which fraction is in simplified form?

(a) 38/28

(b) 36/48

(c) 9/14

(d) 26/39

(e) 2/200

C) X/12 reduced to lowest terms. What could X be?

(a) 2

(c) 4

(b) 3

(d) 5

52/65

(e) 6

D) Simplify 36 / 48 to lowest terms and determine if it is equivalent to 6/8.

E) Tell why 17/51 is not reduced to lowest terms. What fraction does it equal in simplified form?

Give two other fractions that are equivalent to it.

F) Is 3/4 equivalent to 15/20? Which can you imagine more easily: 3/4 or 15/20? Why?

Which makes more sense to you: ¡°simplify a fraction¡± or ¡°reduce to lowest terms¡±? Why?

F3 A) Which fraction is larger than 1/2 ?

1/10

40/100

7/9

7/19

B) The fraction X/10 is smaller than 1/2. What whole number values can X have?

C) Find equivalent fractions with the same denominator for 5/8 and 3/5 and then determine which is larger.

Page 3 of 16

ProficiencyReview.doc

D) Which set of fractions is ordered from smallest to largest?

Set I: 1/2

2/3

3/5

Set II: 3 1/5

(a) Only Set I

(b) Only Set II (c) Both sets I and II

3 2/4

4 1/8

(d) Neither sets

F4. A) Write a mixed numeral for 41/10 and for 80/7.

B) Write improper fractions for these mixed numerals: 3 4/5

F5 A) Estimate the answer without doing the actual computation:

(1) 4 9/10 + 2 8/10 =

a) 6

b) 7

c) 8

d) 9

(2) 100 1/12 ¨C 4 9/10 = a) 90

b) 95

c) 96

d) 105

(3) 3 1/10 x 220 1/3 =

a) 260 b) 660 c) 223 d) 110

(4) 24 1/13

¡Â1

9/11 =

a) 10

b) 12

c) 24

d) 48

and

8 3/10

e)10

e) 2230

e) 480

B) Compute answers:

(a) 2/8 + 4/8 =

(b) 2/3 ¨C 1/2 =

(e) 2 1/2 x 3 1/2 =

(f) 2 2/3

(i) (2/3) 2 + 5/9 =

¡Â1 1/3 =

(j) ( 1 1/3) 2 + 1/3 =

¡Â 1/6 =

(c) 2/3 x 4/5 =

(d) 2/3

(g) 3 2/3

(h) 14 2/6

+ 6 4/6

- 2 1/2

(k) (5 1/4 - 2 3/5) x 20 =

F6 The replacement set for the open sentences here is the set of rational numbers or fractions.

A) Is 3/4 a solution for this equation? 2 1/4 + Z = 3

B) Find W where 3/5 ? W = 18/20

C) Is 2/3 a solution for this inequality? X + 1/12 > 10/12

Tomato

Soup

10 3/4 oz

Mushroom

Soup

10 1/2 oz

D) Is 2/3 a solution for X2 + X ¨C 1 = 0 ? Explain.

E) What is the total weight of the contents in T cans of tomato soup and M cans of mushroom soup?

(c) 10 3/4 ? T + 10 1/2 ? M

(d) T ? M

(a) T + M

(b) (T + M) (10 3/4 + 10 1/2)

F) Which equation does not have a rational number solution?

(a) 2Z = 1

(b) 10Y =2

(c) W2 = 2

(d) 2R = 1/2

F7 A) Irene likes to bike to Brooklyn College from her apartment. She travels 3 6/10 miles one way.

How far does she travel biking to and from Brooklyn College in all, Monday through Friday?

?

B) Lucy loves licorice. She buys a long piece of it 3 1/2 feet long. She munches up 1 2/5 feet of it after lunch. How much does she still

have that she can eat on her way home from school?

C) Ana uses the formula F = 9/5C + 32 to convert Celsius degrees into Fahrenheit. What is F if C = 0? If C = 100?

D) Happy Howard ate 2/3 of a SuperBurger that weighed 15 ounces. How many ounces does he have left to eat?

E) The area of a rectangular fish pond is 15 1/6 square meters. The width is 3 1/2 meters. What is the length of this pond?

F) The area rule for a circle is A = ¡Ç R2 where we can use 22/7 as an approximate value for ¡Ç. What is the approximate area of a circular

swimming pool with a radius of 10 yards? (a) 30 sq. yds.

(b) 300 sq. yds. (c) 60 sq. yds.

(d) 600 sq. yds.

G) IBM stock has been up and down this week. Today it rose $5 1/4 to 95.25. What was the price of IBM at the start of the day?

Page 4 of 16

ProficiencyReview.doc

DECIMALS

D1 A) Write a decimal for the part of the rectangle that is shaded.

B) Which is the best estimate for the how much of the box is shaded? Why?

(a) .48

(b) .62

(c) .81

(d) .93

C) Find the letters that correspond to these decimals and fractions.

.6 _______

4/10 _______

1.5 _______

0

O

0.1 _______

1.7 _______

1

A

B

C

D

E

F

G

H

I

2

J

K

L

M

N

O

D) What letters on the number line above best match these decimals?

.42 _______

0.09 _______

1.91 _______

1.599 _______

D2 A) Write as decimals and also write in words:

B) Write as a fraction:

.75

4/10

.005

Q

R

S

T

U

V

W

X

.40099 _______

5 32/100

32/1000

13/10000

.875

C) Write as a fraction reduced to lowest terms:

.20

D) Write as a decimals:

42/50

4/5

P

15/20

E) Use a calculator. Write as repeating decimals:

.375

2.5

42/500

2/3

7/9

F) What pattern do you notice in changing fractions of the form XY/99

12/99

1/7

36/99

(eg., 12/99) to a decimal? Explain.

D3 A) What is the place value of these digits in the decimal: 123.456709

digit 5: _______ digit 6: _______ digit 2: _______ digit 9: _______

B) Round to the nearest tenth:

0.18 _______

3.79 _______

23.092 _______

C) Round to the nearest hundredth:

.908 _______

3.792 _______

23.0978 _______

D) Which decimal is smaller?

.05 or .1

.123 or .08

E) Order these decimals from smallest to largest.

.6

.34

1.2

.012

1.0000 .09

.19 or 1.2

.04 or .200

1.01

D4 A) Compute (if possible, in your head; check with a calculator if you wish).

(a) .3 + .6 =

(b) .34 + .51 =

(c) .882 - .007 =

(d) .34 + .7 =

(g) 4 x .5 =

(m) .4

(h) .2 x .45 =

¡Â2=

(n) .4

¡Â .02 =

B) Example: estimate this quotient

Estimate each answer:

(1) 3.98 x 7.1234 =

(2) 1.9 + 2.01 + 3.701 + 4.0 =

(3) 4.008

40.998

(4) 9.7 + 10.002 x 1.8

(i) 3.6

¡Â3=

(o) .3 + .2 x .1 =

2.98

21.0101

¡Â .3 =

(p) .6 ¡Â .2 - .1 =

(j) .9

Think: this is about

3

a) 10

a) 10

b)11

b) 11

c) 21

c) 12

d) 28

d) 13

e) 32

e) 14

a) 1

b) 9

c) 10

d) 11

e) 40

a)20

b) 30

c) 50

d) 180

21

(e) 4.84 ¨C 1.2 =

(f) .3 x .2 =

(k) .2 x .3 x .4 =

(l) (.11)2 =

so the answer is about 7

D5 A) Which decimal is NOT a solution for the inequality X + .4 < 4.2 ?

(a) .2 (b) 4.6 (c) 3.6 (d) 1.00

B) Is .5 a solution for X2 + X + .25 = 1 ?

D6 A) A piece of paper is about .03 cm thick. How many centimeters tall is a stack of 1000 sheets?

Page 5 of 16

ProficiencyReview.doc

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