Problem Solving Techniques!!
Start out simple...
Family Fun - From Problem Solving Connections
1. Four families each brought the same number of chairs to a block party. Three more chairs
are needed to seat all 27 of the participants. How many chairs did each family bring?
Consecutive Numbers
2. An example of consecutive odd numbers is 23, 25, 27, and 29. Find four consecutive odd
numbers with a sum of 160. Show your work.
Opening Day
3. It is the grand opening of a local supermarket. Every 5th customer will receive a coupon
for a free turkey and every 7th customer will receive a coupon for a free half gallon of ice
cream. If 400 customers come in on opening day, how many will get a free turkey and a
free half gallon of ice
cream?
Kennel Klub
4. Four dozen dogs live in 6 different colored kennels. The smallest kennel has 6 dogs and the
orange kennel is the largest with 10 canines. The yellow kennel and the green kennel are the
only ones with the same number of dogs. The 13 youngest pups are in the red and blue
kennels with the least number of dogs. The purple kennel has 2 more dogs than the blue
kennel. How many dogs are in each kennel? Make a chart.
This will really challenge you...
Car Rental
5. Sam needs to rent a car for his upcoming trip. CheapWheels charges $20.25 per day plus
$.14 a mile. Easy Rider charges $18.25 a day plus $.22 a mile. San plans to do a lot of driving
on his 3-day trip. Which company should Sam go with? Explain your choice.
Does the difference in cost go up or down as mileage increases? Support your answer.
Measurement
Start out simple...
Flooring
6. Mr. and Mrs. Corkwood were going to tile their den floor. They had tiled the kitchen floor
earlier in the year with ceramic tile. The kitchen was 14ft by 10 feet. The den is 20 ft. by 14
ft. The kitchen tile job cost the Corkwoods $1400 including labor. How will this help them
determine the price of the den floor?
Now try to work this out...
Squares
7. Look at this picture of squares. The area of square F is 16 square units. The area of square
B is 25 square units. The area of square H is 25 square units. Find the area for square D and
square E. Explain how you got your answer.
This will really challenge you...
Try It
8. A rectangular sheet of wood has 4 small squares removed. It is then cut to make a box that
is 5 centimeters by 4 centimeters with a volume of 60 cubic inches. (Four pieces of wood size
A-4 are removed.) Find the area of the original sheet of wood.
Geometry
Start out simple...
The Shape of Things
9. Look around your classroom and find as many different shapes as you can. Make a tally to
help you determine which shapes are used most often. Why do you think this is so?
Now try to work this out...
Map It
10. Willy Wonka went to Kandy Korners to see his friend. His friend told him how the town
was set up and where to find the candy factory.
•
•
•
•
Sweet Street, Rocky Road, and Almond Avenue are all parallel.
Lollipop Lane is perpendicular to Rocky Road and ends at Sweet Street.
Chocolate Boulevard is diagonal to Almond Avenue.
The candy factory is bordered by Sweet Street, Lollipop Lane, Sour Patch Place, and
Chocolate Boulevard.
Draw a map and put in the candy factory.
This will really challenge you...
11. Look at this shape. How would you go about finding the total area of this shape? Show
how you would do this and then find the total area.
Patterns, Algebra, And Functions
Start out simple...
Rafting - From Problem Solving Connections
12. You are pulling a raft up a river. You can make 15 miles a day. At night, while you are
asleep, however, the current pushes the raft 3 miles back downstream. If you set out on
Tuesday to get to a town 85 miles upstream, on which day will you arrive?
Now try to work this out...
Stepping Up
13. Study this picture. How many blocks would you need for a 20-step staircase? Set up an
algebraic equation to solve.
This will really challenge you...
Wacky Walter
14. Walter and his parents are a little unusual. If he does an acceptable job of doing his
chores, he gets paid $3.33 for that day. If he does an outstanding job, he gets $3 more. During
one 10 day period, Walter received $42.30 for his work. How many days did Walter earn the
extra money?
Data, Statistics, And Probability
Start out simple...
Happy Halloween
15. Pete went out for Halloween. He got tons of candy. He put all of his candy in a shopping
bag. He got 80 different things. 30 were chocolate, 20 were gum, 25 were plain, and 5 were
cookies. Make a circle graph to display his loot.
Now try to work this out...
Mean or Median
16. Sam made the following scores on unit tests for the term:
92, 98, 15, 92, 87, 92
Sam's teacher said that his grade would be based on the mean of his grades. Sam argued that
his grade should be based on the median score of his grades. Find the mean and the median of
Sam's grades. Which do you think best reflects Sam's work for the term? Explain your
answer.
This will really challenge you...
Perfection
17. From 11 positive integer scores on a 10-point quiz, the mean is 8, the median is 8, and the
mode is 7. Find the maximum number of perfect scores possible on this test.
Grab Bag
Be Careful
Tony's father is a driver for Brank's Armoured Truck Company. They take money from
businesses to the bank. His last stop is at the largest hotel in the city where he collects a large
sum of money. A map of part of the city showing the location of the hotel and the bank looks
like this:
If Tony's father must take a different route every day to prevent a robbery attempt, how many
different pathsdown and to the left can he take from the hotel to the bank?
Search
December Solutions
Number Theory | Measurement | Geometry |
Patterns, Algebra, and Functions | Data, Statistics, and Probability |
Grab Bag
Remember: These are open-ended problems. There could be other solutions
than the ones we are giving.
Number Theory
1. Each family brought 6 chairs.
2. 37, 39, 41, 43
3. 400/35 = 11 times
4. Orange = 10 dogs, yellow = 8, green =8, red = 6, blue = 7, purple = 9
5. Sam should go with CheapWheels. The difference increases with mileage.
Measurement
6. The area of the den is double that of the kitchen so the cost should double.
7. Square E - If B is 5 by 5, and F is 4 by 4, then E is 1 by 1.
Square D - If B is 5 by 5 and E is 1 by 1, then C is 6 by 6 and D is C + E.
D = 7 by 7 = 49 square units
8. L x W x H = V
5 x 4 x H = 60
H = 60/20 H = 3 so each square is 3 x 3. Original dimensions are l = 11 and w= 10 so
original area is 110 cm squared.
Geometry
9. Answers will vary.
10. Several possible answers.
11. In order to do this problem, you need to divide the shape into smaller shapes. You will
have a large rectangle 35 x 26; a rectangle 35 x 4; 2 rectangles 4 x 26 and 2 right triangles
4 x 26. The total area is 1570 cm squared.
Patterns, Algebra, And Functions
12. Tuesday of next week.
13. 210 blocks
14. 3 outstanding days
Data, Statistics, And Probability
15. Check graphs. Gum is 1/4 of the circle, Plain is about 1/3, Chocolate is a little more
than 1/3 and Cookies are what is left.
16. Mean = 79.3; Median = 92. Decide if student's logic makes sense but should reflect
the idea that all of Samantha's grades were terrific except 1. Therefore, that grade should
not count as heavily.
17. Two
Grab Bag
18. Answers will vary. Check diagrams.
HOW DID YOU DO?
December Problems
Back to Open-Ended Math Index
Mean/Average:
1. Chandra is playing in a tennis doubles tournament. The rules
say that the average age of the pair of players on each side
must be ten years old or younger. Chandra is eight years old.
Her partner must be _____ years old or younger.
2. Jamyang took five spelling tests in the last marking period.
He scored 100% in all but one. His lowest score was 80%. What was his
mean score for the spelling tests in the last marking period?
3. Lhatu bought seven pens. Four of the pens cost a dollar each. Three of
the pens cost 30 cents each. What was the average cost of each pen?
Mean and Median
The mean of a set of numbers is the average. You find the average by
adding all the numbers and dividing the sum by the amount of numbers you
added.
The median of a set of numbers is the middle number. If you list all the
numbers in order, the median is in the middle of the list. If it’s an even
amount of numbers, average the two numbers in the middle.
Examples:
These are Ryan’s spelling test scores:
38, 75, 20, 90, 77
To find the mean (average) score, first you add the scores.
38 + 75 + 20 + 90 + 77 = 300
There are 5 scores, so divide 300 by 5. 300 divided by 5 = 60.
Ryan’s mean score is 60.
To find the median score, put the scores in order.
20,38, 75, 77, 90
or
90, 77, 75, 38, 20
The middle score is 75, so Ryan’s median score is 75.
Mean and Median
Find the mean and median of the following sets of numbers.
1.
8, 400, 12
The mean is ________. The median is ________.
2.
10, 6, 8
The mean is ________. The median is ________.
3.
5, 31, 88, 31, 100
The mean is ________. The median is ________.
4.
10, 20,30, 40, 50, 60
The mean is ________. The median is ________.
The Problem
1.
2.
3.
4.
How many 2-digit numbers are there that contain at least one 2?
How many 2-digit numbers are there that contain at least no 2 at all?
How many 3-digit numbers are there that contain at least one 3?
How many 5-digit numbers are there that contain at least one 5?
1.
2.
3.
4.
5.
6.
7.
Teaching sequence
Introduce the problem by looking at a number, say 1676, and brainstorming all the "features"
of the number. (Its even, divisible by 4, divisible by 6, 4 digits, a 6 in the ones place etc)
Pose the first part of the problem for the students to solve.
Check solutions and approaches used. For this part many may have listed possibilities. Ask if
there is another way to have found the 18 solutions.
Pose the rest of the problem for the students to solve in pairs.
As the students work ask questions that focus on the rules of divisibility and the way they are
"thinking" about the numbers.
Ask the students to record their solutions for each of the parts to share with the rest of the
class.
Share solutions. Encourage the students to reflect on the approaches used by others. Which
ones can they follow? Which do they think are more efficient than the approach they used?
Extension to the problem
How many r-digit even numbers are there?
Solution
1. The 2-digit numbers that contain 2 can be produced by listing them systematically. They are
12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82, 92. There are 18 of them.
Is there another way to do this though? After all, if we were asked to find the number of 7digit numbers that contained 2, we would have to produce a very long list.
2. What about the 2-digit numbers that contain no 2 at all? Now you might think of using a list
to get started here. So we would have 10, 11, 13, 14, 15, 16, 17, 18, 19, 30, 31, 33, 34, 35, 36,
37, 38, 39, 40, 41, 43, … Hang on! First this is getting to be a very long list and second there
is a pattern from here on that will stop us having to write all these numbers down.
Look, there are 9 numbers that start with a 1; there are 9 numbers that start with a 3; there are
…; there are 9 numbers that start with a 9. So we have 9 times 8 numbers altogether. The 8
comes from the fact that there are 8 numbers in the sequence 1, 3, 4, 5, …, 9. That means that
there are 72 numbers that don’t have a 2 in them.
Now think!!
What have we found so far? There are 18 2-digit numbers that have a 2 and there are 72 2digit numbers that don’t. 18 + 72 = 90. Does that ring any bells? Surely there are 90 2-digit
numbers altogether? So we were wasting our time when we started listing and then counting,
the 2-digit numbers without a 2. We could have just taken 18 from 90 and made our life
easier!
3. With the 3-digit numbers we could make a list but it’s clearly going to be more difficult to be
sure we haven’t missed anything. So we should start to think ‘sneaky’. Would it be easier, for
instance to count all 3-digit numbers that didn’t contain a 3? (There’s a hint from (b).)
OK then, we’ll first of all count all 3-digit numbers, then count all 3-digit numbers with no
threes, then subtract the second number from the first.
Right, first the 3-digit numbers. Well, first of all there are 9 possible digits for the first place
(you can’t use 0), 10 for the second place (you can use anything from 0 to 9) and 10 for the
third. That makes 9 x 10 x 10 = 900.
Now for the 3-digit numbers with no threes. Their first (hundreds) digit can be chosen in just
8 ways (no 0 and no 3), their second digit in just 9 (no 3 remember), and their third digit in 9
ways. So there are 8 x 9 x 9 of these. That’s 648 altogether.
So the number of 3-digit numbers with at least one three is 900 - 648 = 252. (That would
have been a long list!)
4. Obviously the same trick can be performed with the 5-digit numbers. So we get 9 x 10 x 10 x
10 x 10 – 8 x 9 x 9 x 9 x 9 = 90000 – 52488 = 37512. (This is a frighteningly long list. How
long would it take to write this list down?)
Solution to the extension
There is obviously going to be a trick to this, as we couldn’t possibly write down a list here.
(Though if you are stuck at this point, then writing down the list for r = 2 and r = 3 might give
you some inspiration.)
When is a number even? Well it is even if it ends in 0, 2, 4, 6, or 8. This means that we have
a choice of 5 numbers for the last digit. Once again we have a choice of 9 for the first digit,
then 10 for the next digit, then 10 for the next, … , and then 5 for the last digit. So we have to
multiply together one 9, one 5 and r – 2 10s. This gives 5 x 9 x 10r-2. That’s 45 with r – 2
zeros.
WORD PROBLEMS ( MONEY ) – 1
1) £ 3.55 + £ 6.94 (£ )
2) 78P + 98p (£ )
3) What is the total cost of five books at £ 3.99 each? (£ )
4) Jim has £ 2.50. He wants to buy a book that costs
£ 5.99. How much more does he need?
5) Sue had £ 25.78. She spent £ 6.12 on her shopping. How
much did she have left?
6) What is the total of £ 2.59, £ 4.56 and 89p?
7) If Paula saves £ 1.00 a week, how many weeks will she
have to save for to buy a pair of trainers that cost £ 45?
8) How much do 4 toy cars cost altogether if they cost
£ 3.45 each?
9) How much change will I have from £ 5.00 if I spend
£ 1.35?
10) Hazel saves 20p per week. How many weeks will she
have to save to buy a magazine that costs £ 2.50?
11) A new kettle costs £ 30.00, how much more money do I
need to find if I only have £ 12.85?
12) A pair of trousers cost £ 20.00, a top costs £ 14.00
and a pair of shoes cost £ 25.00. How much money do I
need to buy all the items?
13) The book that I like is priced at £ 6, how much do I
need to buy the book if there is a 1/3 off?
14) If all 32 children in the class had 3 stars, how many
stars would there be altogether?
15) What would you pay in total if you paid £ 2.67 over 3
weeks?
C. O’Neill.
WORD PROBLEMS ( MONEY ) – 2
1) £ 32.55 + £ 61.94 (£ )
2) 78P + 98p +45p (£ )
3) What is the total cost of five books at £ 6.99 each? (£ )
4) Jim has £ 1.24. He wants to buy a book that costs
£ 5.99. How much more does he need?
5) Sue had £ 67.78. She spent £ 56.87 on her shopping.
How much did she have left?
6) What is the total of £ 27.59, £ 4.56 and 89p?
7) If Paula saves £ 25 a week, how many weeks will she
have to save for to buy a pair of trainers that cost £ 267?
8) How much do 4 toy cars cost altogether if they cost
£ 3.89 each?
9) How much change will I have from £ 20.00 if I spend
£ 13.99?
10) Hazel saves 12p per week. How many weeks will she
have to save to buy a magazine that costs £ 5.00?
11) A new kettle costs £ 29.00, how much more money do I
need to find if I only have £ 12.85?
12) A pair of trousers cost £ 20.99, a top costs £ 14.50
and a pair of shoes cost £ 25.99. How much money do I
need to buy all the items?
13) The book that I like is priced at £ 18, how much do I
need to buy the book if there is a 2/3 off?
14) If all 296 children in the school had 3 stars, how many
stars would there be altogether?
15) What would you pay in total if you paid £ 2.67 over 9
weeks?
C. O’Neill.
....
WALT: Make number stories to represent sums.
You are writing a book on word problems. You have the sums but
you now need to make sentences to go with them.
Example:
Sum; 6.83 x 27 = 184.41
Sentence; 27 CDs at £6.83 each will cost £184.41
Now your turn: (tip; don’t always use CD’s!)
143.5 + 32.45 = 175.95
57.2 – 2.56 = 54.64
448.91 ÷ 53 = 8.47
377 x 58 = 21,866
26 x 3.5 = 91
689 ÷ 137.8 = 5
Maths Challenge
Can you work out which operation (+, – , ÷, x) is missing from
each sum?
? 377 58 = 435
? 377 58 = 6.5
? 377 58 = 319
? 377 58 = 21,866
WALT: Make number stories to represent sums.
You are writing a book on word problems. You have the sums but
you now need to make sentences to go with them.
Example:
Sum; 564 ÷ 8 = 70.5
Sentence; If you cut 8 equal pieces from 564mm of string, each
piece will be 70.5 mm long.
Now your turn: (tip; don’t always use string!)
1435 + 3245 = 4680
38.7 x 24 = 928.8
572 – 25 = 547
25.3 x 84 = 2125.2
26 x 3.5 = 91
689 ÷ 137.8 = 5
Maths Challenge
Can you work out which operation (+, – , ÷, x) is missing from
each sum?
? 319 274 = 593
? 18 6 = 108
? 572 291 = 281
? 228 38 = 6
WALT: Make number stories to represent sums.
You are writing a book on word problems. You have the sums but
you now need to make sentences to go with them.
Example:
Sum; 90 x 4 = 360
Sentence; Four portions of fries at 90p each would cost £3.60 or
360p.
Now your turn: (tip; don’t always use fries!)
435 + 245 = 680
72 – 25 = 47
93 ÷ 3 = 31
23 x 4 = 92
532 + 247 = 779
Maths Challenge
Can you work out which operation (+, – , ÷, x) is missing from
each sum?
? 19 21 = 40
? 72 29 = 43
? 80 6 = 480
? 28 2 = 14
90p 90p 90p 90p
######
Word Problems worksheet 1
1. John goes to the supermarket and spends 15 dhs on sweets, 13 dhs on
fruit and 28 dhs on meat. He took 100 dhs with him. How much change
does he have?
2. Susan spends 68 dhs at Spinneys and 103 dhs at the Megamall. How
much has she spent altogether?
3. Colin goes to the cinema, a ticket costs 35 dhs. He also wants to buy
popcorn which costs 19 dhs, but he only has 38 dhs for both. How
much more money does he need?
15 dhs
28 dhs
13 dhs
100 dhs
68 dhs 103 dhs
35 dhs
19 dhs
38 dhs
NAME:_______________________
Word Problems 2
(PLEASE SHOW ALL WORKING OUT)
1a. Mohammed wanted to play football but he needed 11 people to play on his
team. How many more people are needed?
b. Mohammed wants to enter his team in a competition. In the competition
there are 8 teams. How many people will be playing altogether?
2. There are 45 people in a crowd and they needed to be separated into 9
rows. How many people will there be in each row?
Mohammed’s
team
Crowd of 45
people
1
2
3
4
5
6
7
8
9
How many in each row?
3a. 100 people go to a Blue concert but only half can get in. How many are
allowed in?
b. Half of the people that went to the concert go home by taxi, the other
half walked home. How many went home by taxi?
4. Mary – Jane went to Germany by plane it took 7 hours. It took her half
that time to travel to England after Germany. How much time as she
traveled altogether.
5. How many fingers are there on 6 hands (not including thumbs)?
½ of
100
½ by taxi
½ walk
7 hours ½ of 7 hour
#####
WALT: Solve problems with numbers.
Use a calculator to solve the following problems.
Find two consecutive numbers with a product of;
1) 1332
2) 702
3) 2652
4) 1482
5) 7310
6) 9120
Maths Challenge
Each represents 1, 2, 3, 4, 5, or 6
Use each of the digits 1 to 6 once.
Replace each to make the correct sum.
X=
WALT: Solve problems with numbers.
Use a calculator to solve the following problems.
Find two consecutive numbers with a product of;
1) 182
2) 210
3) 272
4) 462
5) 506
6) 1332
Maths Challenge
Each represents a missing digit.
Replace each to make the correct sum.
X 6 = 6272
WALT: Solve problems with numbers.
Use a calculator to solve the following problems.
Find a pair of numbers with a product of;
1) 54
2) 72
3) 99
4) 121
5) 252
6) 420
Maths Challenge
Each represents a missing digit.
Replace each to make the correct sum.
– = 38
&&&&&
WORD PROBLEMS.
1) There are 1286 children in a secondary school. 249 of them go to a
power station as a school trip. How many children are left at school?
2) 356 cars travelled through Yarm High Street in one hour on a Monday
morning. 12 were grey, 34 were red and 19 were yellow. How many cars
were other colours?
3) John set out 173 chairs in a hall for a meeting. Only 68 people went.
How many empty chairs were there?
4) Sandra is busy reading the entire works of William Shakespeare. She
has read 2671 pages. The book has 4992 pages. How many more pages
does she have to read to reach the end?
5) Eaglescliffe to Worcester takes about 3 and a half hours in a car. The
journey is 173 miles long. If a car drives 53 miles in the first hour and 61
miles in the second hour, how much further does it have to drive to get to
Worcester?
6) Pete bought a pack of 60 biscuits on Sunday. On Monday he ate half of
them. On Tuesday he ate 19. How many biscuits did he have left for
Wednesday?
7) 4385 fleas were hopping around Fido. Fido’s owner got him a flea
collar, which killed 1749 of them. This still left how many fleas?
8) 2673 people went to watch Fenton F.C. The week after, 3754 people
went to watch their next match. What was the difference in crowd
attendance?
9) Ravi had to write an essay of 6000 words. In the first week he wrote
657 words. In the second week he wrote 493 words and in the third week
he wrote 1183 words. How many more words does he need to write?
&&&&&&&
WALT: Solve problems involving money.
You have gone shopping for new clothes. You need to add up different
items to see if you can afford them before you get to the till.
? How much would it cost you to buy a baseball hat, scarf and socks?
? How much would the doggie slippers, top hat and scarf cost
altogether?
? You have £115. Can you afford to buy the top hat and the shirt?
How much change would you get?
? Your mum has given you £50 to buy your sisters birthday present.
Can you afford to buy her the doggie slippers and the scarf?
How much more would you need?
On your way home mum has asked you to go to the supermarket and
buy
food for dinner.
? She has given you £25. You need one ham, one lettuce, one bag of
mushrooms and three ice-cream sundaes. Can you afford to buy them
all? How much change would you get?
Maths Challenge
Use a calculator or written method to solve this problem.
? 4030 people go to a football match. Each ticket costs £4.25.
What is the total cost of all the tickets?
£58.47
£23.19
£32.73
£29.46
£65.82
£14.79
47p
£10.97
£1.73 £2.84
each
WALT: Solve problems involving money.
You have gone shopping for new clothes. You need to add up different
items to see if you can afford them before you get to the till.
? How much would it cost you to buy a baseball hat, scarf and socks?
? How much would the doggie slippers, top hat and scarf cost
altogether?
? You have £40. Can you afford to buy the top hat and the shirt?
How much change would you get?
? Your mum has given you £20 to buy your sisters birthday present.
Can you afford to buy her the doggie slippers and the scarf?
How much more would you need?
On your way home mum has asked you to go to the supermarket and
buy
food for dinner.
? She has given you £15. You need one ham, one lettuce, one bag of
mushrooms and two ice-cream sundaes. Can you afford to buy them
all? How much change would you get?
Maths Challenge
Use a calculator or written method to solve this problem.
? 403 people go to a football match. Each ticket costs £4.
What is the total cost of all the tickets?
£18.47
£13.19
£22.73 £19.46
£15.82
£4.79
47p
£6.97
£1.73 £2.84
each
WALT: Solve problems involving money.
You have gone shopping for new clothes. You need to add up different
items to see if you can afford them before you get to the till.
? How much would it cost you to buy a baseball hat, scarf and socks?
? How much would the doggie slippers, top hat and scarf cost
altogether?
? You have £20. Can you afford to buy the top hat and the shirt?
How much change would you get?
? Your mum has given you £15 to buy your sisters birthday present.
Can you afford to buy her the doggie slippers and the scarf?
On your way home mum has asked you to go to the supermarket and
buy food for dinner.
? She has given you £15. You need one ham, one lettuce, one bag of
mushrooms and two ice-cream sundaes.
Can you afford to buy them all?
How much change would you get?
£8.47
£3.15
£2.75
£9.46 £5.82
£2.14
27p
£2.97
£1.15 £2.25
Each
######
The table below shows the distribution of students’ major areas of study in a particular
college. Which of the following pie charts could be used to represent this data?
Example 2: Daniel has recorded the following data for his basketball team.
Player
Number of Shots Made
Number of Shots Attempted
Sarah
3
5
Jill
8
9
Maria
2
3
Brook
1
2
For each player, make a fraction by putting the number of shots made over the number
of shots attempted. Find the player with the best shooting record by ordering these
fractions from greatest to least. a) Sarah b) Jill c) Maria d) Brook
OGET TEACHER TEST PREP SEMINAR – NORTHERN OKLAHOMA COLLEGE
MATH COMPETENCIES
Example 3: The following graph shows the distribution of test scores on an Algebra
Exam. Which of the following statements do you know to be true? I. The majority of
students scored higher than 60? II. The test was a fair measure of ability. III. The
mean score is probably higher than the median. IV. The test divided the class into
distinct groups. a) I and II only b) I and IV only c) I, III, and IV only d) IV only
Example 4: The distribution of a high school chorus is depicted in the graph below.
There is a total of 132 students in the chorus. Which of the following expression
represents the percentage of freshman and sophomore girls in chorus? a) 2115132+ x
100 b) 2115132++ 100 c) 2115132+ d) 2115132+ x 132
OGET TEACHER TEST PREP SEMINAR – NORTHERN OKLAHOMA COLLEGE
MATH COMPETENCIES
Example 5: Using the line of best fit shown on the scatterplot below, which of the
following best approximates the rental cost per video to rent 300 videos? a) $3.00 b)
$2.50 c)$2.00 d)$1.50 Example 6: The table below shows the number of visitors to a
natural history museum during a 4-day period.
Day
Number of Visitors
Friday
597
Saturday
1115
Sunday
1346
Monday
365
Which expression would give the BEST estimate of the total number of visitors during
this period? a) 500 + 1100 + 1300 + 300 b) 600 + 1100 + 1300 + 100 c) 600 +1100 + 1300
+ 400 d) 600 + 1100 + 1400 + 400 APPLY MATHEMATICAL REASONING SKILLS TO
ANALYZE PATTERNS AND SOLVE PROBLEMS - 0010
• Draw conclusions using inductive reasoning and deductive reasoning.
OGET TEACHER TEST PREP SEMINAR – NORTHERN OKLAHOMA COLLEGE
MATH COMPETENCIES
Example 7: Use the diagram below to answer the question that follows. If the sequence
above continues in the same pattern, how many small triangles would be needed to make
the figure that would occur in Step 5? a) 16 b) 25 c) 36 d)49
Example 8: Find the next term in the following non-arithmetic sequence: 315375,,..... a)
119 b) 71 c) 97 d) 57 Example 9: Find the measure of ∠J. a) 120 0 b) 30 0 c) 60 0 d) 180 0
Example 10: When a student is questioned about his school, he replies that there are at
least as many freshmen as there are juniors and at least as many juniors as there are
sophomores. If the student is correct, which of the following statements must be
true? a) There are just as many sophomores as there are freshmen. b) There are at
least as many sophomores as there are freshmen. c) There are at least as many
freshmen as there are sophomores. d) There are more freshman than there are
sophomores.
OGET TEACHER TEST PREP SEMINAR – NORTHERN OKLAHOMA COLLEGE
MATH COMPETENCIES
Example 11: Which of the following statements is logically equivalent to “If it is
Saturday, then I am not in school.” a) If I am not in school, then it is Saturday. b) If it
is not Saturday, then I am not in school. c) If I am in school, then it is not Saturday. d)
If it is Saturday, then I am in school. WORD PROBLEMS AND ALGEBRAIC METHODS
– COMPETENCY 0011
• Apply combinations of algebraic skills to solve problems.
• Identify the algebraic equivalent of a stated relationship.
• Identify the proper equation or expression to solve word problems involving one and
two variables.
Example 12: Katie babysat for the Wilsons one evening. They paid her $5 just for
coming over to their house, plus $7 for every hour of sitting. How much was she paid if
she babysat for 4 hours? a) $12 b) $35 c) $28 d) $33 Example 13: Which equation
could be used to solve the following problem? “Three consecutive odd numbers add up
to 93. What are they? a) x + x + x = 93 b) x + (x + 2) + (x + 4) = 93 c) x + 3x + 5x= 93 d)
x + (x + 1) + (x + 3) = 93 Example 14: On a trip to the planetarium, Darren’s ticket cost
$18.97. Altogether, Darren spent a total of $53.21. To find how much he spent on
items other than his ticket, solve this equation for x. a) $72.18 b) $34.24 c) $15.27
d)$30.00
OGET TEACHER TEST PREP SEMINAR – NORTHERN OKLAHOMA COLLEGE
MATH COMPETENCIES
Example 15: Four students about to purchase concert tickets for $18.50 for each
ticket discover that they may purchase a block of 5 tickets for $80.00. How much
would each of the 4 save if they can get a fifth person to join them and the 5 people
equally divide the price of the 5-ticket block? a) $1.50 b) $2.50 c) $3.13 d) 12.50
Example 16: An airplane flew by 8 hours at an airspeed of x miles per hour (mph), and
for 7 more hours at 325 mph. If the average airspeed for the entire flight was 350
mph, which of the following equations could be used to find x? a) x + 325 = 2(350) b) x
+ 7(325) = 15(350) c) 8x + 7(325) = 2(350) d) 8x + 7(325) = 15(350) Example 17: Which
equation could be used to solve the following problem? “The sum of three consecutive
integers equals 132.” a) x + x + x = 132 b) x + (x + 2) + (x + 4) = 132 c) x + 2x + 3x= 132
d) x + (x + 1) + (x + 2) = 132 Example 18: Mark tried to compute the average of his 7
exam scores. He mistakenly divided the correct sum of all of his exam scores by 6,
which yielded 84. What is Mark’s correct average exam score? a) 70 b) 72 c) 84 d) 92
COMPUTATION SKILLS BASIC NUMBER SKILLS – COMPETENCY 0012
• Solve word problems involving integers, fractions, decimals, and percentages.
• Solve word problems involving ratio and proportions.
• Solve word problems involving units of measurement and conversions.
Example 19: In scientific notation, 20,000 + 3,400,00 = ? a) 3.42 x 10 6 b) 3.60 x 10 6 c)
3.42 x 10 7 d) 3.60 x 10 7
OGET TEACHER TEST PREP SEMINAR – NORTHERN OKLAHOMA COLLEGE
MATH COMPETENCIES
Example 20: Mr. Brown went grocery shopping to buy meat for his annual office picnic.
He bought 7 ¾ pounds of hamburger, 17.85 pounds of chicken, and 6 ½ pounds of hot
dogs. How many pounds of meat did Mr. Brown buy? a) 32.10 b) 31.31 c) 26.25 d) 22.10
Example 21: On a math test, 12 students earned an A. This number is exactly 25% of
the total number of students in the class. How many students are in the class? a)15 b)
21 c) 30 d) 48 Example 22: Which of the following fractions is equivalent to 28%? a)
1/5 b) 7/25 c) 1/4 d) 5/6 Example 23: One day, 22 students were absent from Tonkawa
Elementary School. If that represents about 4.4% of the students, what is the
population of the school? a) 477 b) 500 c) 96.8 d) 88 Example 24: The maximum speed
limit on the interstate is 70mph. The minimum speed limit on the interstate is 40mph.
Which inequality describes the allowable speeds indicated by the speed limits. a)
7540≤≤x b) 7540<>x c) 4075≤≤x d) 4075<>x Example 25: Which of the following
proportions will not give the same value for x as the proportion 91922=x? a) 22199=x b)
x19229= c) x22919= d) 19229=x
OGET TEACHER TEST PREP SEMINAR – NORTHERN OKLAHOMA COLLEGE
MATH COMPETENCIES
Example 26: A store owner buys a computer for $620. If she sells the computer for
40% more than she paid for it, what is the computer’s final price? a) $635 b) $660
c)$775 d) $868 Example 27: A school district would like to decrease the size of it
kindergarten classes from 32 to 20 students. What percent decrease would this
represent? a) 23.5 b) 31.5 c) 37.5 d) 42.5 GRAPHING AND SOLVING EQUATIONS COMPETENCY 0013
• Graph numbers or number relationships.
• Find the value of the unknown in a given one-variable equation.
• Express one variable in terms of a second variable in two-variable equations.
Example 28: If 6b + 20 = a, and 4b + 30 = a, then b = a) -5 b) -1 c) 1 d) 5 Example 29:
Solve the following equation 5 + 2(x – 3) = x – 14 for x. a) -7 b) 13 c)10 d) -9 Example
30: If 2(x - 5) = -11, then x = ? a) -21/2 b) -8 c) -11/2 d) -1/2 Example 31: Solve the
following equation 4x – 2y =10 for y. a) y = -4x + 10 b) y = 2x + 5 c) y = 2x - 5 d) y = 4x –
10
OGET TEACHER TEST PREP SEMINAR – NORTHERN OKLAHOMA COLLEGE
MATH COMPETENCIES
Example 32: Which of the following is a factor of the polynomial xx220−−? a) x – 5 b) x –
4 c) x + 2 d) x + 5
Example 33: Which point lines of the line m? a) (-1, 1) b) (1, -4) c) (-3, -4) d) (2,-2)
Example 34: The following table below shows values for x and corresponding values for
y.
x
y
21
3
14
2
28
4
7
1
a) y = 1/7x b) y = 7x c) y = x – 6 d) y = x - 18 Example 35: What is the slope of the line
with the equation 2x + 3y = -6? a) -2/3 b) -3 c) -2 d) -6
OGET TEACHER TEST PREP SEMINAR – NORTHERN OKLAHOMA COLLEGE
MATH COMPETENCIES
Example 36: If a = 12 and b = 16, use the Pythagorean Theorem to find c, the length of
the hypotenuse of the right triangle shown below. a) 14 b) 20 c) 400 d) 144
GEOMETRY– COMPETENCY 0014
• Solve problems involving geometric figures – Perimeter, Area, Volume, and Surface
Area
Example 37: A circular area rug has a circumference of 26π inches. What is the radius
of the rug? a) 26inches b) 13 inches c) 26 inches d) 39 inches Example 38: A company
makes boxes that have the shape of a cube measuring 3 inches on each edge. What is
the surface area of each box? a) 9 square in. b) 27 square in. c) 54 square in. d) 81
square in.
OGET TEACHER TEST PREP SEMINAR – NORTHERN OKLAHOMA COLLEGE
MATH COMPETENCIES
Example 39: D&J company estimates that it will cost $2.00 per square foot to paint
the gas tank. Which of the following expressions could be used to determine the total
cost to paint the top and the side of the tank? Example 40: Find the surface area and
volume of the rectangular prism shown in the figure below.
a) (1200π + 100π)(2.00) b) 1200100200ππ+.
c) (1200π + 200π)(2.00) d) 1200200200ππ+.
. Helpers are needed to prepare for the fete. Each helper can make either 2 large cakes per
hour, or 35 small cakes per hour. The kitchen is available for 3 hours and 20 large cakes and
700 small cakes are needed. How many helpers are required?
A. 10
B. 15
C. 20
D. 25
E. 30