1. PROBLEM
SOLVING
STRATEGIES
Dr. Liza Lorena C. Jala
This book outlines the different strategies in solving
Math problems. Although this is not exhaustive, the
most common strategies are presented here.
2. Philippine Copyright, 2016
by
Liza Lorena Casayas-Jala, MST- Mathematics, Ph.D.Ed.-RE
ALL RIGHTS RESERVED
No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form
or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior
written permission of the author.
Printed by:
CRIS’ Print Options
Lapu-lapu City, Philippines
3. ii
PREFACE
The main goal of this book is to present different strategies in solving problems in Math. Although the
strategies presented here are not exhaustiive, but the most common ones are presented here to
encourage students who find difficulty in solving math. The explanations of the concepts are done
simply with the examples.
Exercises for firming their up their learned concepts are given. Writing activities are also assigned after
the firming-up activities so that the users of this book can create their own sets of activity for their
future students. Moreover, the lesson plan activity is also added where they are encouraged to write a
lesson plan.
Liza Lorena C. Jala, Ph.D.Ed.-RE
4. iii
Table of Contents
Module 1 The Four Principles of POLYA in Solving Problems 1
Firming-up 3
Writing Activity 6
Lesson Planning 7
Module 2 The SQRQCQ Road Map 8
Firming-up 14
Writing Activity 17
Lesson Planning 18
Module 3 STAR Strategy 19
Firming-up 20
Writing Activity 22
Lesson Planning 23
Module 4 RIDGES Strategy 24
Firming-up 25
Writing Activity 27
Lesson Planning 28
Module 5 SOLVE Strategy 29
Firming-up 30
Writing Activity 32
Lesson Planning 33
Module 6 Guess & Check Strategy 34
Firming-up 37
Writing Activity 39
Lesson Planning 40
Module 7 Making an Organized List or Table 41
Firming-up 43
Writing Activity 45
Lesson Planning 46
Module 8 Draw a Picture Strategy 47
Firming-up 50
Writing Activity 54
Lesson Planning 55
Module 9 Find a Pattern Strategy 56
Firming-up 58
Writing Activity 61
Lesson Planning 62
Module 10 Divide and Conquer Strategy 63
Firming-up 67
Writing Activity 69
Lesson Planning 70
Module 11 Start at the End Strategy 71
Firming-up 73
Writing Activity 78
Lesson Planning 79
Module 12 Writing a Number Sentence Strategy 80
Firming-up 81
Writing Activity 83
Lesson Planning 84
5. MODULE 1
The Four Principles of POLYA in Solving Problems
Problem Solving
Problem solving has long been recognized as one of the hallmarks of mathematics. The greatest goal
of learning mathematics is to have people become good problem solvers. We do not mean doing
exercises that are routine practice for skill building.
Definition of problem solving
What does problem solving mean? Problem solving is a process. It is the means by which an
individual uses previously acquired knowledge, skills, and understanding to satisfy the demands of an
unfamiliar situation. The student must synthesize what he or she has learned and apply it to the new
situation.
Problem Solving Strategy
A strategy refers to “a plan that not only specifies the sequence of needed actions, but also consist
of critical guidelines and rules related to making effective decisions during a problem solving process
(Ellis & Lenz, 1996). Some features that make strategies effective for students are:
• Memory devices to help students remember the strategy (e.g. first letter mnemonic, which
is created by forming a word from the beginning letters of other words);
• Strategy steps that use familiar words stated simply and concisely and begin with action
verbs to facilitate student involvement (e.g. read the problem carefully)
• Strategy steps that are sequenced appropriately(i.e. students are cued to read the word
problem carefully prior to solving the problem) and lead to the desired outcome (i.e.
successfully solving a math problem)
• Strategy steps that use prompts to get students to use cognitive abilities (i.e. the critical
steps needed in solving a math problem) and
• Metacognitive strategies that use prompts for monitoring problem solving performance
Some strategies combine several of these features.
The POLYA (Four-step) Method
The POLYA Problem-solving Process
As part of his work on problem solving, Polya developed a
four-step problem-solving process similar to the following:
• Understanding the Problem
1. Can you state the problem in your own words?
2. What are you trying to find or do?
3. What are the unknowns?
4. What information do you obtain from the
problem?
5. What information, if any, is missing or not needed?
6. 2
The teacher is to select the question with the appropriate level of difficulty for each student to
ascertain if each student understands at their own level, moving up or down the list to prompt each
student, until each one can respond with something constructive.
• Devising a Plan
Pólya mentions that there are many reasonable ways to solve problems. The skill at choosing an
appropriate strategy is best learned by solving many problems. You will find choosing a strategy
increasingly easy. A partial list of strategies is included:
1. Guess and check
2. Make an orderly list
3. Eliminate possibilities
4. Use symmetry
5. Consider special cases
6. Use direct reasoning
7. Solve an equation
Also suggested:
1. Look for a pattern
2. Draw a picture
3. Solve a simpler problem
4. Use a model
5. Work backward
6. Use a formula
7. Be creative
8. Use your head/noggin
• Carrying out the Plan
1. Implement the strategy in Step 2 and perform any necessary actions or
computations.
2. Check each step of the plan as you proceed. This may be intuitive checking or a
formal proof of each step.
3. Keep an accurate record of your work.
• Looking Back
1. Check the results in the original problem. In some cases, this will require a proof.
2. Interpret the solution in terms of the original problem. Does your answer make
sense? Is it reasonable?
3. Determine whether there is another method of finding the solution.
4. If possible, determine other related or more general problems for which the
techniques will work.
This problem solving method of Polya is helpful on whatever strategy one has to use because each
student is trained to understand first the problem, devise a plan on what to do with the given and
what is asked for, the implement what is in the plan and check the results by looking back at the
problem and assessing whether what is asked for is satisfied by the answer.
7. 3
FIRMING-UP
Name: __________________________________ Score: ___________
Instructor: _______________________________
Directions: Find what is asked using the 4-step POLYA process.
1. For an assignment, Rosa looked at which countries got the most Nobel Prizes in various
decades.
Nobel Prize winners
Country 1960s 1970s
Italy 2 2
Japan 2 2
France 7 3
Germany 10 9
Holland 1 2
Of the countries shown, which country had the most Nobel Prize winners in the 1960s?
2. Look at this pictograph:
U.S. Representatives
Alaska
Arizona
Maryland
Connecticut
Each = 2 representatives
Each = 1 representative
How many representatives does Maryland have?
8. 4
3. Mr. Todd, the band director, counted the number of instruments played by each member of
the band.
Band instruments
Instrument Boys Girls
Clarinet 1 12
Trombone 16 13
Flute 13 9
Drums 9 8
Trumpet 5 7
Do fewer girls play the clarinet or the trumpet?
4. Tanner had 0.61 grams of pepper. Then he used 0.31 grams of the pepper to make some
scrambled eggs. How much pepper does Tanner have left?
5. Edgar runs 9.32 kilometers every morning. One day, he runs 5.1 kilometers and then stops to
take a break. How much further does Edgar have left to run?
6. The poetry club has fewer members than the science club but more members than the
debate club. Which club has the fewest members?
9. 5
7. Anna has $21.00. How much money will Anna have left if she buys an ice cream scoop and a
cookie sheet?
cutting board $9.00
cookbook $4.00
ice cream scoop $7.00
mixing bowl $6.00
baking dish $4.00
cookie sheet $5.00
8. Five of the students in Shawna's grade can swim the breaststroke. 10 students can swim the
sidestroke, and 4 students can swim both the breaststroke and the sidestroke. How many
students can swim the breaststroke or the sidestroke or both?
9. Of the children in Dayton's class, 9 have ice skates. 10 children have a skateboard. 5 children
have both ice skates and a skateboard. How many children have a skateboard but not ice
skates?
10. City Hall is 10 meters shorter than the bank. The bank is 13 meters tall. How tall is City Hall?
12. 8
MODULE 2
The SQRQCQ Road Map
SQRQCQ is a road map to help guide students down the right highway of
understanding to reach their destination of correctly answering word
problems.
L. Fay developed this metacognitive strategy in 1965. The strategy was
created to help students struggling with math word problems. (Roberts, J
2004) Often classroom teachers have witness students bypassing the words within a word problem
racing to the numbers. Without navigating their way through the problems utilizing the written
directions, frequently, they end up lost. The SQRQCQ slows a student down and helps them take the
correct route to their destination.
According to Heidema (2009), “Mathematics is about problem solving, and reading comprehension
is an important component, especially for word problems” (p. 2). Heidema explains a skill model
developed by Polya in 1957. In this model, Polya developed four steps to effectively solve word
problems. The steps include understanding the problem, devising a plan, carrying out the plan, and
looking back. Polya’s conceptual model and the successful SQ3R strategy were blended, and the idea
of SQRQCQ was born. SQRQCQ is a comprehension strategy specifically designed for word problems
in mathematics. Fay presented his idea of SQRQCQ at an International Reading Association
conference in 1965. Fay stated the reasoning behind the SQRQCQ strategy “It is in the problem-
solving phase of mathematics...that the reading study skills have their major applications at both a
general level of study procedure and a more specific level involving vocabulary, comprehension and
interpretation skills”. SQRQCQ is a more directed approach to solving word problems, with very
specific steps.
SQRQCQ has six steps. First, the student should survey the information; this is the first S (Survey).
The student should scan the problem quickly to gain a general idea of what the problem is about.
Next, Q (Question): the student should determine the question the problem is asking. After that, R
(Reread): the student should carefully reread the problem in order to recognize important
information. The fourth step, Q (Question) is to ask another question, “What mathematical
operation(s) do I apply?” Here, students decide how they are going to solve the problem. C
(Computation) is the fifth step. Students compute to find their answer. The final step is Q (Question).
This time, students are asking themselves if the answer is correct and sensible.
Essentially, they are reviewing and checking their work. Fay (1965) states that for this method of
study to be effective: a twofold foundation is needed. The first is mathematical. The student must
understand the number system and know the basic arithmetic facts. The second is vocabulary
foundation that provides the basis for quantitative reasoning and the clues for the use of
mathematical processes...Building upon the mathematical and language foundations, problem
solving demands the application of comprehension and critical reading skills. Although historical
studies are scant in the area of SQRQCQ, there are advocates of the practice, since it aligns with the
theories surrounding comprehension. Heidema (2009) supports using the SQRQCQ strategy by
explaining that it helps students focus on the problem, solve it, and reflect upon their understanding.
13. 9
Here are the steps to the strategy:
v SURVEY First the students survey the problem rather quickly to get a general idea or
understanding of it. Students will read the word problem to get a general feel for what the
problem is asking.
v QUESTION Then they come up with the questions- what they believe the problem is asking
for.
v REREAD The third step is to reread the problem to identify facts, relevant information and
details they will need to solve it. This is when rereading the questions students can get a
better idea of facts and information the problem offers.
v QUESTION Now another question is formulated that focuses on what mathematical
operations(s) to apply.
v COMPUTE The students actually compute the answer – solving the problem.
v QUESTION The question to be asked at this point involves the accuracy of the answer. Is it
correct? Does the answer make sense?
Breaking word problems up into pieces or different parts, allows students to compartmentalize the
problem better resulting in correct answers and better yet understanding of the math word
problem. As word problems can be difficult for students visualize, SQRQCQ helps bring the whole
problem into focus, adding reasoning and lessening frustration that may surround math word
problems. SQRQCQ strategy can alleviate anxiety and brings order and logic to the question of word
problems. While working with students with special needs, the teacher must be mindful of strategies
such as SQRQCQ to strengthen academic success but also emotional state and ultimately self-
efficacy. By outlining the problem the road map to the answer begins be become clear.
What will students learn: Students will learn to compute math word problems by adding structure
and rational to the problem. They will gain confidence and feelings of independence. With SQRQCQ
students will begin to feel empowered by their ability to structuralize a logical answer for
often times frustrating word problems. When learning to solve math word problems students can
begin to understand what math means in a more logical and tangible way. They can also learn how to
attain knowledge of how to collect given information. Students can work in groups or individually,
aiding in communication skill improvement as well as independence.
How does SQRQCQ Strategy help? Students begin to understand the application of math, by adding
structure and easily understandable steps that allow students to feel successful and able. This
strategy helps by giving students and students with learning disabilities the tools to get through
math word problems. After using this strategy students can eventually follow the
step independently of teacher instruction, allowing independence and confidence to develop. With a
little modeling and practice students can begin using the strategy alone.
14. 10
Lesson Plan for a 5th Grade Mathematics Class
SQRQCQ
Topic: Solving word problems using 6 steps
Objectives: The student will solve problems in adding whole numbers
Set Induction: Say to students, “Today I’m going to let you in on a little secret. When it comes to
solving word problems, I have found a plan of attack that, if you practice it, is guaranteed to make
solving word problems almost painless. Who wants to hear it?” The teacher then models SQRQCQ
using one of the student’s word problems for the day.
Activities:
1. Students turn to the assigned problems for the day.
2. Teacher models SQRQCQ through 2 or 3 examples, pointing out to the students how the
strategy provides a way to attack and solve the problem.
3. Next the students try SQRQCQ with a partner on several problems. The teacher circulates
and listens to the interaction, offering suggestion and modeling for those who are having
difficulty.
4. Finally, students work on the assigned problems using SQRQCQ as they work.
Closure: Ask the students to share what SQRQCQ has done for them and how confident they feel in
using it.
Evaluation suggestions: observation of the paired SQRQCQ; observation of students as they work on
the assigned problems (looking for “silent” SQRQCQ behaviors); students ‘success on solving the
assigned problems; students’ journal entries for that day and subsequent days; students’ future
attitude toward word problems; results on test and quizzes which include word problems.
Resources and Materials: SQRQCQ instructions transparency, assigned an example word problems,
student paper/pencils
15. 11
How does this help students who struggle with math word problems?
Let’s look first at why math word problems tend to be
challenging for students! Why do they take the direct route to
computation bypassing the road signs that lead them to their
correct destination?
Students with mathematical deficits often lack “Concept
Imagery” which is their ability to visualize the whole (Gestalt)
picture by creating a mental image. (Bell, 1991)
If a student lacks this ability, they are unable to interpret the
words within the problem correctly which impacts their math reasoning skills and can result in faulty
calculations.
They have difficulty manipulating all of the information in their working memory that has been
presented to them to see the whole picture. What also helps to delay their arrival is slow processing
speed. Frustration and anxiety will frequently gear up their need to make it to the finish line in the
fastest way possible. (I.e. the computation piece)
So… how do we help to get them back on the right track? Provide students with a logical order to
solve math word problems that is just the right amount of direction to guide them. Help them to
look at what math means. Understanding the purpose behind the computation.
SQRQCQ is a metacognitive guide to understanding word problems. Mnemonics like SQRQCQ are
memorable strategies that help students to obtain accurate results in a logical order. (fcit.us.edu,
2011)
Let’s look at a few pros and cons.
PRO’s
1. SQRQCQ provides an outline for problem solving.
2. SQRQCQ can assist with information retrieval.
3. SQRQCQ helps student to make a meaningful connection between the question being asked
in a word problem and how to solve the problem.
CON’s
1. The SQRQCQ process can appear to be long and time consuming.
2. SQRQCQ method requires students to engage and interact with others in order for it to be
effective.
16. 12
SQRQCQ fits within the UDL guidelines providing students are taught how to use the strategy
through modeling and guided practice with the goal being independent performance of the task.
Using one of the SQRQCO templates or participating in group discussions and working through the
steps, the students have multiple means of acquiring information and understanding the process.
During step 1, they focus in on meaning both at the word level and in context. Steps 2, 4 and 6
incorporate self-regulation strategies through reflecting and evaluating the previous step by asking
questions. Each step is a building block to the next where students are encourage to look deeper
into the meaning of the problem. Various techniques can be used within the framework of the
strategy that would lead to multiple means of action, expression and engagement depending upon
the creativity of the teacher. Word problems can be found and or created that can touch upon a vast
array of intellectual strengths to engage all students.
It is worth the time to give the SQRQCQ strategy a try to see if it could help your
students reach their destination of correctly solving word problems and enjoying the
ride! Have a safe trip and enjoy the view along the way.
17. 13
Name: ____________________________________________
Instructor: _________________________________________
Course: ___________________________________________
Module/Lesson: ____________________________________
SQRQCQ for Math
S = SURVEY the story problem: read the problem over quickly for general understanding.
Q = QUESTION: Ask yourself general questions about this problem such as:
1. What am I trying to find out or solve?
2. What important information is provided in the problem
1.
2.
3.
4.
5.
R = READ the problem again. Focus on facts and details that will help you solve the problem.
Q = QUESTION: Ask yourself, what operations must be performed and in what order?
1.
2.
3.
4.
5.
C = COMPUTE the answer based on your questions above.
Q = QUESTION: As yourself, does this answer make sense? If not, repeat the process.
18. 14
FIRMING-UP
Name: __________________________________ Score: ___________
Instructor: _______________________________
Directions: Find what is asked using the template SQRQCQ.
1. The baseball coach bought 7 new baseballs for $3 each. The basketball coach bought 5 new
basketballs for $11 each. How much more did the basketball coach spend than the baseball
coach?
2. Kathleen bought 15 bags of white rice and 18 bags of brown rice. She also bought 4 bags of
potatoes. How many kilograms of rice did Kathleen buy in all?
3. The white dog is lighter than the brown dog. The brown dog is not heavier than the black
dog. Which dog is the lightest?
4. Emily sliced 6 cakes. Each chocolate cake had 4 slices and each lemon cake had 8 slices. If
Emily made 44 slices in total, how many of each type of cake did she slice?
19. 15
5. Julie went on a quiz show. The questions in the first round were worth 1 point. The questions
in the second round were worth 7 points. Julie answered a total of 8 questions and earned 26
points. How many questions did she answer in each round?
6. Kylie bought 3 packs of red bouncy balls, 3 packs of yellow bouncy balls, and 1 pack of green
bouncy balls. There were 18 bouncy balls in each package. How many bouncy balls did Kylie
buy in all?
7. Carter has 17 more blue pencils than orange pencils, and 2 more orange pencils than green
pencils. The green pencils are 3 centimeters shorter than the blue and orange pencils. He has
20 green pencils. How many blue, orange, and green pencils does Carter have in all?
8. A parking garage near Amy's house is 4 storeys tall. There are 32 open parking spots on the
first level. There are 2 more open parking spots on the second level than on the first level,
and there are 9 more open parking spots on the third level than on the second level. There
are 42 open parking spots on the fourth level. How many open parking spots are there in all?
20. 16
9. Clayton and Ben went to lunch at a cafe. They ordered a spinach salad for $6.75, a tuna
sandwich for $8.15, and 2 glasses of lemonade for $0.60 each. The tax was $1.60. They gave
the waiter $20.00. How much change should they have received?
10. The gold ribbon is shorter than the white ribbon but longer than the red ribbon. Which
ribbon is shorter, the white ribbon or the red ribbon?
23. 19
MODULE 3
STAR Strategy
S.T.A.R. Strategy
The STAR strategy was developed in 1998 by Paula Maccini. The steps for STAR include:
1. Search the word problem;
v Read the problem carefully
v Ask yourself questions: “What do I know? What do I need to find?
v Write down the facts
2. Translate the problem;
v Translate the problem into an equation in picture form
3. Answer the problem;
4. Review the solution
v Reread the problem
v Ask yourself questions: “Does the answer make sense? Why?
v Check the answer
Template
Search
Translate
Answer
Review
24. 20
FIRMING-UP
Name: __________________________________ Score: ___________
Instructor: _______________________________
Directions: Find what is asked using the template STAR.
1. Type the next number in this sequence: 1, 3, 7, 13, 21, 31, _______.
2. Brad wants to buy a dictionary that costs $18, a dinosaur book that costs $20, and a children's
cookbook that costs $7. He has saved $25 from his allowance. How much more money does
Brad need to buy all three books?
3. Derick's mom gave him $97 to go shopping. He bought a sweater for $31, a T-shirt for $7, and
a pair of shoes for $28. How much money does Derick have left?
4. Nina's aunt gave her $88 to spend on clothes at the mall. She bought 3 shirts that cost $6
each and a pair of pants that cost $23. How much money does Nina have left to buy more
clothes?
25. 21
5. Eve wants to make 5 bowls of peach punch and 8 bowls of pineapple punch. She needs 3
liters of soda to make each bowl of punch. How many liters of soda will Eve need in all?
6. Kaleen has a cardboard box and a wooden box. The wooden box is 12 centimeters taller than
the cardboard box. The wooden box is 17 centimeters tall. If Kaleen stacks the cardboard box
on top of the wooden one, how many centimeters tall will the stack be?
7. Dakota went on a quiz show. The questions in the first round were worth 8 points. The
questions in the second round were worth 2 points. Dakota answered a total of 4 questions
and earned 26 points. How many questions did she answer in each round?
8. 6 of the children in Johnny's class have a green marble and 4 have a blue playground ball. 3
children have both a green marble and a blue playground ball. How many children have a
green marble or a blue playground ball or both?
28. 24
MODULE 4
RIDGES Strategy
RIDGES Strategy
Developed by Kathleen Snyder in 1988, this is another mnemonic based strategy to help students
figure out how to approach word problems. “Using RIDGES to Solve Word Problems
1. Read the problem. If the problem is not understood, re-read it.
2. Identify all of the information given in the word problem. List the information separately.
After listing all of the information, circle the information that is needed to solve the problem.
3. Draw a picture- Draw a picture of the information in the problem. This may help a student
pick out the relevant information.
4. Goal Statements. The student should express, in his or her own words, the question the
problem is asking.
5. Equation development- The student will write an equation to the problem. (i.e. length +
width + length + width = distance around the field)
6. Solve the equation- The given information is plugged into the equation (i.e.
10+6+10+6=distance around the field)
Source: Snyder, K. (1988) Ridges: A problem-solving math strategy. Academic Therapy, 230), 261-263”
29. 25
FIRMING-UP
Name: __________________________________ Score: ___________
Instructor: _______________________________
Directions: Find what is asked using the template RIDGES.
1. How much money does Alex need to buy a wooden stool, a footstool, and a chest of
drawers?
wooden stool $47
desk $97
footstool $18
chest of drawers $49
2. While sorting some change into piggy banks, Malia put 78 coins in the first piggy bank, 90
coins in the second piggy bank, 102 coins in the third piggy bank, and 114 coins in the fourth
piggy bank. If this pattern continues, how many coins will Malia put in the fifth piggy bank?
3. A new cookbook is becoming popular. The local bookstore ordered 2 copies in September, 6
copies in October, 18 copies in November, 54 copies in December, and 162 copies in January.
If this pattern continues, how many copies will the bookstore order in February?
30. 26
4. The teacher is handing out note cards to her students. She gave 3 note cards to the first
student, 4 note cards to the second student, 6 note cards to the third student, 9 note cards
to the fourth student, and 13 note cards to the fifth student. If this pattern continues, how
many note cards will the teacher give to the sixth student?
5. A hair tie costs $0.78. Lacey bought 10 hair ties. How much did Lacey spend in all?
6. One-quarter of the 16 students in the choir have brown hair. How many students in the choir
have brown hair?
7. Kelsey bought 46 boxes of pens. There were 82 pens in each box. How many pens did Kelsey
buy?
8. Packages of oatmeal cookies cost $4 each. Bonnie buys 4 packages of oatmeal cookies. How
much does Bonnie spend?
9. Five-sixths of the 18 bowls in the cupboard are yellow. How many yellow bowls are in the
cupboard?
10. Justin's father gave him $100. Justin bought 7 books, each of which cost $9. How much
money does Justin have left?
33. 29
MODULE 5
SOLVE Strategy
The SOLVE strategy is a five step strategy that only seems to be found on
Makingmathmeticians.com. Here is their explanation of the strategy
SOLVE is a strategy used to solve word problems. Each letter in SOLVE represents one of the 5 steps
in solving a word problem: Study the problem Organize the facts Line up a plan Verify your plan with
action Examine the results Now let’s look at each letter individually. S stands for Study the Problem.
When you study the problem you need to:
• Highlight or underline the question.
• Answer the question, “What is the problem asking me to find?”
O stands for Organize the Facts. When you organized the facts you need to:
• Identify each fact
• Eliminate unnecessary facts (By putting a line through it)
• List all necessary facts
L stands for Line up a Plan. When you line up a plan you need to:
• Choose and operation (Add, Subtract, Multiply or Divide)
• Tell in words how you are going to solve the problem
V stands for Verify Your Plan with Action. When you verify your plan with action, you need to:
• Estimate your answer
• Carry out your plan
E stands for Examine the Results. When you examine the results you need to ask yourself:
• Does your answer make sense? (Check what the problem was asking you to find)
• Is your answer reasonable? (Check you estimate)
• Is your answer accurate? (Check your work)
34. 30
FIRMING-UP
Name: __________________________________ Score: ___________
Instructor: _______________________________
Directions: Find what is asked using the template SOLVE.
1. For breakfast, Dave bought a muffin for $0.95 and a cup of coffee for $1.70. For lunch, Dave
had soup, a salad, and lemonade. The soup cost $2.25, the salad cost $5.90, and the lemonade
cost $2.00. How much more money did Dave spend on lunch than on breakfast?
2. Gabby bought 18 cartons of ice cream and 5 cartons of yoghurt. Each carton of ice cream cost
$4 and each carton of yoghurt cost $2. How much more did Gabby spend on ice cream than
on yoghurt?
3. The adventure club went on a hike to see a waterfall. To get to the hike, the club members
took 3 cars and 6 vans. There were 3 people in each car and 4 people in each van. How many
people went on the hike?
4. Alice bought 3 packs of red bouncy balls, 9 packs of yellow bouncy balls, and 5 packs of green
bouncy balls. There were 8 bouncy balls in each package. How many bouncy balls did Alice
buy in all?
5. Sasha has a brown ribbon that is 19 centimetres long. She also has an orange ribbon that is 18
centimetres longer than her green ribbon. How long is Sasha's orange ribbon?
35. 31
6. Jeffrey bought 12 boxes of greeting cards, 14 rolls of green gift wrap, and 2 rolls of blue gift
wrap. There were 18 metres of gift wrap on each roll. How many metres of gift wrap did
Jeffrey buy in all?
7. A chef got 7 bags of onions. The red onions came in bags of 4 and the yellow onions came in
bags of 5. If the chef got a total of 30 onions, how many bags of each type of onion did he
get?
8. Owen plans to make 6 litres of blackberry punch and 15 litres of lemon-lime punch for the
class party. How many students will Owen's punch serve?
9. Kari spent less money than Brooke but more money than Curt. Who spent more money,
Brooke or Curt?
10. Brenda and Maki already had 17 shells in their shell collection. Then they went to the beach to
collect even more. Brenda found 10 limpet shells, 4 sand dollars, and 5 conch shells. Maki
found 18 more shells than Brenda did. How many shells do the pair have all together?
38. 34
MODULE 6
Guess & Check Strategy
The student should acquire as much experience of independent work as possible. But if he is left alone with his
problem without any help or with insufficient help, he may make no progress at all. If the teacher helps too much,
nothing is left to the student. The teacher should help, but not too much and not too little, so that the student
shall have a reasonable share of the work.
--George Polya
"Guess and Check" is a problem-solving strategy that students can use to
solve mathematical problems by guessing the answer and then checking
that the guess fits the conditions of the problem.
All research mathematicians use guess and check, and it is one of the most powerful methods of
solving differential equations, which are equations involving an unknown function and its
derivatives. A mathematician's guess is called a "conjecture" and looking back to check the answer
and prove that it is valid, is called a "proof." The main difference between problem solving in the
classroom and mathematical research is that in school, there is usually a known solution to the
problem. In research the solution is often unknown, so checking solutions is a critical part of the
process.
Sample Problem –GUESS & CHECK
1. Put the numbers 2, 3, 4, 5, and 6 in the circles to make the sum across
and the sum down equal to 12. Are other solutions possible? List at least
two, if possible.
2
Here is a possible arrangement: 3 4 5
Can you find another one? 6
Emphasize Polya’s four principles – especially on the first several examples, so that that procedure
becomes part of what the student knows.
1st
. Understand the problem.
Have the students discuss it among themselves in their groups of 3, 4 or 5.
2nd
. Devise a plan.
Since we are emphasizing Guess and Check that will be our plan.
3rd
. Carry out the plan.
39. 35
It is best if you let the students generate the solutions. The teacher should just walk around
the room and be the cheerleader, the encourager, the facilitator. If one solution is found, ask
that the students try to find other(s)
Another possible solutions:
2 3 2
1 3 5 2 1 5 1 5 4
4 4 3
Things to discuss (it is best if the students tell you these things):
• Actually to check possible solutions, you don’t have to add the number in the middle – you
just need to check the sum of the two “outside” numbers.
• 2 cannot be in the middle, neither can 4. Ask the students do discuss why.
4th
. Look back.
Is there a better way? Are there other solutions?
Point out that “Guess and Check” is also referred to as “Trial and Error”. However, I prefer to call it
“Trial and Success”.
Teaching Strategy
Teaching a strategy for problem solving is a long term endeavor, revisited with mathematics from
different dimensions. Students need to be given experiences in solving problems for themselves, and
key points about the strategy can be drawn out from the experience. There is also a place for
students to practice strategies, such as guess-check-improve, which apply to a wide range of
problems.
Activity 1: Mystery number
In this activity students use the guess-check-improve strategy to find unknown numbers. The
importance of the improve phase of the guess-check-improve strategy is highlighted so that students
can recognize that this strategy will provide answers more quickly than random guessing. It is
important to use problems where students cannot immediately see the answer otherwise there is no
need to use the strategy of guess-check-improve.
Choose a problem beyond the range of students’ mental calculation skills, such as the following.
There are many different possibilities.
Ask a student to guess the mystery number in the following problem quickly.
"If you multiply 14 by this number you get 252. What is the number?" [Answer: 18]
Get all students to check if the number offered is the correct answer. Use calculators, so that the
focus is on the strategy, rather than being distracted by by-hand calculation. Discuss whether the
result is too high or too low and what this means about the answer to the problem.
40. 36
For example, if the number selected was 11, then the students will have used their calculators to find
that 14 × 11 = 154, so 11 is too small. This information can be used to improve the next guess. They now
know that the number needed is greater than 11. Focus on the use of the words guess, check and
improve.
Now ask another student for a second guess: it should be a number greater than 11. If they suggest 25
(for example), they multiply 14 by 25, get 350, and note it is too big. Discuss what this means for the
mystery number. The students should now recognize that the number they want to find is between
11 and 25.
Ask for another number, now between 11 and 25, gradually funneling onto the answer of 18. Focus
on the word improve so that students can see that they are using their results at each stage to refine
their guesses.
Provide students with some additional problems with whole number answers, such as:
"What number multiplied by 15 gives 405?" [Answer: 27]
"If you multiply 8 by this number you get 256. What is the number?" [Answer: 32]
"I have 288 eggs in boxes which each hold a dozen eggs. How many boxes?" [Ans: 24]
Discussion should focus on how students used their checking to improve their guesses. Focus on the
cycle, guess-check-improve so that students recognize the importance of using results to inform them
to work towards the required numbers.
NOTES:
• Using division: The problem above can solved immediately by using division. This activity has
been suggested because it provides a simple context and at this level students are unlikely to
identify the division strategy. However, if a student does suggest using division to solve the
problem, show all students that this is correct and the best way, but then challenge the
student to find the answer to the problem without using division.
• Variations: All the examples above involve multiplication, but addition or subtraction
examples may be more appropriate for many students. The examples here also all involve
only one operation, but this too can be varied
I thought of a number, added 12, then added my number again, and the answer was 50. What
was my number? Ans: 19)
41. 37
FIRMING-UP
Name: __________________________________ Score: ___________
Instructor: _______________________________
Directions: Find what is asked using POLYA’s method of solving problems and the “guess and check
strategy”.
1. The sum of 2 consecutive odd numbers is 45. What are the two integers?
2. A kindergarten class is going to a play with some teachers. Tickets cost 5 dollars for children
and 12 dollars for adults. A total of 20 people could go to the play. There must be at least 2
teachers to supervise the children, but no more than 10. Find all possible ways this could be
done. How can the school minimize their cost?
3. I thought of a number, added 12, then added my original number again, and the answer was
50. What was my number?
4. Some galahs were sitting in a tree. Half of them flew away, and then another 5 flew away.
Then 10 more arrived, and there were 16 galahs in the tree. How many were there to start
with?
5. On a farm there are ducks and horses. Altogether the ducks and horses have 108 legs and 36
heads. How many ducks are there on the farm?
6. A farmer has some ducks and some horses. Altogether the ducks and horses have 106 legs
and 39 heads. How many ducks and horses are there on the farm?
7. Some birds and spiders are in a shed. Altogether they have 64 legs and 17 heads. How many
spiders are there?
8. To raise money for a charity, students in a fifth grade class organized a competition at the
school's picnic. They filled a big jar of jelly beans and other students and their parents were
asked to guess how many jelly beans were in the jar. The person whose guess was closest to
the real number received a prize and, of course, the jar of jelly beans. To enter the
competition, they charged adults 1.25 YTL and children 75 kurus. Back in their classroom, they
counted the money and learned that they had earned 60.25 YTL from the competition. A
total of 65 people had made a guess. Their teacher asked them to calculate how many
parents and how many children had entered the competition. They were able to solve the
problem. Can you solve it?
42. 38
9. In a game of darts, the first person to score 150 is the winner. You
must throw four darts and must score exactly 150. If you score
more or less than 150 with your four darts, you must wait until
your opponent tries and then you may try again. To be successful,
you must know all the combinations of four numbers that add up
to 150. For example, if you aim for 50 and miss and score 10
instead, what other numbers must you aim for? Make a table and
enter all the possible combinations of four numbers that add up to 150.
10. Ibrahim Bey decided to take all his children and grandchildren to see a movie in the mall. The
tickets cost 5YTL for children and 7YTL for adults. He spent 116YTL. How many children and
how many adults went to the cinema?
11. Elizabeth visits her friend Andrew and then returns home by the same route. She always walks 2
kilometers per hour (km/h) when walking uphill, 6 km/h when walking downhill, and 3 km/h
when walking on level ground. If her total walking time is 6 hours, then what is the total
distance she walks (in kilometers)?
44. 40
Make a lesson plan for Grade 6 and the lesson is area of a rectangle
applying the guess and check strategy.
45. 41
MODULE 7
Making an Organized List or Table Strategy
The real obstacle when we are faced with an impossible problem is inside us. It is our experiences,
mistaken assumptions, half-truths, misplaced generalities, and habits that keep us from brilliant
solutions.
--Scott Thorpe
There is a common saying, "Don't just stand there, do something!" When solving a
problem, don't just think, write something! Draw a diagram or make a list!
It is often obvious when you should draw a diagram. And it is often obvious when
you should make a list. Take a very simple example: putting things in order.
The letters ABCD, can be put into a different order: DCBA or BADC. How many different
combinations of the letters ABCD can you make? To answer this question, obviously, you have to
make a list.
Teach your students to make a SYSTEMATIC list. For example:
ABCD
ABDC
ACBD
ACDB
ADBC
ADCB
BACD
BADC
BCAD
BCDA
BDAC
BDCA
CABD
CADB
CBAD
CBDA
CDAB
CDBA
DABC
DACB
DBAC
DBCA
DCAB
DCBA
By making a SYSTEMATIC list, students will see every possible combination.
(Later, perhaps, they will learn that the number of permutations of size 4 taken from a set of 4 can
be represented by the formula 4 * 3 * 2 * 1 = 24).
46. 42
Ducks and horses
This activity is useful for showing how systematic recording of results can assist in solving problems
using guess-check-improve where students have to do more than multiply one number.
Start by posing the problem for students.
"A farmer has some ducks and some horses. Altogether the ducks and horses have 40 legs
and 14 heads. How many ducks and horses are there on the farm?"
Many students will guess both the numbers of horses and ducks. For example, they might draw 8
horses and 10 ducks and count the legs (52) and heads (18), and then make another guess of both
numbers.
For example, if a student guesses that there are 8 horses then there must be 6 ducks to get 14
heads. They can draw these animals, or just calculate that there are too many legs. Making a
drawing or diagram is a strategy that will be used by children right through to secondary school, so it
is a useful strategy to learn.
Encourage students to record their results in a table. Discuss how students can improve their guess
now, as they know that the number of horses must be less than 8.
# of horses # of ducks Check 14
heads
No. of
horse legs
No. of duck
legs
Total
number of
legs
Comments
8 6 / 32 12 44 Too many
5 9 / 20 18 38 Too few
6 8 / 24 16 40 Correct
There are some important principles to help students develop when using guess-check-improve:
• recording guesses and outcomes is essential
• using labeled tables is a good way to assist in working system
47. 43
FIRMING-UP
Name: __________________________________ Score: ___________
Instructor: _______________________________
Directions: Solve the following by making an organized list.
1. Three darts hit this dart board and each scores a 1, 5, or 10. The total score is the sum of the
scores for the three darts. There could be three 1’s, two 1’s and 5, one 5 and two 1 0’s, And so
on. How many different possible total scores could a person get with three darts?
1
5
2. Renee, Jessica and Anjali are competing in the finals of the obstacle course. How many
different ways can they finish?
3. Beth, Andrea, Nancy, Tanya, Christine and Helene all want to be the starters on the
basketball team. Five of them can start. How many different combinations of starters can be
chosen from this group?
4. How many ways can you arrange the letters A, B, C?
5. Malia has 3 poster-size pictures that she wants to put on her door, but only 2 pictures will fit.
She has a picture of her dog, a picture of her hamster and of a super cool guitar. How many
different ways can she arrange the poster?
6. Pedar Soint has a special package for large groups to attend their amusement park: a flat fee
of $20 and $6 per person. If a club has $100 to spend on admission, what is the most number
of people who can attend?
10
48. 44
7. Ten people arrive at a party. As they arrive, each person kisses every other person twice,
once on each cheek. How many kisses are there?
8. A linen shop sells sheets, duvet covers, and pillow cases. Linens from the Easirest Company
offer a choice in each color: a plain sheet or a sheet with a pattern, a plain duvet cover or a
duvet cover with a pattern, plain pillow cases or pillow cases with a pattern. Customers may
mix and match the plain or patterned sheets, duvet covers, and pillow cases in any
combination they wish. To help the shop make its next order from the Easirest Company, you
have been asked to make a record of the different combinations that customers have
chosen. How many different combinations are possible?
Some students might be more interested in pizza. The same problem might occur in a
pizzeria.
9. A tree diagram is a useful type of list. For example, the game of dominoes is played with
black and white tiles. Each tile is divided into two halves and on each half a number from 0-6
is represented in the form of dots. Each tile contains a pair of numbers and each pair appears
only once in a complete set. How many tiles are there in a complete set of tiles? The solution
to this problem can take the form of a tree diagram.
10. Melike and Funda are good friends. When Melike got the flu, her doctor wrote a prescription
for twenty 30mg pills, and told her to take two a day, one after breakfast and one after
dinner. Three days later, Funda also got the flu. Her doctor prescribed thirty 20mg pills, and
told her to take a pill every four hours between 9:00 and 21:00. Whose medicine was finished
first?
50. 46
Make a lesson plan for Grade 6 and the lesson is area of a rectangle
applying the “making an organized list” strategy.
51. 47
MODULE 8
Problem Solving: Draw a Picture
A great discovery solves a great problem but there is a grain of
discovery in the solution of any problem. Your problem may be
modest; but if it challenges your curiosity and brings into play your
inventive faculties, and if you solve it by your own means, you may
experience the tension and enjoy the triumph of discovery. Such
experiences at a susceptible age may create a taste for mental work
and leave their imprint on mind and character for a lifetime.
--George Polya
The draw a picture strategy is a problem-solving technique in which students make a visual
representation of the problem. It is important that students follow a logical and systematics
approach to their problem solving.
Drawing a diagram or other type of visual representation is often a good starting point for solving all
kinds of word problems. It is an intermediate step between language-as-text and the symbolic
language of mathematics. By representing units of measurement and other objects visually, students
can begin to think about the problem mathematically. Pictures and diagrams are also good ways of
describing solutions to problems; therefore they are an important part of mathematical
communication.
In order to use the strategy of drawing a diagram effectively, students will need to develop the
following skills and understanding.
Using a Time/Distance Line to Display the Information
A time/distance line helps to show distance, or movement from one point
to another. Students were asked to calculate how far they are from the
city when they are 17 kilometers from the ocean, using the information
on this signpost.
Students should draw a line and on it write the distances.
30 km 65 km
City /17 km Ocean
30 km + (65 km – 17km) = 78 km
City Ocean
30 km 65 km
52. 48
Scale
When students are required to draw a diagram of a large area the diagram will often need to be
scaled down.
For example, in a drawing, one centimeter could have the value of one kilometer. Alternatively, a one
centimeter line could represent ten kilometers or even 500 kilometers, depending on the scale of the
drawing.
Show students how to use scaled down measurements to solve a problem, then convert the solution
to the actual measurements.
Mapping or Showing Direction
Students will often be faced with drawings that require them to have an understanding of direction.
They will also meet problems where they are asked to plot a course by moving up, down, right or left
on a grid.
They will also meet to use the compass points to direct themselves —north, south, east, west, north-
easterly, south-westerly, and so on.
They will also need to become familiar with measurement words which may be unfamiliar to them,
such as pace. Opportunities should be given for the students to work out how many paces it takes to
cover the length and breadth of the classroom or to pace out the playground, so they develop a
means of comparison.
Drawing a Picture
Drawing a picture can help students organize their thoughts and
so simplify a problem.
Example: A frog is at the bottom of a 10-meter well. Each day he
climbs up 3 meters. Each night he slides down 1 meter. On what
day will he reach the top of the well and escape?
53. 49
Using a line to symbolize an object
Example: Marah is putting up a tent for a family reunion. The tent is 16 feet by 5 feet. Each 4-foot
section of tent needs a post except the sides that are 5 feet. How many posts will she need?
Demonstrate that the first step to solving the problem is understanding it. This involves finding the
key pieces of information needed to figure out the answer. This may require students reading the
problem several times or putting the problem into their own words.
16 feet by 5 feet
1 post every 4 feet, including 1 at each corner
No posts on the short sides
1. Choose a Strategy
Most often, students use the draw a picture strategy to solve
problems involving space or organization, but it can be
applied to almost all math problems. Also students use this
strategy when working with new concepts such as equivalent
fractions or the basic operations of multiplication and division.
2. Solve the Problem
Students understand that there are posts every 4 feet. In the
second sample problem, students are asked to organize data
spatially to determine the number of posts Marah will need.
They can draw a picture or a diagram to find the answer.
I drew a rectangle where each long side is 16 feet, and there is
1 post every 4 feet. I drew a circle for each post. I remembered
to draw a post at each end. There are 10 posts total.
3. Check Your Answer
Ask students to read the problem again to be sure they answered the question.
I found that there are 10 posts. Students should check their math to be sure it is correct. 16
divided by 4 is 4. There are 4 sections of 4 feet on each long side. There is a post on each end, so
4 + 1 = 5. There are 2 sides to the tent, and 5 x 2 = 10.
Discuss with students whether draw a picture was the best strategy for this problem. Was there
a better way to solve it? Drawing a picture was a good strategy to use for this problem because
students might forget to count the posts on each corner unless they see them.
4. Explain How You Found the Answer
Students should explain their answer and the process they went through to solve the problem. It
is important for students to talk or write about their thinking. There may be more than one way
to represent a problem visually, and asking students to explain their picture helps to understand
their thinking process and identify errors.
54. 50
FIRMING-UP
Name: __________________________________ Score: ___________
Instructor: _______________________________
Directions: Solve the following problems by using a draw a picture strategy.
1. Tai wants to frame a 3 x 5 picture surrounded by 2 inches of mat. How large will her frame need
to be?
2. The children built a log playhouse in a square shape. They used eight vertical posts on each side
of the house. How many posts did they use altogether?
3. On his last day with Uncle Larry, Travis worked with Mr. Wilson on laying tile on the kitchen floor.
Travis worked hard all morning and he was a bit discouraged when he reached his first break and
realized that he had only finished about one-third of the floor. It had taken Travis two hours to
tile one-third of the floor. He thought about this as he drank from his water bottle and ate an
apple. “If it took me this long to tile one-third, how long will it take me to finish?” Travis
wondered. The floor is divided into 12 sections. If he has finished one-third of them, how many
sections has he completed? This is the number that he completed in the two hours. How many
sections does he have left to complete? About how long will it take him to finish the rest?
4. In a stock car race, the first five finishers in some or der were a Ford, a Pontiac, a Chevrolet, a
Buick, and a Dodge.
• The Ford finished seven seconds before the Chevrolet.
• The Pontiac finished six seconds after the Buick.
• The Dodge finished eight seconds after the Buick.
• The Chevrolet finished two seconds before the Pontiac.
In what order did the cars finish the race? What strategy did you use?
5. Four friends ran a race:
• Matt finished seven seconds ahead of Ziggy.
• Bailey finished three seconds behind Sam.
• Ziggy finished five seconds behind Bailey.
In what order did the friends finish the race
55. 51
6. Tyler has eaten one-fifth of the pizza. If he eats another two-fifths of the pizza, what part of the
pizza does he have left? What part has he eaten in all? How many parts of this pizza make a
whole?
7. Maria decides to join Tyler in eating pizza. She orders a vegetarian pizza with six slices. If she eats
two slices of pizza, what fraction has she eaten? What fraction does she have left?
8. If Tyler was to eat half of Maria’s pizza, how many pieces would that be?
9. If Maria eats one-third, and Tyler eats half, what fraction of the pizza is left?
10. How much of the pizza have they eaten altogether?
11. Teri ran 1 ½ miles yesterday, and she ran 2 ½ miles today. How many miles did she run in all?
12. If John ran 7 miles, what is the difference between his total miles and Teri’s total miles? How
many miles have they run altogether?
56. 52
13. If Kyle ran half the distance that both John and Teri ran, how many miles did he run?
14. If Jeff ran 3 ½ miles, how much did he and Kyle run altogether?
15. What is the distance between Jeff and Kyle’s combined mileage and John and Teri’s combined
mileage?
16. Murat has decided to plant grapes in the garden behind his house. His neighbor Volkan has
grown grapes successfully for a long time and has given Murat advice about how to plant vines.
Volkan told him to plant them three meters apart in rows that are three meters apart. He also
told him to leave at least three meters between each vine and the edge of the garden. Murat has
measured the size of his garden and learned that it’s a rectangle with sides of 25 meters and 35
meters. How many vines should Murat buy?
17. Four holes are drilled in a straight line in a rectangular steel plate. The distance between hole 1
and hole 4 is 35mm. The distance between hole 2 and hole 3 is twice the distance between hole 1
and hole 2. The distance between hole 3 and hole 4 is the same as the distance between hole 2
and hole 3. What is the distance, in millimeters, between the center of hole 1 and the center of
hole 3?*
57. 53
18. Ayşegul is wrapping presents for her friends. She has made 10 rings for 10 friends using brightly
colored polymer clay. She has bought 10 little jewelry boxes and now she is shopping for wrapping
paper and ribbon. She estimates that she needs a rectangle of paper 20cm by 15cm to wrap each
box. She finds lovely silver wrapping paper that is sold in 60cm x 60cm sheets. Since the paper is
expensive, she does not want to buy too much. How many sheets should she buy?
19. The diameter of the earth is 12756 km, and the circumference of the earth is 40075 km. If 15
meter poles were erected all the way around the equator and a wire were stretched from the top
of one pole to the top of the next pole, all the way around the equator, how long would the wire
be?
20. Mrs. Roberts is an enthusiastic gardener. She has made a small pond in which she will keep fish
and grow water plants. The pond has a circular plastic liner, 3 meters in diameter. Around the
pond she will make a cement path, one meter wide. To be certain that the path is strong and will
not crack, she wants the cement to be 20cm deep. A contractor will come to her house and
make the path for 100YTL per cubic meter of cement. How much will the path cost?
59. 55
Make a lesson plan for Grade 6 and the lesson is area of a circle applying the
“draw a picture” strategy.
60. 56
MODULE 9
Problem Solving: Find a Pattern
It's not that I'm so smart, it's just that I stay with problems longer.
--Albert Einstein
One of the most famous patterns in mathematics is known as the Fibonacci series, named for a
mathematician who lived in Italy in the 13h Century. Fibonacci introduced this pattern by posing a
problem:
A pair of rabbits, one male and one female, are put into a pen. After two months they have two
offspring, one male and one female. They continue to have an additional two offspring every month
thereafter, always a pair, one male and one female. This pattern continues: after two months, every
pair of rabbits start to reproduce and every month thereafter they have a pair of offspring. After one
year, how many pairs will there be?
The solution produces a series of numbers now known as the Fibonacci series.
Month Pairs
of
Rabbits
1 1 The original pair (A)
2 1 The original pair (A)
3 2 After two months the original pair produce a pair of offspring (B1)
4 3 The first pair of rabbits produce a second pair (B2)
5 5 The first pair of rabbits produce a third pair (B3). The first pair of
offspring (B1) produce a pair of offspring (C1)
6 8 And so on . . .
7 13 And so on. . . .
If you see the pattern that develops month by month, you can easily predict how many pairs there
will be after 12 months (144) and after 13 months (233). Each succeeding number in the series is the
sum of the previous two numbers.
When you see a pattern you can make a prediction, and that is the essence of the problem solving
strategy: see the pattern, make a prediction.
61. 57
Here is an example of a pattern and a prediction:
The Problem. In their biology class, Ayse and Mehmet learned how to count a population of yeast
cells. Using a special counting microscope, they counted the cells every hour and entered their data
in a table.
Time Yeast Cells
9:00 9
10:00 17
11:00 37
12:00 75
Their teacher told them that the population would stop growing and remain stable at about 500
cells. At what time would Ayse and Mehmet discover that the population had stopped growing?
The pattern. Ayse and Mehmet see a pattern in their data. The population doubles (approximately)
every hour.
The prediction. At 13:00 there will be (approximately)150 cells; at 14:00 there will be (approximately)
300 cell. At 15:00 , if the teacher is correct, there will be (approximately) 500 cells. They will know the
population has stopped growing at 16:00 if there are still (approximately) 500 cells.
62. 58
FIRMING-UP
Name: __________________________________ Score: ___________
Instructor: _______________________________
Directions: Solve the following problems by finding a pattern.
1. A woman is trying to cut down the number of cans of soda she drinks each week. She
makes a plan so that in several weeks she will be drinking only one can of soda. If she
starts with 25 cans the first week, 21 cans the second week, 17 cans the third week, 13
cans the fourth week, and continues this pattern, how many weeks will it take her to
reach her goal?
2. Continue these numerical sequences. Copy the problem and fill in the next three blanks
in each part.
a) 1, 4, 7, 10, 13, _____, _____, _____.
b) 19, 20, 22, 25, 29, _____, _____ _____.
c) 2, 6, 18, 54, _____, _____, ______.
3. Copy and continue the numerical sequences:
a) 3, 6, 9, 12, _____, _____, _____
b) 27, 23, 19, 15, 11, _____, _____, _____
c) 1, 4, 9, 16, 25, _____, _____, _____
d) 2, 3, 5, 7, 11, 13, _____, _____, _____
4. If you build a four-sided pyramid using basketballs and don't count the bottom as a side,
how many balls will there be in a pyramid that has six layers?
63. 59
5. In a high school of 1000 students every student has a locker. Imagine that one student
opens all the doors of all 1000 cupboards.
Then a second student starts at the second locker and closes every second door.
Then a third student starts at the third locker and changes the state of every third door (closes it
if it was open or opens it if it was closed.)
Then a fourth student starts at the fourth locker and changes the state of every fourth door.
Then a fifth student starts at the fifth locker and changes the state of every fifth door.
And so on . . . After 1000 students have followed the same pattern, which doors will be open and
which doors will be closed?
6. If the first, third and thirteenth terms of an arithmetic progression are in geometric
progression, and the sum of the fourth and seventh terms of this arithmetic progression
is 40, find the first term and the (non-zero) common difference.
7. The ninth, thirteenth and fifteenth terms of an arithmetic progression are the first three
terms of a geometric progression whose sum of infinity is 80. The sixteenth term of an
arithmetic progression is equal to the fourth term of the geometric progression.
Calculate the sum of the first sixteen terms of the arithmetic progression.
8. A ladder has 12 rungs that uniformly decrease in length. The bottom rung is 31 inches and
the top rung is 20 inches. If there are 12 rungs, find the total length of board required.
64. 60
9. In a stone race, 14 stones are placed in a row at distances of 6 feet except the first stone
which is 10 feet from the basket. A boy starts from the basket, picks up the stones and
brings them back one at a time to the basket. How far does he travel to finish the stone?
10. A well is to be drilled 80 feet deep. The cost of the first foot is P50. What is the cost for
drilling the whole well, if the cost of each foot after the first is P15 more than the
preceding foot?
11. In a pile of pipes, each layer contains one more pipe than the layer above and the top
contains just two pipes. If there are 90 logs in the pile, how many layers are there?
12. A city has a population of 30, 000 and an annual growth rate of 3%. Find the total
population at the end of 8 years.
13. A right isosceles triangle with equal sides of 4 ft is formed by connecting the midpoints of
the sides of the triangle. Then a similar construction is made to form a right angle from
the newly formed triangle. The process is continued until there are six such right
triangles. Find the sum of the perimeters of the six right triangles.
14. A rolling ball travels 5 meters in the 1st
second, and each subsequent second, the distance
traveled is 20% less than the distance traveled in the preceding second. Find the total
distance traveled.
66. 62
Make a lesson plan for Grade 6 and the lesson is area of a rectangle
applying the “look for a pattern” strategy.
67. 63
MODULE 10
Divide & Conquer Strategy
It's much more wonderful to know what something is really like than to sit there and simply, in
ignorance, say, "Oooh, isn't it wonderful?"
--Richard Feynman
There is a folk tale about a rich farmer who had seven sons. He was afraid that when he died, his land
and his animals and all his possessions would be divided among his seven sons, and that they would
quarrel with one another, and that their inheritance would be splintered and lost. So he gathered
them together and showed them seven sticks that he had tied together and told them that anyone
who could break the bundle would inherit everything. They all tried, but no one could break the
bundle. Then the old man untied the bundle and broke the sticks one by one. The brothers learned
that they should stay together and work together and succeed together.
The moral for problem solvers is different. If you can't solve the problem, divide it into parts, and
solve one part at a time.
An excellent application of this strategy is the magic squares problem. It is well known. You have a
square formed from three columns and three rows of smaller squares.
Part 1: Draw a Diagram
Into these squares you will enter the numbers from 1-9 in such a way that the sum of each column,
each row, and each diagonal is 15. Since we know the way to solve problems is to start, let's start by
guessing and checking.
68. 64
Part 2: Guess and Check
Begin by entering the numbers in order, just to see what happens:
1 2 3
4 5 6
7 8 9
The middle column and the middle row add up to 15 and the two diagonals add up to 15. It has
become obvious that 5 is a good choice for the middle, but we have to adjust the other squares.
Part 3: Make a List
It would be helpful now to identify all the combinations of three digits that add up to 15.
1 + 5 + 9 = 15
1 + 6 + 8 = 15
2 + 4 + 9 = 15
2 + 5 + 8 = 15
2 + 6 + 7 = 15
3 + 4 + 8 = 15
3 + 5 + 7 = 15
4 + 5 + 6 = 15
Part 4: Guess and Check Again
Now we can quickly see which combinations of numbers will solve the puzzle. There are four
combinations that have the number 5, and all four combinations are needed—with number 5 in the
middle.
69. 65
4 3 8
5
2 7 6
When these four combinations are placed correctly, the other three combinations needed to finish
the puzzle are easy to find.
4 3 8
9 5 1
2 7 6
Thus, by dividing the problem into 4 parts, it can be solved systematically.
Mixture Problems. Mixture problems often appear in mathematics text books. Here is an example
of a mixture problem.
A mixture is 25% red paint, 30% yellow paint, and 45% water. If 4 quarts of red paint are added to 20
quarts of the mixture, what is the percentage of red paint in the new mixture? This problem is taken
from the book crossing the River with Dogs and Other Mathematical Adventures, by Ken Johnson and
Ted Herr, a book about problem solving strategies: http://www.keypress.com.
70. 66
A student solving this problem has divided it into four parts:
Part 1: Find the amount of red paint in the original mixture:
Part 2: Find the total amount of red paint:
Part 3: Find the total amount of the whole mixture:
Part 4: Calculate the new percentage:
A problem which at first seems difficult becomes easier if you divide it into parts and solve one part
at a time.
71. 67
FIRMING-UP
Name: __________________________________ Score: ___________
Instructor: _______________________________
Directions: Solve the following problems correctly.
1. Can you solve a magic square problem in which there are five columns and five rows? Use all the
numbers from 1 to 25, and the sum of each row, each column, and each diagonal should be 65.
2. Joe Curry owns a furniture shop. He sets his prices at 20% above wholesale. When he reduces his
prices for a sale, he still wants to make at least 10% profit on each item. The regular price for a
couch was $240. During the sale, he reduces this price by 10%. Will Joe make his 10% profit?
3. A more challenging version of Problem 2 omits the price of any one item. Everything is priced at
20% over wholesale. During the sale, Joe reduces everything by 10%. Will he make at least a 10%
profit on each item?
4. When Laura goes to the gym, she jogs for 20 minutes on the treadmill, equivalent to a distance of
2.5 kms. In good weather, instead of going to the gym she jogs from Arnavutkoy to Rumeli Hisar,
a distance of 3 kms. If she wants to jog at the same speed as in the gym, how long should it take
her to go from Arnavutkoy to Rumeli Hisar and back again?
5. Paul went to the car dealer to buy a car. He wanted the same car that his friend Barbara had
bought the day before, which had a sticker price of $15,000. The salesman said he could give a
discount and offered Paul a significantly reduced price. But Paul knew that Barbara had received
a 30% discount, and the salesman was offering him only a 20% discount. When he pointed out that
his friend had received a 30% discount the day before, the salesman took another 10% off the 20%
discounted price. Paul was satisfied with the new price and bought the car, thinking he had paid
the same price as Barbara. Was he right? Did they both pay the same price?
72. 68
6. The furniture in a classroom consists of tables and chairs. The homeroom teacher is making a
seating plan. If 2 students sit at each table, 8 students will be left without a place. If 3 students sit
at each table, 4 tables will be left empty. How many students are there in the homeroom?
7. After making a parquet floor in an office building, the carpenters had left-over pieces of wood in
the shape of right triangles with sides of 1, 2, and Ö5. The architect would like to use these pieces
for a parquet floor in his own house. He wants to know: can he make a perfect square from 20 of
these triangles? If so, what would it look like?
8. Mrs. Summersby has an antique circular table with a diameter of 1.5 meters. It is big enough to
accommodate six people for dinner. The table is divided in the middle so that leaves can be
added to make the table bigger, thus creating a rectangle with two semi-circular ends.
Unfortunately the leaves for making the table bigger have been lost. Mrs. Summersby has asked
a carpenter to make her new leaves so that she can accommodate ten people for dinner. Each
leaf should be 35 centimeters wide and 1.5 meters long. To seat ten people, the perimeter of the
table should be at least 6.8 meters and the area at least 3 square meters. How many leaves
should she ask the carpenter to make?
74. 70
Make a lesson plan for Grade 7 applying the “divide and conquer” strategy.
15.
75. 71
MODULE 11
Start at the End Strategy
Problem solving is an individual's capacity to use cognitive processes to
confront and resolve real, cross-disciplinary situations where the
solution path is not immediately obvious.
OECD Organization for Economic Co-operation and Development
PISA Programme for International Student Assessment
Sometimes in order to accomplish something you have to start at the
end. Athletes see themselves winning even before the competition
begins. It is called visualizing success. Engineers make drawings of finished products even before
they know how to build them. Stephen Covey in his famous book 7 Habits of Highly Effective People
says that highly effective people “start with the end in mind.” In Understanding by Design, a book
about teaching and learning, Grant Wiggins and Jay McTighe describe a method called “backwards
design”: you start by asking what you will ask your students to do to show that they understand . . .
and then you plan to teach them how to do it. Very often, the road to success starts at the end and
not at the beginning.
So it is with problem solving. To solve some problems, you start at the end and work backwards.
However, the directions for going backwards are not exactly the same as the directions for going
forwards. Imagine leaving the school to go to the Post Office and then returning to the school.
FORWARDS to go to the Post Office:
• Turn left out of the school (Independence
Avenue)
• Take the 3rd right turn (National Avenue)
• Take the 2nd left turn (Station Street)
• Cross two streets on Station Street
• Turn left into the Post Office
BACKWARDS to the school
• Turn right out of the Post Office (Station
Street)
• Take the 3rd right on to National Avenue
• Take the 2rd left turn on to Independence
Avenue
• Cross two streets on Independence Avenue
• Turn right into the school
Here is a well-known problem that can be solved by starting at the end.
76. 72
THE MANGOES PROBLEM
One night the King could not sleep. He went to the royal kitchen, where he found a bowl full of
mangoes. Being hungry, he took 1/6 of the mangoes in the bowl.
Later that same night, the Queen could not sleep, and she was hungry. She found the mangoes and
took 1/5 of what the King had left in the bowl.
Still later, the youngest Prince awoke, went to the kitchen, and ate 1/4 of the remaining mangoes.
Even later, the second Prince ate 1/3 of what his younger brother had left.
Finally, the third Prince, the heir to the throne, ate 1/2 of what his younger brothers had left, and
then there were only three mangoes left in the bowl.
How many mangoes were in the bowl when the King found them?
This problem and its solution can be found on the Illuminations website of the National Council of
Teachers of Mathematics http://illuminations.nctm.org/LessonDetail.aspx?ID=L264
To solve the Mangoes Problem, start with the 3 mangoes left in the bowl after the King, the Queen,
and the three Princes have all eaten their share—and work BACKWARDS from there.
The third Prince ate 1/2 of the mangoes he found in the bowl and left 3. So he must have found 6
mangoes in the bowl.
The second Prince ate 1/3 of the mangoes that he found in the bowl and left 6. Therefore 6 = 2/3 of
the mangoes he found, and 1/3 = 3. The second Prince must have found 9 mangoes in the bowl.
The youngest Prince ate 1/4 of the mangoes he found in the bowl, leaving 3/4. Therefore 3/4 = 9, and
1/4 = 3. The youngest Prince must have found 12 mangoes.
That means the Queen left 12. Since she ate 1/5, 4/5 = 12 and 1/5 = 3. Therefore the Queen found 15
mangoes in the bowl.
Since the King left 15 mangoes after eating 1/6, 5/6 = 15 and 1/6 = 3. Therefore there were 18 mangoes
in the bowl when the King found them.
You started at the end and worked BACKWARDS to the beginning!
77. 73
FIRMING-UP
Name: __________________________________ Score: ___________
Instructor: _______________________________
Directions: Solve the following problems by working backwards.
1. Mr. and Mrs. Atkins had friends from Canada come to visit them. They decided to take
their friends to their favorite restaurant for dinner. In addition to the cost of the dinner,
Mr. Atkins had to pay some extra expenses. He paid $12 for parking, $18 for tax, and he
left a tip of $30 for the waiters. When they got home, Mrs. Akins asked Mr. Atkins how
much the dinner had cost. “Well,” he said, looking in his wallet. “I know I started with
$300, and now I have $15.”
What will he tell Mrs. Atkins? How much was the dinner?
2. Later today, you mother will take you to the doctor’s office for a check-up. “When do
you think we should leave?” she asks. “Help me decide.” Since she always has errands to
run, so you ask her, “What do we have to do on the way? “She answers, “I’d like to go to
the dry-cleaners in the mall, and then let’s have lunch at the restaurant in the mall. Then
we can get dog-food at the pet shop and money at the bank. And then we can go see
the doctor.” “OK,” you tell her, “let’s say that it takes 20 minutes to drive to the mall and
park, and ten minutes to get the dry cleaning, then 45 minutes for lunch, 10 minutes to
get dog food and 10 minutes to get money at the bank. After that we’ll need 20 minutes
to drive to the doctor’s office. What time is our appointment?” “Our appointment is at
2:00 p.m., so when should we leave?”
78. 74
3. Town Planning Problem
This is a street map of New Town. The Town Planning Commission wants to know how many
different ways you can drive a car from A to B, going only North and/or East.
4. Jack walked from Santa Clara to Palo Alto. It took 1 hour 25 minutes to walk from Santa
Clara to Los Altos. Then it took 25 minutes to walk from Los Altos to Palo Alto. He arrived
in Palo Alto at 2:45 P.M. At what time did he leave Santa Clara?
5. Sarah got on the school bus. At the stop after Sarah’s, 7 students got on. Five students
got on the bus at the next stop. At the last stop before the school, 9 students got on.
When the bus arrived at school, 38 students got off. How many students were already on
the bus when Sarah got on?
79. 75
6. Mrs. Allen baked some cookies for the school bake sale. Franco bought 3 of the cookies
and Chandra bought 2. Mr. Walker bought 1 dozen of the cookies. William and Starr each
bought 6 cookies. Then Ms. Porter bought 4 of the cookies. That left only 3 cookies for
Scott to buy. How many cookies did Mrs. Allen bake for the sale?
7. Dave, Nora, Tony, and Andrea are members of the same family. Dave is 2 years older than
Andrea, who is 21 years older than Tony. Tony is 4 years older than Nora, who is 7 years
old. How old are Dave, Tony, and Andrea?
8. Paula went shopping at a department store. She bought 2 CDs on sale for $8.95 each, a
notebook for $4.29, and a bottle of shampoo for $2.58. When Paula paid for her
purchases, the cashier gave her $5.23 in change. How much money did Paula give the
cashier?
9. Brian gave 10 stamps from his collection to both Sam and Rob. Then he gave 14 stamps to
Kathy and 6 stamps to Grace. He still had 275 stamps. How many stamps were in Brian’s
collection to begin with?
10. Sarah got on the school bus. At the stop after Sarah’s, 7 students got on. Five students
got on the bus at the next stop. At the last stop before the school, 9 students got on.
When the bus arrived at school, 38 students got off. How many students were already on
the bus when Sarah got on?
11. Mrs. Allen baked some cookies for the school bake sale. Franco bought 3 of the cookies
and Chandra bought 2. Mr. Walker bought 1 dozen of the cookies. William and Starr each
bought 6 cookies. Then Ms. Porter bought 4 of the cookies. That left only 3 cookies for
Scott to buy. How many cookies did Mrs. Allen bake for the sale?
12. Dave, Nora, Tony, and Andrea are members of the same family. Dave is 2 years older than
Andrea, who is 21 years older than Tony. Tony is 4 years older than Nora, who is 7 years
old. How old are Dave, Tony, and Andrea?
80. 76
13. Paula went shopping at a department store. She bought 2 CDs on sale for $8.95 each, a
notebook for $4.29, and a bottle of shampoo for $2.58. When Paula paid for her
purchases, the cashier gave her $5.23 in change. How much money did Paula give the
cashier?
14. Brian gave 10 stamps from his collection to both Sam and Rob. Then he gave 14 stamps to
Kathy and 6 stamps to Grace. He still had 275 stamps. How many stamps were in Brian’s
collection to begin with?
15. The figure below shows twelve toothpicks arranged to form three squares. How can you
form five squares by moving only three toothpicks?
16. Sixteen toothpicks are arranged as shown. Remove four toothpicks so that only four
congruent triangles remain.
82. 78
Make a lesson plan for Grade 6 and the lesson is area of a rectangle
applying the “working backwards” strategy.
83. 79
MODULE 12
Writing a Number Sentence Strategy
Writing a number sentence is a strategy of transforming the
worded problems into number sentences by understanding the
key words.
You can write a Number Sentence to solve most problems. Use
the Write a Number Strategy when
1) There is only one possible answer.
2) You can add, subtract, multiply, or divide to solve the
problem.
3) You can use a formula to solve the problem.
When you write a Number Sentence, you can use the letter n to
stand for the number you need to find. This is called a variable.
Use common sense to decide whether you need to add,
subtract, multiply, or divide. Then solve the number sentence to
solve the problem. In the number sentence the variable can be
put anywhere, but it is easier if it is at the end.
Example. Tony practices his trumpet 3 times a week, for a total of 4 hours. Each practice session is
the same length. How long is each practice session?
Understand. What do you need to find? You need to find how long each practice session is.
Plan. How can you solve the problem? First, find the total number of minutes Tony practices in one
week. Then find how many minutes he practices in each session. You can write number sentences.
Solve. Change 4 hours to minutes.
Think:
1 hour = 60 minutes
4 x 60 minutes = 240 minutes
Then divide the total number of minutes by the number of practice sessions.
240 minutes ÷ 3 = 80 minutes ------Tony practices for 80 minutes each session.
Look Back. Tony practices over an hour each session. 80 minutes is 1 hour 20 minutes, so the answer
makes sense.
Number sentences are used to introduce students to concepts of structure and algebra prior to
more formal study.
84. 80
FIRMING-UP
Name: __________________________________ Score: ___________
Instructor: _______________________________
Directions: Write the number sentences to help you solve the following exercise.
1. Tony plays 3 concerts in a month. Each is the same length. He plays for a total of 2 hours 15
minutes. How long is each concert?
2. Tony takes 2 music lessons each week, for a total of 1 hour 20 minutes. Each is the same
length. How long is each lesson?
3. It takes Tony 15 minutes to get to his music lessons and the same amount of time to get
home. How much time does Tony spend traveling to and from his lessons each week?
4. Darius baked 40 treats for the 10 dogs staying at his kennel this week. He gave each dog the
same number of treats. How many treats did Darius give each dog?
5. Mrs. Sheppard bought 16 books. Each book cost $5. Which number sentence could be used
to find the total cost of the books?
85. 81
6. Steve read that 128 players entered the basketball tournament in his town. The organizers
split the players into 16 teams, with an equal number of players on each team. What number
sentence can be used to determine P, the number of players on each team?
7. Kyle has 350 paperclips. He will put the same number of paperclips into 5 boxes. Which
number sentence could be used to determine the number of paperclips Kyle will put in each
box?
8. Katy paced out a garden plot that measured 9 feet by 14 feet. What is the area of her garden
plot?
87. 83
Make a lesson plan for Grade 6 and the lesson is area of a rectangle
applying the “working backwards” strategy.
88. 84
FINALS
Name: __________________________________ Score: ___________
Instructor: _______________________________
Directions: Solve the following problems using any of the strategies you have learned. Show your
solutions.
Easter egg hunt
1. There are five eggs under Tim’s bed. There are four times as many chocolate eggs as
marshmallow eggs. How many marshmallow eggs are under Tim’s bed?
2. Danielle found 23 eggs. She smiled broadly because she had found nine more eggs than
Chris. Jennie smiled even more. She had found exactly as many eggs as Chris and Danielle
together. How many eggs did Jennie find?
3. Lee Ann is mixing blue dye and yellow dye so that she can have some green eggs. She will use
an equal amount of each color (blue and yellow). She has four blue dye tablets and seven
yellow dye tablets. Which one of the following statements can you be sure is NOT true.
Justify your answer.
a) Lee Ann needs more blue tablets to make the green dye.
b) Lee Ann has more yellow tablets than she needs to make the green dye.
c) Lee Ann must get green tablets to make green dye.
d) Lee Ann wants to dye eight eggs.
89. 85
Pumpkin Problems
1. Paula Pumpkin is heavier than Peter Pumpkin. Patrick Pumpkin weighs less than Peter
Pumpkin. Which of the following statements is NOT true?
a) Paula Pumpkin weighs more than Patrick Pumpkin.
b) Peter Pumpkin weighs less than Paula Pumpkin.
c) Patrick Pumpkin weighs more than Paula Pumpkin.
2. Pretty Pammy Perry prepared twenty perfect pumpkin pies. She placed one half of them in a
package and put the rest of them in her car. She drove the pies in her car to Pittsburgh. She
sold half of those pies and ate the rest of them. How many pumpkin pies did pretty Pammy
Perry eat?
3. Ryan had 74 pumpkin seeds. He put as many seeds as possible into eight bags, being careful
to make sure that each bag had the same number of seeds. He could not put _______ seeds
into the bags.
4. There are 5,097 pumpkins in Gwen’s pumpkin patch. There are 7,000 pumpkins in Andy’s
pumpkin patch. How can we figure out how many more pumpkins are in Andy’s pumpkin
patch than in Gwen’s?
a) It’s impossible to calculate from the given information.
b) Add 5,097 and 7,000.
c) Multiply 5,097 by 7,000.
d) Subtract 5,097 from 7,000.
90. 86
Mean Problems
Remember, mean = average.
1. Sandra 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. Sandra is eight years old. Her
partner must be _____ years old or younger.
2. Juan has played in four baseball games this season. He struck out an average of twice per
game. In the last three games, he didn’t strike out at all. How many times did he strike out in
the first game of the season?
3. Jerome 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?
4. Lucy 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?
91. 87
Video Time
1. Charlene wants to show her favorite video movie. It lasts for 130 minutes. Sam wants to
watch a video that lasts exactly two hours. Charlene’s video is _______ minutes longer than
Sam’s.
2. We've decided that we only have enough time to watch half of Sam’s video. That will take
_______ minutes.
3. Kelly showed four videos at her slumber party. They ran for 105 minutes, 180 minutes, 120
minutes, and 95 minutes. What was the average length of each video?
4. The videos at Kelly’s party ran _______ hours and 20 minutes altogether.
92. 88
Halloween Scene
1. Cassandra Witch was mixing up a cauldron of her favorite brew. She needed 104 ounces of
lemon flavored bat saliva for the recipe. However, she only had three quarts of the delicious
liquid. How much more lemon flavored bat saliva did she need?
2. Diego is buying his Halloween costume. He will either buy a gray dinosaur suit for nine dollars
or a green teacup outfit for $ 14.95. How much will he save by buying the less expensive
costume?
3. Patty is selling the 180 pumpkins she’s grown. She will charge the same price for each
pumpkin. She wants to collect at least 200 dollars from selling them all. How can she figure
out the minimum (least) price she needs to charge for each pumpkin?
a) She should subtract $ 180.00 from $ 200.00.
b) She should multiply 180 times $ 200.00.
c) She should divide $ 200.00 by 180.
d) She should divide 200 dollars by 180 dollars.
4. Ryan used up seven bags of candy corn at his Halloween party last year. There were 15
people at that party. This year, there will be 30 people at his Halloween party. What is a
reasonable estimate of the number of bags he’ll need for this year’s party?
5. Anne is dressing up as a sore throat for Halloween. Her costume will cost $ 43.99. She has a
twenty dollar bill, a ten dollar bill, a five dollar bill, and a one dollar bill. She also has a half
dollar. How much more money does she need to but the costume?
93. 89
Place Value and Palindrome Riddles
1. I am a three digit number. I am less than 500. I am greater than 200. All my digits are odd.
2. If you take each of my three digits and add them together, they equal 5. What number am I?
3. I am a four digit number. I have a one in my thousands place, and a two in my hundreds
place. I am a palindrome. (A palindrome reads the same, forwards and backwards. The words
“pop” and “level” are palindromes. The numbers “747" and “842248" are palindromes.)
What number am I?
4. I am a palindrome. I am >11 (greater than eleven) and <50 (less than fifty). I am an odd
number. What am I?
5. I am also a palindrome. I am greater than the number of days in a year and less than the
product of 19 and 20. What number am I?
6. I’m a seven digit number. Six of my digits are zeros. I am the greatest number possible with
those characteristics. What number am I?
94. 90
Riddles
1. I am a sum. My addends are five different whole numbers. All my addends are greater than
zero and less than eleven. All my addends are odd numbers. What am I?
2. I am a product. I have two factors. One of my factors is the last year of the twentieth century.
My other factor is half of a pair. What am I? Clue: Read the definition of “century” in your
dictionary.
3. I am a whole number represented by three digits. If you double me, I will still be represented
by three digits. That’s true of other numbers as well, but I am larger than any of them. What
am I?
4. I am a whole number represented by three digits. I am the smallest three digit whole number
that does not contain any 0, 1, 2, or 3. What am I?
5. I am a four digit whole number. Each digit is an even number. All the digits are different. I am
the greatest number that can be described that way. What am I?
95. 91
Assorted Problems
1. There are thirteen thousand thirty-three thirsty thinkers thinking of drinking seven hundred
seventy-seven salty sodas. They want to drink one salty soda each. How many more salty
sodas do they need?
2. Sometimes stunningly small Stephie strains so she can do seventy-six sweaty sit-up. If she
successfully does that each day in September, she will do a lot of sweaty sit-ups. To figure
out exactly how many sit-ups she’ll do, you need to __________.
a) Add 76 and 12.
b) Add 76 and 30.
c) Multiply 30 times 76.
d) Multiply 31 times 76.
e) Divide 76 by 31.
f) Subtract 30 from 76.
3. Freddy and Frank fry fabulous fish for Friday’s famous fish fry. Last month, Freddy and Frank
faithfully fried flounder for four Fridays. Altogether, they fried 4,544 flounder. On average (as
if they fried the exact number each day), how many flounder did Freddy and Frank fry each
Friday?
4. There was a young lady named Kay who brushed her teeth nine times each day. Her teeth,
they would last, but toothpaste went fast with this many brushings in May:
a) 40
b) 279
c) 270
d) 39
e) 229
96. 92
Rounding Riddles
1. I am a number. If you round the number of days in October to the nearest ten and round the
number of days in February to the nearest ten, I am half of the product of those two
numbers. What number am I?
2. I am an amount of U.S. money. I am the cost of five $ .88 hamburgers rounded to the nearest
dollar. How much money am I?
3. I am a number. I am the difference between 800 rounded to the nearest 10 and 800 rounded
to the nearest 100. How much am I?
4. I am a number. I am the missing number from each of two of the equations below. I am also
the sum of 237 and 240 rounded to the nearest 10. Which two equations do I complete?
a) 1,782 - 800 =
b) 8 X 60 =
c) 480 + 20 =
d) 5,322 - 4,842 =
5. I am a number. I am the smallest number that can become 500 when it’s rounded to the
nearest 10. What number am I?
6. I am a number. I am the largest whole number that must equal two thousand when rounded
to the nearest thousand. What number am I?
97. 93
7. I am a number. I am the sum of 1,270 rounded to the nearest 10, 1,270 rounded to the nearest
100, and 1,270 rounded to the nearest 1,000. What number am I?
8. I am a number. I am a multiple of nine. I am 30 when I’m rounded to the nearest 10. What
number am I?
9. I am a U.S. state. I am one of the 48 contiguous states. I border the Pacific Ocean. If you
round the number of days in two weeks to the nearest 10, you will get the number of letters
in my name. The last two letters in my name are o-n. What state am I?
10. I am a number. I am the year of Columbus’ famous voyage rounded to the nearest 1,000.
What number am I?
11. I am a number. I am the number of days in a decade rounded to the nearest thousand. What
number am I?
12. I am a day of the week. To find the number of letters in my name, multiply 78 by 88, round
that product to the nearest thousand and then divide that number by 1,000. What day am I?
