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Math Trail 2002--Thinking Logically, Solving Problems
Solving problems logically is part of the mathematical thinking skills students are to acquire during the intermediate elementary grades. Most of all, these are fun as well as challenging. Students love solving these problems, particularly when they know the "tricks" to getting the right answer--the use of diagrams! My students loved writing these problems and the demand for clear thinking and careful writing were very much in evidence.
Students have written several logic problems that reflect the situations that they encounter regularly at school. Read the description of each of the types of problems and then work the problems that students have designed. You will need to draw the diagram to solve the problem on scratch paper. If you are not familiar with how to solve table, circle and matrix problems, read the hints about How To Solve them.
Janet Beyersdorfer
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Group Written Problem: This problem demands that students determine the placement of six rooms down one long hallway. You may find the chart a useful starting point after you read through the clues. You can solve the problem by carefully reading all the clues and labeling the rooms. Some clues will be very specific. Some clues must be used in connection with another clue. Read each clue each time another piece of information is recorded in the chart. The additional information may help you solve the problem.
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Hallway Problem
Clues
1. Ms Kahn’s room is four rooms to the north of the art
room.
2. The music room is the last room to the south.
3. Mrs. Henderson’s
room is between Ms. Kahn’s room and Mrs. Zima’s.
4. Mrs. Henderson and Ms.
Kahn teach the same grade level students.
5. Mrs. Zima’s room is the only
third/fourth multiage class in that hallway.
6. Mrs. Armstrong teaches the
youngest students in this hallway, grade two students.
7. The art room is
next to the Music room.
8. Mrs. Henderson’s is adjacent to Mrs. Zima’s room.
Mrs. Zima teaches some students of the same grade level as Mrs. Henderson. There
are no straight fourth grade classes in this hallway.
Problems Written by Pairs of Students:
Studying Insects (Circle Problem)
Ants, crickets, and grasshoppers are fun to study. Decide which student is studying which insect.
Clues:
1. Karen is studying crickets and that’s all.
2. Hans
studies all of the insects.
3. Sarah does not like to study any
insects.
4. Robert does not study any insect except ants.
5. Mara is
studying crickets and grasshoppers.
6. Jason likes to study grasshoppers and
ants.
Class Schedules (Matrix Problem)
Mrs. Zima’s class has different classes on Monday, Tuesday, Wednesday, Thursday, and Friday. The classes are music, gym, computer, art, and library. Decide which class is on which day.
Clues:
1. Music is on a day that starts with T.
2. Gym is on
the fourth day of the school week.
3. Computer is not in the middle of the
school week.
4. Art never comes at the beginning of the school week.
5.
Library is right in the middle of the week.
Favorite Foods (Matrix Problem--two variables)
Mac and cheese, pizza and tacos are some kids’ favorite foods. Sue, Jack, Ann, Sam, Lucy, and Ben all like different meals. Use the correct diagram to identify which food each person likes.
Clues:
1. Sue likes everything except pizza.
2. Jack does
not like mac and cheese but likes everything else.
3. Ann only likes
pizza.
4. Sam likes everything.
5. Lucy likes everything but tacos.
6.
Ben does not like anything.
Seating in the Art Room (Table Problem)
Corrie, Patrick, Kelly, and Ben attend art class together. In the art room, they have assigned seats. Each table has ten chairs. Two chairs are on each short end of the table. There are three chairs on each of the two long ends. Which chair is assigned to which student?
Clues:
1.Corrie is the only one who sits on an
end.
2.Patrick sits 4 seats to the left of Corrie.
3.Ben sits 5 spaces to
the right of Patrick.
4.Kelly sits three seats to the left of Ben.
Time to Read! (Matrix Problem--two variables)
Ben, Jason, Karen, and Laura are reading different books. The books are
How To Draw Faces, Dogs, Harry Potter, and Nothing But The Truth. Try
to match the student to the book he or she is
reading.
Clues
1. Jason really likes to draw.
2. Ben
doesn't like dogs at all.
3. Karen doesn't like Harry Potter
4. Jason and
Laura don’t like dogs.
5. Ben doesn't like Nothing But The Truth.
Lunch Time Friends (Table Problem)
Ben, Karen, Laura, and Jason are sitting around a table eating lunch. Match their first name to their last name. You also have to figure out where they sit.
Clues
1.Laura sits across from Ben.
2.The four last
names are Woodson, Elwind, Halberg, Newland
3.Laura`s last name starts with
the letter N.
4.Jason Elwind is on Ben`s left.
5.Laura sits next to Karen
Woodson.
6.Ben sits on the right of the table.
Student
Problems
Line Up in Squad
Order! (Modified Table George, Lucas, Chen, Liz, Ben, Sam, Kelli, and Lisa are
asked to line up in squad order but they forgot the squad order! Help the kids
to get back into squad order. Squads line up on the horizontal blue line in the
gym. Use the clues to help you.
Clues:
1. Kelsey is at the end of the line.
2. Lisa is between Sam and
Kelli.
3. Sue is at the opposite of Kelsey.
4. Lisa is four people behind
Sue.
5. George is two people in front of Ben.
6. Chen is between Ben and
George.
7. Lucas is behind Sue.
8. Liz is six people in front of Ben.
9. The front of the line is on the left.
10. Ben is the second from the
end of the line.
Picnic Time (Matrix Problem--three variables)
We are having picnic at school. Kyle,Yusuke, Sarah, and Karen are going to the picnic. They are eating one of the following sandwiches: turkey, baloney, P B and J, or cheese. They will eat at different times the times are 10:00, 11:00, 12:00 and 12:3 0. Decide when each ate lunch and what kind of sandwich eat enjoyed.
Clues:
1 . Kyle eats before Karen.
2 . Yusuke hates
baloney.
3 . Sarah eats before Karen, but after Kyle.
4 . Karen never eats
at 12:00.
5 . Sarah hates P.B. J.
6 . Kyle does not eat at 11.00.
7 .
Yusuke eats before Sarah and after Kyle.
8 . Karen likes baloney.
9 . Kyle
eats turkey.
10 . Yusuke is a vegetarian
11 . Kyle hates jelly.
12 .
Sarah hates meat.
Math Content: Logic Problems, Relationships
Math Objectives: Students will think logically and analyze information to solve logic problems. Students will construct their own logic problems to further their ability to organize and analyze information to arrive at logical conclusions.
Goals: Illinois State Goal 8: Use algebraic and analytic methods to identify and describe patterns and relationships in data, solve problems, and predict results. Interpret and describe numerical relationships using variables and patterns; using tables, graphs and symbols.
How to Solve These Logic Problems:
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How to Solve a Table Problem: 1. Read the introduction and all the clues for the problem to identify the variables/categories. 2. There are two rules to remember: partners sit across from one another; people always sit facing the table. 3. Write each person's first name on the top line and the last name on the bottom line. 4. Read each clue each time another piece of information is recorded in the chart. The additional information may help you solve the problem. 5. Make sure that each person has a first and last name as well as being seated with the correct partner and those on the left and right. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
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How to Solve a Circle Logic Problem: 1. Read the introduction and all the clues for the problem to identify the variables/categories. 2. Label the circles by placing the first category as a name for the left circle, the second category as the name for the right circle, and the third category as the name for the lowest circle. 3. Remember that overlapping circles indicate shared category items. 4. Identify the item by using the number inside the portion of the circle that corresponds to the item's characteristics. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
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How to Solve a Matrix Problem: 1. Read the introduction and all the clues for the problem to identify the variables/categories. 2. Write the names for the variables/categories in the matrix. (In this diagram there are 3 variables represented by an item number, the alphabet letters A--E, and the symbols. 3. Locate the clues that provide specific information which answer Yes. Write the letter Y or an X in that category's box. Fill in the remaining boxes with N's or O's in the boxes within that category's row/column. 4. Find the clues that give some information which when used together will lead to a Yes answer. 5. Review your answers to make certain that you have provided the information requested by the problem. |
Answers:
Group Problem
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room 1 Ms. Kahn Gr. 3 |
room 2 Mrs. Henderson Gr. 3 |
room 3 Mrs. Zima Gr. 3/4 |
room 4 Mrs. Armstrong Gr. 2 |
room 5 Mrs. Meyers Art |
room 6 Mrs. Evens Music |
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Robert--1 Karen--2 Jason--4 Hans--5 Mara--6 Sarah--no numbers |
Monday--Computer |
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Studying Insects |
Class Schedules |
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Jason is reading How to Draw Faces |
Use compass points to determine answer for this square
table problem:
North: Karen Woodson |
| Time to Read | Lunch Time Friends |
| Ann--1 Jack--2 Sue --6 Sam--5 Lucy--4 Ben--no numbers |
Names of students lined up from left to right: Sue, Lucas, Liz, Sam, Lisa, Kelli, George, Chen, Ben, Kelsey. |
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Favorite Foods |
Squad Order |
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Kyle eats turkey at 10:00 Yusuke eats PB&J at 11:00 Sarah eats cheese at 12:00 Karen easts balony at 12:30 |
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Seating in the Art Room |
Picnic Time |