Problem: Marcus, Alex and Victor went on a picnic. They each brought some food. Altogether they had peanut butter sandwiches and cheese sandwiches, bananas and apples, bags of corn chips and bags of potato chips. Draw a picture to show the different lunches they could make if for each lunch they took 1 sandwich, 1 piece of fruit and 1 bag of chips.
Math Topic/Concept: Counting, Number sense, organizing data
Materials: Paper, pencil
Classroom Use: (Introductory/Developmental/Evaluation)
Grade: 5
Grade Cluster: (EarlyElem/LateElem/MS-Jr.High/EarlyHS/LateHS)
Illinois Goal: 6B.2 and 10C.2
Standard: 6B.2 and 10C.2
Applied? (1-4): 3
Source: Ten Minute Math Mind Stretchers by Laurie Steding ISBN0-590-86563-3
Answer: 8 different lunches
Strategies Listed: Make an organized list, Make a drawing.
Solution: There are four different combinations with a peanut butter sandwich: (1) Peanut butter sandwich, banana, corn chips, (2) Peanut butter sandwich, banana, potato chips, (3) Peanut butter sandwich, apple, corn chips and (4) Peanut butter sandwich, apple, potato chips. There are four different combinations with cheese sandwiches: (1) Cheese sandwich, apple, potato chips, (2) Cheese sandwich, apple, corn chips, (3) Cheese sandwich, banana, potato chips, and (4) Cheese sandwich, banana, corn chips.
Intended rubric or assessment method: Informal Observation
Write-up submitted by: Ann Hulsizer, 5th Grade, Monmouth
Problem: It was Cheryl's first day at school. The teacher suggested that it would be a good idea for each child to meet every other child in the class. The teacher said, "When you meet, please shake hands and introduce yourself by name." If there were 15 children in the class, how many total handshakes were there? It is assumed that every child shakes hands with every other child once and only once.
Math Topic/Concept: Combinations, Number sense, addition.
Materials: pencil, paper
Classroom Use: (Developmental)
Classroom use comments*: It would be fun to act this out, too.
Grade: 5
Grade Cluster: (LateElem)
Illinois Goal: 6.B.2, 10.C.2
Standard: 6.B.2, 10.C.2
Applied? (1-4): 3
Source: http://www.syvum.com
Answer: 105
Strategies Listed: Make a chart, look for a pattern, use a simpler problem, repeated additions.
Solution: The class has 15 children. The first child
shakes hands with the other 14 children. The second child has already shaken
hands with the first child, and so has to shake hands with only the other
13 children. In this manner, the second-last child has to shake hands with
only one child, and the last child has already met all the children.
Thus, the number of handshakes is
14 + 13 + ……. + 2 + 1 =105.
Other solution methods (if any)*: Students could also make a chart. “15 Choose 2” is 105.
Extensions or related problems*: many possibilities to this handshake problem.
Intended rubric or assessment method: Grade 5 "Student Friendly" Mathematics Scoring Rubric found at http://www.isbe.state.il.us/isat/rubric5.htm
Write-up submitted by: Ann Hulsizer, 5th Grade, Monmouth
Problem: Mike was dealing out the cards for Julie. After he dealt out 4 cards, he told Julie to turn them over. “Now,” he said, “how many different ways can you arrange your four cards in a stack?” Julie had these four cards: 3, 5, 7, and 9. How many ways are there?
Math Topic/Concept: Patterning, permutations
Materials: Decks of cards
Classroom Use: (Evaluation)
Classroom use comments*: I would use this as a problem solving activity to practice for the ISAT test.
Grade: 4
Grade Cluster: (LateElem)
Illinois Goal: 10
Standard: 10C2
Applied? (1-4): 2
Source: The Problem Solver 4 by Judy Goodnow and Shirley Hoogeboom, Creative Publications
Answer: There are 24 ways:
Strategies Listed: Chart or table, Guess and check
Solution:
3579 5379 7359 9357
3597 5397 7395 9375
3759 5739 7539 9537
3795 5793 7593 9573
3957 5937 7935 9735
3975 5973 7953 9753
Other solution methods (if any)*: 4x3x2x1 = 24
Extensions or related problems*: You could have the students use 5 different cards and find all the ways to stack them.
Intended rubric or assessment method: Student friendly ISAT Rubric
Write-up submitted by: Kathy Erlandson
Problem: For this problem you need 5 different colored slips of paper. Find how many ways you can arrange the slips of paper in 2 groups. A group can have 1 or more slips of paper in it. Draw pictures or make a table to record the different groups.
Math Topic/Concept: Number sense, combinations, partitions.
Materials: 5 different colored slips of paper per child
Classroom Use: (Introductory)
Classroom use comments*: Students could work in pairs.
Grade: 5
Grade Cluster: (LateElem)
Illinois Goal: 10
Standard: 10.C.2
Applied? (1-4): 2
Source: Ten Minute Math Mind Stretchers by Laurie Steding ISBN 0-590-86563-3
Answer: There are 15 different ways to arrange the slips of paper into 2 groups.
Strategies Listed: Manipulating the colored slips of paper into groups.
Solution: The 15 ways to arrange the colored paper into groups is as follows- 5 ways to have 1 slip of paper in 1 group and 4 slips in the other, and 10 ways to have 2 slips in 1 group and 3 slips in the other.
Extensions or related problems*: What if 6 different colored slips were used? What if you grouped them in 3's?
Intended rubric or assessment method: Informal observation
Write-up submitted by: Ann Hulsizer, 5th Grade, Monmouth
Problem: "Lucky Ducks" is a popular game at Lincoln School's
carnival. Here's how is works. Students pick a duck out of
a large tub of water filled with plastic ducks that are floating around.
If the bottom of the duck is marked Prize, a prize is awarded. If
the duck is not marked, the student receives a duck sticker. The
duck is then returned to the tub.
A large display board shows that so far, 43 people have received stickers
and 21 people have won prizes.
Ricky thinks he might try his luck, but first he wants to know his
chances of winning a prize. His friend Alex has suggested that he
can figure out how likely it is he'll win a prize from the results on the
board.
What are Ricky's chances of winning a prize? Would you consider
his chances favorable or unfavorable? Explain your thinking.
Math Topic/Concept: Probability
Materials: Paper, pencil, could also have paper ducks cut out with "prize" written on them or plain.
Classroom Use: (Introductory)
Grade: 5
Grade Cluster: (LateElem)
Illinois Goal: 10.C.2a
Standard: 10.C.2a
Applied? (1-4): 2
Source: Explain It! Grades 5-6 Creative Publications ISBN0-7622-1598-4
Answer: One out of every 3 people who tried the game won a prize. The chances of winning a prize are pretty good. And remember, there is a sticker as a consolation prize.
Strategies Listed: make a chart, guess and check, manipulate the cut-out ducks
Solution: If 21 people won a prize and 43 people got a
sticker, then 64 people played the game. That means 21 out of 64
people won prizes, and the fraction 21/64 is about the same as the fraction
1/3. Ricky has 1 chance out of 3 to win a prize. I think the
chances are favorable.
People that won / People that played = 21/64
= 1/3
Other solution methods (if any)*: Since 43 people got stickers, that's about two times as many as the 21 people that got prizes. That means that for every 2 people that got a sticker, 1 got a prize. Ricky's chance of winning a prize are good.
| Sticker | Sticker | Prize |
Intended rubric or assessment method: Grade 5 "Student Friendly" Mathematics Scoring Rubric found at http://www.isbe.il.us/isat/rubric5.htm
Write-up submitted by: Ann Hulsizer, 5th Grade, Monmouth
Problem: It was dark in the morning when Jesse was getting dressed. He reached into his sock drawer, where he kept 10 pairs of white socks and 5 pairs of black socks. (Of course, they weren't together in pairs!) What is the fewest number of socks Jesse would need to pull out of his drawer before he could be positive he'd have two matching socks of the same color?
Math Topic/Concept: Probability
Materials: paper, pencil
Classroom Use: (Introductory)
Classroom use comments*: Students could use colored pieces of paper cut out like socks to act this out.
Grade: 5
Grade Cluster: (LateElem)
Illinois Goal: 10.C.2a
Standard: 10.C.2a
Applied? (1-4): 2
Source: Ten Minute Math Mind Stretchers by Laurie Steding ISBN 0-590-86563-3
Answer: 3 socks
Strategies Listed: Students could draw or write down possible socks pulled out of the drawer. For example, they might draw a sock and label it with a "W" for white, the next one labeled "B" for black and so on. If they are acting this out, they could pull black or white paper socks from an imaginary drawer.
Solution: The first sock pulled out would be either black or white. The second sock might match the first sock or it might not. The third sock will either match the first sock, the second sock, both socks or neither sock, but in that case, the first 2 socks would have to match.
Intended rubric or assessment method: Informal observation
Write-up submitted by: Ann Hulsizer, 5th Grade, Monmouth
Problem: Kendra’s father likes to design unusual dartboards.
He just completed the four designs shown below.
Kendra’s friend, Maurice, would like to buy one of these new dartboards; but first, he wants to make sure that the board is fair. A board is fair when a player has an equal chance to win whether he/she chooses to aim for the shaded areas or the white areas.
Are all of these boards fair? If you find one that is unfair, tell which color has the advantage. Then explain how you might change the board to make it fair.
Math Topic/Concept: Probability
Materials: Paper and pencil
Classroom Use: (Evaluation)
Classroom use comments*: Some questions you might want to ask the students are: How can you determine if the shaded areas and unshaded areas offer an equal chance to win? Could you add squares to the board and make the board fair? Could you make the board fair by removing squares?
After doing a unit on probability, I would use this as an activity to assess their knowledge of probability.
Grade: 3-4
Grade Cluster: (LateElem)
Illinois Goal: 10
Standard: 10C2a
Applied? (1-4): 3
Source: Explain It Grade 3-4 by Creative Publications
Answer: Board A, B, and Dare fair. Board C is not fair.
Strategies Listed: Drawing or Model
Solution: First I checked to see if all the sections on a board were the same size. I counted how many sections there were. Then I checked to see how many were shaded and how many were white. To be a fair board, there should be the same number of each kind.
I found that Board C was not fair. It has 15 squares, 8 are shaded and 7 are white. That gives an advantage to the person choosing the shaded squares.
If you add another row of squares to the board there will be 20. Then
fill in the pattern and you’ll have 10 white squares, and the board will
be fair.
Other solution methods (if any)*: If you match the shaded sections
with the white sections, for dartboards A, B, and D the sections match
evenly. Dartboard C has 1 extra shaded square, so it gives an advantage
to the person picking shaded squares.
One way to fix this dartboard is to take off the last column of squares.
Then there will be a white square for every shaded square and the board
will be fair.
Extensions or related problems*: You could ask students to design their own dartboard that is a fair one.
Intended rubric or assessment method: Student friendly ISAT Rubric
Write-up submitted by: Kathy Erlandson
Problem: The Spears family is planning a summer vacation.
They are trying to decide
whether to travel by plane, train, or car.
They will go to Yosemite National
Park, Yellowstone National Park, or the Grand
Canyon. They can stay in a
motel or campground. What are all the
different trips that the Spears could
plan?
Math Topic/Concept: Logical reasoning, Multiplication principle
Materials: Paper, pencil
Classroom Use: (Evaluation)
Grade: 4th grade
Grade Cluster: (LateElem)
Illinois Goal: 10
Standard: 10.C.2a
Applied? (1-4): 4
Source: The Problem Solver 4 by Judy Goodnow and
Shirley Hoogeboom – Creative
Publications – ISBN 0-88488-584-4
Answer: 18 different trips
Strategies Listed: Organized list
Solution: make an organized list
Intended rubric or assessment method: Student-Friendly ISAT rubric
Write-up submitted by: Donna Spears
Problem: At the beginning of each math class, ask, “What are your chances of having your name drawn from the can today?”
Math Topic/Concept: concepts of probability, ratio, and patterns
Materials: 3 colors of index cards, marker, lamination or clear contact, and coffee can
Classroom Use: (Introductory)
Classroom use comments*: This is a daily activity to be used in conjunction with another activity, such as “How’s the Weather?” Teacher will label 3 index cards with each student’s name (3/student), laminate, and place cards in large coffee can (or box).
Grade: 4 (Developmental at 5)
Grade Cluster: (LateElem)
Illinois Goal: 10
Standard: 10.C.2a and 10.C.2c
Applied? (1-4): 3
Source: Rebecca Cummins
Answer: varied from day to day.
Strategies Listed: look for pattern, use logical reasoning, make a table, draw a diagram or chart
Solution: Students will respond to the question variously, depending on whether or not their names have been drawn and depending on how many other names have been drawn and how many total number of name cards are left in the can. For example, a student may still have all three of his name cards in the can, and if there are 35 total number of cards left in the can (only one name drawn from the original 36—representing the total of 12 students in math class), then the student would respond that his chances are 3 in 35.
Extensions or related problems*: Students can interpret their chances in terms of percents as well as fractions. For a challenge, occasionally ask students what are their chances for their blue name card being drawn today.
Intended rubric or assessment method: NA
Write-up submitted by: Rebecca Cummins (Westmer CUSD
203)
Problem: There are three types of ice cream at Robbins-Carlson ice cream shop: vanilla, chocolate, and strawberry. How many different double-dip cones can be ordered?
Math Topic/Concept: Combinations
Materials: Three different-colored pencils, crayons or markers per group, three different-colored cubes, about 25 per student or group, recording sheet.
Classroom Use: (Developmental/Evaluation)
Classroom use comments*: Expect to discuss whether a chocolate with vanilla on top is the same or different from a cone with vanilla with chocolate on top. The solution given below treats these as different. If you wish to consider them the same, then your solution will differ from the one given.
Grade: 1 - 5
Grade Cluster: (EarlyElem/LateElem)
Illinois Goal: Goal 6, Goal 8, and Goal10
Standard: 6B1, 6B2, 6B3a, 6B3b, 8A2a, 8B1, 8B2, 8D3a, 10C1
Applied? (1-4): 2
Source: First Grade Academic Expectations, Galesburg Public Schools, Galesburg, IL
Answer: 9.
Strategies Listed: With the appropriate materials, the children would be expected to draw, or otherwise model, all the possible double-dip cones.
Solution: Three choices for each. 3x3 = 9.
Extensions or related problems*: The problem can be posed with a different number of flavors or with a different number of scoops or both. Using four flavors with two scoops is a simpler problem than using three flavors with three scoops.
Intended rubric or assessment method: A possible rubric
for first grade could be:
Exceeds: Student finds all nine cones. Student is able to explain approach
and justify solution.
Meets: Student finds most of the cones. Student has a fairly well-organized approach to finding the solution and can explain the approach.
Does not meet: Student finds some of the cones. Has some repeats or has no organized approach to finding solutions.
Write-up submitted by: Melfried Olson
James R. Olsen, Western Illinois University
E-mail: jr-olsen@wiu.edu
updated Aug. 18, 2001