Problem: John likes to count things. He recently got a bag of marbles and counted them. This is what he told his mother. When I count by 2’s, there is one left over; by 3’s there is one left over; by 4’s there is one left over; by 7’s it comes out even. How many marbles were in John’s bag?
Math Topic/Concept: Problem solving using number facts, division, remainders
Materials: Manipulative such as marbles or plastic chips)
Classroom Use: (Introductory/Developmental/Evaluation)
Classroom use comments*: Students were told to solve their problems and write their solutions so that a stranger “off the street” could pick it up, understand what they did and why they did it. Give the students a couple of days to work on as homework and use class time to discuss any questions.
Grade: 5
Grade Cluster: (EarlyElem/LateElem/MS-Jr.High/EarlyHS/LateHS)
Illinois Goal: 6
Standard: 6B, 6C
Applied? (1-4): 2
Source: www.isbe.state.il.us/ils/benchmarking/PDF/cubes.pdf
Answer: 49 marbles
Strategies Listed: Guess and check, counting marbles, make an organized list
Solution: There are 49 marbles in all. One way to approach this problem is to make a chart. The clues in the problem said that the numbers 2, 3, and 4 went into the number with one left over. The number seven went in evenly. Since the number seven went in evenly I knew that the number of marbles had to be a multiple of seven. I filled in the chart with a yes or no according to if they went into the number properly. When I came to a column where all the answers were yes I knew it was the answer. That row was 7x7 or 49.
Other solution methods (if any): Since I realized the answer was a multiple of seven, I listed all the multiples of seven up to 210. I compared all the multiples that fit the requirements of counting by 2, 3, or 4. I found that both 49 or133 could be the answer. Investigating further from the resource of counting cubes from our MPAAC workshop, I found that 49 plus the multiples of 84 gives correct answers. I continued listing these possibilities as follows: 49. 133, 217, etc. John’s bag holds a maximum of 200 marbles. The possible answers are 49 or 133.
Intended rubric or assessment method: Analytic Scoring Scale (Jim Olsen - W.I.U.)
Write-up submitted by: Carl Carlson – Westmer School
Problem: Pose this situation: Bill and his dad, Mr. Jones, filled the Chevy Blazer tank at the Mobile station in the Quad Cities at 8:00 a.m. and drove to Chicago for a ball game. They crossed interstate 94 at 11:00 a.m. Mr. Jones was surprised because it usually takes him 3 and 1/2 hours to make that trip. Bill noted that the odometer recorded 228 miles. (Guide students in generating questions they might like to answer such as those which follow.)
* What was the average rate of speed he drove to get there?
* If gas cost $1.62 per gallon, and Mr. Jones used 15 gallons
for the trip,
how much did it cost him to drive to Chicago?
* How many miles per gallon did his Blazer get?
Students: Remember to ESTIMATE results (and record estimates) before solving! After solving, compare your results with a partner or your group, then compare your own estimates with results. Write a response to explain why your estimate is close or not close to the actual answer and describe how you chose information from the problem to use.
Math Topic/Concept: understanding and using concept of rate (with these questions, students will find miles/hour, gallons/mile, and cost/gallon but others are optional)
Materials: pencil paper and calculators
Classroom Use: (Developmental)
Classroom use comments*: Review the kinds of rate, concept of rate, and rate formulas with students prior to this activity.
Grade: 4-5
Grade Cluster: (LateElem)
Illinois Goal: 6
Standard: 6.B.2, 6.C.2a, 6.C.2b, 6.D.2
Applied? (1-4): 3
Source: adapted from Thinking Multiplicatively by J.R. Olsen
Answer: For the questions posed in the problem as given:
76m/h, $24.30 and 15 1/2 gal./mi
(Answers will vary according to questions generated by teacher with
students.)
Strategies Listed: estimation, computation, application of prior knowledge, use of calculator
Solution: Students apply formulas for solving problems of rate, label units, and write an explanation discussing the relationship between their estimates and actual result.
Other solution methods (if any)*: students may use repeated addition or subtraction, they may draw pictures, make charts or tables
Extensions or related problems*: Add questions such as: “If Bill’s dad had driven 65 miles/hour, how long would it have taken to drive the 228 miles to Chicago?” Add a task: such as: “Create a graph which demonstrates a comparison between two or three pieces of data.” Another possible option: use a computer program to manipulate the data.
Intended rubric or assessment method: Analytical Scoring Scale (Jim Olsen, WIU)
Write-up submitted by: Rebecca Cummins (Westmer CUSD 203)
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: Seth is having a birthday party and is expecting 28 people to attend. Seth asks his mother if she would make her homemade chocolate cake. Since she knew how much he liked it she agreed on one condition. The condition was that Seth had to make sure she would have enough ingredients to make enough cake for everyone. How will Seth know how much cake to make? How much of each ingredient will be necessary?
Recipe: 3 cups of flour, 2 cups of sugar, 1/3 teaspoon baking cocoa, 2 teaspoons baking soda, 1 teaspoon salt, 2 cups of water, 3/4 cup vegetable oil, 2 teaspoon vanilla extract, 2 teaspoons vinegar. Ingredients for the cream cheese frosting: 1package cream cheese-softened, 1/4 cup butter or margarine-softened, 2 cups of powdered sugar, 1/3 cup baking cocoa, dash of salt, 3 tablespoons milk, 1/2 teaspoon vanilla extract. This recipe yields 12-15 servings.
Math Topic/Concept: Ratios and Proportions, Multiply whole numbers and fractions.
Materials: Copy of original recipe
Classroom Use: (Introductory/Developmental/Evaluation)
Grade: 4-5
Grade Cluster: (EarlyElem/LateElem/MS-Jr.High/EarlyHS/LateHS)
Illinois Goal: 6
Standard: 6B2, 6C2
Applied? (1-4): 3
Source: www.isbe.state.il.us/ils/benchmarking/mathactivities.htm
Answer: Seth doubled the original recipe and make two cakes. The amount of ingredients are as follows: 6 cups of flour, 4 cups of sugar, 2/3 cup baking cocoa, 4 teaspoons baking soda, 2 teaspoons salt, 4 cups of water, 11/2 cups vegetable oil, 4 teaspoons vanilla extract, 4 teaspoons vinegar. Recipe for the cream cheese frosting: 2 packages cream cheese-softened, 1/2 cup of butter or margarine-softened, 4 cups of powdered sugar, 2/3 cup baking soda, 2 dashes of salt, 6 tablespoons milk, 1 teaspoon vanilla extract. This recipe yields 24-30 servings.
Strategies Listed: Make an organized list.
Solution: Seth read the recipe and noticed that the original recipe would make enough to serve 12-15 people. Since he knew that 28 people were coming he figured that two cakes would be enough for everyone. So he doubled all the ingredients and made two cakes instead of one. This would make enough cake to serve 24-30 people.
Intended rubric or assessment method:
Assessment ISAT Mathematics Grade 5 Student-Friendly Rubric
www.isbe.state.il.us/isat/rubric5.htm
Write-up submitted by: Carl Carlson – Westmer School
Problem: Boris Bear called a meeting of the bears and marmots of northern Yosemite, to talk about the problems with campers. The number of campers was increasing, and they were making more noise and leaving more trash. Altogether 178 bears and marmots attended the meeting, and there were 44 more bears than marmots. How many marmots and how many bears came to the meeting?
Math Topic/Concept: number sense, addition, subtraction, division
Materials: paper and pencil
Classroom Use: (Introductory/Developmental/Evaluation)
Grade: 5
Grade Cluster: (EarlyElem/LateElem/MS-Jr.High/EarlyHS/LateHS)
Illinois Goal: 6
Standard: 6.B.2
Applied? (1-4): 2
Source: The Problem Solver 5 by Creative Publications, 1987 (ISBN #0-88488-585-2)
Answer: 111 bears and 67 marmots
Strategies Listed: Logical reasoning or Guess and check
Solution: Altogether there are 178 bears and marmots. If the groups were equal, it would be 178 divided by 2 = 89. Since there has to be a difference of 44, by dividing 44 by 2 =22 and adding 22 + 89 = 111 then 89 – 22 = 67. 111 + 67 =178 and 111 – 67 = 44.
Other solution methods (if any)*: Using guess and check, the student would have to keep trying different combinations of numbers until they came up with the correct answer.
Extensions or related problems*: Change the numbers
Intended rubric or assessment method: Informal assessment
Write-up submitted by: Jonna Young
Problem: The kitten climbed its first tree and got stuck on the top branch. First it went up the trunk of the tree and on up to the 6th branch. A big squirrel scared the kitten and it climbed down 3 branches. A bird flew at the kitten and scared it again. Now it climbed up 10 branches. The kitten climbed back down 2 branches and then went up 4 branches to the very top of the tree. How many branches were in the tree?
Math Topic/Concept: Logical reasoning, adding and subtracting.
Materials: vertical line chart, paper, pencil
Classroom Use: (Developmental)
Grade: 4th grade
Grade Cluster: (LateElem)
Illinois Goal: It does not fit any goal specifically, but
definitely fits the Applications of
Leaning for Math in the Illinois State Goal. Perhaps Goal 6.
Standard: 6.B.2
Applied? (1-4): 2
Source: Problem Solver 4 by Judy Goodnow and Shirley Hoogeboom,
- Creative
Publication – ISBN 0-88488-584-4
Answer: 15 branches
Strategies Listed: Make a picture or diagram
Solution: Make one vertical line and then mark off lines
for the branches. When you
begin your diagram, you will show 6
branches.
Intended rubric or assessment method: Teacher observation
Write-up submitted by: Donna Spears
Problem: You need to decorate the 15- inch edge of a box using
a mixture of tiles in various patterns. The tiles are 2 inches, 3
inches, and 5 inches long.
· Find six ways to use the 2, 3, and 5- inch tiles to decorate
the edge of a box.
· Start by varying the number of 5-inch tiles you use.
Math Topic/Concept: Number sense, division, multiplication, addition, and measurement.
Materials: tiles
Classroom Use: (Introductory/Developmental)
Grade: 3
Grade Cluster: (LateElem)
Illinois Goal: 6, 7.
Standard: 6.B.2, 7.B.2.
Applied? (1-4): 2
Source: Facts That Last, Division, Creative Publications, 1999
Answer:
5-inch tiles 3-inch tiles 2-inch tiles
3
0
0
2
1
1
1
2
2
0
5
0
0
3
3
0
1 6
Strategies Listed: Guess and check. Make an organized list.
Solution: What is the fewest number of tiles that can be
used? 3 tiles
What is the greatest number of tiles that can be used? 7 tiles
Extensions or related problems*: You may vary the length and size of tiles.
Notes*: The problem can be simplified by asking for less than six ways. Then you can ask how many ways are possible.
Write-up submitted by: Diana Kent
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: Seth is having a birthday party and is expecting 28 people to attend. Seth asks his mother if she would make her homemade chocolate cake. Since she knew how much he liked it she agreed on one condition. The condition was that Seth had to make sure she would have enough ingredients to make enough cake for everyone. How will Seth know how much cake to make? How much of each ingredient will be necessary?
Recipe: 3 cups of flour, 2 cups of sugar, 1/3 teaspoon baking cocoa, 2 teaspoons baking soda, 1 teaspoon salt, 2 cups of water, 3/4 cup vegetable oil, 2 teaspoon vanilla extract, 2 teaspoons vinegar. Ingredients for the cream cheese frosting: 1package cream cheese-softened, 1/4 cup butter or margarine-softened, 2 cups of powdered sugar, 1/3 cup baking cocoa, dash of salt, 3 tablespoons milk, 1/2 teaspoon vanilla extract. This recipe yields 12-15 servings.
Math Topic/Concept: Multiply whole numbers and fractions, comparing quantities, Ratios and Proportions.
Materials: Copy of original recipe
Classroom Use: (Developmental)
Classroom use comments*: I listed the ingredients in the problem so that the class would get uniformed answers. A suggestion would be to allow them to bring in their own recipes or supply them with a copy of various recipes to choose from.
Grade: 4/5
Grade Cluster: (LateElem)
Illinois Goal: 6
Standard: 6.B.2, 6.C.2
Applied? (1-4): 3
Source: www.isbe.state.il.us/ils/benchmarking/mathactivities.htm
Answer: Seth doubled the original recipe and make two cakes. The amount of ingredients are as follows: 6 cups of flour, 4 cups of sugar, 2/3 cup baking cocoa, 4 teaspoons baking soda, 2 teaspoons salt, 4 cups of water, 11/2 cups vegetable oil, 4 teaspoons vanilla extract, 4 teaspoons vinegar. Recipe for the cream cheese frosting: 2 packages cream cheese-softened, 1/2 cup of butter or margarine-softened, 4 cups of powdered sugar, 2/3 cup baking soda, 2 dashes of salt, 6 tablespoons milk, 1 teaspoon vanilla extract. This recipe yields 24-30 servings.
Strategies Listed: Multiply whole numbers and fractions, comparing quantities and proportions
Solution: Seth read the recipe and noticed that the original recipe would make enough to serve 12-15 people. Since he knew that 28 people were coming he figured that two cakes would be enough for everyone. So he doubled all the ingredients and made two cakes instead of one. This would make enough cake to serve 24-30 people,
Extensions or related problems*: How many cakes would have to be made so that every fifth grader in school would get a piece? The answer is dependent upon how many fifth graders are in your school.
Intended rubric or assessment method:
Assessment ISAT Mathematics Grade 5 Student-Friendly Rubric
www.isbe.state.il.us/isat/rubric5.htm
Write-up submitted by: Carl Carlson – Westmer School
Problem: Each of the six boats in the race has one of these four-digit numbers on its sail.
6347 8356 5744
9123 8162 1287
· The boat with the highest number on its sail finished last.
· The sum of the digits on the first two boats was the same.
· The numbers on the boats in 2nd and 3rd place both ended with
an odd digit.
· The boats in 3rd and 5th place had all but one of the same
digits, but in different order.
From first place to last, in what order did they cross the finish line?
Math Topic/Concept: Logical Reasoning
Materials: Paper and pencil
Classroom Use: (Developmental)
Classroom use comments*: If I used this activity at the beginning of the year, I would have the students work with a partner to find the solution.
Grade: 3-4
Grade Cluster: (LateElem)
Illinois Goal: 6
Standard: 6B2
Applied? (1-4): 2
Source: http://eduplace.com/brain
Answer: From first place to last place, the boats finished in this order:
1st place: 5744
2nd place: 6347
3rd place: 1287
4th place: 8356
5th place: 8162
6th place: 9123
Strategies Listed: Chart or table
Solution: See answer
Intended rubric or assessment method: Student friendly ISAT Rubric
Write-up submitted by: Kathy Erlandson
James R. Olsen, Western Illinois University
E-mail: jr-olsen@wiu.edu
updated Aug. 20, 2001