Problem:  The students at Central Elementary School had a bake sale.  They gave the money they earned to the local library.  Mrs. Emery’s third-grade class raised $28.55 at the sale.  Ms. Gomez’s third-grade class raised $32.25.  How much more money did Ms. Gomez’s class raise?
 A  $4.70
 B  $3.70
 C  $16.30
 D  $50.80
 E.  $60.80

Math Topic/Concept: Number concepts and skills, subtraction

Materials: If a worksheet is desired, one can be made using the questions under classroom use comments.

Classroom Use: Developmental

Classroom use comments*:
· What facts do you know from the problem?
· What does the problem ask?
· What operation will you use to solve this problem?
· What numbers will you use to solve the problem?
· Show how you solved the problem.
· What is your answer to the problem?
· Does your answer make sense?  How can you tell?
How can you use a different operation to check if your answer is correct?
I would use the overhead and model a similar problem.  Then put this one up and have the students explain the steps.  They would write down the information the class discusses.  The rubric would then be given out to see if the class has followed all the steps for mastery.

Grade:  Third

Grade Cluster: Early Elementary

Illinois Goal:  Mathematics State Goal Number 6

Standard:  6.C.2a

Applied? (1-4):  3

Source:  Bridging the Test Gap in Mathematics-Book 3
 1999 Curriculum Associates, Inc.  CA8053-Single

Answer:  The Answer is (B) $3.70.  The students had to subtract and find the difference.  It dealt with money, so they had to watch dollar sign and decimal point.

Strategies Listed: Pick an operation

Solution:  They had to find the question and decide what operation was needed to solve it.  Then line up their figures, decimal points, and dollar signs.

Intended rubric or assessment method:  The rubric I will use is the scoring rubric for short answer questions for all grade levels.  It has clear BUZZ WORDS that will remind my students of what they will need to score well on the developmental level.

Write-up submitted by: Cheri Stanton

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:  The students in the fourth grade classes at Lincoln Intermediate School were
      asked to bring empty aluminum cans to school for the recycling program.  The
      cans were stored in large plastic bags for pick up.  Each bag held about 100
      cans.  Here are the results.
  Mrs. Spears’ class  562 cans
  Mrs. Young’s class 445 cans
  Mrs. Stewart’s class 462 cans
  Mrs. Paulsgrove’s class 375 cans
  Mrs. Leath’s class 318 cans
Estimate the number of cans the students brought in.  Explain how you found
                 your estimate.  About how many bags were filled with cans?

Math Topic/Concept:  Estimation, Number relationship, Rounding

Materials:  Paper, pencil

Classroom Use: (Evaluation)

Grade:  4th grade

Grade Cluster: (LateElem)

Illinois Goal: 6

Standard:  6.C.2

Applied? (1-4):  4

Source:  Explain it!  (Grades 3-4) By Creative Publications – ISBN 0-7622-1597-6

Answer:  2,200 cans,   22 bags

Strategies Listed:  Rounding

Solution:  I used rounding to find my answer.
  562 – 600
  445 – 400
  462 – 500
  375 – 400
  318 - 300
------------------
         2,200
      They probably filled about 22 bags, but one bag might only be half full.
     Because 600 is bigger than 562, 500 is bigger than 462, 400 is bigger than 375,
     my answer is bigger than the real answer.

Other solution methods (if any)*: I think they filled about 21 bags.  I think that because
         I added the numbers in the hundreds column.
         500 + 400 + 400 + 300 + 300 = 1900.
         But the numbers were really bigger, so I thought it
         could be another two hundred.
        1900 + 200 = 2100

Intended rubric or assessment method: ISAT  Rubric

Write-up submitted by:  Donna Spears

Problem: Ann, Trevor, Brendon, Melissa, and Nina are reading a book that has 98 pages.

      Anna has 36 pages left.

· Trevor has 4 pages more to read than Anna.
· Brendon has 11 fewer pages to read than Trevor.
· Melissa has read 2 more pages than Brendon.
· Nina has read 5 more pages than Anna.

                 What page is each person on?

Math Topic/Concept: Logical Reasoning

Materials: Paper and pencil

Classroom Use: (Introductory/Developmental/Evaluation)

Classroom use comments*: I would use this activity at the beginning of the year to review basic addition and subtraction facts.

Grade: 3-4

Grade Cluster: (EarlyElem/LateElem/MS-Jr.High/EarlyHS/LateHS)

Illinois Goal: 6

Standard: 6C2a

Applied? (1-4): 3

Source: http://eduplace.com/math/brain

Answer: The page that each person is on is:
 Anna – p. 62
 Trevor – p. 58
 Brendon – p. 69
 Melissa – p. 71
 Nina – p. 67

Strategies Listed: Guess and Check, use logic

Solution: Anna: Anna has 36 pages left, out of a total of 98 pages. 98-36=62

Trevor: Subtract 4 from 62, because Anna read 62 pages and Trevor read 4 pages less than she did. 62-4=58

Brendon: If Brendon has 11 fewer pages left to read than Trevor, he has read 11 pages more already. 58+11=69.

Melissa: She has read 2 more pages than Brendon. Brendon is on page 69. 69+2=71.

Nina: Nina has read 5 more pages than Anna. Anna is on page 62. 62+5=67.

Intended rubric or assessment method: Percentage based assessment

Write-up submitted by: Kathy Erlandson

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.

Intended rubric or assessment method:  Analytic Scoring Scale (Jim Olsen - W.I.U.)

Write-up submitted by: Carl Carlson – Westmer School

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

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: 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.

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:  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:  Complete the circle puzzle following these directions.  Write a number in each of the empty sections.  The numbers must be placed so that each number is opposite a number which is double its value or half its value.  The sum of all the numbers in the puzzle should be 36.

Math Topic/Concept:  Number sense, addition, subtraction

Materials: Circle puzzle and pencils

Classroom Use: (Introductory)

Grade:  5

Grade Cluster: (LateElem)

Illinois Goal:  6C.2b

Standard:  6C.2b

Applied? (1-4):  1

Source:  Ten Minute Math Mind Stretchers  by Laurie Steding  ISBN  0-590-86563-3

Answer: 

Strategies Listed:   Guess and check

Extensions or related problems*:  You could also give your students a blank circle puzzle and see what they come up with.

Intended rubric or assessment method:   Informal observation

Write-up submitted by:   Ann Hulsizer, 5th grade, Monmouth
 
 

James R. Olsen, Western Illinois University
E-mail: jr-olsen@wiu.edu
updated Aug. 20, 2001