Problem: Jonna brought home a strip of wood 120 cm in length
and 2 cm in width. She instructed her daughter, Diana, to make her
a rectangular picture frame and to use all of the wood. She wanted
the frame to be made as shown below.
What are several sizes of frames Diana could make?
What frame would give Diana the largest enclosed area?
Frame Style:
Math Topic/Concept: Area and perimeter.
Materials: Pencil, paper, graph paper, calculators (optional)
Classroom Use: (Developmental/Evaluation)
Classroom use comments*: It will likely take some discussion to decide that the outside perimeter and the insider perimeter of the design is not 120. The design is not the strongest design, but has been chosen so the problem is easier to solve. Students may need to spend time thinking about how the boards need to be cut so that the rectangular frame can be made.
Grade: 4 - 6
Grade Cluster: (LateElem/MS-Jr.High)
Illinois Goal: Goal 7 and Goal 6
Standard: 7A3b, 6B3a, 6B3b, 6C3a, 6C3b
Applied? (1-4): 2
Source: original
Answer: There are several rectangles that can be made. The rectangle with an outside perimeter of 32 cm and an inside perimeter of 28 cm gives the largest area for the picture. This area is 576 cm2.
Strategies Listed: Making an organized list.
Solution: See answer. It is important to note that the area on the inside of the frame is desired. The width of the wood matters here as well.
Extensions or related problems*: The length or the width of the board can be varied. Different ways to make the frame could also be discussed.
Intended rubric or assessment method: The following rubric
could be used.
Exceeds: Student finds a pattern to the length and width combinations
and lists possible combinations. Clear explanation and/ or work to show
understanding. Finds the combination that makes the largest area
and explains why it is the largest. May use decimals or fractions in the
pattern and identify an infinite number of possibilities.
Meets: Student inds a pattern to the length and width combinations and lists combinations with perhaps a few errors. Finds the frame dimensions that make the largest area but does not provide justification. Otherwise, explains work fairly well.
Does not meet: Student may find some patterns or sizes for the frame. Work contains major errors. May not find the rectangle with the largest area. Does not explain or gives poor explanations.
Write-up submitted by: Melfried Olson
Title: You are what you eat!
Problem: 
How many grams of carrots are in the entire container?
How many calories are in the entire container?
How many calories are there in one gram of carrots?
Math Topic/Concept: Rates
Materials: Paper/pencil, handout with nutrition facts
Classroom Use: (Introductory/Developmental/Evaluation) Developmental
Classroom use comments: This would be a good problem to use after introducing basic rate information. Students would also need to be familiar with multiplying and dividing decimals.
Grade: 6th.
Grade Cluster: Middle School
Illinois Goal: 7: Estimate, make and use measurements of objects, quantities and relationships and determine acceptable levels of accuracy.
Standard: Measure and compare quantities using appropriate units, instruments and methods.
Applied? (1-4): 4
Source: Original. Nutrition information from www.nutrition.gov
Answer: 212.5 grams of carrots, 112.5 calories, about .5294 calories per gram
Strategies Listed: Choose pertinent information, multiply, divide.
Solution: There are 2.5 servings per container. There are 85 grams of
carrots in a serving. 2.5*85=212.5 grams of carrots. There are 45 calories per
serving. 2.5*45=112.5.
112.5/212.5 OR 45/85 = .5294 calories per gram.
Other solution methods (if any):
Extensions or related problems: Find grams of sugar per container, miligrams of sodium per container, grams of fiber per container, etc.
Intended rubric or assessment method:
1 point for correctly setting up the problem.
1 point for calculating properly.
Write-up submitted by: Megan Orton
Title: It’s a Splash
Problem: The Sherman family has a pool 30 ft. long and 25 ft. wide. There is a walkway 4.5 ft. wide around the pool. A) What is the perimeter of the pool? B) What is the perimeter of the outside of the walkway?
Math Topic/Concept: Measurement, Perimeter
Materials: Straightedge, plain paper
Classroom Use: (Developmental)
Classroom use comments*: Use in the unit on perimeter/area.
Grade: 7
Grade Cluster: (MS-Jr.High)
Illinois Goal: 7
Standard: 7.A.3b
Applied? (1-4): 2
Source: Transition Mathematics (Scott Foresman)
Answer: A) 110 ft. B) 146 ft.
Strategies Listed: Computation, Draw a Picture
Solution: A) Use perimeter formulas or add all the sides. B) Each side increases by 9 ft. (4.5 ft. is on each side).
Extensions or related problems*: Find the areas of the pool and walkway. If given yard dimensions could the pool fit?
Notes*: Maybe need to do similar problems before so they understand the concept of increasing on each side.
Write-up submitted by: Denise Mann and Jenni Robinson
Problem: Rebecca is wrapping presents for people at the shopping mall. She had 8 ½ yards of ribbon when she began. She used 2 1/3 yards of ribbon to wrap one present. She needs to wrap 2 more presents of the same size. Does she have enough ribbon?
Math Topic/Concept: Measurement, Fractions
Materials: Optional: Ribbon, yardstick
Classroom Use: (Developmental)
Classroom use comments*: Use after you have introduced addition, subtraction, and multiplication of mixed numbers.
Grade: 7
Grade Cluster: (MS-Jr.High)
Illinois Goal: 6 and 7
Standard: 6.C.3a and 7.A.3a
Applied? (1-4): 3
Source: Hot words, Hot topics (Creative Publications) ISBN 0-7622-0629-2
Answer: Yes
Strategies Listed: Computation
Solution: 2 1/3 x 3 = 7 or 2 1/3 + 2 1/3 + 2 1/3 = 7
Other solution methods (if any)*: If they don’t compute they must explain how they physically did the measuring.
Extensions or related problems*: How much paper is left
over? 8½ - 7 = 1½
Related: A ribbon is 8 yards long. A dressmaker wants to
make bows, each of which requires exactly ¾ foot of ribbon. With
no waste how many bows can she make. (32)
Intended rubric or assessment method: Worksheet with related addition and subtraction problems of mixed numbers. Set the problems as story problems.
Write-up submitted by: Jenni Robinson and Denise Mann
Title: Heart Beats in a Lifetime
Problem: Consider a person whose hearts beats 70 beats per minute and lives to be 85 years old. Estimate the number of times the person’s heart beats during his or her life. Do not acknowledge leap years. Write your answer in decimal form and in scientific notation.
Materials: Paper and pencil
Classroom Use: Developmental
Classroom use comments: Discuss with the students the importance of scientific notation and where might they see it outside of math class.
Grade: Algebra I
Grade Cluster: jr-high/early high school
Illinois Goal: 7.A.3b apply concepts and attributes of time to practical situations; 6.A.3 represent in scientific notation
Applied Level: Level 3 real life application of mathematics
Source: 2001, McDougal Littell, Algebra I, Houghton Mifflin Company, Larson, Boswell, Kanold, & Stiff
Answer: 3,127,320,000 and 3.12732 x 109
Strategies listed: Unit conversion. conversion of seconds to years; conversion of decimal form to scientific notation
Solution: 60 seconds x 24 hours x 365 days x 85 years = 3,127,320,000 and 3.12732 x 109
Extensions or related problems: Any type of real life application that deals with extremely large or extremely small numbers.
Intended Rubric: +6 points : 4 points for each conversion of time
1 point for the correct decimal answer
1 point for the correct scientific notation answer
Notes: Remind students when the original number is larger than 10 or equal to, the exponent is positive. If the original number is less than 1, the exponent is negative.
Write up submitted by: Sarah Schnowske
James R. Olsen, Western Illinois University
E-mail: jr-olsen@wiu.edu
updated June 27, 2001