Problem:  Pose these questions:  What is the biggest angle you can create by spreading your fingers apart?  Are there any obtuse angles?  Any acute?  Any right?  How many of each kind of angle?  Are your angles different sizes than those of your classmates?  What is the measure of the biggest angle you can make?  What is the measure of the biggest angle someone in this class might be able to make?  (Direct students to make and record estimates of answers for each of the posed questions, and for any other questions generated during this introductory discussion.)

Math Topic/Concept:  estimation and measurement of angles

Materials:  white paper, pencil, rules, and protractors (optional: math journals)

Classroom Use: (Developmental/Evaluation)

Classroom use comments*:  Discuss with students the variance of angles they can create with their spread fingers.  Also point out the angles which could be created if one or more of their fingers were folded out of view.

Grade:  4 (developmental)  5 (evaluation)

Grade Cluster: (LateElem)

Illinois Goal:  7

Standard:  7.B.2a and 7.B.1b

Applied? (1-4):  2

Source:  www.isbe.state.il.us/ils/benchmarking/mathactivities.htm#LE

Answer:  various, according to students’ physical meausre and their chosen “spreads”

Strategies Listed:   estimation, measurement, computation, illustration

Solution:   Students stretch fingers to examine possible angles, estimate and record responses to the questions posed in introductory discussion, place hand on white paper and trace, use ruler to extend lines to form vertices, measure angles with protractor, and record results next to estimates.  Compare results with estimates and then compare final measures with other students’ results.  Students then write responses to describe how they estimated angles and how they determined accuracy in measurement.

Other solution methods (if any)*:   Students could lay “angle sketchers” over the drawing of their hands.  (Angle sketchers can be made by attaching two straight 6” strips of paper together with a brad.)

Extensions or related problems*:  Students can examine the angles which the hands of a clock make.  Possible inquiries include the following:  how many times will a 90 degree angle be formed, at what times will acute angles be formed, at what times will obtuse angles be formed, what times will represent 180 degree angles, what will be the exact measure of 10:22, and others.  Let students generate 5 of their own questions and provide a circle chart on which they can create a clock with accurate increments.

Intended rubric or assessment method:  Analytic Scoring Scale (Jim Olsen, WIU)

Write-up submitted by:  Rebecca Cummins (Westmer CUSD 203)

Problem: The students in the Fourth grade classes at Jay Elementary 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 holds about 100 cans.
 

 Here are the results:
  Mr. Horn’s Class  385 cans
  Mr. Gray’s Class  462 cans
  Mrs. Gladstone’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:  Estimating number relationships

Materials:  If a worksheet is needed, one can be made using the problem and the questions in classroom use comments.  I would advise you to make an overhead copy to work out with the students giving you comments.  You will also need a pencil.

Classroom Use: Developmental

Classroom use comments*:
· Try to decide which method you will need to solve this problem?
· Try rounding off to 100.
· Try front-end estimation.
· Can you think of any other ways to solve this problem.

Grade:  Third

Grade Cluster:  Early Elementary

Illinois Goal:  Mathematics State Goal 7

Standard:  7.B.1a

Applied? (1-4):  3

Source:  Explain it!  Grades 3-4   (Answering Extended Response Math Problems)
 Louis Lepore  2001 Creative Publications   ISBN 0-7622-1597-6
 
 

Answer:
A range of 11 to 12 bags is acceptable.  If the students round off to the nearest 100, they would come up with an estimate of 1,200 cans.  That would require 12 bags.
Some students might use front-end estimation and have 300 + 400 + 300 and then add an extra 100 for the extra cans.   That would make a total of 1,100 cans using 11 bags.
 

Strategies Listed:  Rounding off by 100’s or front-end estimation.

Solution:  The correct answer for rounding off would be 12.  The correct answer for front-end estimation would be 11.  They have to write out their solution using time order words and explain.

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
 
 

James R. Olsen, Western Illinois University
E-mail: jr-olsen@wiu.edu
updated Aug. 15, 2001