Problem:  At basketball practice, T.J. shot the ball 8 times and made 6 baskets.  His
      friend Andy took 12 shots and made 9 baskets.  Who did a better job at
      making baskets, T.J. or Andy?  Explain how you found your answer.  Include
     a drawing.

Math Topic/Concept:  Comparing fractions

Materials:  2 different color of counters, or fraction pieces

Classroom Use: (Introductory)

Grade:  4th grade

Grade Cluster: (LateElem)

Illinois Goal: 6

Standard:  6.D.2

Applied? (1-4):  3

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

Answer: 6/8 and 9/12 are equivalent amounts, both boys did equally well at making their
    baskets.  (Andy did maintain the shooting average for a longer period.)

Strategies Listed:  drawing or models

Solution:  You might consider supplying students with 2 colors of counters or with
                 fractions pieces to have them work on this problem.  By placing counters
                 in arrays, it is easy to see that both 6/8 and 9/12 are the same as 3/4 and
                 therefore each represents an equal part of the whole amount.

Solution: Other solution methods (if any)*: Another way  to look at this is that in both
              cases, for every 4 shots, there are 3 baskets and 1 miss.

Intended rubric or assessment method: Teacher observation

Write-up submitted by:  Donna Spears

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:  For one of the activities at the summer sports clinic, students throw a football
      and try to have it land in a marked area.  At each practice, each student gets 40
      chances to throw the ball.   During his first practice, Harold threw the ball 40
      times, and 5 of the balls landed in the marked area.  In his second practice, out
      of 40 balls thrown, 11 landed in the marked area.  In his third practice, he
      managed throw 16 balls into the area.  If he continues to improve at the same
      rate, about how many balls could Harold expect to be able to land in the
      marked area for each of his next two practices?

Math Topic/Concept: Patterns, predictions, sequence, estimating

Materials:  paper, pencil

Classroom Use: (Developmental)

Classroom use comments*:  They did not give you enough information to determine a
 pattern.  You may want to add more information to the
 problem to help the students determine a pattern.

Grade:  4th grade

Grade Cluster: (LateElem)

Illinois Goal:  6

Standard:  6. D.2

Applied? (1-4):  2

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

Answer:  For his next practice Harold should get 21 or 22 balls into the area. On the
                following practice, 26 or 27 balls should be expected to land in the marked area.

Strategies Listed:    chart or table

Solution:  Harold improved by 6 throws, then by 5 throws.  By adding either 5 or 6 to his result, an estimate can be made.  It would be reasonable to accept a range of 4 – 7 balls added on the last total.
 

Practice  #	1	2	3	4	5
Balls landed	5	11	16

Intended rubric or assessment method:  Teacher observation

Write-up submitted by: Donna Spears
 
 

James R. Olsen, Western Illinois University
E-mail: jr-olsen@wiu.edu
updated June 27, 2001