Problem:  Mr. and Mrs. Hippo want to get new paint and a new rug for their home.  The Hippo Home Store sells blue paint and yellow paint.  The Happy Rug Store sells green rugs and orange rugs.  Mr. and Mrs. Hippo can fix their home with new paint and a new rug.  What are the four different ways Mr. and Mrs. Hippo can fix up their home?

Math Topic/Concept:  Organizing data and finding combinations of items.

Materials:  Yellow, blue, green and orange crayons.  A worksheet showing paint cans and rugs for the students to color can be provided.

Classroom Use: Introductory

Grade:  1

Grade Cluster: Early Elementary

Illinois Goal:  10

Standard:  10 C. 1b

Applied? (1-4): Level 2

Source:  The Problem Solver 1, page P2, Creative Publications, 1987

Answer:  blue paint / green rug  blue paint / orange rug
      yellow paint / green rug  yellow paint / orange rug

Strategies Listed: Make a list

Solution:  The students will color paint cans and rugs in different combinations.

Extensions or related problems*:  Add a different color of paint, and/or a rug.

Write-up submitted by:  Jodi Johnson and Cyndi Fisher

A worksheet can be made from the following:

Problem:  Katie Kangaroo is going to school!  She takes out her shoes and socks.  Katie has a pair of red shoes and a pair of yellow shoes.  She has one pair of blue socks and one pair of green socks.  What are the four different sets of shoes and socks that Katie can put on today?

Math Topic/Concept: Organizing data and finding combinations of items

Materials: Worksheet below. Red, yellow, blue, and green crayons, if desired.

Classroom Use: Introductory

Classroom use comments*:  Some of the combinations will be obvious to most of the students.  You will need to possibly give hints to those students that are missing combinations.  Working with a partner or as a group would be a good idea for this lesson since it is introductory.

Grade:  1

Grade Cluster: Early Elementary

Illinois Goal:  10

Standard:  10 C.1b

Applied? (1-4):  Level 1

Source:  The Problem Solver 1, T7, Creative Publications, 1987

Answer:  red shoes / blue socks    red shoes / green socks
     yellow shoes / blue socks      yellow shoes / green socks

Strategies Listed:  Make an organized list

Solution: Students will color the shoes and socks in different combinations.

Extensions or related problems*:  Add a third color of shoes, socks, or both to the problem.

Write-up submitted by:  Cyndi Fisher and Jodi Johnson

A worksheet can be made from the following:

Katie Kangaroo is going to school!  She takes out her shoes and socks.  Katie has a pair of red shoes and a pair of yellow shoes.  She has one pair of blue socks and one pair of green socks.  What are the 4 different sets of shoes and socks that Katie can put on today?
 

Color to show the 4 different sets of shoes and socks that Katie can put on.

Problem:  "Lucky Ducks" is a popular game at Lincoln School's carnival.  Here's how is works.  Students pick a duck out of a large tub of water filled with plastic ducks that are floating around.  If the bottom of the duck is marked Prize, a prize is awarded.  If the duck is not marked, the student receives a duck sticker.  The duck is then returned to the tub.
A large display board shows that so far, 43 people have received stickers and 21 people have won prizes.
Ricky thinks he might try his luck, but first he wants to know his chances of winning a prize.  His friend Alex has suggested that he can figure out how likely it is he'll win a prize from the results on the board.
What are Ricky's chances of winning a prize?  Would you consider his chances favorable or unfavorable?  Explain your thinking.

Math Topic/Concept:  Probability

Materials:  Paper, pencil, could also have paper ducks cut out with "prize" written on them or plain.

Classroom Use: (Introductory)

Grade:  5

Grade Cluster: (LateElem)

Illinois Goal:  10.C.2a

Standard:  10.C.2a

Applied? (1-4):  2

Source:  Explain It!  Grades 5-6 Creative Publications  ISBN0-7622-1598-4

Answer:  One out of every 3 people who tried the game won a prize.  The chances of winning a prize are pretty good.  And remember, there is a sticker as a consolation prize.

Strategies Listed:  make a chart, guess and check, manipulate the cut-out ducks
 
 
 
 

Solution:  If 21 people won a prize and 43 people got a sticker, then 64 people played the game.  That means 21 out of 64 people won prizes, and the fraction 21/64 is about the same as the fraction 1/3.  Ricky has 1 chance out of 3 to win a prize.  I think the chances are favorable.
People that won  =  21   =  1
     People that played         64        3
 

Other solution methods (if any)*:  Since 43 people got stickers, that's about two times as many as the 21 people that got prizes.  That means that for every 2 people that got a sticker, 1 got a prize.  Ricky's chance of winning a prize are good.
 
 

Sticker

Sticker

Prize

Intended rubric or assessment method:  Grade 5 "Student Friendly" Mathematics Scoring Rubric found at http://www.isbe.il.us/isat/rubric5.htm

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

Problem: There are three types of ice cream at Robbins-Carlson ice cream shop: vanilla, chocolate, and strawberry. How many different double-dip cones can be ordered?

Math Topic/Concept:  Combinations

Materials: Three different-colored pencils, crayons or markers per group, three different-colored cubes, about 25 per student or group, recording sheet.

Classroom Use: (Developmental/Evaluation)

Classroom use comments*:  Expect to discuss whether a chocolate with vanilla on top is the same or different from a cone with vanilla with chocolate on top. The solution given below treats these as different.  If you wish to consider them the same, then your solution will differ from the one given.

Grade:  1 - 5

Grade Cluster: (EarlyElem/LateElem)

Illinois Goal:  Goal 6, Goal 8, and Goal10

Standard: 6B1, 6B2, 6B3a, 6B3b, 8A2a, 8B1, 8B2, 8D3a, 10C1

Applied? (1-4):  2

Source: First Grade Academic Expectations, Galesburg Public Schools, Galesburg, IL

Answer:  9.

Strategies Listed:  With the appropriate materials, the children would be expected to draw, or otherwise model, all the possible double-dip cones.

Solution:  Three choices for each.  3x3 = 9.

Extensions or related problems*:  The problem can be posed with a different number of flavors or with a different number of scoops or both.  Using four flavors with two scoops is a simpler problem than using three flavors with three scoops.

Intended rubric or assessment method:  A possible rubric for first grade could be:
Exceeds: Student finds all nine cones. Student is able to explain approach and justify solution.

Meets: Student finds most of the cones. Student has a fairly well-organized approach to finding the solution and can explain the approach.

Does not meet: Student finds some of the cones. Has some repeats or has no organized approach to finding solutions.

Write-up submitted by:  Melfried Olson

Problem: With cubes of two different colors available. How many different three-story towers can you make?

Math Topic/Concept:  Combinations

Materials:  Connecting cubes or Unifix cubes and paper for recording purposes.

Classroom Use: (Developmental/Evaluation)

Classroom use comments*:  For a simpler version, two-story towers could be made.  For a more advanced version, four- or five-story towers can be made.  Students might need guidance to determine the conditions being used to determine if towers are different.

Grade:  4

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

Illinois Goal:  Goal 8 and Goal10

Standard: 8B1, 10B1b, 10B1c, 10C1

Applied? (1-4):  2

Source:  Melfried Olson, "How Many Towers?", Teaching Children Mathematics, Volume 6, Number 1, Page 30.  Solution in Volume 6, Number 9, Pages 591- 592.

Answer:  There are eight possible three-story towers.  There are 4 possible two-story towers, 16 possible four-story towers, and 32 possible five-story towers.

Strategies Listed:  Make an organized list, look for patterns, and use a formula are possible strategies that students may use.

Solution:  After making all the two- and three-story towers students may observe the following pattern and predict the number of towers for any given number of stories.

Stories   Number of Towers
------------------------------------
2                 4
3                 8
4                 16
5                 32
6                 64
n                 2^n
 

Extensions or related problems*:  The problem could be posed with a different number of colored cubes.

Intended rubric or assessment method:  ISAT Rubric

Write-up submitted by:  Melfried Olson

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