Tricky Triangles

Patricia Priestas, Educator, Mathematics & Science Center

Adapted from Trying Triangles, Pieces and Patterns, A Patchwork in Math and Science, AIMS Activities Grades 5-9

 

Developed with funding from Capitol One Financial Services


Major Understanding

The sum of the two shorter straws must be greater than the longest straw in order to form a triangle of the straws. 

 

 

Grade/Subject

Math 6-8; Geometry & Measurement; Statistics

 

 

Objectives

Determine whether three numbers (which represent lengths of line segments) will form a triangle.

 

 

Discover and identify a pattern.

 

 

Collect, analyze and interpret data.

 

Compare a whole number to the sum of two whole numbers using concrete materials.

 

 

Take a list of all possible outcomes and determine the probability of the occurrence of an event within the list.

 

 

Make inferences and predictions based on the analysis of a set of data.

 

Time

Anticipatory Set

5 min

Background Information

5 min

Demonstration

5 min

Introduction of Graphing Calculator & Random Integer

     Generation

   15 min

Activity I & Tricky Triangles Handout 1

15 min

 

Discussion and Interpretation of Data

10 min

 

Activity II & Tricky Triangles Handout 2

15 min

 

Discussion, Interpretation & Analysis of Data

10 min

 

Ratio Decimal Percent Discussion – Theoretical Probability

10 min

 

Materials

 

For each student:

28 Graphing calculators (TI-83 or TI-83+)

30 Zip bags with sets of pre cut straws

     (Three 1”, three 2”, three 3”, three 4”, three 5”, three 6”)

3   24” pieces of string and a protractor

30 Tricky Triangle Handout 1

30 Tricky Triangle Handout 2  

 

For the class:

3 Normal dice

1 Zip bag with set of pre-cut straws, string and protractor

 

For the teacher:

1 Overhead projector

1 Transparency of Student Handouts 1 and 2

1 Teacher’s model graphing calculator (TI83 or TI83+)

1 View Screen for graphing calculator

1 Laptop, with spreadsheet and graphing software (optional)

1 Computer projection system (optional)

State and National Correlations

Virginia Standards of Learning: Grade 6 (6.4, 6.14, 6.20); Grade 7 (7.18)

 

National Math Education Standards:  Work flexibly with fractions, decimals and percents to solve problems; Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules; Use geometric models to represent and explain numerical and algebraic relationships; Use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations; Build new mathematical knowledge through problem solving; Make and investigate mathematical conjectures;  Organize and consolidate mathematical thinking through communication; Communicate their mathematical thinking coherently and clearly to peers, teachers, and others; Use the language of mathematics to express mathematical ideas precisely.

Instructional Strategies

1.      Anticipatory Set
Explain to the students that this lesson deals with probability. 
Ask the students whether or not three numbers rolled or randomly generated, will make a triangle?

2.      Background Information
Define a triangle as a closed, three sided figure.

3.      Demonstrate
Place a set of straws on the overhead against a protractor, showing the different lengths that correspond to the set of dice (3).  Each length of straw represents a numeral on the set of dice.  Roll the set of dice (3).
Place the straws that represent the numerals on the dice on a piece of string.  Loop the string and tighten to see if the straw segments form a triangle.

 

4.      Activity I
Distribute Tricky Triangles Handout 1.  Review the handout, filling in the first row with the numerals from the demonstration.
Emphasize the need to list the numerals from least to greatest on the worksheet.  Have the students record in the last column of the worksheet, whether or not a triangle was formed.

5.      Introduction of Graphing Calculator & Random Integer Generation
Distribute a graphing calculator to each student.  Give a quick tour on how to use the graphing calculator.  Give instructions on how to randomly generate integers:  For TI-83 or TI-83+: (ON, CLEAR, MATH, PRB, #5, 1, 6, 3, ENTER, ENTER, ENTER…).

6.      Activity I
Distribute Tricky Triangles Handout 1.  Have students complete the handout, generating the combinations using the dice or graphing calculators.

7.      Discussion and Interpretation of Data
Discuss handout and interpret data.  Make a class list on the overhead
transparency.  Look for patterns.  Have the class determine a number
sentence that shows the relationship between the shorter sides of the
triangle and the longest side (if a + b > c, then it’s a triangle).  Discuss the number of possible arrangements that may occur when rolling 3 dice (6 x 6 x 6 = 216).  Point out that the actual list of all the possible outcomes (arrangements) is called the “sample space.”  Ask the students whether or not they rolled all of the possible arrangements (probably not).  Of these 216 arrangements, many are simple rearrangements of the same digits.  There are 56 unique combinations.

8.      Activity II
Distribute Tricky Triangles Handout 2 and use a transparency copy on the overhead.  Review the handout noting the 56 unique combinations. Have the students complete the top portions of the handout, including the total number of combinations.

9.      Discussion, Interpretation & Analysis of Data
Discuss Handout 2.  34 of the 56 combinations result in triangles, 22 of the 56 arrangements will not form triangles.

10.  Ratio Decimal Percent Discussion – Theoretical Probability
Have the students determine the percentage of combinations that form  triangles, using the calculators and filling in the table at the bottom of the handout.  Discuss the results and discuss theoretical probability. 
      34/56 = .6071428 or .61 x 100 = 61% make triangles
      22/56 = .3928571 or .39 x 100 = 39% do not make triangles

 

Practice

Guided Practice:  Have students conduct a simulated coin toss activity, using the TI-83+ graphing calculator and the APPS key, under 6:  Prob Sim or using the following NCTM link:  Simulating Probability Using Box Models.  Students first determine the theoretical probability of tossing heads or tails.  Then students, working in pairs, do the simulation using either the TI-83+ or the NCTM link, determining their experimental probability.  Percentages and a class average can be determined.  Discussion of theoretical versus experimental probability.

Independent Practice:  Students investigate games (board, electronic, computer, graphing calculator or other of their choosing), to determine the probabilities associated with the particular game.  For example, Yahtzee uses a set of five (5) dice.  Students determine the number of combinations possible and the probability of rolling a specific combination.  The following links will be of interest:

·       Probability of Childhood Games

·       Games and Probability

 

Closure

Restate lesson objectives and relate them to the class experience.  Discuss the difference between theoretical and experimental or simulation
probability.

Extensions

1.      Using a laptop with spreadsheet and graphing software such as Excel or ClarisWorks and a computer projection system, place the theoretical percentages for forming a triangle and for not forming a triangle in a spreadsheet and display in a graph.  Have the students determine the simulation percentages from Handout 1.  Determine a class average and represent those percentages in a graph on the computer for the number of combinations that made triangles and for those that did not.  Compare the experimental or simulation probability with the theoretical probability.

2.      Teach the Triple Triangles Lesson and Squared Triangles Lesson.

3.      Have students design and create a game for the classroom based on probability.

 

Assessment

A sample of items is provided for use in assessing students’ understanding:

·        Paper-Pencil Test: Tricky Triangles

·        Product Task: Poster Showing Straw Combinations

·        Rubric: Poster Showing Straw Combinations

 

The following table shows how the assessment items are related to specific objectives.

 

 

Objective

Paper-Pencil Test

Product/
Performance

Determine whether three numbers, which represent lengths of line segments, will form a triangle.

2

 

Discover and identify a pattern.

6

 

Collect, analyze and interpret data.

3

 

Compare a whole number to the sum of a pair of whole numbers.

1

 

Make a list of all possible combinations and determine the probability of the occurrence of an event within the list.

7, 8, 9

 

Make inferences and predictions based on the analysis of a set of data.

4, 5

 

The sum of the two shorter straws must be greater than the longest straw in order to form a triangle of the straws.

 

Product Task & Rubric:  Poster

 

Teaching Tips

 

For additional information on teaching this lesson, go to the following links:

·        Answer Key: Paper-Pencil Test: Tricky Triangles

·        Frequently Asked Questions

·        Teaching Tips

 

References

AIMS Education Foundation.

This site contains integrated hands-on math-science activities. Tricky Triangles is adapted from Trying Triangles, a lesson in Pieces and Patterns A Patchwork in Math and Science.

http://www.aimsedu.org

 

Mathematics & Science Center

Visit the Center’s sight to learn all that is new and happening at the Mathematics & Science Center, a regional consortium of K-12 school divisions located in central Virginia.  Register for student and teacher programs.

http://mathsciencecenter.info

 

MathInScience

Visit this educational resource site to acquire web-based lessons and resources for K-12 students and teachers.  The site is the on-line educational arm of the Mathematics & Science Center.

http://MathInScience.info

 

National Council of Teachers of Mathematics

This site contains the Principles and Standards for School Mathematics.

http://www.nctm.org/standards