Introduction: A break from routine can enliven the spirit and kindle a creative wave of energy. That’s the goal of this post. Three enrichment units from number theory are presented. The first unit is suitable for pre-third graders. The latter units require more advanced knowledge. So postpone them until the required concepts are mastered.
Concept attainment strategies (CA) are used in the first unit to introduce the triangular numbers. Young learners love CA strategies. Enjoy the spirited learning environment this unit kindles.
Briefly, concept attainment strategies require teachers to prepare a two-column table. The first column – labeled the YES column – contains examples of the concept to be learned. The second column – labeled the NO column – contains non-examples of the concept. They do not have the characteristics of the concept. Using the examples, students formulate a hypothesis of the concept that the YES examples represent. The lesson ends when students can append five correct YES examples to the YES column.
The second unit is a worked example.[1] It gives a derivation of the formula for the nth triangular number. The long-term objective of this unit is to open the learner to the potential of a geometric solution to a non-geometric problem. Before introducing this unit, learners should know that the total count in a rectangular pattern is one half of the product of the counts along its length and the count along its width.
The third unit has applications of triangular numbers designed for student participation in the learning process.
UNIT 1
Concept Attainment: Triangular Numbers
Subject Area: Mathematics
Specific Content: Number Theory
Grade Level:4th Timing:45-55 minutes
Instructional Objective:
1. To introduce triangular numbers.
2. Learners list the first 5 triangular numbers and their pictorial representations.
Long-Term Objective:
1. Learners to derive the formula for the nth triangular number.
2. Learner to see geometric ideas used to discover properties of triangular numbers.
Prerequisite Knowledge and Skills:
- Experience with Concept Attainment
- Basic arithmetic
- Elementary geometric facts
Why is the Content of This Lesson Relevant?
This content uses geometric and counting ideas to derive the formula for the nth triangular number. Thereby illustrating the potential of using geometric ideas to discover elementary properties of triangular numbers.
Materials:
1. Graph paper, pencil, eraser
2. Paper to record which the first 5 triangular number
3. Table of example and non-examples ( See pictures 2,3,4 and 5 at the beginning of the article.)
Slide 2 is the top of the the table of examples
and Slide 3 is the remainder of the table of examples
Model of Teaching:Concept Attainment
Procedures
The procedures according to the phases of Concept Attainment
1. State the objectives of the lesson. “I’m thinking of numbers that have a certain geometric property. You are to discover the numbers from the examples representing them.”
2. The examples are presented in a two-column table. The first column is labeled the YES. It contains examples of the concept. The second column is labeled NO. The examples in the NO list do not have the properties of the concept.
3. Teacher: After the first line of the table, present only one line at a time.
4. Direct learners to read the first YES and NO examples.
5. Instruct learners to formulate a hypothesis of the concept using the examples in the YES list.
Say: “The examples in the YES list are a representations of numbers I am thinking about. Do you see a common property they have? What name would you use to classify this number?“
Also state: Use the examples in the NO list to help you to fine tune you thoughts of the concept?
6. Direct learners to count the objects that make up the example. And ask, “Can you make the NO examples look like the Yes examples?”
7. Introduce another YES/No example from the table and repeat 4, 5, 6.
8. Continue to exhaust the table. Add more if needed. When the students get the concept, direct them to add five additional YES examples to the table.
9. Ask: Do you see a ways to express the YES examples as numbers? If so, what numbers represents the first 7 examples in the YES list?
10. State: It is the convention to make the first number of YES list to be one.
11. How would you explain your arrival at the name and property of the numbers in the YES column?
Counseling Aids: Use these comments to redirect struggling students. Remind learners that the geometric examples in the YES list represent a number. To ensure learners do not think the table contains all triangular numbers, ask, “Is there a largest YES number?”
Assessment Criteria:
What evidence will demonstrate your students’ learning today? Learners who give the correct geometric pattern for a given triangular number will have learned the concept.
Identify cultural concerns and describe how you will address them. To ensure learners understand what a triangle is: Review triangles and rectangles before introducing this unit.
Technology – What technology might enhance this lesson?
Small white boards can be helpful.
UNIT 2
The Formula for the Nth Triangular Numbers
Subject Area: Mathematics
Specific Content: Number Theory
Grade Level: 5th
Timing: 45-55 minutes
Instructional Objective:
1. Derive the formula for the nth triangular number.
2. Learners compute five triangular numbers using the formula.
Long-Term Objective:
Learners discover properties of triangular numbers using geometric and algebraic properties.
Prerequisite Knowledge and Skills:
1. Basic algebraic properties and
2. Elementary geometric facts
Why is the Content of This Lesson Relevant?
It demonstrates a geometric method to compute the nth triangular number. Further, it presents a useful approach to explore other properties of triangular numbers.
We will use Tn to represent the nth triangular number.
The Approach
1. Discuss whether the properties of the image representing a triangular number are affected if its representation is rotated?”
2. Direct learners to draw the third triangular number.
3. Ask them to make a copy of the geometric representation of T3 on the same page but rotated through 180 degrees.
Learners should have the image of Slide 4.
4. Direct learners to merge the original representation of T3 and the rotated representation of T3 into a rectangle.
Learners should have the image of Slide 5.
So 2 times T3 is the count in the rectangle, which is 3 times 4 =12.
So T3 = 3(3+1)/2
Ask: Do you think this formula will hold for the 4th triangular number? That is, if the “3” is replaced with a “4” in the formula, will the formula give a correct answer?
Direct them to make the substitution and verify.
Ask them to replace the “3” in the formula with a “5”. Does the formula give the correct count for T5 ?
Ask: “Do you think that replacing “3” with “15”, will gives the correct count for the 15th triangular numbers?”
Ask for an explanation of their belief.
Discuss whether replacing the “3” with an “n” limits in the derivation of the formula limits the conclusion.
UNIT 3
Some Applications
1. If three hundred people gather and each person shakes the hand of every other person in the group exactly once, the total number of handshakes will be T299.
Teacher: To help learners understand this fact, lead them through this example. It clearly reveals the secrets of the solution.
The Example: Consider a group of four people. Call them Able, Baker, Charlie and Dan. A visual representation of the group is four dots labeled A, B, C and D.
Represent a handshake between two people in the group by a line segment joining the points that represent the two people. Thus, a line segment connecting A and B means Able and Baker shook hands. We will count the number of handshakes when all group members shake hands only once.
Direct students to connect A to B, A to C and finally A to D. Ask the question: What do these connections represent? These connections account for three of the total number of handshakes when these group members shake hands once.
Have students to connect B to C and B to D. Ask for the meaning of these connections. Then ask them, “Why don’t we connect B to A?” These connections account for 2 more connections within the total handshakes.
Finally have students to connect C to D. This connection accounts for the final handshake in the group.
The total number of connections among the four points is 3+2+1 =T(4-1)= T3 .
So the total number of handshakes that takes place in the group when each person shakes the hand of every other person exactly once is T3.
Let’s go a little deeper.
If a group of “n” objects have a single connection among themselves, then the number of connections is Tn-1. This is just a restatement of the handshake problem.
The important concepts in the handshake problem are:
1. The interpretation of the word ” handshake”,
2. The number of objects in the group, “n” and
3. Each pair has only one connection.
To complete the generalization of the situation, have students to complete the following table.
Then instruct them to add their contribution of “objects in a group ” and “an action to represent a handshake” to the table. See the first four table entries for ideas.
Remind them that the action can only take place once between the objects.
Direct them to state the meaning of these substitutions.
Remind them that under all circumstances, the total number of actions taking place in the group is Tn-l.
Objects in a group Action among once
People————————————touch each other once
People————————————telephone each other once
Trees————————————- in a grove grow to each other
Cancer cells in an organ————-stick to another in one place
Towns————————————————-?
Airports———————————————–?
2. Read Ezekiel 1: 4 – 8. How many connections are there between these four creatures? To see the scripture:
Go to http://www.biblegateway.com and in the space for scripture lookup, supply Ezekiel 1:4-8
3. Certain birds tend to migrate in the geometric pattern used to represent triangular numbers.
[1] John Sweller and Graham Cooper, The Use of Worked Examples as a Substitute for Problem Solving in Learning Algebra, Cognition and Instruction 1985.
[2] Romelia Morales,Valerie Shute and James Pellegrino, Developmental Differences in Understanding and Solving Simple Mathematics Word Problems, Cognition and Instruction 1985.
Further readings: Fascinating triangular number: http://www.shyamsundergupta.com/triangle.htm
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