Purpose:
Participants will explore relationships between two numbers and their greatest
common divisor
and least common multiple
Overview:
Pairs of participants will identify relationships between the product of two
numbers and the
product of the GCD and LCM of the two numbers . This investigation will require
them to prime
factor each of the two numbers, the GCD, and the LCM. Participants will look for
patterns within
the data.
TExES Mathematics 4-8 Competencies. The beginning
teacher:
1.002.C Uses a variety of concrete and visual
re presentations to demonstrate the connections
between operations and algorithms .
I.003.A Demonstrates an understanding of ideas from number theory (e.g., prime
factorization, greatest common divisor) as they apply to whole numbers,
integers,
and rational numbers , and uses these ideas in problem situations.
I.003.E Applies properties of the real numbers to solve a variety of theoretical
and applied
problems.
1.004.A Uses inductive reasoning to identify, extend, and create patterns using
concrete
models, figures, numbers, and algebraic expressions .
1.004.C Makes, tests, validates, and uses conjectures about patterns and
relationships in
data presented in tables, sequences, or graphs .
TEKS Mathematics Objectives. The student is
expected to:
4.4.B Represent multiplication and division situations in picture, word, and
number form.
4.4.C Recall and apply mutiplication facts through 12 x 12.
4.4.D Use multiplication to solve problems involving two-digit numbers.
4.4E Use division to solve problems involving one-digit divisors.
5.3.B Use multiplication to solve problems involving whole numbers (no more than
three
digits times two digits without techno logy ).
5.3.C Use division to solve problems involving whole numbers.
5.3.D Identify prime factors of a whole number and common factors of a set of
whole
numbers.
5.5.B Use lists, tables, charts , and diagrams to find patterns and make
generalizations.
5.5.C Identify prime and composite numbers using concrete models and patterns in
factor
pairs.
6.1.D Use prime factorizations using exponents .
6.1.E Identify factors and multiples including common factors and common
multiples.
6.2.C Use multiplication and division of whole numbers to solve problems.
6.5 Formulate an equation from a problem situation.
7.2.E Simplify numerical expressions involving order of operations and
exponents.
7.2.F Select and use appropriate operations to solve problems and justify the
selections.
8.2.A Select and use appropriate operations to solve problems and justify the
selections.
Terms.
Factor, divisor, multiple, exponent, prime factorization, prime factors, least
common multiple,
greatest common divisor
Materials.
• Transparencies
• Activity Sheets
• Calculator for each participant
Transparencies.
• Transparency: GCD and LCM
• Transparency: Solution
Activity Sheet(s).
• Activity Sheet: GCD and LCM
Procedure:
| Steps |
Questions/Math Notes |
1. Display the
transparency GCD and LCM
on the overhead projector and allow
participants time to ask questions about
the table. |
At first glance this
activity appears to be a plug
and chug exercise. Inform participants that the
first few examples are fairly straightforward to
allow participants to discover patterns. The
latter problems are quite challenging and
require a higher degree of understanding. |
2. Introduce Venn
Diagrams as a means for
de termining the GCD and LCM of two
numbers. |
See explanation of how
to use Venn Diagrams
after the Procedure section. |
3. Have participants
work in small groups on
the Activity Sheet: GCD and LCM using
calculators to assist them in identifying
patterns.
Encourage participants to find multiple
answers for the last three problems. |
What information do you
need to fill in each
blank in the table?
Is there additional information that you need? |
4. Have each group
create a list of patterns
they discovered. Have them post their
lists on the wall. |
What patterns did you
observe?
How did you discover the pattern? |
5. Debrief the activity
by having each group
present a different problem. |
How can you determine
the two numbers when
you know their GCD and LCM?
How can you determine the second number
when all you know is one number and the
LCM? |
Using Venn Diagrams to Determine GCD and LCM of Two
Numbers.
Example: Find the GCD and LCM of 24 and 60.
Prime factor each number.
48 = 2 x 2 x 2 x 2 x 3 and 60 = 2 x 2 x 3 x 5
Draw two intersecting circles for your Venn Diagram.
Place the factors of 48 in one circle and the factors of 60 in the other circle
with the common
factors in the intersecting part of the circles. The GCD is product of the
factors inside the
intersecting part; the LCM is the product of all the factors in the Venn
Diagram.
GCD (48, 60) = 2 x 2 x 3 = 12
LCM (48, 60) = 2 x 2 x 2 x 2 x 3 x 5 = 240
Solution.
List patterns you discovered.
A x B = GCD (A, B) x LCM (A, B)
For example, patterns emerge when solving problem #10:
