Download Day 10: Precious Conjectures Grade 7

Day 10: Precious Conjectures
Grade 7
Description
some conjectures about powers and primes.
• Examine factors, multiples and products of prime numbers, square numbers,
and odd numbers as they prove or disprove a variety of conjectures.
• Explore
Materials
• BLM 10.1, 10.2
Assessment
Opportunities
Minds On ...
Action!
Consolidate
Debrief
Concept Practice
Whole Class Æ Discussion
Describe how some mathematicians spend years attempting to prove or
disprove a conjecture or theory. Sometimes practical applications of these
theories are not evident for years afterward. Students may recognize Einstein’s
Theory of Relativity, E = MC2 as an example of a conjecture.
Lead a discussion on Goldbach’s Conjecture: “Every even number greater than
2 can be written as the sum of two prime numbers.” Students provide numeric
examples to prove or disprove the conjecture as they answer the three questions
on BLM 10.1.
Examples to support Goldbach’s Conjecture:
4=2+2
6=3+3
8=3+5
10 = 3 + 7
Students work in pairs to prove or disprove further conjectures. To do so, they
must understand certain terminology.
Bronzebach’s Conjecture requires knowing that perfect squares are 1, 4, 9,
16, 25, 36, 49, … and that these can be written as 12, 22, 32, 42, 52, …
Tinbach’s Conjecture refers to the prime numbers 2, 3, 5, 7, 11, 13, 17…
Brassbach’s Conjecture refers to factors or divisions of a number.
Ensure that students understand that every number has 1 and itself as factors.
Pairs Æ Activity
Curriculum Expectations/Observation/Mental Note: Observe students’
mental arithmetic skills, appropriate use of calculators, and problem solving
skills as they work on the activity.
Working in pairs or groups of three, students complete BLM 10.2 to prove or
disprove each of the remaining conjectures. Students complete each of the three
questions for each conjecture, as modelled for Goldbach’s Conjecture.
Whole Class Æ Debrief
Students present their findings. Lead a discussion about what would be enough
to prove a conjecture true or to disprove or refute a conjecture. Most students
will agree that you can prove a conjecture if you show all possible cases to be
true. Many students will not think it possible to establish a proof if there are an
infinite number of cases. A few students may be interested in researching
“proof by mathematical induction.”
The following counter-example is sufficient to disprove or refute Brassbach’s
Conjecture: 16 has 5 factors 1, 2, 4, 8, 16, not just 3 factors.
A conjecture is a
statement that may
appear to be true,
but has not yet
been proven.
In honour of
Goldbach,
mathematicians
humorously named
several other
prime number
conjectures after
metals and
minerals.
(Tinbach’s,
Copperbach’s,
Aluminumbach’s
Conjecture)
A prime number
has only 2 different
factors: 1 and
itself.
“There are an
infinite number of
prime numbers” is
a conjecture that
many
mathematicians
have worked on.
Prime numbers are
of particular
interest to those
who create
security codes.
Select a student to
add vocabulary to
the Word Wall:
power, prime,
product, and
factor.
Home Activity or Further Classroom Consolidation
Complete worksheet 10.2 for Precious Conjectures about Powers and Primes.
TIPS: Section 3 – Grade 7
© Queen’s Printer for Ontario, 2003
Page 46
10.1: Goldbach’s Conjecture
Name:
Date:
“Every even number greater than 2 can be written as the sum of two prime
numbers.”
4=2+2
1. Verify that the conjecture is true for the first ten numbers in the pattern.
2. Select five additional numbers (not necessarily consecutive or immediately following the
first 10) and verify that the conjecture is also true for them.
3. Suppose that you were able to write valid statements for the first 1 000 000 numbers in the
pattern. Would this be enough to prove that the conjecture is true? Explain your reasoning.
TIPS: Section 3 – Grade 7
© Queen’s Printer for Ontario, 2003
Page 47
10.2: Precious Conjectures about Powers and Primes
Name:
Date:
For each of the conjectures listed:
• verify that the conjecture is true for the first ten numbers in the pattern.
• select five additional numbers and verify that the conjecture is true for them. These numbers
do not have to be consecutive or immediately follow the first ten numbers.
• suppose that you were able to write valid statements for the first 1 000 000 numbers in the
pattern. Would this be enough to prove that the conjecture is true? Explain your reasoning.
Conjectures
Bronzebach’s Conjecture
Every natural number is the sum of four or fewer perfect squares.
10 = 32 + 12
Tinbach’s Conjecture
Every number can be expressed as a difference of two prime numbers.
14 = 17 – 3
Copperbach’s Conjecture
The product of any number of prime numbers is odd.
5 × 3 × 7 = 105
Aluminumbach’s Conjecture
Every odd number can be expressed as the sum of three primes.
11 = 3 + 3 + 5
Challenge
Brassbach’s Conjecture states that “Every square number has exactly 3 factors.” One example
is 22, with factors 1, 2, and 4. Investigate this conjecture using the first 10 perfect squares. Write
a brief explanation of your results and explain how they confirm or refute the conjecture.
TIPS: Section 3 – Grade 7
© Queen’s Printer for Ontario, 2003
Page 48