6th Grade - Making Good Guesses

 
     
 
     
 
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6th
Reasoning
Making Good Guesses
Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed.
Make a conjecture (generalization, educated guess or prediction) about a mathematical question or problem posed, based on a general description of the problem and question or problem posed. Show conjectures as an algebraic expression or a statement. Give evidence for a mathematical conjecture. Use mathematical reasoning and relationships to derive mathematical conjectures and provide evidence to support them. Test conjectures and use, revise or reject them.
 

Sample Problems

(1)

Kylie has made the generalization that multiples of ten are even. Is her generalization reasonable? Is her statement a conjecture? Explain.

(This is a reasonable conjecture, because the first five multiples of 10 are 10, 20, 30 ,40 , and 50, which shows that each multiple ends in 0. I know that all numbers that end in O.)

(2)

Write a conjecture for the multiples of 2. Provide evidence to justify why your conjecture is reasonable.

(Sample conjecture: The multiples of 2 are even. My evidence is that the multiples of two can be found by counting by twos, when you do this, you always get even numbers.)

(3)

Prove the statement true or false. Explain.

All multiples of 10 are also multiples of 30.

(False, this is not a conjecture. The first three multiples of 10 are 10, 20, and 30. Only 30 is also a multiple of 30, so it is not true that all multiples of 10 are also multiples of 30.)

(4)

Rewrite the statement so that it is a conjecture.

All multiples of 10 are also multiples of 30.

(All multiples of 30 are also multiples of 10.)

(5)

Is the statement below a reasonable conjecture? Explain.

If a fraction has a prime number in its numerator and denominator, it is in simplest form.

(This is a reasonable conjecture, because prime numbers only have two factors, 1 and themselves. So, there will be no way to reduce these fractions any lower.)

Learning Tips

(1)

Sixth graders need to understand that a conjecture is a generalization or prediction that is based on evidence, and thus believed to be true. A conjecture can be written as a statement or as an algebraic expression. In order to children to determine if a generalization is a conjecture, they will need to evaluate evidence surrounding the problem to determine if it appears to be true. If the all evidence suggests that the generalization is true, it can be called a conjecture. However, if evidence suggests that the generalization is not true, the generalization can be either revised to make it true, or dismissed altogether.

(2)

The first step in evaluating a generalization to determine if it is a conjecture is to gather as much data as possible to support the generalization. We will use the generalization, “All fractions that have a one in the numerator and prime number in the denominator are in simplest form,” to show how to do this.

1. Find evidence to support the generalization and list examples.

1/3, 1/5, 1/7, 1/ 11, 1/13….

2. Determine if there is any evidence that opposes the generalization.

There is no evidence that opposes the generalization, because prime numbers are divisible by one and themselves only.

3. If you’ve found supporting evidence in step 1 and no opposing evidence in step 2, you have a conjecture. However, if there was not strong evidence in step 1 or opposing evidence in step 2, the generalization must either be revised to become a conjecture or rejected completely.

(3)

Sixth graders will need to be able to revise a generalization to make it a conjecture. Let’s say we have the generalization, “All multiples of 5 are also multiples of 10.” We would begin by using the steps in lesson tip 2 to determine if the generalization is a conjecture. In doing this, we would find that some multiples of 5 are not multiples of 10 (i.e. 5, 15, 25). However, we would also notice that some multiples of 5 are multiples of 10. This is where mathematical reasoning comes in. We know that a conjecture needs to be a generalization that is believed to be true all of the time. So, this generalization will need to be revised.

1. Think about what is true all of the time when looking at your data.

5: 5, 10, 15, 20, 15, 30, 35, 40, 45, 50, 55, 60…

10: 10, 20, 30, 40, 50, 60

The evidence shows that all multiples of 5 end in a 5 or a zero. It shows that all multiples of 10 end in a zero. So, we have disproved the statement that, “All multiples of 5 are also multiples of 10.”

2. Find a way to rephrase the statement so that it is true.

All multiples of 10 are also multiples of 5.”

This is a conjecture. As we look at the data for the multiples of 10, we see these numbers also in the multiples of 5.

(4)

A question can be used to help you make a conjecture. For example, if you are given the question, “What is the product of a negative integer and a positive integer?” it could be used to form a conjecture.

1. Answer the question by gathering data.

-2 x 2 = -4, -5 x 10 = 50, -6 x 6 = -36….

2. Look at the data and write a generalization based on your evidence.

The product of a negative integer and a positive integer is always negative.

3. Confirm that your conjecture always holds true by trying to find opposing evidence. If you can’t find any, your conjecture is good.

(5)

Visual learners may need to draw pictures or create a table, list or other visual to help organize the data from each generalization as they try to prove it. It may also be helpful to have them highlight or color-code information as they go. One easy way to do this is to have them color-code each step.

(6)

Kinesthetic learners will benefit from first working with tangible topics. For example, you can have your child look at a generalization such as, taller people have longer legs. You child can then measure the legs of different people in your house to compare the heights to the leg length. As your child measures, he/she should gather data. The final step would be to use the data gathered to support or disprove the conjecture.

Extra Help Problems

(1)

Test the generalization. Explain why it is a conjecture or an incorrect generalization.

The difference between two odd numbers is always even.

(Conjecture, sample evidence that supports this conjecture is 7-3=4, 3-1=2, 103 – 101= 2…)

(2)

Test the generalization. Explain why it is a conjecture or an incorrect generalization.

If two angles form on a straight angle, then the angle measures add up to 180.

(Conjecture, the measurement of any straight angle is 180, so it is reasonable to believe that two angles that are formed on a straight angle are also 180>)

(3)

Test the generalization. Explain why it is a conjecture or an incorrect generalization.

The sum of a positive integer and a negative integer is always negative.

(Incorrect generalization, sample opposing evidence: 5 + -1 = 5, 10 + -2 = 8, -6 + 16 = 10….)

(4)

Test the generalization. Explain why it is a conjecture or an incorrect generalization.

Two congruent chords of a circle are equally distant from the center of the circle.

(Conjecture, supporting evidence: a chord is a straight line segment that reaches from one edge of a circle to another edge, congruent chords have the same size and shape. Due to the nature of a circle, it is reasonable to believe that congruent chords would be the same distance from the center of the circle, no matter where they may be on the circle.)


(5)

Test the generalization. Explain why it is a conjecture or an incorrect generalization.

Every prime number is the sum of prime number and an even number.

(Incorrect generalization, opposing evidence: 1 + 0 = 1, zero is neither prime nor composite.)

(6)

Test the generalization. Explain why it is a conjecture or an incorrect generalization.

The product of two prime numbers is never an even number.

(Incorrect generalization, opposing evidence: 2 times any number is an even number and two is prime. Ex. 2 x 3 = 6)

(7)

Test the generalization. Explain why it is a conjecture or an incorrect generalization.

The sum of two odd numbers is an even number.

(Conjecture, supporting evidence: 1+3=4, 43+7 =50, 19 +1= 20...)


(8)

Test the generalization. Explain why it is a conjecture or an incorrect generalization.

All multiples of 4 are also the multiples of 2.

(Conjecture, supporting evidence: 4:4,8,12,16,20…. Multiples of 4 are even and all multiples of 2 are even.)

(9)

Revise each generalization to make it a conjecture.

All multiples of 3 are also multiples of 9.

(All multiples of 9 are also multiples of 3.)

(10)

Revise each generalization to make it a conjecture.

All perfect square numbers have exactly 3 factors.

(All perfect square numbers have an odd number of factors.)

(11)

Revise each generalization to make it a conjecture.

A GCF cannot always be found for a set of whole numbers.

(A GCF of at least 1 can always be found for a set of whole numbers.)

(12)

Revise each generalization to make it a conjecture.

A fraction with two odd terms is always in simplest form.

(A fraction with two prime terms is always in simplest form.)

(13)

Revise each generalization to make it a conjecture.

If two angles are adjacent, they are congruent.

(If two angles are vertical, they are congruent.)

(14)

Write a conjecture for the question.

What is true for all multiples of 6?

(Sample: All multiples of 6 end in 0, 2, 4, 6, 8.)

(15)

Write a conjecture for the question.

What happens when you multiply an odd number with an even number?

(Sample: The product of an odd number with an even number is always even.)

(16)

Write a conjecture for the question.

What happens when you add two odd numbers?

(Sample: The sum of two odd numbers is always even.)

(17)

Write a conjecture for the question.

What is true for all fractions?

(Sample: All fractions have equivalent fractions.)

(18)

Write a conjecture for the question.

What is always true for the product of two negative numbers?

(Sample: The product of two negative numbers will always be positive.)


(19)

José made the conjecture that the product of two odd numbers is always an odd number. Test his conjecture and explain why it is correct or incorrect.

(José’s conjecture is correct. Some sample evidence is 3 x 5 =15, 9x9=81, 7x3=21, 21x103=2,163…)

(20)

Talia made the conjecture that composite numbers have an even number of factors. Is her conjecture reasonable? Explain.

(Talia’s conjecture is not reasonable. Some opposing evidence is that the number 4 has an odd number of factors: 1, 2, 4 (3 factors). )

 

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