The Math Blog 19-20

  • Finding the LCM

    Posted by Mark Rinaldi on 11/12/2019

    First we used Venn diagrams to compare the prime factors of numbers to help us determine the Greatest Common Factor.  Now we are going to see how that same set up can help us find the Least Common Multiple.  LCM is the smallest multiple that two numbers have in common.  If the numbers are small enough, we can just skip-count by the numbers and look for the first number they share.

     
    Example:  LCM (4,6) = ?
     
    4 -  4, 8, 12, 16, 20, 24, ...
    6 - 6, 12, 18, 24, ...
     
    If we list the multiples of 4 and 6, we see the first number they have in common is 12 so the LCM of 4 and 6 is 12.
     
    Sometimes the numbers are bigger and we can't skip count as easily.  We have another strategy for this situation.  To use a Venn diagram for this, we find the prime factors of two numbers and see what they have in common.

    36 = 2 * 2 * 3 * 3
    54 = 2 * 3 * 3 * 3

    If we compare their factors in a Venn diagram in would look like this:

     
    To find the LCM, we multiply everything found in whole diagram..  

    LCM(36,54) = 2 x 2 x 3 x 3 x 3 = 108
     
    Khan Academy links:
    Video review - Least Common Multiple
    Practice problems - Least Common Multiple Practice
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  • Finding the GCF

    Posted by Mark Rinaldi on 11/6/2019

    We have been working on finding the GCF.  GCF or greatest common factor is the biggest factor that two numbers have in common.  One way to find the greatest common factor is to list out all the factor pairs of the two numbers.  Let's look at an example:

    GCF(36, 54) = _____

    36

    1x36

    2x18

    3x12

    4x9

    6x6

     

    54

    1x54

    2x27

    3x18

    6x9

    You can see that the common factors of 36 and 54 are 1, 2, 3, 6, 9, and 18.  Therefore, GCF(36, 54) = 18

    We can also use the prime factors of numbers to help us determine the Greatest Common Factor.We find the prime factors of two numbers and see what they have in common.

    36 = 2 * 2 * 3 * 3
    54 = 2 * 3 * 3 * 3

    To find the GCF, we multiply all of the prime factors they have in common. 

    GCF(36,54) = 2 * 3 * 3 = 18

     
    The biggest number that would divide evenly into 36 and 54 would be 18.
     
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  • Prime Factorization

    Posted by Mark Rinaldi on 10/25/2019

    We have been working with factoring numbers into their prime factors.  Every number has a unique set of prime factors.  To factor a number into its prime factors, we can use the divisibility rules and our multiplication facts.  Let's look at an example by using the number 48.

    prime factors of 48

    In this example we split 48 into 8 x 6.  Then, since neither of them is prime, we continue to break down the numbers by splitting them into new branches.  We keep going until the end of every branch is a prime number.  We then list out our prime factors in order from smallest to biggest.

    48 = 2 x 2 x 2 x 2 x 3

    We can also rewrite those using exponents for the repeating factors and write our answer like this:

    48 = 2x 3

    The interesting thing about prime factorization is that no matter how you split the number, the final answer will be the same.  Let's look at 24 and split it two different ways.

    24 split two ways

    Notice both ways give us the same end result.

    24 = 2 x 2 x 2 x 3

    You can check out the Prime Factorization section on Khan Academy for some videos and practice problems.

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  • Exponents

    Posted by Mark Rinaldi on 10/23/2019

    When students first learned about multiplication, they were shown that it was a quicker way to show repeated addition.  They learned that instead of writing 3+3+3+3+3+3+3+3, it was much quicker to write it as 8x3.  Similarly, exponents are a quicker way to show repeated multiplication.  

    Let's look at an example:

    4 x 4 x 4 x 4 x 4

    In this example, we see that 4 is the number being repeated, so we make that the base.  It is being repeated 5 times, so we make that the exponent.

    4 x 4 x 4 x 4 x 4 = 4

    We would read 45 a couple different ways.  We can read it as four to the power of five, or four to the fifth power. 

    Anything that is to the power of two can also be called squared and anything to the power of three can be called cubed.

    72 can be called seven squared.

    83 can be called eight cubed.

    We can find the value of a number in exponential form by solving the multiplication problem.

    4= 4 x 4 x 4 x 4 x 4 = 1024

    Notice we do not just multiple the base by the exponent.

    Also, 23 ≠ 32.  They are not interchangeable.  If you solved those, you would see that 2 x 2 x 2 does not equal 3 x 3.  8 does not equal 9.

    You can check out some videos and practice problems on Khan Academy in their grade 6 exponents section.

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  • Divisibility Rules

    Posted by Mark Rinaldi on 10/18/2019

    When trying to develop a number sense, one of the things that is good to look at is the Divisibility Rules.  When saying one number is divisible by another number, that means you would be able to set up a division problem and get a whole number answer with no remainder.  For example, 12 is divisible by 3 because if you did 12 ÷ 3 = 4 with no remainder.  Mathematicians have developed divisibility rules for all sorts of numbers, but we were looking primarily at two through ten.  Honestly, the main rules I tell the students to memorize are 2, 3, and 5.  The students should have these written down as a reference in their binders.

     Divisibility Rules

    2 - A number is divisible by 2 if the last digit is even (0, 2, 4, 6, 8).

    3 - A number is divisible by 3 if the sum of the digits is a multiple of 3.  (Example: The number 117 is divisible by 3 because the sum of the digits 1+1+7 is 9 and 9 is a multiple of 3.)

    4 - A number is divisible by 4 if the last two digits are a multiple of 4. (7312 is divisible by 4 because the last two digits are 12 and 12 is a multiple of 4.)

    5 - A number is divisible by 5 if the last digit is 5 or 0.

    6 - A number is divisible by 6 if it is divisible by both 2 and 3.

    7 - I don't bother teacher the divisibility rule for 7 because it's an odd rule and it is usually more work than just doing the division problem.  You can Google it and find it, though.

    8 - A number is divisible by 8 if the last three digits are divisible by 8.

    9 - A number is divisible by 9 if the sum of the digits is a multiple of 9.

    10 - A number is divisible by 10 if the last digit is 0.

     

    You can check out a Khan academy video about the rules here or try some practice problems here

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  • Number systems Unit Test review

    Posted by Mark Rinaldi on 10/9/2019

    We are nearing the end of our opening unit.  We will be having a test on the content we have covered.  Looking back over your homework assignments we have done is one way to review for the test.  Another way to review would be to check out the Negative Numbers section at Khan Academy.  They review everything from introducing negative numbers to working on a coordinate grid.  Videos and Lessons are on the left under the "Learning" heading.  Practice problems are on the right under the "Practice" heading.  Towards the end of the section, they even have a unit test.

    Topics:

    • Classifying and Ordering (<, >) Integers and Rational Numbers
    • Absolute Value
    • Graphing on a Coordinate Plane
    • Reflections and Distance on a Coordinate Plane

     

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  • Distance on a Coordinate Grid

    Posted by Mark Rinaldi on 10/8/2019 3:00:00 PM

    The other skill we were looking at was finding the distance between two points that are on the same vertical or horizontal number line.  Again, I try to get them to focus on patterns to help remember the procedure.  Let's look at a couple examples.

    • How far apart are the points (-3,8) and (-3,-4)?

    The first thing we do is figure out if the points are in the same quadrant or not.  To do that, we just look at their signs.  For two points to be in the same quadrant, their signs have to be the same.  In this case, the first point is (-,+) and the second point is (-,-).  They are in different quadrants.  Because they are in different quadrants, we are going to add their distances together.  Looking at the points, we see that the y-coordinate changed, so we are going to look for the distance between the y-coordinates.  To do that, we need to find their distance from zero, or absolute value

    |8| = 8 and |-4| = 4

    Because they are different quadrants we add those values together:

    8 + 4 = 12

    The distance between the two points is 12 units.

    Now lets look at a second example.

    • How far apart are the points (2,-3) and (9,-3)?

    Again, we look to see if they are in the same quadrant.  Same signs (+,-) for both points means same quadrant. This time, the x-coordinate changed so that's what we are going to look for.  Again, we need to take the absolute value to work with their distance.

    |2| = 2 and |9| = 9

    Now, because they were in the same quadrant, we are going to subtract.  (Remember, "same signs" and "subtract" both start with S.)  Set up the subtraction problem with the larger number first to ensure the answer is positive.

    9 - 2 = 7

    The distance between the two points is 7 units.

     

    Summary of Procedure:

    1.  Are they in the same quadrant or different quadrants? (Look at the signs.)

    2.  Find the absolute value of the coordinate that changed.

    3.  Add or subtract those distances based on quadrants.

    • Same quadrant --> Subtract
    • Different quadrants --> Add

     

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  • Reflections on a Coordinate Grid

    Posted by Mark Rinaldi on 10/8/2019 2:00:00 PM

    We were focusing on two main skills on a coordinate grid.  The first was the idea of reflections, and in our case we were looking at simple reflections using the x-axis or y-axis as the point of reflection.  What that means is if we are reflecting a point across the x-axis, we are visualizing folding the grid along the x-axis and seeing where the new point would end up.  If we took the point (3,2) and flipped it over the x-axis, it would end up at (3,-2).  We noticed the pattern that flipping across the x-axis changes the y-coordinate to its opposite.  When we flip the point across the y-axis, the x-coordinate changes to its opposite.  The point becomes a mirrior image of the original point.

     

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  • Coordinate Grids

    Posted by Mark Rinaldi on 10/7/2019

    We have been working on coordinate grids in class lately.  We have been focusing on the parts of the grid and the patterns that apply.  The coordinate grid is created with two perpendicular number lines.  The horizontal number line is called the x-axis and the vertical number line is the y-axis.  It is important to look at the scale of the number lines.  The scale means looking at the value of each line.  On these number lines, the scale would be 1 because each line is equal to 1.  Where the interesect is call the origin

    The two axes split the grid into four main sections we call quadrants.  The upper right grid where both the x-coordinate and the y-coordinate are positive is called Quadrant I. (They use roman numerals for the quadrant labels.)  It then goes counterclockwise around for Quadrant II, Quadrant III and Quadrant IV.

    coordinate grid

    To find a point on the coordinate grid, we use what is called an ordered pair.  An ordered pair is a set of coordinates (x,y) where the first, or x-coordinate is always the distance to the left or right and the second, or y-coordinate is always the distance up or down.

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  • Types of Numbers

    Posted by Mark Rinaldi on 9/16/2019

    We have been looking at the different groups of numbers in our number system.  The main groups we will be looking at are listed below:

    • Whole numbers - 0, 1, 2, 3,...
    • Integers - The integers are defined as the whole numbers and their opposites (..., -2, -1, 0, 1, 2,...)
    • Rational numbers - All of the integers as well as everything in between such as the fractions and decimals.  Rational numbers can all we written in the form a/b where both a and b are integers and b does not equal zero.

    We have been working on ordering them and comparing them using inequality symbols such as greater than (>) and less than (<).

     

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