Our next unit of study will be devoted to rates, ratios, and proportions. Fraction and decimal notation will be used to express rates and ratios and to solve problems.
Ratios compare quantities that have the same unit. These units cancel each other in the comparison, so the resulting ratio has no units. For example, the fraction 2/20 could mean that 2 out of 20 people in a class got an A on a test or that 20,000 out of 200,000 people voted for a certain candidate in an election.
Another frequent use of ratios is to indicate relative size. For example, a picture in a dictionary drawn to 1/10 scale means that every measurement in the picture is 1/10 the corresponding measurement in the actual object. Students will use ratios to characterize relative size as they examine map scales and compare geometric figures.
Rates, on the other hand, compare quantities that have different units. For example, rate of travel, or speed, may be expressed in miles per hour (55 mph); food costs may be expressed in cents per ounce (17 cents per ounce) or dollars per pound ($2.48 per pound).
Easy ratio and rate problems can be solved intuitively by making tables. Problems requiring more complicated calculations are best solved by writing and solving proportions. Students will learn to solve proportions by cross multiplication. This method is based on the idea that two fractions are equivalent if the product of the denominator of the first fraction and the numerator of the second fraction is equal to the product of the numerator of the first fraction and the denominator of the second fraction. For example, the fractions 4/6 and 6/9 are equivalent because 6 * 6 = 4 * 9. This method is especially useful because proportions can be used to solve any ratio and rate problem and will be used extensively in algebra and trigonometry.
Students will apply these rate and ratio skills as they explore nutrition guidelines. The class will collect nutrition labels and design balanced meals based on recommended daily allowances of fat, protein, and carbohydrate. You might want to participate by planning a balanced dinner together and by examining food labels while shopping with your child. Your child will also collect and tabulate various kinds of information about your family and your home and then compare the data by converting them to ratios. In a final application lesson, your child will learn about the "Golden Ratio" - a ratio found in many works of art and architecture.
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Do-Anytime Activities:
1. Look with your child through newspapers and magazines for photos and check to see if a size-change factor is mentioned in the caption. (Such as 2X for an enlarged photo or 1/2 X for a photo reduced by half). Have your child explain to you what that size-change factor means.
2. Encourage your child to read nutrition labels and use a proportion to calculate the percent of fat in the item.
3. Help your child distinguish between part-to-part and part-to-whole ratios. When comparing a favorite sports team's record, decide which ratio is being used. For example, wins to losses (such as 5 to 15 for 5 wins and 15 losses) are part-to-part ratios. Part-to-whole ratios are used to compare wins to all games played (such as 5 to 20 for 5 wins out of 20 games played).
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View the Unit 8 Outline:
http://apps.gcsc.k12.in.us/blogs/jellars/files/2010/08/Unit-8-Outline.pdf
Wednesday, March 28, 2012
Sunday, March 25, 2012
Ratio and Proportion
Our classes finished the Probability Unit with the unit test before spring break. Scores should be online at this time. There are two scores for the unit evaluation: the unit 7 test and the computation checkup both dated 3/23/12. A small "M" in the grade slot indicates the test is "missing". Unfortunately several papers were turned in with no names and a few weren't turned in at all. We'll attempt to match students with nameless papers after break. If a student can't be matched with a finished test, there will be an opportunity for a retake of an equivalent examination to avoid reciving a score of zero on missing papers. Of course, students who were absent on March 23 will be given the opportunity to take the original exam when we return for full credit.
The next unit of study will explore proportional thinking through the study of rates and ratios. The skills focused on throughout the unit will include:
1. Using rate tables and unit rates to solve problems;
2. Using cross multiplication to solve proportions;
3. Solving percent problems;
4. Setting up and solving ratio problems;
5. Identifying pairs of corresponding sides in similar polygons;
6. Using a size-change factor to solve problems;
7. Evaluating expressions using the order of operations (PEMDAS);
8. Solving perimeter and area problems (including circles and using pi);
9. Dividing decimal numbers.
The next unit of study will explore proportional thinking through the study of rates and ratios. The skills focused on throughout the unit will include:
1. Using rate tables and unit rates to solve problems;
2. Using cross multiplication to solve proportions;
3. Solving percent problems;
4. Setting up and solving ratio problems;
5. Identifying pairs of corresponding sides in similar polygons;
6. Using a size-change factor to solve problems;
7. Evaluating expressions using the order of operations (PEMDAS);
8. Solving perimeter and area problems (including circles and using pi);
9. Dividing decimal numbers.
Wednesday, March 21, 2012
Zero
At a young age we enter into the world of numbers. We learn that 1 is the first number and that it introduces the counting numbers - 1, 2, 3, 4, 5 ... Counting numbers do exactly what their name implies; they count real things - apples, oranges, sheep, etc. It's only later that we learn how to count the number of apples in an empty box.
Even the early Greeks, who brought about giant leaps in science and mathematics, and the Romans, the greatest engineers in the ancient world, didn't have a way to deal with the number of apples in an empty box.They both failed to give "nothing" a numerical name. The Romans had their own ways of combining I, V, X, L, C, D, and M but where was 0? They never tried to count "nothing."
The use of a symbol to desinate "nothingness" probably originated thousands of years ago. The Maya of Meso-America used zero in various forms. The astronomer Claudius Ptolemy, influenced by Babylonians, used a symbol similar to our modern 0 as a placeholder in his number system. As a placeholder, zero could be used to distinguish between numbers such as 75 and 705 instead of relying on context as in Babylonian mathematics. This is somewhat like the introduction of the comma into written language as a way of clarifying meaning. And like the comma, zero comes with its own set of rules for usage.
The seventh century Indian mathematician Brahmagupta treated zero as number, not merely a placeholder and set up rules for dealing with it. These included "...the sum of zero and a positive number is positive" and "...the sum of zero and zero is zero." The Hindu-Arabic numbering system that included zero as both a number and a placeholder was introduced into Western mathemics in the early 1200's by Leonardo of Pisa, better known as Fibonacci.
The use of zero in our number system posed a problem that Brahmagupta had addressed: how was this new number to be treated? How could zero be integrated into into the existing system of arithmetic algorithms in a precise way? When it came to addition and multiplication, 0 fitted in neatly, but was more problematic in subtraction and division.
Adding and multiplying with zero is straightforward and presented little challenge. While you can add (i.e. attach) 0 to 10 to get 100; when used in an algorithm as a number, 0 + a number leaves the number unchanged as you're adding nothing to it and 0 * a number = 0 as you are combining a given number of sets with nothing in them. Subtraction is a simple operation, but may lead to negative numbers, 7-0=7, but 0-7=-7.
Division using zero raised some difficulties. Imagine a length of 24 feet to be measured with a standard 3 foot yardstick. Division involves finding out how many yardsticks we can lie along that 21 foot path. The answer is 24 divided by 3, or symbolically, 24/3=8
What, then, can be made of 0 divided by 3? Using algebraic symbology, that could be written as 0/3 = x. By using cross multiplication, this expression is equivalent to 0= 3 * x. Since this is the case, the only possible value for x is zero itself because if the product of two numbers is zero, one of them has to be zero.
This still isn't the main difficulty with zero. The sticking point is division by zero. If we treat 3/0 in the same way that we treated 0/3, we would have the equation: 3/0 = x. Cross multiplication would produce 0 * x = 3 leading to the nonsensical conclusion that 0=3. By admitting the possibility that 3/0 is a number opens the door for mathematical mayhem! The way out was to say that 3/0 is undefined. We can't arrive at a sensible answer by dividing any nonzero number by zero, so we simply don't allow the operation to take place. This is similar to not allowing a comma to be placed in the middle of a word to avoid the creation of linguistic nonsense.
The 12th century Indian mathematician Bhaskara suggested that a number divided by zero was infinite. He asked how many times could one go to a man with five stones in his hand and take nothing away from him? By adopting this form of reasoning, we become sidetracked into the concept of infinity. This offers no mathematical resolution. Infinity doesn't conform to the usual rules of arithmetic and isn't a number in the usual sense.
Considering 0/0 is also an interesting avenue of exploration. By using cross multiplication on the equation 0/0 = x we arrive at the conclusion that 0 = 0. While this isn't nonsense, neither is it particularly illuminating. It also leads to the conclusion that the value of x could be anything. So we use the term "indeterminate" to describe 0/0. All in all, when we consider dividing by zero we arrive at the conclusion that it's best to exclude that operation from our system of algorithms. Math can get along perfectly well without it.
So what use is zero? Simply put, we can't get along without it. It is one of our base concepts that makes the number system, algebra, geometry, along with all their related disciplines functional. On the number line 0 is the number that separated positive and negative integers. In the decimal system 0 makes it possible to write and use both huge and microscopically small numbers.
When 0 was introduced it must have seemed like an exceedingly odd idea, but mathematicians have a habit of fastening onto strange concepts which are proved to be useful much later. The acceptance and utilization of zero was just such a concept that became one of the greatest inventions of man.
Even the early Greeks, who brought about giant leaps in science and mathematics, and the Romans, the greatest engineers in the ancient world, didn't have a way to deal with the number of apples in an empty box.They both failed to give "nothing" a numerical name. The Romans had their own ways of combining I, V, X, L, C, D, and M but where was 0? They never tried to count "nothing."
The use of a symbol to desinate "nothingness" probably originated thousands of years ago. The Maya of Meso-America used zero in various forms. The astronomer Claudius Ptolemy, influenced by Babylonians, used a symbol similar to our modern 0 as a placeholder in his number system. As a placeholder, zero could be used to distinguish between numbers such as 75 and 705 instead of relying on context as in Babylonian mathematics. This is somewhat like the introduction of the comma into written language as a way of clarifying meaning. And like the comma, zero comes with its own set of rules for usage.
The seventh century Indian mathematician Brahmagupta treated zero as number, not merely a placeholder and set up rules for dealing with it. These included "...the sum of zero and a positive number is positive" and "...the sum of zero and zero is zero." The Hindu-Arabic numbering system that included zero as both a number and a placeholder was introduced into Western mathemics in the early 1200's by Leonardo of Pisa, better known as Fibonacci.
The use of zero in our number system posed a problem that Brahmagupta had addressed: how was this new number to be treated? How could zero be integrated into into the existing system of arithmetic algorithms in a precise way? When it came to addition and multiplication, 0 fitted in neatly, but was more problematic in subtraction and division.
Adding and multiplying with zero is straightforward and presented little challenge. While you can add (i.e. attach) 0 to 10 to get 100; when used in an algorithm as a number, 0 + a number leaves the number unchanged as you're adding nothing to it and 0 * a number = 0 as you are combining a given number of sets with nothing in them. Subtraction is a simple operation, but may lead to negative numbers, 7-0=7, but 0-7=-7.
Division using zero raised some difficulties. Imagine a length of 24 feet to be measured with a standard 3 foot yardstick. Division involves finding out how many yardsticks we can lie along that 21 foot path. The answer is 24 divided by 3, or symbolically, 24/3=8
What, then, can be made of 0 divided by 3? Using algebraic symbology, that could be written as 0/3 = x. By using cross multiplication, this expression is equivalent to 0= 3 * x. Since this is the case, the only possible value for x is zero itself because if the product of two numbers is zero, one of them has to be zero.
This still isn't the main difficulty with zero. The sticking point is division by zero. If we treat 3/0 in the same way that we treated 0/3, we would have the equation: 3/0 = x. Cross multiplication would produce 0 * x = 3 leading to the nonsensical conclusion that 0=3. By admitting the possibility that 3/0 is a number opens the door for mathematical mayhem! The way out was to say that 3/0 is undefined. We can't arrive at a sensible answer by dividing any nonzero number by zero, so we simply don't allow the operation to take place. This is similar to not allowing a comma to be placed in the middle of a word to avoid the creation of linguistic nonsense.
The 12th century Indian mathematician Bhaskara suggested that a number divided by zero was infinite. He asked how many times could one go to a man with five stones in his hand and take nothing away from him? By adopting this form of reasoning, we become sidetracked into the concept of infinity. This offers no mathematical resolution. Infinity doesn't conform to the usual rules of arithmetic and isn't a number in the usual sense.
Considering 0/0 is also an interesting avenue of exploration. By using cross multiplication on the equation 0/0 = x we arrive at the conclusion that 0 = 0. While this isn't nonsense, neither is it particularly illuminating. It also leads to the conclusion that the value of x could be anything. So we use the term "indeterminate" to describe 0/0. All in all, when we consider dividing by zero we arrive at the conclusion that it's best to exclude that operation from our system of algorithms. Math can get along perfectly well without it.
So what use is zero? Simply put, we can't get along without it. It is one of our base concepts that makes the number system, algebra, geometry, along with all their related disciplines functional. On the number line 0 is the number that separated positive and negative integers. In the decimal system 0 makes it possible to write and use both huge and microscopically small numbers.
When 0 was introduced it must have seemed like an exceedingly odd idea, but mathematicians have a habit of fastening onto strange concepts which are proved to be useful much later. The acceptance and utilization of zero was just such a concept that became one of the greatest inventions of man.
Sunday, March 18, 2012
Moving from Arithmetic to Algebra
One of the important goals of a sixth grade math curriculum is the beginning of the transition from arithmetic to algebra. To that end, we spend a lot of time talking about some different ways of thinking about problem solving.
Algebra gives us a distinctive way of solving problems, a deductive method with a twist. That twist is "backward thinking." Consider the problem of taking the number 25, adding 17 to it, and getting 42. This is "forward thinking." We're given the numbers and we just add them together. But, instead, suppose we were given the answer 42 and asked a different question. We now want the number which when added to 25 gives us 42. This is where backward thinking comes in. We want the value of x which solves the equation 25 + x = 42 and we subtract 25 from 42 to give it to us. (Working from back to front.)
Word problems which are meant to be solved by algebra have been given to students for centuries. I'm sure we all remember the two cars travelling toward each other at different speeds and having to determine when they would meet. Or the one that says:
My niece Michelle is 6 years old, and my age is 40 . When will I be three times as old as her?
This could be solved by trial and error, but algebra is more economical. In x years from now Michelle will be 6 + x years old and I will be 40 + x. I will be three times as old as her when 3 * (6 + x) = 40 + x.
Multiply out the left hand side of the equation and you'll get 18 + 3x = 40 + x, and by moving all the x's over to one side of the equation and the numbers to the other, we find that 2x = 22 which means that x = 11. When I'm 51 Michelle will be 17 years old... Magic! (These kinds of equations are called linear equations. They have no exponents and when plotted on a graph will create a straight line.)
Mathematics underwent a big change when it passed from the science of arithmetic to the science of symbols, or algebra. To progress from numbers to letters is a mental jump but the effort is certainly worthwhile.
Algebra gives us a distinctive way of solving problems, a deductive method with a twist. That twist is "backward thinking." Consider the problem of taking the number 25, adding 17 to it, and getting 42. This is "forward thinking." We're given the numbers and we just add them together. But, instead, suppose we were given the answer 42 and asked a different question. We now want the number which when added to 25 gives us 42. This is where backward thinking comes in. We want the value of x which solves the equation 25 + x = 42 and we subtract 25 from 42 to give it to us. (Working from back to front.)
Word problems which are meant to be solved by algebra have been given to students for centuries. I'm sure we all remember the two cars travelling toward each other at different speeds and having to determine when they would meet. Or the one that says:
My niece Michelle is 6 years old, and my age is 40 . When will I be three times as old as her?
This could be solved by trial and error, but algebra is more economical. In x years from now Michelle will be 6 + x years old and I will be 40 + x. I will be three times as old as her when 3 * (6 + x) = 40 + x.
Multiply out the left hand side of the equation and you'll get 18 + 3x = 40 + x, and by moving all the x's over to one side of the equation and the numbers to the other, we find that 2x = 22 which means that x = 11. When I'm 51 Michelle will be 17 years old... Magic! (These kinds of equations are called linear equations. They have no exponents and when plotted on a graph will create a straight line.)
Mathematics underwent a big change when it passed from the science of arithmetic to the science of symbols, or algebra. To progress from numbers to letters is a mental jump but the effort is certainly worthwhile.
Saturday, March 17, 2012
Fascinating Nines
Here's a fascinating pattern with nines. Evaluate each expression to uncover the pattern!
Pattern with Nines
Pattern with Nines
March 19-23
Mini-Economy Auction:
Our sixth grade students have been working hard since January on our Mini-Economy project. It officially comes to an end Tuesday, March 20 with the auction. Students will be able to use the money they've been earning for positive behavior, work habits, and academic performance to bid on a wide variety of items that day.
Dodgeball Attendance Tournament:
Also on Tuesday, the homeroom classes with the best March attendance records will participate in a dodgeball tournament.
Check this week's assignment outline:
http://apps.gcsc.k12.in.us/blogs/jellars/?page_id=8
Our sixth grade students have been working hard since January on our Mini-Economy project. It officially comes to an end Tuesday, March 20 with the auction. Students will be able to use the money they've been earning for positive behavior, work habits, and academic performance to bid on a wide variety of items that day.
Dodgeball Attendance Tournament:
Also on Tuesday, the homeroom classes with the best March attendance records will participate in a dodgeball tournament.
Check this week's assignment outline:
http://apps.gcsc.k12.in.us/blogs/jellars/?page_id=8
Tuesday, March 13, 2012
Sunday, March 11, 2012
Outline of Probability Unit
Our sixth grade math class has recently begun a new unit studying probability and discrete mathematics. Click below for an outline of topics, activities, assessments, and suggested extended activities.
Probability Unit Outline
Probability Unit Outline
Sunday, March 4, 2012
March 5-9
This week we're starting the first round of ISTEP testing, beginning a unity exploring probability, and reviewing work with fractions.
http://apps.gcsc.k12.in.us/blogs/jellars/files/2010/08/March-5-9.pdf
ISTEP Schedule:
Monday 9:25-11:10
Tuesday 8:10-9:10
Wednesday 9:30-10:45
Thursday 8:10-9:20
Everyone is prepared and ready to show what they can do! There's no doubt that we're going to have some remarkable scores. Just remember to get plenty of rest and have a good breakfast. So, no late night texting or Game Boy and get up in time to get fueled up for the day at the breakfast table.
Since our math classes need to share calculators for the testing with all the other classes, anyone who can bring a calculator along for the testing will help out very much! All students should bring pencils and a book to read. (Just in case you finish early!)
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Parents - My apologies if you'd had some extra paper airplanes flying around the house recently. The RTI classes were experimenting with paper airplane design this past week and have been keeping track of data results on which planes were able to fly the furthest.
http://apps.gcsc.k12.in.us/blogs/jellars/files/2010/08/March-5-9.pdf
ISTEP Schedule:
Monday 9:25-11:10
Tuesday 8:10-9:10
Wednesday 9:30-10:45
Thursday 8:10-9:20
Everyone is prepared and ready to show what they can do! There's no doubt that we're going to have some remarkable scores. Just remember to get plenty of rest and have a good breakfast. So, no late night texting or Game Boy and get up in time to get fueled up for the day at the breakfast table.
Since our math classes need to share calculators for the testing with all the other classes, anyone who can bring a calculator along for the testing will help out very much! All students should bring pencils and a book to read. (Just in case you finish early!)
---------------------------------------------------------------------------------------
Parents - My apologies if you'd had some extra paper airplanes flying around the house recently. The RTI classes were experimenting with paper airplane design this past week and have been keeping track of data results on which planes were able to fly the furthest.
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