PROPORTIONAL RELATIONSHIPS
EXPRESSING FRACTIONS AS DECIMALS
A fraction is a way to examine a portion of a number that is taken from a whole. The top of a fraction is called the numerator (it represents the number of parts being considered) and the bottom number is called the denominator (the total number of parts).
We can convert a fraction into a decimal by using long division if we do not have a calculator handy. For a review of how to complete long division check out the following video clip below or visit the following weblink that includes two interactive practice sites "Stork's Long Division" and "Long Division with Help":
http://www.freetrainingtutorial.com/longdivisiongames.html
A handy mneumonic to remember the steps for long division is: Dracula Must Suck Blue Blood (Divide, multiply, subtract, bring down, begin again)
We can convert a fraction into a decimal by using long division if we do not have a calculator handy. For a review of how to complete long division check out the following video clip below or visit the following weblink that includes two interactive practice sites "Stork's Long Division" and "Long Division with Help":
http://www.freetrainingtutorial.com/longdivisiongames.html
A handy mneumonic to remember the steps for long division is: Dracula Must Suck Blue Blood (Divide, multiply, subtract, bring down, begin again)




Reminders:
Fraction Tipping  To rewrite a fraction into long division format you must take the numerator and move it to the right of the denominator.
Repeating Decimal  A decimal in which a block of one or more digits eventually repeats in a pattern. Example, 25/99 = 0.252 525..
Terminating Decimal  A decimal that is complete after a certain number of digits, with no repetition.
For example, 29/40 = 0.725
Greater Than (>) and Less Than (<) The open end of the symbol always points towards the larger number. For example, 8 > 4 whereas 2 < 10.
Fraction Tipping  To rewrite a fraction into long division format you must take the numerator and move it to the right of the denominator.
Repeating Decimal  A decimal in which a block of one or more digits eventually repeats in a pattern. Example, 25/99 = 0.252 525..
Terminating Decimal  A decimal that is complete after a certain number of digits, with no repetition.
For example, 29/40 = 0.725
Greater Than (>) and Less Than (<) The open end of the symbol always points towards the larger number. For example, 8 > 4 whereas 2 < 10.
MULTIPLYING AND DIVIDING DECIMALS
Today the students worked on the multiplication and division of decimal numbers. They were reintroduced to the methods of long multiplication and division that was taught to them in previous years.
Reminders:
1. Dividend is the quantity to be divided
2. Divisor is the number by which the dividend is to be divided, or the number of parts into which the dividend is to be divided.
3. Quotient is the answer to a division question.
** When the divisor is a decimal, change it to a whole number by moving the decimal point to the right. Balance the change in the divisor by moving the decimal point in the dividend an equal number of places to the right. **
4. Long Multiplication  For a review of the steps of long multiplication check out the following weblink:
http://bit.ly/rR2sCB It uses animation to show how to work through multiplication with decimals.
4. Long Multiplication  For a review of the steps of long multiplication check out the following weblink:
http://bit.ly/rR2sCB It uses animation to show how to work through multiplication with decimals.




INTRODUCTION TO RATIOS
A MATH LOVE STORY
Concept 3
Proportions are built from ratios. A "ratio" is just a comparison between two different things. For instance, someone can look at a group of people, count noses, and refer to the "ratio of men to women" in the group. Suppose there are thirtyfive people, fifteen of whom are men. Then the ratio of men to women is 15 to 20.
Notice that, in the expression "the ratio of men to women", "men" came first. This order is very important, and must be respected: whichever word came first, it's number must come first. If the expression had been "the ratio of women to men", then the ratio would have been reversed "20 to 15".
Expressing the ratio of men to women as "15 to 20" is expressing the ratio in words. There are two other notations for this "15 to 20" ratio:
Notice that, in the expression "the ratio of men to women", "men" came first. This order is very important, and must be respected: whichever word came first, it's number must come first. If the expression had been "the ratio of women to men", then the ratio would have been reversed "20 to 15".
Expressing the ratio of men to women as "15 to 20" is expressing the ratio in words. There are two other notations for this "15 to 20" ratio:
 ratio notation: 15 : 20
fractional notation: 15/20
Today we watched the video below entitled "Bad Date" which explores the pursuit of a true 1:1 relationship. We then completed the Bad Date handout (see below).
bad_date_learner.pdf  
File Size:  1296 kb 
File Type: 


SOLVING PROPORTIONS
Reminders:
A ratio is a comparison of two numbers. We generally separate the two numbers in the ratio with a colon (:). Suppose we want to write the ratio of 8 and 12.
We can write this as 8:12 or as a fraction 8/12, and we say the ratio is eight to twelve.
Each number in a ratio is referred to as a term. For example, 5 and 9 are the terms in the ratio 5:9.
Equivalent Ratios are formed when two ratios have the same simplest form. For example, 5:20 and 1:4 both have a simplest form of 1:4.
A Scale Factor is the number that you multiply or divide a ratio by to make ratios equal. For example, to make 1:4 equivalent to 5:20 you must multiply each term in the first ratio by 5 (1 x 5 and 4 x 5). Therefore, 5 is the scale factor.
A proportion is written in the form where one ratio is equal to another. To find how much larger or smaller the ratio is in proportion to another we must multiply or divide by the scale factor.
A ratio is a comparison of two numbers. We generally separate the two numbers in the ratio with a colon (:). Suppose we want to write the ratio of 8 and 12.
We can write this as 8:12 or as a fraction 8/12, and we say the ratio is eight to twelve.
Each number in a ratio is referred to as a term. For example, 5 and 9 are the terms in the ratio 5:9.
Equivalent Ratios are formed when two ratios have the same simplest form. For example, 5:20 and 1:4 both have a simplest form of 1:4.
A Scale Factor is the number that you multiply or divide a ratio by to make ratios equal. For example, to make 1:4 equivalent to 5:20 you must multiply each term in the first ratio by 5 (1 x 5 and 4 x 5). Therefore, 5 is the scale factor.
A proportion is written in the form where one ratio is equal to another. To find how much larger or smaller the ratio is in proportion to another we must multiply or divide by the scale factor.


CALCULATING SCALE FACTOR
Today we furthered our understanding of scale factor and how to calculate it in relation to proportions. We watched the Math Snacks video entitled Scale Ella. This entertaining video explores "practical" uses of scale factor. We then completed the worksheets below which reinforced the concepts from the video.
scaleella_learner.pdf  
File Size:  764 kb 
File Type: 
Mid Chapter Review
Reminders
Reminders
 To find out if a fraction is terminating, look at the denominator and find its prime factors.
 If the prime factors are 2 and/or 5's only, then it is terminating
 If not then it is nonterminating
RATES
Concept 6
 Speed  the rate at which a moving object changes position over time
 Rate  a comparison of two quantities measured in different units, and includes units
 Unit Rate  a rate in which the second number is 1. Ex 60km/h (here the h stands for one hour)
 We need to find the unit cost or the unit rate to compare two or more quantities.
Due: Wednesday, November 30th
REPRESENTING PERCENT
Concept 7
Whenever we have a whole of an object we say we have 100 percent of it.
Percents are rates or fractions that are compared to 100. The word "cent" derives from a Latin word meaning 100. It is found in words like "century" that means 100 years. When we add the prefix "per" to the front of "cent" we are essentially saying what would ______ if it was per 100.
As we found yesterday, a rate compares two quantities of different units. When we consider percents we are talking about how a rate would compare if it were increased or decreased out of 100.
We can use percent grids to get a visual representation of what a percent looks like. Watch the following video clip to see how to use percent grids to model percents and fractions:
Concept 7
Whenever we have a whole of an object we say we have 100 percent of it.
Percents are rates or fractions that are compared to 100. The word "cent" derives from a Latin word meaning 100. It is found in words like "century" that means 100 years. When we add the prefix "per" to the front of "cent" we are essentially saying what would ______ if it was per 100.
As we found yesterday, a rate compares two quantities of different units. When we consider percents we are talking about how a rate would compare if it were increased or decreased out of 100.
We can use percent grids to get a visual representation of what a percent looks like. Watch the following video clip to see how to use percent grids to model percents and fractions:
Now that you can visualize what is meant by "a fraction of a whole", then we can start to apply this to calculating percents that involve whole numbers or decimals, and whole number percents that are greater than 100%.
Example A  St. Mary school has a population of 800 students from grades 712. About 15% of the population is in grade eight. What is the population of the grade eights at St. Mary?
1) To start we should use a percent grid to help us to visualize what 15% looks like. In this case we would shade in 15 boxes out of the total of 100 boxes in one of the grids. This represents 15% or "15 out of 100".
2) Now we need to consider how many students are represented by each box on the grid. To do this we take the entire (100%) population of St. Mary students (800) and divide it by the total number of boxes in one grid (100).
800 students / 100 boxes gives us a rate of 8 students/box. This number is called our scale factor.
3) Now we can calculate how many grade eight students there are by referring to how many boxes we shaded in earlier. In this case, we shaded in 15 boxes. If each box represents 8 students there are about (15 x 8) 120 grade eight students at St. Mary.
Example B  Last year there were 15 students who took part in the school talent show. This year, there is 120% more students participating in the show. How many more students are participating this year and how many are participating in total?
1) To start use a percent grid to visualize what 120% looks like. In this case, you would shade in 120 boxes (or one whole grid and 20 boxes on the next grid). This represents 120% or "120 out of 100".
2) Now consider how many students are considered by each box on the grid. Take the entire population of students who participated in the talent show last year (15) and divide it by the total number of boxes in one grid (100).
15 students / 100 boxes gives us a rate of o.15 students/box. This number is our scale factor.
3) Now we can calculate how many students there are in this years talent show by referring to how many boxes we shaded in earlier. In this case, we shaded in 120 boxes. If each box represents 0.15 students there are (120 x 0.15) 18 more students performing at the talent show this year or (15 + 18) 33 total students performing this year.
Example A  St. Mary school has a population of 800 students from grades 712. About 15% of the population is in grade eight. What is the population of the grade eights at St. Mary?
1) To start we should use a percent grid to help us to visualize what 15% looks like. In this case we would shade in 15 boxes out of the total of 100 boxes in one of the grids. This represents 15% or "15 out of 100".
2) Now we need to consider how many students are represented by each box on the grid. To do this we take the entire (100%) population of St. Mary students (800) and divide it by the total number of boxes in one grid (100).
800 students / 100 boxes gives us a rate of 8 students/box. This number is called our scale factor.
3) Now we can calculate how many grade eight students there are by referring to how many boxes we shaded in earlier. In this case, we shaded in 15 boxes. If each box represents 8 students there are about (15 x 8) 120 grade eight students at St. Mary.
Example B  Last year there were 15 students who took part in the school talent show. This year, there is 120% more students participating in the show. How many more students are participating this year and how many are participating in total?
1) To start use a percent grid to visualize what 120% looks like. In this case, you would shade in 120 boxes (or one whole grid and 20 boxes on the next grid). This represents 120% or "120 out of 100".
2) Now consider how many students are considered by each box on the grid. Take the entire population of students who participated in the talent show last year (15) and divide it by the total number of boxes in one grid (100).
15 students / 100 boxes gives us a rate of o.15 students/box. This number is our scale factor.
3) Now we can calculate how many students there are in this years talent show by referring to how many boxes we shaded in earlier. In this case, we shaded in 120 boxes. If each box represents 0.15 students there are (120 x 0.15) 18 more students performing at the talent show this year or (15 + 18) 33 total students performing this year.
SOLVING PERCENT PROBLEMS USING PROPORTION
Concept 8
Yesterday we learned how to visualize percentages before solving them using percent grids. Today we will use proportions (?/100 = part/whole) to help to solve percent problems.
Any given percent can be written as a fraction. For example, 65% is the same as 65/100. This means then, that we can use given information in a word problem to help us write a propottion whenever percents are involved.
Example A  What is 16% of 90?
1) To solve this we first need to write this as a proportion. The proportion would look like this:
16/100 = ?/90 (The ? refers to the unknown number).
2) We can then use corresponding terms in each of the fractions to calculate the scale factor.
To find the scale factor, divide the larger (100) of the corresponding terms by the smaller (90) of the corresponding terms.
100 / 90 = 1.11 (The answer  1.11  is called the scale factor).
3) We can then use the scale factor to find the unknown value. In this case we will divide 16 by the scale factor.
16 / 1.11 = 14.4 (14.4 is the answer or rather 16/100 = 14,4/90)
Example B  A dog has a litter of 3 male pups and 5 female pups. What percent of the pups in the litter are male?
1) To solve this we first need to write this as a proportion. There are 3 male pups out of a total of 8 pups. What percentage of the pups are male?
The proportion in this case would look like this:
3/8 = ?/100
2) We can then use the corresponding terms in each of the fractions to calculate the scale factor. In this case, we would divide the 100 by 3.
100/8 = 12.5 (The number 12.5 is the scale factor)
3) We can now use the scale factor to find the unknown value. In this case, we will multiply 3 by the scale factor.
3 x 12.5 = 37.5 (37.5% is the answer or rather 3/8 = 37.5/100)
Of the litter of pups, 37.5% were male.
Concept 8
Yesterday we learned how to visualize percentages before solving them using percent grids. Today we will use proportions (?/100 = part/whole) to help to solve percent problems.
Any given percent can be written as a fraction. For example, 65% is the same as 65/100. This means then, that we can use given information in a word problem to help us write a propottion whenever percents are involved.
Example A  What is 16% of 90?
1) To solve this we first need to write this as a proportion. The proportion would look like this:
16/100 = ?/90 (The ? refers to the unknown number).
2) We can then use corresponding terms in each of the fractions to calculate the scale factor.
To find the scale factor, divide the larger (100) of the corresponding terms by the smaller (90) of the corresponding terms.
100 / 90 = 1.11 (The answer  1.11  is called the scale factor).
3) We can then use the scale factor to find the unknown value. In this case we will divide 16 by the scale factor.
16 / 1.11 = 14.4 (14.4 is the answer or rather 16/100 = 14,4/90)
Example B  A dog has a litter of 3 male pups and 5 female pups. What percent of the pups in the litter are male?
1) To solve this we first need to write this as a proportion. There are 3 male pups out of a total of 8 pups. What percentage of the pups are male?
The proportion in this case would look like this:
3/8 = ?/100
2) We can then use the corresponding terms in each of the fractions to calculate the scale factor. In this case, we would divide the 100 by 3.
100/8 = 12.5 (The number 12.5 is the scale factor)
3) We can now use the scale factor to find the unknown value. In this case, we will multiply 3 by the scale factor.
3 x 12.5 = 37.5 (37.5% is the answer or rather 3/8 = 37.5/100)
Of the litter of pups, 37.5% were male.
SOLVING PERCENT PROBLEMS USING DECIMALS
Concept 9
Definitions:
The following list of definitions will be helpful in working through the word problems in today's questions.
a) Interest = The charge for borrowing money from a financial institute (usually a bank) or the return for lending it.
b) Loan = A sum of money that is borrowed that is expected to be paid back with interest.
c) Commission = An amount of money (usually a percent) that is paid to a sales person upon the completion of a sale for a third party
d) Sales tax = A tax collected by all retailers when they make a retail sale. This money is given to the government (Federal, Provincial or Municipal) to be used to provide services for the citizens.
e) Discount = An amount or percentage of money that is deducted from the regular price of an object.
The following web link provides a good summary of the types of questions assigned to students today and how to answer them by converting the percent to a decimal.
http://eduplace.com/math/mathsteps/6/c/index.html
The video below provides a description of how to convert percents to decimals in questions involving discounts. It also reviews yesterdays concept of how to solve these same types of problems using proportions instead of decimals.
Concept 9
Definitions:
The following list of definitions will be helpful in working through the word problems in today's questions.
a) Interest = The charge for borrowing money from a financial institute (usually a bank) or the return for lending it.
b) Loan = A sum of money that is borrowed that is expected to be paid back with interest.
c) Commission = An amount of money (usually a percent) that is paid to a sales person upon the completion of a sale for a third party
d) Sales tax = A tax collected by all retailers when they make a retail sale. This money is given to the government (Federal, Provincial or Municipal) to be used to provide services for the citizens.
e) Discount = An amount or percentage of money that is deducted from the regular price of an object.
The following web link provides a good summary of the types of questions assigned to students today and how to answer them by converting the percent to a decimal.
http://eduplace.com/math/mathsteps/6/c/index.html
The video below provides a description of how to convert percents to decimals in questions involving discounts. It also reviews yesterdays concept of how to solve these same types of problems using proportions instead of decimals.
This video provides a description of how to use decimals to solve questions involving simple interest:
CHAPTER REVIEW
In preparation for the unit test, the students will complete a review of the concepts we have covered.
In preparation for the unit test, the students will complete a review of the concepts we have covered.
UNIT TEST