October 12, th Grade. Percents

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2 6th Grade Percents

3 Table of Contents Click on the topic to go to that section Percents Percents & Fractions Percents & Decimals Using Percents Glossary & Standards Teacher Notes

4 Percents Return to Table of Contents

5 Percent When broken down, percent means out of 100. Per = out of Cent = 100 This means that it is a ratio which is always based on the total being 100. There are different ways to visually model this relationship found throughout nature. Math Practice These models can help when working with percent problems throughout mathematics.

6 10 x 10 Grid Model One way of looking at percents is using a 10 x 10 grid model. This allows us to visually compare different quantities to the 100 squares within the grid.

7 10 x 10 Grid Model There are 100 total squares. Click How many are shaded? 45

8 10 x 10 Grid Model There are 100 total squares. How many are shaded? Since there were 45 shaded squares, how could we write this as a fraction? Click Click Because percent means out of 100, we can say the shaded area is Click Click 45 or 45% 100

9 10 x 10 Grid Model There are 100 total squares. What percent are shaded? Click 70%

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11 Tape Diagram Another way of looking at percents is using a tape diagram. This allows us to visually compare different parts of a ratio and the different parts of a percent.

12 Tape Diagram There are 5 total squares. How many are shaded?

13 Tape Diagram % 20% 40% 60% 80% 100% There are 5 total squares. How many are shaded? Click 3 out of 5 Click Since there were 3 out of 5 shaded squares, how could we compare this to 100? We can divide 100 into 5 equal boxes, making each box of the tape 20%. Click We can then say that the shaded area is also 60%.

14 Tape Diagram % 20% 40% 60% 80% 100% There are 5 total squares. What percent is shaded? Click 80%

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16 Tape Diagram Tape diagrams can help us solve word problems! A survey was taken that asked a total of 1,000 participants whether or not they were happy with their job. An overall score was given. 30% of the participants were unhappy while the rest of the participants were happy with their job. How many of the participants were unhappy? Math Practice From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

17 Tape Diagram A survey was taken that asked a total of 1,000 participants whether or not they were happy with their job. An overall score was given. 30% of the participants were unhappy while the rest of the participants were happy with their job. How many of the participants were unhappy? First we need to draw the tape diagram and decided what information is important to the problem. # of Participants Percent From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

18 Tape Diagram A survey was taken that asked a total of 1,000 participants whether or not they were happy with their job. An overall score was given. 30% of the participants were unhappy while the rest of the participants were happy with their job. Then can further label both sides of our tape using the given information! # of Participants ,000 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Percent From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

19 Tape Diagram A survey was taken that asked a total of 1,000 participants whether or not they were happy with their job. An overall score was given. 30% of the participants were unhappy while the rest of the participants were happy with their job. We can now use the tape to find what # of participants matches up with our given percent! # of Participants ,000 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Percent From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

20 Tape Diagram A survey was taken that asked a total of 1,000 participants whether or not they were happy with their job. An overall score was given. 30% of the participants were unhappy while the rest of the participants were happy with their job. 300 of the total participants said they were unhappy with their job. # of Participants ,000 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Percent From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

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22 Double Number Line Another way of looking at percents is using a double number line. This allows us to visually compare specific numbers with specific percents.

23 Double Number Line There are 20 total tick marks on each number line. At what number is the blue circle at?

24 Double Number Line 25%35% 50% 75% 100% There are 20 total tick marks on each number line. Click At what number is the blue circle at? 7 out of 20 Math Practice Click Since the blue circle is at 7 out of 20, how could we compare this to 100? Click Since 7 is in between 5 and 10, we can mark common percentages we know, like 5 = 20% and 10 = 50%. Click We can then back track to the 7 by 5% intervals and Click determine that the blue circle is also at 35%.

25 Double Number Line There are 20 total tick marks on each number line. At what percent is the blue circle at? Click 80%

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27 Percents & Fractions Return to Table of Contents

28 Looking more closely at the grid model, we can see how percents and fractions are related. For example, Percent out of The total number of boxes is the denominator of the fraction.

29 Fraction to Percent Notice that we are able to go from fraction to percent because the denominator of the fraction is 100. out of The denominator needs to be 100 in order to be turned into a percent. What if the denominator is not 100?

30 Fraction to Percent is equal to what percent? We must force the denominator to be 100 to rewrite as a percent.

31 Fraction to Percent is equal to what percent? We can also look at a tape diagram: % 20% 40% 60% 80% 100% The model supports that is equal to 40%.

32 Percent to Fraction We can also use the fact that the denominator has to be 100 to go from a percent back to a fraction. What fraction is 75%? However, we always must reduce a fraction to simplest form! Therefore,

33 Percent Notice that when going from fraction to percent it needs to have a denominator of 100 before it can be turned into a percent, but when going from percent to fraction, the fraction should be reduced to simplest terms. Math Practice

34 Fractions & Percents Match the percent with the equivalent fraction by touching two cards.

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36 Percents & Decimals Return to Table of Contents

37 Percent Percents can also be expressed as decimals. Since percent means out of 100, we can use place value to help us. Both can be read as thirty-six hundredths. Since percent is out of 100 any decimal that ends in the hundredths place can be written by removing the decimal and adding a percent sign = 13% 0.25 = 25% 0.87 = 87% = 96%

38 Percent In truth, we are moving the decimal two places to the right when changing from a decimal to a percent = 63% 0.86 = 86% 0.02 = 2% This is important to note when a decimal does not end in the hundredths place % because the decimal must be moved two places. 0.3 = 30% 0.9 = 90% 1.34 = 134% = 2.5%

39 Decimal to Percent Hint: The letter D (for decimal) comes before P (for percent). Move to the right when changing from a decimal to a percent. D P

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41 Percent to Decimal To go from a percent to a decimal, move two decimal places to the left. Remember, if there is no decimal written, it is at the end of the number. 34% = % = % = 0.95 Keep in mind that if the percent is more than or fewer than two digits, the decimal still gets moved two places. 5% = % = % = 0.005

42 Percent to Decimal Hint: The letter D (for decimal) comes before P (for percent). Move to the left when changing from a percent to a decimal. D P

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44 Using Percents Return to Table of Contents

45 Using Percents Sometimes we need to find the percent of a number to help us solve a problem. That percent becomes a part of a total ratio. There are many ways to find that part of a ratio given the percent and the total. When the total, is 100 it is very easy to solve. What is 46% of 100? Since a percent is out of a total of 100, it would be 46.

46 Using Percents But what if the whole total is not 100? What if you wanted to find 30% of 50? Math Practice

47 Using Percents What if you wanted to find 30% of 50? Turn the percent into a fraction over 100 Create a proportion with the fraction and the total number Figure out the relationship between the denominators and do the same for the numerators. Solve We now know that 30% of 50 is 15.

48 Using Percents What if you wanted to find 30% of 50? We can also look at a tape diagram: % 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% The model supports that 30% of 50 is 15.

49 Using Percents Click What is 15% of 20? 3 Click What is 32% of 25? 8

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51 Using Percents Sometimes percents can be more than 100%. Treat them the same as you would any other percent. What is 250% of 50? 125 is 250% of 50

52 Using Percents 130% of 10. Click % of 220 Click 715

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54 Using Percents Using the same technique, we can find the total of a ratio given the percent and the part. 20% of the sixth grade students prefer chicken nuggets to pizza. There are 40 students who prefer chicken nuggets. How many students are in the whole sixth grade?

55 Using Percents 20% of the sixth grade students prefer chicken nuggets to pizza. There are 40 students who prefer chicken nuggets. How many students are in the whole sixth grade? Create an equivalent ratio. Make an equivalent fraction and solve There are 200 students in the sixth grade.

56 Using Percents 75 is 25% of what number? 75 is 25% of 300.

57 Using Percents Click 48 is 96% of what number? 50 Click 60 is 20% of what number? 300

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59 Selina bought a shirt on sale that was 20% less than the original price. The original price was $5 more than the sale price. What was the original price? Explain or show work. Math Practice Problem is from Illustrative Mathematics 6.RP Shirt Sale Click for link to commentary and solution.

60 Glossary & Standards Teacher Notes Return to Table of Contents

61 Percent A ratio which is always based on the total being 100. When broken down, percent means out of 100. Per = out of 1 =1/100 25% Cent = 100 or one hundredth $ Back to Instruction

62 10 x 10 Grid Model A square grid made of 100 squares (10 rows and 10 columns) that can be used to represent different part to whole situations 43 out of 100 squares shaded = = 43% 25 out of 100 squares not shaded = = 25% Back to Instruction

63 Tape Diagram A visual model that uses rectangles to represent the parts of a ratio % 20% 40% 60% 80% 100% 2 1/ 2 miles 4 = 80% $14 + $14 + $14 = $42 5 Back to Instruction

64 Double Number Line A diagram that is used to compare two quantities with different units. 25%35% 50% 75% 100% ounces grams 7 20 = 35% 12 ounces = 21 grams Back to Instruction

65 Standards for Mathematical Practices MP1 Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 Attend to precision. MP7 Look for and make use of structure. MP8 Look for and express regularity in repeated reasoning. Click on each standard to bring you to an example of how to meet this standard within the unit.

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