Chapter 7 – Decimals, Ratio, Proportion, and Percent



Decimals, Ratios, Proportion, and Percent Unit

Decimals

(class period before we will do the Base 10 exploration of decimals activity)

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Exploring Decimal Numbers with Base 10 blocks

MA318

1. If the Cube is the unit (whole), then what number is represented by 2 Blocks, 3 Flats, 0 Longs, and 8 Cubes?

2. Draw a representation of the number 4,236 as base 10 pieces.

3. Can you represent 46, 321 with the base 10 pieces? Why or why not?

4. Can you represent 8 ½ with the blocks you have? Why or why not?

5. Name the other base 10 pieces when the given piece is considered ‘one unit’.

|Block |Flat |Long |Cube |

| | |One | |

| |One | | |

|One | | | |

6. How would you represent 0.3? (draw below and be sure to indicate what the ‘unit’ is)

7. How would you represent 4.3? (draw below and be sure to indicate what the ‘unit’ is)

8. How many tenths are in four wholes? In forty wholes?

9. What do you have to add to 0.9 to have one whole? (draw below and be sure to indicate what the ‘unit’ is)

10. The number 4.5 is ________ ones and __________ tenths, or ___________ tenths.

11. If the flat is the unit, which of the following are equivalent to one flat and four longs: 14? 1.4? 140? 14 longs? 140 cubes?

12. If 6.40 can be described as 0.64 tens or 6.4 ones or 64 tenths or 640 hundredths, then 18.04

can be described as ____________ tens or ____________ ones or ___________ tenths or

_________ hundredths.

13. Can you think of a way to represent 0.018 using base 10 pieces? Draw your answer below and indicate which piece is the unit.

14. Which number is smaller 12.17 or 12.4? Draw a representation to demonstrate this and indicate your unit.

Notes on Decimals

I. What is your understanding of decimals?

II. Decimals are representations of fractions using our place value system

A. (Discussion of activity using base 10 blocks to model decimals)

1. If Flat is the unit, then Long is 1/10 and Cube is 1/100

2. 4F, 3L, 2C = 4 + 3/10 + 2/100 = 432/100

3. 4F, 3L, 2C = 4.32 (Note decimal point)

4. We read this as 4 and 32 hundredths (note it could be read 4 and 3 tenths plus

2 hundredths.

5. Depending on how you represent the unit (whole), base 10 blocks can be used

to represent a variety of decimal numbers.

B.

|[pic] |[pic] |[pic] |[pic] |Decimal Point |[pic] |[pic] |[pic] |

|1000 |100 |10 |1 |. |[pic] |[pic] |[pic] |

|Thousands |Hundreds |Tens |Ones |(And) |Tenths |Hundredths |Thousandths |

C. Write 259.371 in expanded form, then read the number correctly

1. 2(100) + 5(10) + 9(1) + 3(1/10) + 7(1/100) + 1(1/1000)

2. Say two-hundred, fifty-nine and three hundred, seventy-one thousandths

D. Number line representation of decimals.

(---------------------------------------------------------------------------------(

1. Locate 0.53

2. Locate 0.791

3. Which is larger 0.751 or 0.76 ?

E. Rewrite the fractions as decimals and say the numbers correctly

1. 3/100 = 0.03

2. 25 5/8 = 25.625

F. Terminating decimals – Let a/b be a fraction in simplest form. Then a/b has a

terminating decimal representation if and only if b has only 2’s and/or 5’s in its prime

factorization.

1. Express [pic] as a decimal without using a calculator.

III. Ordering decimals – 4 methods

A. Hundreds square – works best for tenths or hundredths

B. Number Line

C. Place Value Method – matching place values from left to right until a place value is

found in which the digits are different. Which digit is bigger in that place value? That is

the larger number. Exp. Which is bigger 0.125 or 0.13?

D. Fraction methods – convert decimals to fractions with common denominators,

compare numerators.

IV. Mental Math and Estimation with decimals.

A. Use properties to aid with mental math problems (p.268)

B. Use fraction equivalents for common decimals/fractions (table p. 268)

1. 68 x 0.5 = 68 x ½

2. 0.25 x 48 = ¼ x 48

C. Multiplying and dividing by powers of 10

1. This is one of the major strengths of place value decimals.

2. 3.75 x 10^4 = 37500 (multiplying – move decimal point right)

3. 127.9/10^2 = 1.279 (dividing – move decimal point left)

D. Estimation

1. Range

2. Front-end

3. Rounding to the nearest whole or half

a. Rounding rules

b. Round 27.9521 to the nearest tenth? The nearest hundredth?

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Algorithms for Operations with Decimals

I Addition

A. Fraction Approach

1. 1.57 + 2.7

B. Decimal Approach

1. 1.0218 + 3.12

II. Subtraction

A. Fraction Approach

1. 0.297 – 0.01

B. Decimal Approach

1. 2.13 – 0.0018

III. Multiplication

A. Fraction Approach

1. 2.52 x 1.3

B. Decimal Approach – multiply as whole numbers, the number of digits to the right of

the decimal point in the answer is the sum of the number of digits to the right of the

decimal points in the numbers being multiplied.

1. 2.52 x 1.3

C. Alternative to Decimal Approach – estimate the answer, use the estimation to place the decimal point.

IV. Division of decimals

A. Fraction Approach

1. [pic]

2. [pic] is the same as [pic] (maintaining the quotient)

The zero was obtained in the divisor when a common denominator was found.

Also note that [pic]

B. Decimal Approach

1. Use the long division algorithm on the problem using the whole number

divisor – it is important to stress place value when teaching this algorithm

C. Terminating Decimals have denominators of the form [pic]

D. Repeating Decimals

1. [pic], [pic]

2. period of the decimal (number of the repeated digits)

3. the repeated digits are called the repetend.

4. If the denominator has factors other than 2 or 5 it will repeat.

E. Converting repeating decimals to fraction form.

1. [pic]

2. If n = repetend of a decimal then 10n = power of 10 needed to use this

algorithm.

[pic]

3. Every fraction has a decimal representation and every repeating decimal has a

fraction representation.

Ratio and Proportion

I. Ratios are everywhere – can you give examples?

A. Def. Ratio – an ordered pair of numbers written a:b, with b not equal to zero.

1. Allows us to compare the relative sizes of 2 quantities

B. Ratios are like, but not like fractions

1. They can compare part-to-whole or whole-to-part relations

Exp. Boys: All children or All children: Boys

2. They can be used to compare part-to-part

Exp. Boys: Girls

3. The part-to–whole ratios can be expressed as fractions

C. Ratios always represent relative, not absolute, amounts

E. Equality of ratios [pic].

1. [pic] can also be written a:b = c:d. The quantities a and d are

called the extremes and the quantities b and c are called the means.

2. Ratios are equal if and only if the product of their means equals the product of

the extremes (restatement of E).

II. Proportion – an equation setting 2 ratios equal.

A. Used in problem solving

1. Grape juice concentrate is mixed with water in a ratio of 1 part concentrate to 3

parts water. How much grape juice can be made from a 10 oz. can of concentrate?

2. A crew clears brush from ½ acre of land in 3 days. How long will it take the same crew to clear the entire plot of 2 ¾ acres?

a. Should we use fractions or decimals?

b. Be consistent when setting up ratios and proportions.

B. Ratios involving different units are called rates.

1. Rates that relate one unit of some amount to another amount are sometimes

called unit rates.

Exp. 15 miles per 1 gallon

$0.39 per 1 dozen

Percent

I. Percent – per hundred – special kind of ratio – can you give examples?

N% = [pic]

A. Fractions are related to decimals are related to percents.

4/100 = .04 = 4%

63/100 = .63 = 63%

120/100 = 1.20 = 120%

1/5 = 20/100 = .20 = 20%

II. Solving Percent problems

A. Proportion approach

1. 4200 lb automobile contains 357 lb of rubber. What percent of the car’s total

weight is rubber? n =8.5%

B. Grid approach

1. Visualize the problem using a 10 x 10 grid.

2. A car was purchased for $13,000 with a 20% down payment. How much was

the down payment?

C. Equation approach

1. 20% of 180 is what number?

2. 20% of what number is 16?

3. What percent of 240 is 30?

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