Chapter 7Decimals, Real Numbers &

Proportional Reasoning

Position Names

Expanded Form

• 25.3 = 2x101 + 5 x 100 + 3 x 10 -1

• 136.72 = 1x102 + 3x101 +6x100 +7x10-1 +2x10-2

• 1.235 = 1x100 + 2x10-1 + 3x10-2 +5x10-3

Squares, Strips & Flats

Base Ten Blocks

Activity 2c

Activity 1 &2

Negative Exponents

• 10-2 = 1/102

• 10-3 = 1/103

• 10-4 = 1/104

• 5-2 = 1/52

• A negative exponent means I over.• Usually a negative exponent means a fraction.

Converting Decimals to Fractions

• 17/40 = 17 ÷ 40

• 40 17.00000.925

Converting Decimals to Fractions

.355 = 355 = 5 x 71 = 71 1000 5 x 200 200

.256

.58

Repeating Decimal

• .3333…

• .242424…

Ordering Decimals

• To order positive decimals. Line up the decimal points and determine the first digit that is greater. (Add zeros at the end of the decimaqls to have the same number of decimal places.)

• .256• .238• .2563

Irrational Numbers

• Non-terminating, non-repeating decimals.

• .0123456789101112131415….• .0100100010000100001……

• The square root of any non perfect square.

Homework

• Pg 431 # 1,2,3,5,6,7,8,10,31,32

Adding & Subtracting

• To Add or Subtract Decimals write the numbers in vertical style, lining up the decimal points and then add or subtract just as we add integers. BRING THE DECIMAL POINT STRAIGHT DOWN.

• 37.6 + 12.45• 37 + 12 is 49. Our answer should be around

49

• 37.6• +12.45• 50.05

• 23.47 + 7.81

• 351.42 – 217.815

Multiplying & Dividing Decimals

• .9 x .25 =• 9/10 x 25/100 =• 9 x 25 / 10 x 100 =• 225 / 1000 = .225• 2.3 x 3.2• 2.3 x 1.8• Activity 5

2.5 x.19 225 25 .475

1 digit to the right of the decimal point

2 digits to the right of the decimal point

3 digits to the right of the decimal point

2.56 1.2 512 256 3072• Think about what the answer should be.

Where would you put the decimal point? Why?

2.56 1.2 512 256 3.072

2 digits to the right of the decimal point

1 digit to the right of the decimal point

3 digits to the right of the decimal point

471.2 X 2.3 14136 9424108376Think about what the answer should be. (2x471)Where would you put the decimal point? Why?

537.6 ÷ 2.56 = ÷ = x

= 210

210. 2.56 537.60 We don’t divide by a decimal, so we make the

divisor a whole number. We multiplied the divisor by 100, so we have to multiply the dividend by 100 (move the decimal 2 places to the right.)

• We don’t divide by a decimal, so we make the divisor a whole number. We multiplied the divisor by 100, so we have to multiply the dividend by 100 (move the decimal 2 places to the right.)

Homework

• Pg 441 # 1,2,4,10,32-35

Ratios & Proportions

• A comparison of 2 numbers by division.• Ratio(nal) number• Should express ratios in simplest form.• Joe missed 12 free throws out of 45 attempts.• Ratio of:• Shots made to number attempted 33:45• Shots missed to number attempted 12:45 =4:15• Shots made to number missed 33:12 = 11:4

285 Boys & 228 Girls

• Ratio of:• Boys to Girls• Boys to students

One seventh of the students are nonswimmers.

• What is the ratio of nonswimmers to swimmers.

• Divide the students into 7 groups.

• 1 group will be non-swimmers.

Proportions

• A statement that says 2 ratios are equal.

• Then axd = cxb

• 28 x 21 = 49 x

• 12 = x

• Marci liked to make mixed nuts by combining 6 cups of cashews with 2 cups of pecans.

• If she has 5 cups of pecans, how many cups of cashews does she need?

• If she has 20 cups of cashews, how many cups of pecans does she need?

The ratio of boys to girls in Mr. A’s class is 5 to 7The ratio of boys to girls in Mr. B’s class is 2 to 5

• What fraction of Mr. A’s class are boys?• What fraction of Mr. B’s class are girls?• There are 24 students in Mr. A’s class. How

many are girls?• If Mr. A’s class is bigger than Mr. B’s class,

what would be a reasonable number of students in Mr. B’s class?

Similar Triangles

• Page 455- 7.21

Homework

• Pg 457 # 1 all, 2,3,4,5,637,38,39

Percents

• Percent from the Latin meaning per hundred.

• 25% = 25/100

• Pg 463• Activity 11

3 Basic Problems

• Find x• % of # =x• 15% of 1750• % of x = #• 15% of x is 262.50• X% of a # is another #• x% X 1750 = 262.50

% of #

• By Multiplying• 15% of 1750• 15% X 1750 =• .15 X 1750 = 262.50

% of #

• By Proportion• 15% of 1750

% of #

• By Diagram:

% of x = #

• By equation• 92% of x is 23• .92 X = 23• X = 23 / .92• X = 25

% of x = #

• By Proportion

• 92 X = 23 x 100• X = 2300 / 92• X = 25

X% of a # is another #x% of 35 is 28

• By Equation• X x 35 = 28• X = 28 / 35• X = .8 = 80%

X% of a # is another #x% of 35 is 28

• By Proportion

• 35 x = 28 x 100• X = 2800 / 35 = 80%

• 45% of the students at a college are males.• If the college has an enrollment of 22,500

students, how many are male? Use the 100 grid in your solution.

Homework

• Page 470 # 1,2,3,4,5,6all,7,9,10,37,38