Decimals

Children should be led to make their own investigations, and to draw upon their own inferences. They should be told as little as possible which can produce unlimited learning potential.

Herbert Spencer

Intellectual Moral and Physical

1864

McDonald’s Menu….I’m Lovin It!!

Double Cheeseburger: $ .99 Big Mac Value Meal: $ 4.79 Chicken McNuggetts Meal: $ 3.80 Small Drink: $ .99 McFlurry: $ 1.97 Salad: $ 4.80 2 Cheeseburger Meal: $ 3.70 Ice Cream Cone: $ .87

Order Up!Least Expensive to Most Expensive Ice Cream Cone .87 Double Cheeseburger: .99 Small Soft Drink: .99 McFlurry: 1.97 2 Cheeseburger Meal: 3.70 Chicken McNuggetts Meal: 3.80 Big Mac Value Meal: 4.79 Chicken Salad: 4.80

What Do I Mean Compare Decimals? When we compare we use terms such as:

Less than < Greater than > Equal to =

Comparing decimals is similar to comparing whole numbers. 45<47 150>105

When we compare decimals we use place value or a number line.

Place Value

1,000 100 10 1 0.1 0.01 0.001 0.0001

Compare Sara’s score with Danny’s score.

1. Line Up Decimal Points Sara: 42.1 Danny: 42.5

2. Start at the left and find the first place where the digits differ. Compare the digits

1 < 5 42.1 < 42.5 This means Sara’s score was

lower than Danny’s score.

Sara 42.1

Danny 42.5

Ross 42.0

Bethany 40.7

Jacob 46.1

Diving Results

Let’s Try Using A Number Line

Sara 42.1Danny 42.5Ross 42.0

Bethany 40.7Jacob 46.1

42.0 42.1 42.5

Numbers to the right are greater than numbers to the left. Since 42.5 is to the right of 42.1 we have:

42.5 > 42.1

42 43

Equivalent Decimals

Decimals that name the same number are called equivalent decimals.

0.60 and 0.6

Are these the same???

0.6 0.60=

Annexing Zeros

This means placing a zero to the right of the last digit in a decimal.

0.6

Although we added a zero, the value of the decimal did not change!!

Annexing or adding zeros is useful when ordering a group of decimals.

0.60

Ordering Decimals

We can order decimals from least to greatest or we can order from greatest to least.

Let’s try an example:

Order 15, 14.95, 15.8, and 15.01 from least to greatest

First, line up the decimal points 15 14.95 15.8 15.01

15, 14.95, 15.8, 15.01

15, 14.95, 15.8, 15.01 Next, annex zeros so that each number has the

same number of decimal places

15.00 14.95 15.80 15.01

Finally, use place value to compare the decimals. Always start from the left!!

14.95, 15, 15.01, 15.8

One More Example

Order these numbers from greatest to least

35.06, 35.7, 35.5, 35.849

35.0635.735.535.849

35.06035.70035.50035.849

35.06, 35.5, 35.7, 35.849

Rounding Decimal Numbers

Sometimes close enough is good enough!

Molly has $16 in her pocket

She buys:

3 hotdogs @ $1.89 each

3 milkshakes @ $2.19 each

3 Large French fries @ $ $1.79 each

Does she have enough money?

Gas costs $1.79 per gallon

I’m running on empty and I have a 20 gallon tank on my truck……

About how much will it cost to fill up?

You have one hour to take a 50 question test.

About how much time can you spend on each question?

Close enough is good enough!

The Rounding Poem

Find your number

Look right next door

4 or less, just ignore

5 or more add one more!

Rounding Whole Numbers

Round 474 to the closest hundred.

474 Find your number

Look right next doorFour or less, just ignore

Five or more, add one more

500

Round 474 to the closest 10’s

474

Just ignore

470

Try These

4,215

394

27

127,591

638

What about decimal numbers

The same rules (and poem) apply!

Round 4.167 to the closest hundredths

4.167 Find your number

Look right next door

4.17 Five or more, add one more

Remember

If you are asked to round to the nearest tenth…….your

answer stops at the tenth place!

Example: Round 42.73 to the closest tenth would be 42.7

Special Case!Rounding a 9

Round 2.197 to the closes hundredths

Since there is a “9” in the hundredths place consider the tenth and hundredths together – 19 – do you leave it at 19 or move it to 20?

Round 4.95 to the closest tenth

Since there is a “9” in the tenth spot consider the ones and tenths together…… should you leave it at 49 or move it to 50?

4.95 rounded to the nearest tenth is 5.0

Try these

Round to tenths:

4.56 4.6

5.32 5.3

0.98 1.0

Round to hundredths

18.784 18.78

9.456 9.46

9.997 10.00

Addition and Subtraction

Model using base ten blocks and an open number line.

a) 13 + 18

b) 1.3 + 1.8

Show 2 different representations of the mathematical thinking as you evaluate each expression.

What Does Addition Look Like?

13

Model using base ten blocks and an open number line.a) 97 − 58b) 9.7 − 5.08

Show 2 different representation of solutions for each expression.

9.7

What Does Subtraction Look Like?

Reflect1. Describe the strategies you used to

solve addition problems.2. Describe the strategies you used to

solve subtraction problems.3. a) How is adding whole numbers and

decimal numbers similar? b) Different?

4. a) How is subtracting whole numbers and decimal numbers similar? b) Different?

Compare your solutions. How are they similar? different?

a)Solve this problem in 2 different ways.

b)Show your work.

Developing Conceptual Understanding

The veterinarian told Camilla that the weight of her puppy increased by 3.5 lb in the last month. If the puppy weighs 15.6 lb now, what was its weight a month ago?

Alternative Algorithms

Traditional Addition

Partial-Sums Addition

1 1

348

+583

931

34.8+ 58.3 80.0 12.0 1.1 93.1

348+ 583 800 120 11 931

1 1

3.48

+5.83

9.31

Adding-Up Subtraction

7.24

– 3.79 3.79 + 0.01

3.80 + 3.00

6.80 + 0.40

7.20 + 0.04

3.45

724

– 379 379 + 1

380 + 300

680 + 40

720 + 4

345

Alternative Algorithms

439.56

+ 88.10

439.56

+ 88.10

439.56

+ 88.10417.66 411.110 41117.66

PracticePractice

439.56

– 88.10

439.56

– 88.10451.46 451.46

Sharing Strategies

1. What strategies do you use to add and subtract whole numbers mentally?

2. What strategies do you use to add and subtract decimal numbers mentally?

257 + 39257 − 39

25.7 + 3.925.7 – 3.9

Mental Math for Addition

Adding On

136 + 143

136 + 100 = 236236 + 40 = 276276 + 3 = 279

Compensation

236 + 297

236 + 300 = 536536 – 3 = 533

Constant Sum

153 + 598

Take 2 from 153 and add 2 to 598151 + 600 = 751So, 153 + 598 = 751

236 533 536136 236 276 279

100 40 3

Partial Subtraction387 – 146

387 – 100 = 287287 – 40 = 247247 – 6 = 241

Compensation

547 – 296

547 – 300 = 247247 + 4 = 251So547 – 296 = 251

Constant Difference598 – 153

Add 7 to 153 to make it 160. So, add 7 to 598.605 – 160 = 445Check: 153 + 445 = 598

247 251 547241 247 287 387

6 40 100

Mental Math for Subtraction

Apply These Strategies

Joan is wondering whether the class’ hot lunch sales were above or below the actual cost of the lunches, $637.45. Here are the numbers she is using. Calculate using different mental math strategies:

1. 637.45 + 219.18

2. 637.45 – 219.18