Decimals 1

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 2

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 Ones Tens Tenths Hundreds Thousands Hundredths Thousandths 1,000 100 10 1 0.1 0.01 0.001 0.0001 Tens Ones Tenths Hundreds Thousands Hundredths Thousandths Ten-thousandths

Compare Sara’s score with Danny’s score. Sara 42.1 Danny 42.5 Ross Diving Results Compare Sara’s score with Danny’s score. Line Up Decimal Points Sara: 42.1 Danny: 42.5 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

Let’s Try Using A Number Line Sara 42.1 Danny 42.5 Ross 42.0 Bethany 40.7 Jacob 46.1 42 43 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

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

15, 14.95, 15.8, 15.01 First, line up the decimal points 15 14.95 15.8

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 35.060 35.700 35.500 35.849 35.06 35.7 35.5 35.849 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?

About how much will it cost to fill up? 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. 500 474 Find your number Look right next door Four or less, just ignore Five or more, add one more

Round 474 to the closest 10’s 474 470 Just ignore

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 Five or more, add one more 4.17

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 hundredths Round to tenths: 4.56 4.6 18.784 18.78 5.32 5.3 9.46 9.456 10.00 0.98 1.0 9.997

Addition and Subtraction

What Does Addition Look Like? Model using base ten blocks and an open number line. 13 + 18 1.3 + 1.8 Show 2 different representations of the mathematical thinking as you evaluate each expression. 13

What Does Subtraction Look Like? Model using base ten blocks and an open number line. 97 − 58 9.7 − 5.08 Show 2 different representation of solutions for each expression. 9.7

Reflect Describe the strategies you used to solve addition problems. Describe the strategies you used to solve subtraction problems. a) How is adding whole numbers and decimal numbers similar? b) Different? a) How is subtracting whole numbers and decimal numbers similar?

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? Solve this problem in 2 different ways. Show your work. Compare your solutions. How are they similar? different?

Alternative Algorithms Partial-Sums Addition Traditional Addition 1 1 3.48 +5.83 9.31 1 1 348 +583 931 348 + 583 800 120 11 931 34.8 + 58.3 80.0 12.0 1.1 93.1

Alternative Algorithms Adding-Up Subtraction 724 – 379 379 + 1 380 + 300 680 + 40 720 + 4 345 7.24 – 3.79 3.79 + 0.01 3.80 + 3.00 6.80 + 0.40 7.20 + 0.04 3.45

Practice 439.56 + 88.10 439.56 + 88.10 439.56 + 88.10 417.66 411.110 41117.66 439.56 – 88.10 439.56 – 88.10 451.46 451.46

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 + 39 257 − 39 25.7 + 3.9 25.7 – 3.9

Mental Math for Addition Adding On 136 + 143 136 + 100 = 236 236 + 40 = 276 276 + 3 = 279 Compensation 236 + 297 236 + 300 = 536 536 – 3 = 533 Constant Sum 153 + 598 Take 2 from 153 and add 2 to 598 151 + 600 = 751 So, 153 + 598 = 751 136 236 276 279 100 40 3 236 533 536

Mental Math for Subtraction Partial Subtraction 387 – 146 387 – 100 = 287 287 – 40 = 247 247 – 6 = 241 Compensation 547 – 296 547 – 300 = 247 247 + 4 = 251 So 547 – 296 = 251 Constant Difference 598 – 153 Add 7 to 153 to make it 160. So, add 7 to 598. 605 – 160 = 445 Check: 153 + 445 = 598 241 247 287 387 6 40 100 247 251 547

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