Study Guide 3: Addition of Whole Numbers Category 2: Computation and Algebraic Relationships Vocabulary Addition Addends Missing addend Sum Total Plus Number sentence Equation Regroup Estimate Estimation Mental math Compatible numbers Commutative property Associative property Number patterns an operation that gives the total number when you put together two or more numbers (also can be called add or adding) a number added to find a sum a missing number being added in an equation the number that is the answer when two or more numbers are added another word for sum, an answer to an addition problem name of the symbol (+) used in an addition problem a math statement that has numbers, an operation symbol, and an equal or inequality sign a number sentence that uses an equal (=) sign to name a number in a different way without changing its value a number that is close to or about the same as the exact answer using approximate number values that are easier to work with solving number problems in your head instead of finding an answer with paper and pencil numbers that are easy to compute mentally and are close to the real numbers a rule that states that the order in which the numbers are added will not change the sum a rule that states that the sum of a set of numbers is the same no matter how the numbers are grouped a regular repeating design or sequence of shapes or numbers. C & C Educational Materials, all rights reserved 61
Concepts Adding Whole Numbers Addition uses numbers to put two or more quantities together. When the parts of a set are known, addition is used to name the whole in terms of the parts. This action involves joining the two parts or putting together the two parts. For example, the student has 5 red marbles and 3 blue marbles. How many marbles does the student have? The 5 and 3 are the parts (5 + 3) and the answer is the whole set, or 8. Addition with Base Ten Blocks Without Regrouping Using Place Value The third grade class sold 125 tickets to the carnival. The fourth grade class sold 214 tickets. How many tickets did both classes sell? You model both numbers with the blocks. You begin with the unit cubes. You add the 5 ones in 125 with the 4 ones in 214 to get a total of 9 ones. Then, you add the 2 tens in 125 with the 1 ten in 214 to get a total of 3 tens. In the final step you add the 1 hundreds block in 125 and the 2 hundreds blocks in 214 to get 3 hundreds for a total of 339. + Below shows the place value expanded form and the standard algorithm. 100 + 20 + 5 125 + 200 + 10 + 4 + 214 300 + 30 + 9 339 62 C & C Educational Materials, all rights reserved
Addition with Base Ten Blocks With Regrouping Using Place Value The third grade class sold 165 tickets to the carnival. The fourth grade class sold 256 tickets. How many tickets did both classes sell? You model both numbers with the blocks. Begin with the unit cubes. Add the 5 ones in 165 with the 6 ones in 256 to get a total of 11 ones. You cannot have more than 10 in the ones place, so you must regroup the 11 ones into a tens rod and 1 unit. Carry the tens rod to the tens place and place the 1 unit cube in the ones place in your answer. Then, add the 6 tens in 165 with the 5 tens in 256, plus the 1 ten you carried, to get a total of 12 tens. You cannot have more than 10 rods in the tens place, so you must regroup the 12 tens into a hundreds block and 2 tens in your answer. Carry the hundreds block to the hundreds place and place the 2 rods in the tens place. In the final step add the 1 hundreds block in 165 and the 2 hundreds blocks in 256, plus the 1 hundred you carried to get 4 hundreds for a total of 421. + Below shows the place value expanded form and the standard algorithm. 1 1 100 + 60 + 5 1 6 5 + 200 + 50 + 6 + 2 5 6 300 + 110 + 11 = 421 4 2 1 C & C Educational Materials, all rights reserved 63
Addition Using Place Value One strategy for addition is to use what you know about place value. You add the numbers using place value with the ones, tens, and hundreds. Here is an example problem. There were 267 balloons for the carnival and 547 stickers. How many balloons and stickers were there in all? 1 1 267 Add the ones 7 + 7 = 14 + 547 60 + 40 = 100 814 200 + 500 = 700 814 Order of Operations for Addition and Subtraction The order rule for addition and subtraction is to do the computation from left to right. Here is an example problem, 8-1 + 6. Working from left to right you take 8 1 = 7. Then add 7 + 6 = 13 If you did not work from left to right you would add 6 + 1 = 7. Then you would subtract 8 7 = 1. This gives you the wrong answer. When an equation has both addition and subtraction operations, you always work the problem from left to right. Commutative Property of Addition In an equation with addition and no subtraction, you do not have to add the three numbers from left to right. You can use the commutative property of addition. The commutative property of addition states the order in which the numbers are added will not change the sum. 6 + 5 = 5 + 6 6 + 5 = 11 and 5 + 6 = 11 Another example is 243 + 325 = 325 + 243 243 + 325 = 568 and 325 + 243 = 568 The commutative property can make the addition of three numbers easier. When adding 25 + 147 + 75, it is easier to change the order and add the 25 and 75 first. 25 + 75 = 100, then add 100 + 147 = 247 The numbers 25 and 75 are called compatible numbers because they are easy to add. Associative Property of Addition The associative property of addition states that the sum of two numbers is the same no matter how the numbers are grouped. (18 + 6) + 24 = 18 + (6 + 24) Addition is always done inside the parentheses first. It is easier to add the compatible numbers 6 + 24 = 30 Then add 30 + 18 = 48 64 C & C Educational Materials, all rights reserved
The associative property of addition can be used when adding the two numbers 243 + 325 by breaking apart a number.. 243 + 325 is equivalent to 243 + (300 + 25) Using the associative property it can be written (243 + 300) + 25 Add in the parentheses 243 + 300 = 543. Then add 543 +25 = 568 Addition Based on the Relationship of Addition and Subtraction Subtraction is the opposite of addition. You can subtract from one number and then add what you subtracted to the other number to make the problem easier. There are 567 fourth and fifth graders at Taylor Elementary. There are 250 third graders. How many third, fourth and fifth graders are there at Taylor Elementary? In this strategy you subtract 50 from 567 to get 517. Then you add 50 to 250 to get 300. It is easier to add 517 + 300 = 817. 567 250 517-50 + 50 + 300 517 300 817 Addition Using a Number Line This strategy uses a number line and counting by tens to add numbers. In the number sentence 25 + 37, you start counting at the first addend. Count 10 to 35, count 10 more to 45, count 10 more to 55,and then count 7 more to get the sum of 62. 20 25 30 35 40 45 50 55 60 65 70 Another way to count this on the number line would be to count your 7 first to get 32 and then count 42, 52, and 62. 20 25 30 35 40 45 50 55 60 65 70 C & C Educational Materials, all rights reserved 65
Estimate Sums with Rounding An estimate is a number close to the exact answer. You can estimate sums by rounding the addends, or you can estimate by using compatible numbers for the addends. Remember, you can use a rule to round numbers. Look at the digit to the right of the place you are rounding to. If the digit is 5 or greater, round up. If it is less than 5, round down. Here are some examples. To round 45 to the nearest ten, look at the number to the right of the tens place which is a 5 in the ones place. You will round up to 50 because the digit is 5 or greater. To round 349 to the nearest hundred, look at the number to the right of the hundreds place, which is a 4 in the tens place. You will round down to 300 because the number is less than 5. Here is an example of estimating a sum using rounding. A store was selling a stereo for $579 and the speakers for $344. Is $1,000 enough money to buy both the stereo and the speakers? Estimate to Hundreds Estimate to Tens $579 rounds up to $600 $579 rounds up to $580 + $344 rounds down to $300 + $344 rounds down to $340 $900 < $1,000 $920 < 1,000 A two digit number will round to tens. A three digit number can be rounded to the nearest hundred or the nearest ten like the two above examples. Estimate Sums with Compatible Numbers You can also use compatible numbers to estimate in addition and subtraction. Compatible numbers can be added and subtracted easily. Numbers that make tens and hundreds are the most common examples. However, numbers ending in 5, 25, 50, and 75 are also compatible numbers because they can also be added and subtracted easily. A store was selling microwaves for $177 and dishes for $143. About how much money will you need to buy the microwave and dishes? Estimate Using 10s or 100s Estimate Using 5, 25, 50, or 75 $177 changes to $200 $177 changes to $175 $143 changes to $100 $143 changes to $150 $300 $325 66 C & C Educational Materials, all rights reserved
Number Patterns in Addition A pattern is a regular, repeating design or sequence of shapes or numbers. In addition, a pattern may be a sequence of numbers that have a common addend between any one number and the next number in the pattern. Number patterns can have a rule. The rule must be correct for all the numbers in the pattern. In this example the rule is to add 6 to each of the numbers to get the next number. 7 13 19 25 31 37 The table below shows the relationship between the number of blue stars and the number of green stars a teacher puts on different posters. Number of Blue Stars 7 10? 19 Number of Green Stars 28 31 35 40 Based on the table, how would you find the missing number? You can do the addition vertically and you will get the missing number of 14. For example, 7 + 21 = 28, 10 + 21 = 31, and 19 + 21 = 40. What number + 21 = 35? The number sentence would be 14 + 21 = 35. The number 14 is the correct answer. An addition pattern was used for this table. You add 3 to the numbers in the first column (7 + 3 = 10 and 28 + 3 =31). This gives you the numbers in the second column. You add 4 to the numbers in the second column (10 + 4 = 14 and 31 + 4 = 14). The missing number is 14. However, to be positive that this is the answer, add 5 to the missing number of 14 and the 35 to see if you get the numbers in the last column (14 + 5 = 19 and 35 + 5 = 40). Your answer of 14 is the correct missing number. C & C Educational Materials, all rights reserved 67