Chapter 3 Addition and Subtraction part 1 Interpretation

Chapter 3: Addition and Subtraction - part 1 • Interpretation of Addition and Subtraction • The Commutative and Associative Properties of Addition and Subtraction

3. 1. Interpretations of Addition and Subtraction Combining and Taking Away If A and B represent two numbers, then the sum A+B represents the total number of objects we will have if we start with A objects and then get B more objects The numbers A and B are called terms, addends, or summands

Basic Meaning of Addition The basic meaning of addition is combining: We can represent the sum 149+85 as the total number of toothpicks we will have if we start with 149 toothpicks and we get 85 more toothpicks.

Basic Meaning of Subtraction If A and B are two numbers, the difference A - B represents the total number of objects we will have if we start with A objects and take away B of those objects. We represent 4 -2 as the number of witch hats if we start with 4 hats and give away 2

Relating Addition and Subtraction We can also view the subtraction (or difference of two objects) as the inverse of addition.

The addition equation, A + B = C, gives rise to a problem when one of A, B, or C is unknown, as in the following equations: The subtraction equation, A - B = C, gives rise to a problem when one of A, B, or C is unknown, as in the following equations:

Types of problems that can be created to fit with these equations • Change Problems • Part-Whole Problems • Comparing Problems

Change Problems are addition and subtraction problems that involve a change over time Add To Problem (Change Unknown): Rachel had 12 CDs. After Rachel got some more CDs, she had 21 CDs. How many CDs did Rachel get? Take Away Problem (Start Unknown): Rachel had some CDs. After she gave 9 away, she had 12 CDs left. How many CD’s Rachel have at first? Take Away Problem (Change Unknown): Rachel had 21 CDs. After she gave some away, she had 12 left. How many CDs did Rachel give away?

Part-Whole Problems are addition and subtraction problems that involve two distinct parts that make a whole but don’t involve change over time. Add to Problems (Result Unknown): Rachel has 12 movie CDs and 9 music CDs (but no other CDs). How many CDs does Rachel have in all? Add to Problems (Change or Start Unknown): Rachel has 21 CDs. Some of the CDs are movies and the rest are music. Rachel has 12 movie CDs. How many music CDs does Rachel have?

Compare Problems are addition and subtraction problems that involve the comparison of two quantities: • Bigger Unknown Rachel has 12 CD’s. Benny has 9 more CD’s than Rachel. How many CD’s does Benny have? • Smaller Unknown Rachel has 21 CD’s. Benny has 12 fewer CDs than Rachel. How many CD’s does Benny have? • Smaller Unknown Rachel has 12 CD’s. Benny has 21 CD’s. How many more CDs does Benny have than Rachel? • Difference Unknown Rachel has 12 CD’s. Benny has 21 CD’s. How many fewer CDs does Rachel have than Benny?

Using Simple Diagrams to Help Understand Solve Problems It is important to realize that the use of keywords is not always reliable for solving word problems. We can always use strip diagrams and number lines to help with the solving of the problems.

3. 2 The Commutative and Associative Properties of Addition, Mental Math, and Single -Digit Facts The Commutative Property of Addition: For any two real numbers A and B A+B=B+A

The Associative Property of Addition: for any three real numbers A, B, and C (A+B)+C=A+(B+C) Note: The associative and commutative properties tell us that whenever we add two or more numbers, it does not matter in what order we add the numbers.

Combining The Commutative and Associative Properties Combined, the Associative and Commutative Properties allow for great flexibility in calculating sums: • Starting with grade 7 and 8, you can apply these properties to abstract algebraic calculations: • From grades 1 and 2 we can use them to teach students mental math and fluency with single-digit addition facts.

The Associative Property and Derived Fact Methods The make a ten method Students in grades 2 and 3 are using this method to help them solve addition/subtraction problems from 1 to 20. Example. Find 8+7. Example. Find 13 -8 by make ten method

Proper Use of the Equal Sign and Writing Equations • An equation is a mathematical statement saying that two numbers or expressions are equal to one another • “=” means equals or equal to. • When showing steps or a method of calculation we string several equations together. This is for the purpose of concluding that the Left Hand Side (LHS) expression of the equation is equal to the Right Hand Side (RHS) expression of the equation. • When we use the equality sign make sure that the expression on the right is indeed equal to the on the left. Otherwise we get a false statement.

Mental Methods for Multi-Digit Addition and Subtraction Make-a-Round Number Method The method works just like making a 10, but instead of making a 10 we are making round numbers that are easier to manipulate. Example.

Mental Methods for Multi-Digit Addition and Subtraction Rounding and Compensating Example

Subtraction problem as an Unknown Addend Problems Is like giving change. For example, if you gave the teller a $50. 00 bill for a $37. 00 purchase, the teller should count back to you a 3+10 which is 13. Thus $50. 00 -$37. 00=$13. 00 Example. Solve 684 -295=?

Example. Tom has learned the following facts: All the sums of whole numbers that add to 10 or less (“forwards and backwards”). For example he knows not only that 5+2 is 7, but also that 7 decomposes into 5+2 or 2+5. as well as 5+2. He knows the doubles: 1+1, 2+2, . . , 10+10; He also knows how to add 10+1, 10+2, 10+3, …, 10+10 For the sum of 6+7 describe at least three different ways that Tom could use reasoning and facts that he already knows to give the right answer.

What about properties of subtraction? Is there a commutative property of subtraction? Is there an associative property of subtraction?

What about properties of subtraction? Is there a commutative property of subtraction? No: Is there an associative property of subtraction? No:

Problem Solving Sections 3. 1 & 3. 2

Section 3. 2, Problem 1 Many teachers have a collection of small cubes of different colours that can be snapped end-to-end to make a “trains” of cubes. a) Describe in detail how you could use snap-to-together cubes in different colours to demonstrate the commutative property of addition. b) Describe in detail how you could use snap-to-together cubes in different colours to demonstrate the associative property of addition.

Section 3. 2, Problem 3 The figure indicates how to make-a-ten method for adding 6+8. a) Write equations that correspond to make-a-ten method for adding 6+8 depicted in the figure above. Your equations should make careful and appropriate use of parentheses. Which property of arithmetic do your equations and the picture above illustrate?

Section 3. 2, Problem 3 b) Draw a picture for 7+5 that illustrate make-a-ten method. Write equations that correspond to the strategy indicated in your picture, making careful and appropriate use of parentheses.

Section 3. 2, Problem 5 Give an example of an arithmetic problem that can be made easy to solve mentally by using the associative property of addition. Write equations that show your use of the associative property of addition. Your use of the associative property of addition must make the problem easier to solve.

Section 3. 2, Problem 7 Describe a way to calculate 304 – 81 mentally, by using reasoning other than the common subtraction algorithm. Then write a coherent sequence of equations that correspond to your reasoning.

Chapter 3: Addition and Subtraction - part 2 • Why the Common Algorithms for Adding and Subtracting in the Decimal System Work • Adding And Subtracting Fractions • Adding And Subtracting Negative Numbers

What is an algorithm? An algorithm is a method or procedure for carrying out a calculation (or other tasks) Addition Algorithm: • We put the numbers one under the other, lining up ones over ones, tens over tens, hundreds over hundreds …; and lining up the decimal points where applicable. • Then we add numbers in each column, regrouping as needed

Example. Calculating 37+26: Note: Regrouping is also called carrying over.

Example. Calculating 149+85:

Subtraction Algorithm: • we put numbers one under the other lining up ones over ones, tens over tens, hundreds over hundreds …; and lining up the decimal points where applicable • then, we subtract column by column, regrouping when needed so we can subtract numbers in a column without resulting to negative numbers

Example. Calculating 142 -83:

Note: Regrouping in subtraction is called borrowing or trading Problem: Allie solves the subtraction problem 304 - 9 as follows: Not correct Explain the correct method using regrouping method with equations in expanded form.

Solution:

Problem: On a space shuttle mission, a certain experiment is prepared 2 days, 14 hours, and 30 minutes. The experiment itself takes 1 day, 21 hours, and 47 minutes to run. When will the experiment be completed? Give your answer in days, hours, and minutes. Solution: 1 day has 24 hours, and 1 hour has 60 minutes.

Problem Solving – Section 3. 3 Problem 2. Describe how to use bundled toothpicks (or bundles of objects) to explain regrouping in the addition problem 167 + 59. Draw simplified pictures to help with your explanation.

Problem 7. Matteo is 4 feet 3 inches tall. Nico is 3 feet 11 inches tall. How much taller is Matteo than Nico? Sarah solved the problem as follows: So Sarah gave 2 inches as the answer. Is Sarah right? If not, explain what is wrong with her method, and show you modify her method of regrouping to make it correct.

Adding And Subtracting Fractions 1. Adding And Subtracting Fractions with Like Denominators In general, when adding or subtracting two or more fractions with the same denominator, we add and subtract the numerators and leave the denominator unchanged

2. Adding and Subtracting Fractions with different Denominators When adding and subtracting fractions with unlike denominator we • first, find the common denominator • second, write the equivalent fractions with same denominator. • third, add and subtract the fractions with like denominator

Example. Find 5/6 + 3/4 The lowest common denominator is 12:

Writing Mixed Numbers as Improper Fractions A mixed number (or mixed fraction) is a number written in the form where A, B, and C are whole numbers, and B/C is a proper fraction. The mixed number stands for the sum of its whole number part and its fractional part:

Problem Solving – Section 3. 4

Problem 21. What fraction of each square is shaded?

Adding And Subtracting Negative Numbers: Adding a Number to Its Negative: How can we interpret (-3)+3? It was -3°C at dawn. In the meantime, the temperature went up 3°C. Now what is the temperature? (-3)+3=0 In general, (-N)+N=0, for any number N. This equation states that N and –N are additive inverses.

Interpreting A + (-B) as A-B How can we interpret 5 + (-3) ? We can use the associative property of addition: Another way to look at 5 + (-3), is to consider

Interpreting A - (-B) = A + B Using a comparison of temperatures to explain why 5 − (− 3) = 5 + 3: The temperature was 5°C in Toronto. At the same time the temperature was – 3°C in North Bay. How much warmer was Toronto than North Bay?

Number Line Meaning of Addition and Subtraction Number Line Meaning of Addition (A+B): 1. Go to A on the number line. 2. Move a distance of B units • To the right if B is positive • To the left if B is negative 3. The resulting point is the location of A+B Using a number line to show that 2 + (− 3) = − 1 Number Line Meaning of Subtraction (A-B): 1. Go to A on the number line. 2. Move a distance of B units • To the left if B is positive • To the right if B is negative 3. The resulting point is the location of A-B Using a number line to show that ( − 2) − ( − 5 ) = 3

Problem Solving – Section 3. 5

Write a compare problem that fits naturally with the equation: (− 7) + ? = − 1 Solve the problem and explain why the solution makes sense

Write a compare problem that fits naturally with the equation: (− 7) + 6 = ? Solve the problem and explain why the solution makes sense.