Learning Math: Number and Operations
Meanings and Models for Operations Part A: Meanings and Relationships of the Operations (40 minutes)
In This Part: Addition
We will begin our look at the various meanings for each of the operations and the laws that govern the operations by examining addition.
Addition is the simplest of the four operations. The operation, however, may look quite different depending on whether a problem has an unknown result, starting point, or change. We can describe addition as a merger or joining of two or more things; we can also describe it as combining parts of a whole, with the whole or one of the parts unknown.
The following table gives an example of each kind of addition problem:
Starting Point Unknown
The merger or joining concept always requires some sort of combining action, whereas the parts-of-a-whole concept is static. See Note 2 below.
One of the most important facts about addition is that no two quantities can be added unless they are measured or reported in the same units. For example, you cannot add 2 tens and 3 ones, or 2 halves and 3 fourths, and expect to get 5 of anything. These quantities can only be combined if we can somehow find a common unit with which to measure or label them. See Note 3 below.
Label each of the addition problems with the correct situation label, and identify the units involved:
MR: Merger, result unknown
MS: Merger, starting point unknown
MC: Merger, change unknown
PW: Parts of a whole, whole unknown
PP: Parts of a whole, one of the parts unknown
a. Moisha has 7 cars. Three are red, and the rest are blue. How many blue cars does she have?
Write an equation and see if you can match it with any of the above descriptions.
b. Jake read 5 mystery books and 8 adventure books. How many books did he read?
c. Bret has $5, and Wendy has $4. How much will they have if they pool their money?
d. Natasha had 4 rabbits. One of her rabbits had babies, and now she has 7 rabbits. How many babies did the rabbit have?
e. Reed’s parents gave him $5 for his birthday. He then had $12. How much money did he have before?
Part A: Meanings and Relationships of the Operations adapted from Chapin, Suzanne and Johnson, Arthur (1999). Math Matters: Understanding the Math You Teach (pp. 40-72). © 2000 by Math Solutions Publications.
In This Part: Subtraction
As in addition, no two quantities can be subtracted unless they are measured or reported in the same units. Thus, you cannot subtract 7 hundreds from 9 tens and expect to get 2 of anything. A quantity can only be subtracted from another quantity if we can first find a common unit between the two.
The operation of subtraction can be thought of as:
|•||a separator, when the result, starting point, or change is unknown (also known as “take-away”)|
|•||a comparison, when the result, starting point, or change is unknown|
|•||a missing addend problem, where one of the parts is unknown – See|
This table gives an example of each kind of subtraction problem:
Starting Point Unknown
The missing addend problems are written as addition problems, but the procedure to solve these problems requires the use of some subtraction strategy. The separating concept always requires some sort of separating action, whereas the comparison concept is static.
When negative numbers are introduced, we can more clearly understand the concept of subtraction as the addition of the inverse. Thus, we can write 13 – 6 as the equivalent of 13 + (-6 ), because -6 is the additive inverse of 6; i.e., 6 + (-6) = 0. Similarly, we can represent 13 – (-6) as 13 + 6, since 6 is the additive inverse of -6. So again, subtracting a number (13 – (-6)) is the same as adding its inverse (13 + 6).
Label each of the subtraction problems with the correct situation label, and identify the units involved:
SR: Separator, result unknown
SS: Separator, starting point unknown
SC: Separator, change unknown
CR: Comparison, result unknown
CS: Comparison, starting point unknown
CC: Comparison, change unknown
MS: Missing addend, starting point unknown
MC: Missing addend, change unknown
a. Moisha has 7 cars. She gave 3 away. How many cars does she have left?
b. Jake read 5 mystery books. He read 3 more adventure books than mysteries. How many adventure books did he read?
c. Bret has $5. Bret has $2 more than Wendy. How much money does Wendy have?
d. Natasha has 7 rabbits. She gave some rabbits to Joshua, and then she had 3. How many rabbits did she give away?
e. Reed’s parents gave him some money for his birthday. He now has $12. He had $7 before his birthday. How much money did they give him?
f. Elyse had some candy bars. She ate 3, and she has 5 left. How many did she have to start?
In This Part: Multiplication
There are several ways in which we can think of multiplication:
- Multiplication is often thought of as repeated addition of equal groups. While this definition works for some sets of numbers, it is not particularly intuitive or meaningful when we think of multiplying 3 by 1/2, for example, or 5 by -2. In such cases, it may be helpful to widen the idea of grouping to include evaluation of part of a group. This concept is related to partitioning (which, in turn, is related to division).
For example, three groups of five students can be read as 3 • 5, or 15 students, while half a group of 10 stars can be represented as 1/2 • 10, or 5 stars. These are examples of partitioning; each one of the three groups of five is part of the group of 15, and the group of 5 stars is part of the group of 10.
- A second concept of multiplication is that of rate or price. For example, if a car travels four hours at 50 miles per hour, then it travels a total of 4 • 50, or 200 miles; if CDs cost eight dollars each, then three CDs will cost 3 • $8, or $24.•A third concept of multiplication is that of multiplicative comparison. For example, let’s say that Sara has four CDs, Joanne has three times as many as Sara, and Sylvia has half as many as Sara. Thus, Joanne has 3 • 4, or 12 CDs, and Sylvia has 1/2 • 4, or 2 CDs.
Two additional situations require multiplication:
- Finding the area of a rectangle using rectangular arrays. For example, an array with three rows by five each will have 3 • 5, or 15, square units in all. This model is often used to introduce multiplication.
- Finding the number of possible combinations using a Cartesian product. For example, with two shirts and three pairs of pants, you could have 2 • 3, or 6, different shirt-pant combinations.
In the first three scenarios, one factor was clearly the multiplier (the number of groups), and the other factor was clearly the number being multiplied (the number of items or individuals in each group). These types of problems are called asymmetrical, because the factors are so different; exchanging the roles of the factors results in an entirely different scenario. For example, 3 CDs at $8 each is different from 8 CDs at $3 each, even though the “answer” is the same.
The two last scenarios present two examples of symmetrical problems, because it isn’t important which of the two factors is the multiplier. The quantities are interchangeable.
The asymmetrical nature of some problems explains why the commutative law of multiplication is less intuitive than the commutative law of addition. The multiplication problem 3 • 4, interpreted as a grouping problem, means three groups of four items. This corresponds to the addition problem 4 + 4 + 4. In contrast, the multiplication problem 4 • 3, also interpreted as a grouping problem, means four groups of three items, which corresponds to the addition problem 3 + 3 + 3 + 3. It may not be obvious at first glance that 4 + 4 + 4 and 3 + 3 + 3 + 3 would give the same sum, or why. Manipulatives and other visual clues discussed later in this session can be helpful in showing the relationship between the two operations.
One of the most important facts about the operations of multiplication and division is that the units of the quantities being multiplied or divided do not have to be the same for the operations to function properly. You can multiply 2 tens by 3 ones, and the result will be 6 tens. Likewise, you can divide 6 tens by 3 ones and get 2 tens. The unit for the answer is found in the same way that you found the numerical answer. For example, if you divided the numbers, then you must divide the units as well. Similarly, if you multiplied the numbers, then you must multiply the units as well. See Note 5, below.
Consider the following multiplication situations. For each one, identify the multiplication problem, the units involved, whether the problem is symmetrical or asymmetrical, and which multiplication concept it is demonstrating:
a. Dinner at the Ritz costs 4 times as much as dinner at the Savoy. My bill at the Savoy was $10. What would dinner cost me at the Ritz?
b. Bob & Jimmy’s Ice Cream offers 6 different ice cream flavors and 5 different sundae toppings. How many different kinds of sundaes can be made using 1 flavor of ice cream and 1 topping per sundae?
c. My shower flows at 3 gallons per minute. How much water would a 6-minute shower use?
d. There are 9 content sessions in this course, and you have completed 1/3 of them. How many sessions have you completed?
e. My yard is 20 meters wide and 33 meters from front to back. What is its area?
In This Part: Division
All of the meanings of multiplication can be used for division, since if the product and one of the factors is known, division can be used to find the other factor. But for the asymmetrical example of equal groups, the process feels different depending on which factor is known — the multiplier or the number in each group.
As you will see, there are two very different concepts of division:
- If the number in each group is known, and you are trying to find the number of groups, then the problem is referred to as a quotative division problem. Quotative division may also be called measurement, or repeated subtraction. You are, in effect, counting or measuring the number of times you can subtract the divisor from the dividend. Long division (remember long division?!) uses this concept.
- If the number of groups is known, and you are trying to find the number in each group, then the problem is referred to as a partitive division problem. Partitive division may also be called equal groups, or sharing and distribution. You are, in effect, partitioning the dividend into the number of groups indicated by the divisor and then counting the number of items in each of the groups.
The following example demonstrates the distinction between the two types of division problems: See Note 6 below.
a. Draw a diagram that represents 153 as a partitive problem.
b. Draw another diagram that represents 153 as a quotative problem.
c. Write a problem for each diagram.
a. Which type of division, quotative or partitive, would be most efficient for computing 10050? Why?
b. Which would you use for 1002?
In this video segment, Susan and Jeanne explore the different notions of quotative and partitive division problems. They challenge their understanding with new insights. Watch this segment after you’ve completed Problems A4 and A5.
Think about which method is easier to do in a particular division problem.
You can find this segment on the session video approximately 4 minutes and 9 seconds after the Annenberg Media logo.
When division problems do not work out evenly, the context of the problem dictates the answer. Sometimes we may need to round the answer up or down to the next integer, and sometimes we may need the exact decimal value of the division.
a. Write a problem that uses the computation 434 and gives 10 as the correct answer.
b. Write a problem that uses the computation 434 and gives 11 as the correct answer.
c. Write a problem that uses the computation 434 and gives 10.75 as the correct answer.
Another important concept to remember, especially when working with rational numbers, is that division can be thought of in terms of multiplying by the inverse. This can be particularly useful when dividing by fractions. Thus, we could show that 122 = 12 • 1/2, where 1/2 is the multiplicative inverse of 2, and 12(1/2) = 12 • 2, where 2 is the multiplicative inverse of 1/2. In these cases, you can see that the multiplicative inverse of every number except 0 is the reciprocal of that number, and that the product of a number and its reciprocal is 1.
a. The problem to solve is 3 + x = 7. This is a PP problem, since we are working with parts of a whole and the unknown is one of the parts. Here, the units are cars.
b. The problem to solve is 5 + 8 = x. This is a PW problem, since we are working with parts of a whole and the unknown is the sum of the parts. Here, the units are books.
c. The problem to solve is 5 + 4 = x. This is an MR problem, since we are merging two things (Bret and Wendy’s money) and the unknown is the result. Here, the units are dollars.
d. The problem to solve is 4 + x = 7. This is an MC problem, since we are merging two things (rabbits) and the unknown is the change (the number of babies). Here, the units are rabbits.
e. The problem to solve is x + 5 = 12. This is an MS problem, since we are merging two things (Reed’s money with his parents’) and the unknown is the starting point (the amount of money Reed had before). Here, the units are dollars.
a. The problem to solve is 7 – 3 = x. This is an SR problem, since we are separating two things and the unknown is the result. Here, the units are cars.
b. The problem to solve is x – 3 = 5. This is a CS problem, since we are comparing two things and the unknown is the starting point. Here, the units are books.
c. The problem to solve is 5 – 2 = x. This is a CR problem, since we are comparing two things and the unknown is the result. Here, the units are dollars.
d. The problem to solve is 7 – x = 3. This is an SC problem, since we are separating two things and the unknown is the change. Here, the units are rabbits.
e. The problem to solve is 7 + x = 12. This is a missing addend problem (MC) since the change is unknown. Here, the units are dollars.
f. The problem to solve is x – 3 = 5. This is an SS problem, since we are separating two things and the unknown is the starting point. Here, the units are candy bars.
a. The problem to solve is 4 • $10 = x. The units are dollars. This is an asymmetrical problem, demonstrating a multiplicative comparison.
b. The problem to solve is 6 • 5 = x. The units are flavors, toppings, and sundaes. This is a symmetrical problem, demonstrating the use of a Cartesian product.
c. The problem to solve is 6 • 3 = x. The units are minutes, gallons per minute, and gallons. This is an asymmetrical problem, demonstrating a rate.
d. The problem to solve is 1/3 • 9 = x. The units are sessions. This is an asymmetrical problem, demonstrating partitioning.
e. The problem to solve is 20 • 33 = x. The units are meters, and the result is in square meters. This is a symmetrical problem, demonstrating the use of a rectangular array to find the area of a rectangle.
c. Answers will vary. Here are two examples:
- For the partitive model, Graphic (a): Matt deals a total of 15 cards to 3 players (including himself). Each player gets the same number of cards. How many cards does each player get? (Here, you know the number of groups and need to find the number in each group.)
- For the quotative model, Graphic (b): Nicole has 15 cans of soda. She gives 3 cans of soda to each of her friends. How many friends got the soda? (Here, you know the number in each group and need to find the number of groups.)
a. The quotative problem is easier to solve. It is equivalent to “How many 50s would you need to get 100?” Meanwhile, the partitive problem is “If 100 items are separated into 50 equal groups, how many are in each group?”
b. The partitive problem is easier. It is equivalent to “If 100 is separated into two equal groups, how many are in each group?” Meanwhile, the quotative problem is “How many twos would you need to get 100?”
Answers will vary. Here are some examples:
a. Arvind wants to buy some ice cream for his coworkers. Each ice cream cone costs $4, and Arvind has $43. How many cones can Arvind buy?
b. Michelle needs 43 batteries to keep her handheld organizer running during a long trip. Batteries come in packs of 4. How many packs of batteries will Michelle need to buy?
c. Lilian bought 4 cakes for a Tuesday night party. She paid $43 for the cakes. How much money did she pay for each cake?
Session 1 What Is a Number System?
Understand the nature of the real number system, the elements and operations that make up the system, and some of the rules that govern the operations. Examine a finite number system that follows some (but not all) of the same rules, and then compare this system to the real number system. Use a number line to classify the numbers we use, and examine how the numbers and operations relate to one another.
Session 2 Number Sets, Infinity, and Zero
Continue examining the number line and the relationships among sets of numbers that make up the real number system. Explore which operations and properties hold true for each of the sets. Consider the magnitude of these infinite sets and discover that infinity comes in more than one size. Examine place value and the significance of zero in a place value system.
Session 3 Place Value
Look at place value systems based on numbers other than 10. Examine the base two numbers and learn uses for base two numbers in computers. Explore exponents and relate them to logarithms. Examine the use of scientific notation to represent numbers with very large or very small magnitude. Interpret whole numbers, common fractions, and decimals in base four.
Session 4 Meanings and Models for Operations
Examine the operations of addition, subtraction, multiplication, and division and their relationships to whole numbers. Work with area models for multiplication and division. Explore the use of two-color chips to model operations with positive and negative numbers.
Session 5 Divisibility Tests and Factors
Explore number theory topics. Analyze Alpha math problems and discuss how they help with the conceptual understanding of operations. Examine various divisibility tests to see how and why they work. Begin examining factors and multiples.
Session 6 Number Theory
Examine visual methods for finding least common multiples and greatest common factors, including Venn diagram models and area models. Explore prime numbers. Learn to locate prime numbers on a number grid and to determine whether very large numbers are prime.
Session 7 Fractions and Decimals
Extend your understanding of fractions and decimals. Examine terminating and non-terminating decimals. Explore ways to predict the number of decimal places in a terminating decimal and the period of a non-terminating decimal. Examine which fractions terminate and which repeat as decimals, and why all rational numbers must fall into one of these categories. Explore methods to convert decimals to fractions and vice versa. Use benchmarks and intuitive methods to order fractions.
Session 8 Rational Numbers and Proportional Reasoning
Begin examining rational numbers. Explore a model for computations with fractions. Analyze proportional reasoning and the difference between absolute and relative thinking. Explore ways to represent proportional relationships and the resulting operations with ratios. Examine how ratios can represent either part-part or part-whole comparisons, depending on how you define the unit, and explore how this affects their behavior in computations.
Session 9 Fractions, Percents, and Ratios
Continue exploring rational numbers, working with an area model for multiplication and division with fractions, and examining operations with decimals. Explore percents and the relationships among representations using fractions, decimals, and percents. Examine benchmarks for understanding percents, especially percents less than 10 and greater than 100. Consider ways to use an elastic model, an area model, and other models to discuss percents. Explore some ratios that occur in nature.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants) who have adapted their new knowledge to their classrooms.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants) who have adapted their new knowledge to their classrooms.