3 What Do You Think? Representation was not one of the standards in the 1989 NCTM Curriculum Standards. Why do you think it was made a separate standard in the 2000 Standards? How does making connections add to your toolbox? What connections among concepts and problem-solving strategies have you seen thus far in the chapter?
4 Representation and Connections A representation can take many forms: diagram, graph, table, sketch, equation, words, etc. Most problems and most mathematical concepts can be represented in different ways. For example, there were several ways to represent the pigs-and-chickens problem. Students commonly ask me which representation is “best,” but this is just not a useful question.
5 Representation and Connections A more useful question is whether this representation fits the purposes before us. Representing the problem with 24 circles (representing 24 bodies) and then putting legs on the bodies is a very appropriate representation for this problem with children. This notion of multiple representations is also related to the standard on Connections.
6 Investigation A – How Long Will It Take the Frog to Get out of the Well? A. A frog is climbing out of a well that is 8 feet deep. The frog can climb 4 feet per hour but then it rests for an hour, during which it slips back 2 feet. How long will it take for the frog to get out of the well? B. What if the well was 40 feet deep, the frog climbs 6 feet per hour, and it slips back 1 foot while resting?
7 Investigation A – Discussion A. One of the amazing things about this problem is that, we can often see 10 or more different valid representations of the problem. Below are two representations that both lead to the same answer. (1)(2)
8 Investigation A – Discussion Both representations show the frog’s progress for each hour and that the frog reaches 8 feet after 5 hours. Let us look more closely: How are representations (1) and (2) alike and how are they different? Alike Different 1. They have line segments. 1. In the first strategy the line segments are not vertical and in the second they are. 2. They have numbers: 2. The second strategy 4, 2, 6, 4, 8. has a number line at the left. cont’d
9 Investigation A – Discussion Alike Different 3. The second strategy has arrows. B. As we found in the pigs-and-chickens problem, some representations can be “scaled up” and others cannot. Each of the two representations shown could be used to solve B, but they would be somewhat tedious. cont’d
10 Investigation A – Discussion In this case, we look for a more efficient representation, one of which is shown below: Even this representation is a bit tedious. However, if we are always looking out for patterns, we can actually get the answer by making only part of the table. cont’d
11 Investigation A – Discussion We can see that the numbers when the hours are even are simply multiples of 5. We can then count by 2s to get close to the 40-foot height, or we can see that the height (in even hours) is always times the number representing the hour. So we can jump to 14 hours when the frog has climbed 35 feet and know that on the 15 th hour the frog will get out. cont’d
12 Representation and Connections Connections: Making connections is at the heart of the NCTM Standards. The mathematics curriculum is generally viewed as consisting of several discrete strands. As a result, computation, geometry, measurement, and problem solving tend to be taught in isolation. It is important that children connect ideas both among and within areas of mathematics. Without such connections, children must learn and remember too many isolated concepts and skills rather than recognizing general principles relevant to several areas.
13 Making Connections Between the Problem and What Is Inside Your Head
14 Making Connections Between the Problem and What Is Inside Your Head You had to create in your head a model of the problem that made sense and that connected the relevant information to your mathematical knowledge. Developing the ability to connect the given information to your mathematical knowledge and to your problem-solving toolbox is one of the central objectives of this course.
15 Connecting New Concepts to Old Concepts
16 Connecting New Concepts to Old Concepts One important kind of connection is the connection between new ideas and something that is familiar to you. Consider the following example. Students were asked to select the decimal equivalent to 12 percent and then to select the decimal equivalent to.9 percent. The results are given in Table Table 1.13
17 Connecting New Concepts to Old Concepts First, they could have connected percents to rational numbers, reasoning that 12 percent means 12/100. Then they could have connected the fraction to a decimal: 12/100 = Following this reasoning for the second question,.9 percent means 0.9/100, which converts to Alternatively, they could have applied the algorithm “Move the decimal point two places.” In other words,.9 becomes.009. Of course, they had to realize that they had to put zeros to the left of.9 in order to be able to move two decimal places!
18 Making Connections Among Different Concepts
19 Making Connections Among Different Concepts Many, if not most, students have come to view mathematics as a collection of separate topics. Looking at the four basic operations—addition, subtraction, multiplication, and division—we can make the following connections: Addition and subtraction are inverse operations, and multiplication can be seen as repeated addition. Likewise, multiplication and division are inverse operations, and division can be seen as repeated subtraction.
20 Connecting Different Models for the Same Concept
21 Connecting Different Models for the Same Concept In mathematics, many concepts can be represented in different ways. For example, consider the ways of representing shown in Figure One of them is not a valid representation of what we mean by. Figure 1.13
22 Connecting Conceptual and Procedural Knowledge
23 Connecting Conceptual and Procedural Knowledge Most of you have probably learned a variety of standard procedures, also called algorithms. However, fewer of you are likely to know why they work. For example, although you can probably convert a mixed number to an improper fraction—for example, —do you know why we multiply the whole number by the denominator, add the numerator, and then put the whole thing over the denominator?
24 Making Connections Between Mathematics and “Real Life” and Between Mathematics and Other Disciplines
25 Making Connections Between Mathematics and “Real Life” and Between Mathematics and Other Disciplines Think of how many political issues are connected to mathematics: The current and future effects of the federal deficit—How many people realize what a really big number 11 trillion is? The issue of income inequality—Women earn an average of 80¢ for every dollar men earn.
26 Making Connections Between Mathematics and “Real Life” and Between Mathematics and Other Disciplines The cost of illiteracy and innumeracy to the United States—In 1990 it was estimated that American employers spent about $40 billion to train their employees in the basic reading, writing, and arithmetic skills they should have learned in school, and that number has only risen since then.
27 Investigation B – How Many Pieces of Wire? A jewelry artisan is making earring hoops. Each hoop requires a piece of wire that is inches long. If the wire comes in 50-inch coils, how many -inch pieces can be made from one coil, and how much wire is wasted? Solve this problem. Discussion: Strategy 1: Divide Some people quickly realize that you can divide “to get the answer.” If you use a calculator, it shows If you use fractions, you get.
28 Investigation B – Discussion Many people interpret these numbers to mean that you can get 13 pieces and you will have inch wasted. Unfortunately, that is not correct. One of the reasons why math teachers stress the importance of labels is that they illustrate the meaning of what we are doing. The meaning of is 13 whole hoops and of a hoop. That is, what we have left would make of a hoop. Because one whole piece is inches long, of a piece is of ; that is, inches is wasted. cont’d
29 Investigation B – Discussion One way to check would be to multiply 13. This would tell us the length of the 13 whole pieces. If this number plus equals 50, then our answers are correct. In fact, and. Strategy 2: “Act It Out’’ Some people understand the problem better when they represent the problem with a diagram like that in Figure cont’d Figure 1.14
30 Investigation B – Discussion Strategy 3: Make A Table Yet other people solve the problem by starting with one piece and building up as shown in Table cont’d Table 1.14