Melinda MacLeish CCLM Project 2 July 15, 2011 Part One: Standard(s) The Number System: 7.NS.1 Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-­‐world contexts. c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-­‐world contexts. d. Apply properties of operations as strategies to add and subtract rational numbers. DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐Milwaukee. (07.15.2011) Part Two: Interpretation of the Standard Grade 7 CCSS-­‐M Domain: The Number System Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. Standard 7.NS.1a Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. (a) Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. Standard 7.NS.1b Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. (b) Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-­‐world contexts. Use teacher friendly language to interpret concepts, key terms, and phrases within the standard. Provide examples of the mathematics in the standard. Students should be able to describe real world situations in which two quantities combine to make zero. These quantities would then be opposites, or additive inverses. If it is ten degrees below zero and the temperate rises ten degrees the resulting temperature would be zero. If a business’s income is equal to its expenses then the business’s profit will be zero. If a student gets five dollars for their allowance and they owe their friend five dollars they are left with zero dollars. Students will be able to add with positive and Use a number line to show the following: negative numbers. Students will be able to demonstrate and explain the process of using 3 + -­‐2 a number line to add with rational numbers. -­‐4 + 5 Students will be able to show that the sum of a number and its opposite is zero. They may 6 + -­‐6 use a number line to do this. -­‐3 + -­‐2 Students will be able to describe real world situations that involve adding with rational Describe real world situations involving rational numbers. numbers. For example students may create problems involving temperature or money. -­‐16 + 10 = ? The temperature at 6 am was 16 degrees below zero. By noon the temperature had risen ten degrees. What was the temperature at noon? DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐
Milwaukee. (07.15.2011) Standard 7.NS.1c Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. (c) Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-­‐world contexts. Standard 7.NS.1d Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. (d) Apply properties of operations as strategies to add and subtract rational numbers. Students will recognize that subtraction is the same as adding the opposite. They will be able to rewrite subtraction problems as addition problems. Students will be able to use a number line to show subtraction problems. Use a number line to show the following: 4 – (-­‐3) 4 + 3 2 – 5 2 + -­‐5 Students will use the associative property of addition and the commutative property of addition to add and subtract rational numbers. Students will apply appropriate strategies to solve problems involving the addition and subtraction of rational numbers. 22 + -­‐14 + 78 = 22 + 78 + -­‐14 = 100 + -­‐14 = 86 (Associative Property of Addition) -­‐48 + 21 = (-­‐27 + -­‐21) + 21 = -­‐27 + (-­‐21 + 21) = -­‐27 (Decomposing and Associative Property of Addition) DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐
Milwaukee. (07.15.2011) Part Three: School Mathematics Textbook Program The School District of South Milwaukee recently adopted a new textbook program for middle school mathematics. At the start of the 2010/2011 school year we began using the Math Thematics program produced by McDougal Littell. I feel that this program is a combination of what would be considered a “traditional” program and a “reform” program. Students are able to learn mathematics through discovery in many lessons, although there are still a fair amount of teacher directed lessons. Math Thematics is a spiraling curriculum so material that is learned at the start of the school year will continue to appear throughout the year. Textbook Development The sixth grade students are initially introduced to the concept of negative numbers through a discussion on temperature. This gives students a real world example of why we would need numbers that are less than zero. Prior to this the students have only dealt with numbers that are greater than zero. Following the discussion on the use of negative numbers in regards to temperature students are shown a few more applications of negative numbers. These applications include discussions on money, elevation, and the introduction of the number line containing values less than zero. Sixth graders only encounter negative numbers one more time in the curriculum. One of the last sections in the text introduces the concept of addition of integers (rational numbers). Students first play a game where they move their game piece forward one space for each “+” they roll and back one space for each “–“ that they roll (these symbols are drawn on opposite sides of beans). They then record the total number of “+” symbols, “-­‐“ symbols, the total number of spaces that they moved, and which direction that they moved their game piece. This activity is done in hopes that students will see that the symbols essentially “cancel” each other. This will then lead them to develop strategies for adding a positive integer and a negative integer. In the next section students use the “bean” model to subtract integers. Students in the sixth grade spend a total of four class periods working with rational numbers. Seventh grade students begin to work with rational numbers within the first month of school. They begin my simply comparing integers. This gives them the opportunity to become familiar with the number line as well as to review the concept of negatives. They also focus on finding the absolute value of a number as well as finding the opposite of a number. While students are asked to identify the opposite of a number, they are not asked to put this concept into a real world context. In order to learn the concept of adding integers to the seventh graders students are asked to literally walk along a number line. The students build large number lines on the ground and then spin multiple spinners to determine starting point, direction, and how many spaces to move. This is the students’ first exposure to using the number line to add and/or subtract with rational numbers. In the following section students once again use the number line to subtract rational numbers. They are also strongly encouraged to rewrite subtraction problems as addition problems. While students only spend DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐Milwaukee. (07.15.2011) about eight class periods dedicated to learning about rational numbers they will continue to use rational numbers throughout the school year. Students in the eighth grade get a quick review of adding and subtracting with rational numbers at the beginning of the year. The assumption is that the students have learned the material in previous years, and it is not a main focus of the eighth grade curriculum. Students spend about two days reviewing the concept of adding and subtracting with rational numbers. Conclusions The Math Thematics textbook series does not spend a large amount of time on the concept of adding and subtracting with rational numbers. In fact I did not find any substantial sections that focused on the addition and subtraction of non-­‐integer rational numbers. While all of the key concepts found in the standards are covered in the text, they are not covered in depth, and could definitely use more development. One part of the standard that the text program seems to skim over is the concept of opposites. It seems that the text simply states what an opposite is, and then has students name numerous values and their opposites. The text does not provide examples of opposites that occur in real world situations, nor does it encourage students to create their own. While the text explains what an opposite is, it does not provide an explanation of how or why they work the way that they do. Students are not asked to do much work with opposites in combination with the number line, aside from showing that a number and its opposite have the same absolute value. The activity that occurs in the seventh grade where students are asked to “walk” the number line is extremely helpful in developing the concepts of addition and subtraction of rational numbers. By completing this activity students are exposed to a visual representation of the concept of adding and subtracting rational numbers. Unfortunately this activity is only done one time, and then is only briefly mentioned again. In order for students to fully understand the concept, as well as fully benefit from the visual model they need to work with it more than a couple of times. When dealing with subtraction problems involving rational numbers students are encouraged to rewrite them as addition problems as indicated in the standard. This concept is touched upon in the extension of the walking activity, but the text does not spend a lot of time on the reason why subtraction problems can be rewritten as addition problems. It would be helpful if students were asked to complete number lines to show that the two forms of the problem really did result in the same answer. Aside from the bean model in sixth grade and the number line model in seventh grade the Math Thematics program does not introduce any additional strategies to the students. This task falls DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐Milwaukee. (07.15.2011) completely on to the teacher. The fourth part of this standard states that students should use the properties of addition and subtraction to solve problems involving the addition and subtraction of rational numbers. This standard also states that students should use prior knowledge and understanding to solve these problems. In order for students to develop these additional strategies the teacher will need to facilitate these discussions. Overall I think that the Math Thematics text does a good job of introducing the concept of adding and subtracting with rational numbers. All of the key concepts from the standard are present in the text. In my opinion the downfall of the Math Thematics program is that it does not go far enough with the material. While all of the key concepts are introduced, the text does not spend enough time developing the information. It is more of an overview of the concept prior to arriving at the “rules” for adding and subtracting with rational numbers. Students need to spend more time developing the understanding on their own to really understand how and why addition and subtraction with rational numbers works. Suggestions I am concerned that the sixth grade students will not be exposed to rational numbers prior to seventh grade because the sections on rational numbers are at the very end of the sixth grade book. I feel it is important for the seventh graders to have at least some knowledge of rational numbers prior to the seventh grade curriculum. It may be necessary for sixth grade teachers to rearrange their curriculum to ensure that the students have the opportunity to learn about rational numbers. If this is not an option then I feel the activities in the sixth grade book should occur in the seventh grade classroom. Some students will understand the “bean” model better than the number line model so it is important that they are exposed to both. The number line activity in the seventh grade book is extremely important. The standard mentions the number line model a number of times, so it is clearly essential for students to be able to work with the number line fluently. Unfortunately the pacing guide in the text recommends only spending one class period on the number line activity. This is simply not enough time for students to fully understand the model. The teacher also needs to ensure that students are not simply going through the motions completing the activity. The teacher can do this by facilitating meaningful discussions while students are completing the activity as well as after. It is tempting to simply blindly follow the text program that we are given. It is easy to assume that the authors of the text have written a program that will accomplish the goals set forth in the CCSSM. Although all of the key concepts from the standard are touched upon in the text program, it is important for us as teachers to ensure that the material is developed completely. In order for students to fully understand the concepts covered in this standard we will need to add additional opportunities for students to work with the models and develop the mathematics. DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐Milwaukee. (07.15.2011) Part Four: A “Check Point” Formative Assessment Task That Reveals Student Thinking Answer the following questions. Be sure to include any diagrams or pictures that you find helpful in solving the problems. Explain all of your answers completely. 1. Write a question in which two values are added together to make zero. Explain your reasoning. 2. Solve the following problems. a. 5 + -­‐12 b. 5 – (-­‐3) c. 23 + -­‐32 3. Write a problem for the expression 10 + -­‐16. Then solve the related equation. DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐Milwaukee. (07.15.2011) DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-­‐Milwaukee. (07.15.2011)