project 2: ccssm interpretation guide...draft document, unedited copy.this material was developed...

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)

Paul Buckholt Common Core Leadership in Mathematics, Project 2

July 15, 2011

Project 2: CCSSM Interpretation Guide

Part 1: Standard 7.NS. 1 a and b 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 charges 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.

Part 2: Key concepts and Terms:

• Part (a) is describing the role of additive inverse as it relates to rational numbers. Additive inverse is where two opposing distances or quantities that have a sum of zero for canceling each other out. It is also referring to adding or subtracting discrete number or objects to get back to the origin.

• A rational number is any number that can expressed as the quotient or fraction a/b of two integers, with the denominator” b” not equal to zero. Since “b” may be equal to 1.

• Examples of rational numbers include the following: 7, 2.8, .002, 2/7, ¾, 1/3, .325325

• Non-example of ration numbers include; a/0, 5/0 , 3.4/0,

• Rational number theory actually begins with cardinality, which means a one to one correspondence with counting. Rational numbers include counting (real) numbers, whole numbers, integers and as a ratio (fraction) of two numbers.

• The number ¾ is a terminating number. When presented in decimal form of .75, it can be written as 75/100. The last two numbers listed are rational numbers because they have repeating pattern. 1/3 if converted to decimal h as a repeating pattern and still can be written as 1/3. If there is a repeating pattern such as .725725 it can be shown to be a ratio of whole numbers. Any rational number can be written as a repeating decimal.

• In our discussion of rational n umbers, it’s also important to understand irrational numbers. Irrational numbers or non-repeating decimals that continue without terminating, examples are pi and the square root of 2.

• Real-world situations where positive and negative rational numbers combine to make 0.


Possible confusing aspects:

• Rational numbers can be written in decimal forms. For example, 1.25 is also the same as 1 ¼ or 5/4, therefore 1.25 is also rational number.

• Rational numbers are not only counting numbers in the numerator and denominator (4.3/9.6)

• That all the operations that apply to real numbers can also be applied to rational numbers (associative, commutative, distributive, and identity properties).

• There are an infinite number of rational numbers between 0 and 1. One inch can be divided up into an infinite number of groups.

• Repeating numbers are rational numbers. This concept can be confusing. One method to clarify this is to have the students reflect upon 1/3 or 2/3, where both numbers repeat (.3 and .6), but can be written in fraction form.

• A misconception is when you multiply a rational number the answer is always larger or when you divide a rational number the answer is always smaller.

Teacher friendly language: Students will be able to identify numbers as rational before they enter the seventh grade.

• They will be able to apply rational numbers to a real-world context by using a horizontal or vertical number line to solve problems where two opposing distances or quantities that have a sum of zero for canceling each other out.

Example #1: A man travels down 5 levels below ground level in a parking garage. How many levels up must he drive to reach ground level? (-5) + 5 = 0

Explanation: Example #1, describes a real-world situation where traveling below ground 5 levels, then traveling back up to the ground level cancel the distance traveled for a net gain of 0. This is also an example of a vertical number line.


Example #2: John owes Bob 3 specific baseball cards that he accidentally damaged. John gives him back all 3 of the cards. How many cards does he still owe his friend? (-3) + 3 = 0

Explanation: Example# 2, describes a real-world situation where Bob is paid back the quantity of cards that were owed to him for a net gain of 0. This is also an example of a horizontal number line. Part 3: School Mathematics Textbook Program: Montessori Program K3-6 grade

A. Textbook Development:

Students at Fernwood Montessori School from k3-6th grade are immersed in the Montessori curriculum. In the 7th and 8th grade the CMP2 curriculum is taught. I am not trained in Montessori mathematics; therefore most of the textbook development will be based upon the 6th through 8th grade CMP2 curriculum.

Rational numbers implies an ever increasing knowledge that builds upon itself from Pre-K through the 8th grade and beyond. The progression starts with cardinality or the concept of one to one correspondence in kindergarten. Then students proceeds through the primary grades with the following concepts being taught:

• Counting number • Whole numbers • Integers • Operations applied to each of these number systems, such as, The associative,

commutative, distributive, identity, and zero properties • The meaning of fractions • The meaning of multiplication • Then the realization that rational numbers encompass all of these concepts and

more. The Montessori materials and lessons cover many of these concepts such as:

• Golden beads-cardinality concepts • Cards and counters-Cardinality concepts • Small, large, flat bead frames-Cardinality, subtraction, addition concepts • Spindle boxes-cardinality concepts • Checker Board-multiplication • Bead bars-adding, subtracting, multiplication • Fraction Decimal board-division, decimals


• Bead chains-addition, multiplication • Squaring and cubing material-multiplication • Metal insets-fractions • Fraction material- all four operations • Snake game-Signed numbers with all four operations

This list is not meant to be all inclusive. It’s meant to give a general idea of the Montessori materials that promote and teach rational numbers. School Mathematics Textbook Program: CMP2 6th, 7th and 8th grade CMP2 Philosophy: is to focus on real-world applications, working with a partner or small groups, and constructive responses are expected to assure comprehension of the content. CMP2 lessons start out with a launch where the teacher will ask pointed questions around a real-world situation or problem. The students then explore the situation in pairs or small groups. Lastly, the teacher summarizes the main concepts of the lesson. The students also engage in application, connection, and extension problems that reinforce the concepts in real-world applications.

A) Textbook Development: In the 6th grade CMP2 students are expected to be able to complete and understand the following concepts that directly support success with 7.NS.1a & b: Bits and Pieces 1:

• Model situations involving fractions, decimals and percents • Understand and use equivalent fractions to reason about situations. • Compare and order fractions and decimals • Move flexibly among fractions, decimals and percent representations. • Use benchmarks, such as 0, ½, 1 ½ and 2, to help estimate the size of a

number or sum. • Develop and use benchmarks that relate different from of rational numbers

(for example, 50 % is the same a ½ or .5) • Use context, physical models, drawings, patterns or estimation to help

reason about situations involving rational numbers.

Bits and Pieces 11: • Use benchmarks and other strategies to estimate the reasonableness of

results of operations with fractions • Develop ways to model sums, differences, products and quotients,

including the use of area, fraction strips and number lines. • Use your knowledge of fractions and equivalence of fractions to develop

algorithms for adding, subtracting, multiplying and dividing fractions. • Recognize when addition, subtraction, multiplication or division is the

appropriate operation to solve a problem.


• Write facts familiar to show the inverse relationship between addition and subtraction, and between multiplication and division.

• Solve problems using operations on fractions. Bits and Pieces 111:

• Use your knowledge of fractions to learn about operating on decimals • Estimate the results of operations on decimals • Use your knowledge of place value in working with decimals • Know when to use each operation in a situation involving decimals • Recognize real-world situations where people often choose to use

decimals instead of common fractions • Develop algorithms for solving a variety of types of percent problems

Example of where this standard (7.NS.1) is taught- this particular standard is not taught in the 6th grade curriculum (bits and pieces 1, 2, & 3). These books do provide a solid base for the students to have a strong understanding of rational numbers. Each book builds upon each other in a logical sequence that prepares them for more advanced mathematics. Textbook Development:

7th grade CMP: The students are expected to be able to complete and understand the following concepts that directly support success with 7.NS.1a & b:

Accentuate the Negative:

• Use the appropriate notation to indicate positive and negative numbers • Compare and order positive and negative rational numbers (fractions,

decimals, and zero) and locate them on a number line. • Understand the relationship between a positive or negative number and its

opposite ( additive inverse) • Develop algorithms for adding, subtracting, multiplying and dividing positive

and negative numbers. • Write mathematical sentences to show relationships • Write and use related fact families for addition/subtraction and

multiplication/division to solve simple equations. • Use parentheses and rules for the order of operations in computations • Understand and use the commutative Property for addition and multiplication • Apply the Distributive Property to simplify expressions and solve problems • Graph points in four quadrants • Use positive and negative numbers to model and answer questions about

problem situations •

Example of where this standard (7.NS.1) is taught- Investigation 1.2 teaches the concept of additive inverse. The lesson asks for the opposite of a set of numbers. These numbers


are positive and negative and implicitly ask for the additive inverse. It concerns me that the book doesn’t explicitly describe the activity as additive inverse and a real-world application is not used to describe the concept.

Conclusions: The Montessori materials increase the knowledge and understanding of rational numbers up to a certain point. In kindergarten the materials focus on cardinality and the operations of addition, subtraction and multiplication. The lower elementary grades (1st-3rd), continue with more advanced material that focus on all four operations and are introduced to addition and subtraction of fractions. Upper elementary grades (4th-6th), they continue to become proficient in all four operations, including rational numbers and the various properties that can be applied to all numbers. The majority of the curriculum of rational numbers is taught in the 6th grade curriculum using CMP2 textbook. This coincides with the CCSSM. The 6th grade CMP2 curriculum covers 3 clusters and eight standards relating to rational numbers. The 7th grade curriculum and CCSSM is an extension of the material that is taught by the curriculum and is expected to be mastered by the CCSSM. The 6th grade curriculum for the CMP2 uses three books to meet the requirements of the CCSSM. These include Bits and Pieces 1, 11, and 111. The 7th grade CMP2 curriculum covers one cluster and three standards that repeat the previous 6th grade standards and extend upon them. The 8th grade curriculum doesn’t explicitly teach rational numbers, but does go into irrational numbers. By 8th grade, rational numbers are expected to be mastered and applied in the 8th grade curriculum. All the operations with whole numbers are the same with rational numbers and are infused into the 8th grade curriculum. Students are expected to apply and understand these concepts that are explicitly taught, such as linear, exponential, and quadratic equations, linear systems, and Pythagoras. In creating pacing guides for Milwaukee Public Schools, the current 7th grade CMP2 aligns with the CCSSM. This is before the publishers aligned the curriculum to the CCSSM. The only standard that was lacking was the 7.RP.1, which consisted of complex numbers. We did find one problem on page 46, #34 that was a complex number. Therefore, choosing this curriculum would help increase state testing proficiency scores because CMP2 addresses the content demands of the CCSSM. Suggestions:

• The Montessori program encourages a strong conceptual understanding of rational numbers but it doesn’t go in-depth or reflect upon real world applications. I would like to see the 6th grade CMP2 taught along with the Montessori curriculum, especially in the context of rational numbers.

• In reviewing current classroom assessments bases on standards (CABS), are lacking in the 6th grade. There should be more formative assessments created to align to the new CCSSM.

project 2: ccssm interpretation guide...draft document, unedited copy.this material was developed...

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