Sara Summ
CCLM^2 Project
Summer 2012
DRAFT DOCUMENT. This material was developed as part of the
Leadership for the Common Core in Mathematics (CCLM^2) project at the University of Wisconsin-Milwaukee.
CCSSM Analysis: 3.NF.2ab Part 1: Standard Grade: 3 Domain: Number and Operations – Fractions / 3.NF.2ab Cluster: Understand a fraction as a number on the number line; represent fractions on a number line diagram. Standard: a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each p art has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. Part 2: Explanation and Examples of Standard Work with the number line diagram in grade 3 is the first time students work with a number line for numbers that are between two whole numbers. Explanation: 3.NF.2a The student will be able to represent a unit fraction on a number line diagram by defining the interval from 0 to 1 as the whole and by partitioning it into equal parts. They will know that each equal part is the same length. Example: To construct a unit fraction on a number line diagram the students must participation the unit interval into 4 intervals of equal length and recognize that each part has the length ¼. Students will be able to locate the number ¼ on the number line. 𝟏
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It is important to note that success with this standard is dependent on the success of 3.NF.1m where students are introduced to fractional concepts by using fraction strips, over time the fraction strips can be represented using a number line as a representational method for understanding fractions. For 𝟒
students to have an understanding of this concept they also must understand what a number line is (infinite ruler). Additional vocabulary students will have to understand is: endpoint, numerator, denominator, number line, and unit fraction. Explanation: 3.NF.2b The student will be able to represent a non-­‐unit fraction on a number line diagram by partitioning the non-­‐unit fraction into the unit fractions. Students will be able to recognize that the resulting interval is the size of the non-­‐unit fraction and its endpoint locates the non-­‐unit fraction on the number line. !
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= number of intervals of parts. *It should be noted that this standard applies to fractions larger than 1. For a student to be able to have this knowledge of this representational way of modeling and understanding the relationship between a unit and non-­‐unit fraction they must have exposure to other representations such as area models, tape diagrams, and fraction strips. The vocabulary words that need understanding are: partitioning, whole and unit / non-­‐unit fractions. Within this standard, teachers should also pay close attention to precision within their modeling and language so students are gaining the true knowledge and not misconceptions. Part 3: School Mathematics Textbook Program Textbook Development in Scott Foresman – Mathematics Grade 3 Text The progression of using representational models in the third grade text starts with shapes, where students are naming the parts of a whole, over the course of two lessons they move into finding equivalent fractions using pre-­‐labeled fraction strips. The students create equivalent fractions in relation, starting with the larger unit fraction and building equivalency with smaller unit fractions. Within this one lesson, there is no vocabulary development in regards to unit and non-­‐unit fractions. Students are introduced to the concept of equivalent fractions in the third lesson, there is building on the understanding of both the numerator and denominator; however there is no mention of the difference between a unit fraction and non-­‐unit, and in fact they refer to them as strips and not parts (shares) of a whole. The lesson following creating equivalent fractions is one that has students comparing and ordering fractions, again students are using pre-­‐labeled fraction strips to represent the fractions in this lesson. In this lesson, unit fraction is introduced. The book moves directly into creating “rules” when comparing fractions without allowing the students to make conjectures and conclusions based on their own mathematical knowledge. After comparing, the next lesson introduces fractions on the number line. It focuses on finding fractions on a number line, with their fraction strips students are asked to create a number line and are given specific directions on how to do so. It has a brief activity where they create the number lines and answer 6 questions regarding “how many sixths are the same as 1”; the language is quite different than what the CCSSM states. The student then moves into labeling missing fractions for 4 different number lines, all to 1 not over 1 whole. The only other time you will see fractions on a number line is later in the same chapter when they have students measure objects using a ruler to the nearest ¼ and ½ of an inch. Grade 4 Text The fourth grade text starts with a brief review on fractional concepts and discusses the parts of a region; they also build on the understanding of what a numerator and denominator are. From there the students talk about the parts of a set, the third lesson is on fractions, length and the number line. The main activity in this lesson has the student naming and showing part of a length, they are comparing the length of the whole (again using the labeled fraction strips) and making comparisons with the unit fractions and the whole. The students begin using their fraction strips to draw a number line showing: fourth, thirds, tenths, and eighths. From there the lesson has students locate and name fractions on a number line. It will have them draw a non-­‐unit fraction on a number line, prompting the student to draw the equal shares; the book even tells the students how to label the number line. It does show the number line as a specified number of equal parts (dependent on the numerator), and will label the endpoint or non-­‐unit fraction as 4 of 5 equal parts or 4/5. The book reiterates that the name of a point on a number line tells the length from 0 to the point. Aside from one other lesson that uses a number line as a strategy for comparing and ordering fractions, this is the only lesson in grade 4 that represents fractional concepts on a number line. Grade 5 Text The fifth grade text starts the fraction chapter with a lesson on naming a fraction as a point on a number line. It shows the students how to plot a fraction on a number line; it also re-­‐emphasizes the use of a numerator and denominator when naming fractions. From that lesson it has students identify and locate fractions and mixed numbers on a number line. The book builds on their knowledge of identifying fractions on a number line to using a number line to compare and order fractions, including mixed numbers. Conclusions and Suggestions: Based on my text analysis my thoughts have been confirmed that there is not a strong alignment of our current text book to the CCSSM. Our text does have few lessons on using the number line with fractions; however I do not feel that the concepts are presented enough where students can have a deep understanding of the mathematics. Our current text is very traditional and will briefly show a strategy, and very abruptly go into the algorithm. The structure of our text doesn’t support the levels of students’ mathematical thinking of CRA (concrete – representational – abstract). Our text often goes directly to the abstract with very little meaning or understanding from the student. I not only looked at our current text book but also at our current district math curriculum, both of which do not place a strong emphasis on deep mathematical understanding, the emphasis is placed on accuracy through algorithms. To be able to have our students successful with the implementation of the CCSSM, I believe that we must provide professional development to vertical teacher teams in regards to the progression of fractions within the new standards. In addition to providing professional development to our teachers, we must provide resources they can use to effectively implement the strategies named in the standard. As a district coordinator, I feel that to more effectively implement the CCSSM we will need a new text book to assist us. In looking for a new text I would like one builds in the awareness of a child’s mathematical development and need to truly investigate and explore mathematical concepts while building on their understanding of a given topic. I am aware that many textbook companies still need supplemental material to address the CCSSM, with that I plan to pull in resources for our teachers from About Teaching Mathematics by Marilyn Burns, Young Mathematicians at Work by Catherine Fosnot and Maarten Dolk, as well as Teaching Student-­‐Centered Mathematics by John Van de Walle and LouAnn Lovin. The use of the number line is a tool used to reinforce fractions, to be able to use this tool successfully our teachers will need training, support, and collaborative discussions involving how this tool can and should be used. Knowing the progression of the Number and Operations – Fractions standards in grade three the students should be afforded an opportunity to manipulate, make connections, question and justify their reasoning about fractions. From the beginning foundation that is laid in 3.NF.1, students can build on their knowledge of fractions on the number line. Teachers must be precise in their language and modeling to make this successful for our students. Along with this shift in mathematical content instructional change will also bring a change in which many teachers currently structure their math block. As we move towards implementation of the CCSSM, teachers will be transforming their math blocks from a traditional, whole group model to one where the teacher asks as a facilitator with skilled questions to elicit knowledge and push students towards mathematical proficiency.