Ann Schmitz-Buechler
CCLM^2 Project , Summer 2012
This material was developed for the Leadership for the Common Core in Mathematics (CCLM^2) project
at the University of Wisconsin-Milwaukee.
CCSSM Analysis: 5.NF.1 Part 1: Standard Grade: 5th Domain: Numbers and Operations-­‐Fractions 5.NF Cluster: Use equivalent fractions as a strategy to add and subtract fractions. Standard: 5.NF.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3+4/5=8/12+15/12= 23/12. (In general, a/b+c/d=(ad+bc)/bd.) Part 2: Explanation and Examples of Standard a. Explanation: Students use their understanding of the operations of addition and subtraction of whole numbers and apply this concept to fractions that have unlike denominator. First, students will need to understand how to add and subtract fractions with like denominators. Then students use their knowledge of equivalent fractions to replace fractions that have unlike denominators. Once the denominators are the same then numerators can be added or subtracted like whole numbers. Students will need to know how to make multiples of the denominators and numerators to make equivalent fractions. Students also need to understand that a fraction represents a quantity that is a portion of a whole. The fraction can be decomposed into equal shares (denominator). These shares can then be added to compose the whole. Ex. 1/4+1/4+1/4+1/4= 4/4 = 1 and 1/4+1/4+1/4+1/4+1/4+1/4+1/4+1/4+1/4= 9/4=4/4+4/4+1/4=2 and 1/4 It is also possible to add shares to make more than 1 whole. b. Examples: Visual Model 2/3 + 4/6=? Recognize that there are unlike denominators which means different sized shares. Replace the given fraction(s) with an equivalent fraction(s) to get like denominators. To get equivalent fractions with like denominators, students can use visual models. 2/3 is the same as 4/6. So, I can replace 2/3 with 4/6 in the equation. 4/6+4/6=8/6 Common Denominator Strategy Find a like (common) denominator by multiplying the unlike denominators. Use this product as the new like (common) denominator. a/b+c/d= (axd)+(bxc)/(axd) 2/3 + 4/5+? Find a like denominator by multiplying unlike denominators, 3x5=15. Make an equivalent fraction using 15 as a denominator. 2/3=?/15 Think: 2/3,4/6,6/9,8/12,10/15 4/5,8/10,12/15 4/5=?/15 2/3 is equivalent to 10/15 and 4/5 is equivalent to 12/15 Solve: 10/15+12/15=22/15 22/15= 15/15+7/15=1+7/15=1 and 7/15, a mixed number Part 3: School Mathematics Textbook Program I will be examining how addition and subtraction of fractions with unlike denominators is developed in 4th, 5th and 6th grade in our textbook. 4th and 5th grade have the same program(textbook), but 6th grade has a different textbook. Standard 4th Grade 4.NF.3a 4.NF.3b 4.NF.3c 4.NF.3d What are students expected to do? Representations, diagrams, contexts, strategies, and examples to support understanding. Students are expected to: -­‐understand addition and subtraction of fractions as joining and separating(composing and decomposing) parts of the same whole. -­‐decompose fractions with like denominators -­‐add and subtract mixed numbers with like denominators by replacing mixed number with an equivalent fraction and /or using properties of operations and relationships between addition and subtraction -­‐solve word problems involving addition and subtraction of fractions that refer to the same whole and having like denominators(e.g. Using visual models and equations to represent the problem) 1/4 + 1/8 +1/8= 1/8+1/8+1/8+1/8=4/8=1/2 Helena had ¾ bag of marbles and she gave ¼ to her brother. How much of the bag is left? XXXXX X 3/4-­‐1/4=2/4=1/2 Use of numberline model: 2 7/8-­‐ 1 5/8=1 2/8= 1 1/4 Standard 5th Grade What are students expected to do? Representations, diagrams, contexts, strategies, and examples to support understanding. Students are expected to: -­‐add and subtract fractions and mixed numbers with unlike denominators by replacing given fractions with equivalent fractions to produce an addition or subtraction problem with like denominators -­‐solve word problems using the expectation from above Visual models such as the clock face, area model, and numberline are used in the text. Ex. The clock face is used to represent twelfths and equivalent fractions to twelfths (halves,fourths,sixths) 5.NF.1 An area model using a 4x6 or 5x12 model: 1/3 + 5/12 = 4/12 + 5/12 = 9/12 = 3/4 Numberline model: 2 7/8 – 1 5/8 = 2 7/8-­‐ 1= 1 7/8 – 5/8 = 1 2/8 = 1 ¼ All these models are reinforces with games played frequently in class. -­‐ Roll Around the Clock -­‐ Fraction Tracks Example of a word problem: -­‐There are 6 brownies on a plate. Ann ate 1 ½ brownies. Sam are 2 ¼ brownies. Ty ate 1 ¾ brownies. How many brownies are left on the plate? Standard 6th Grade In the CCSSM, there are no standards that address addition and subtraction of fractions. However, in reviewing our district text, I did find addition and subtraction of fractions and mixed numbers being taught in 6th grade. What are students expected to do? Students are expected to: -­‐ add and subtract fractions and mixed numbers with unlike denominators by finding common denominators to make equivalent fractions so as to make a problem with like denominators. -­‐solve word problems using the expectation above Representations, diagrams, contexts, strategies, and examples to support understanding. Area models of folded paper(fraction strips) are used to support concept of equivalent fractions. However, the algorithm of finding a common denominator is the expected strategy to add or subtract fractions and mixed numbers. Ex. 3/4 -­‐ 5/8 = Find common denominator, 3/4 = ?/8= 6/8 6/8 – 5/8= 1/8 Very few word problems were given in the lessons about fractions. More were given when mixed numbers were introduced. Ex. Kele is 2 7/8 in. taller than Sharon. Rosa is 2 ½ in. shorter than Kele, but 5/8 in. taller than Gary. Who is taller, Sharon or Gary? By how much? Find the difference between Kele’s height and Gary’s height. Conclusions and Suggestions After reviewing my district’s textbooks for grades 4,5,and 6th grades, I find the Investigations text in grades 4 and 5 to be aligned for development and teaching of the one standard 5.NF.1. However, I was surprised to find in the CCSSM that 6th grade did not have any standard addressing addition and subtraction of fractions and mixed numbers. Our 6th grade text teaches finding the common denominator as the only strategy to making equivalent fractions to solve addition and subtraction problems using fractions and mixed numbers with unlike denominators. I find this to be problematic since the term “common denominator” is not used in 4th or 5th grade. The concept of common denominator is taught using many strategies to find equivalent fractions to make problems that have fractions with like denominators. The Math Leadership is aware of the problem and has begun having discussions between 5th grade and Middle School teachers who are on the Math Leadership Team. We know this problem needs to be addressed. I hope this class has shed some light for us for our discussions. If it was up to me, I would continue teaching all the strategies to find equivalent fractions, as we do, but make sure that the term “common denominator” and the algorithm to find the common denominator be added to the end of the instruction sequence. Students need to connect the strategies of finding equivalent fractions to the algorithm of finding a common denominator. Leadership might also want to discuss whether or not addition and subtraction of fractions needs to be taught in 6th grade. It seems redundant and time might be spent better with just a quick review.