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Teaching Middle School Mathematics Fractions, decimals and percentages Ratios, rates and proportions Work out the problem on your card, then find 3 other people who have the same number as you do. Sit with them to work a collaborative problem. Goals for the four sessions Identify tasks and activities that are effective for teaching the content of each session Use visual representations and the concreterepresentational-abstract approach to enhance students’ understanding of the content of each session Identify mathematical language and develop its use in students Use mathematical discourse to promote engagement and deep processing Provide tasks that promote student engagement and mathematical reasoning within the content of each session Assess students’ understanding and proficiency in order to provide useful feedback and make needed changes to instruction Common Core for fractions Read through each standard, marking “solve word problems” and “use visual fraction models” Match each problem to a standard Anything surprise you about the CCSS? Common Core Standards Components of Mathematical Proficiency 1. Conceptual understanding 2. Procedural skill and fluency 3. Application Adding or subtracting fractions Conceptual understanding Apples and Oranges (same size pieces of the whole) Procedural Skill and Fluency 1 2 + 3 1 4 = 1 3 1 + 2 = Application A pitcher contains 5 8 3 2 4 pints of orange juice. After you pour of a pint into a glass, how much is left in the pitcher? 5.4 – 3.25 = Learning Progression within Conceptual Understanding Concrete - Representational - Abstract Objects 1 2 1 Pictures Draw the essential insight that allows Use fraction circles, bars students to add or pattern blocks. fractions of different size pieces. +3 Symbols 1 2 + 3 = ? becomes 1 3 6 +6=6 2 5 Read “The Role of Representations…” p. 494. Find a “Golden Sentence”. Multiplying fractions 1. 2. 3. Conceptual Understanding Procedural Skill Application Conceptual Understanding Procedural Skill Application Learning Progression within Conceptual Understanding Concrete - Representational - Abstract Objects 2 4x5 Make 4 equal groups, add them. Pictures Draw this on a number line. Symbols Notice that the size of the fraction pieces stay the same.We’re only multiplying the numerator times the number of groups: 4 x 2 = 8. 4 x 2/5 = 8/5 Learning Progression within Conceptual Understanding Concrete - Representational - Abstract Objects 1 2 2 x5 Make ½ of a group. Read the multiplication 1 2 as of . 2 Pictures Bar model: Area model: 5 NLVM.usu.edu Symbols The area model shows that we made pieces of a new size – tenths. The multiplication results in 2 tenths. Notice that this is the result of multiplying the numerators (as in the earlier example) and the denominators. See pp. 23-25 Learning Progression within Procedural Skill Acquisition - Fluency - Generalization C-R-A Practice* Extensions * Guided practice with feedback is critical One extension Generalize this process to multiplication involving mixed numbers. 1 1 ×1 2 3 Draw a visual fraction model. Connect to the procedure. Fun problems: If the rectangle has a value of 1 , show 1. 3 Learning Progression within Application Near Transfer – obvious connection to previous problems to establish the “type” Far Transfer – problem-solving skills are required: 1. What “type” of problem is this? 2. What do I know that I can use? (KWL) 3. Is there a drawing or chart that will help? 4. Other problem-solving strategies 3/4 of a pan of brownies was sitting on the counter.You decided to eat 1/3 of the brownies in the pan. How much of the whole pan of brownies did you eat? A cake mix uses 1 1 2 1 2 2 cups of flour. You want to make recipes of this cake. How much flour do you need? Learning Progressions The Common Core gradually increases complication of working with fractions. The Operations with Fractions packet steps students through these learning progressions carefully and systematically. Interlude… Alternate Algorithms 6 )234 -120 114 -60 54 -30 24 -24 0 20 10 5 4 39 Fluently divide multi-digit numbers using the standard algorithm. 6.NS.2 This type of division is called repeated subtraction “When I reflect on this past unit, I think that learning the alternate algorithms was extremely helpful for me. I chose the scaffolding algorithm as my algorithm of choice for a good reason. Growing up through elementary school, middle school, and high school, I always struggled with long division. I never really got the grasp of an algorithm that made sense to me.” “This scaffolding method has made long division unbelievably easier for me. I finally understand how to solve those problems and can do them on my own now. I originally was taught how to carry the one and cross out certain numbers. But really I had no idea what my teacher was talking about. This scaffolding method not only helps me with my long division, but it also helps me with my multiplication tables, as well as adding. This scaffolding method will stay with me forever, and I truly do believe I will use this for the rest of my life.” This type of division is called fair shares, or partitioning Partial Products Algorithm How would this work for 2.3 x 1.8? So what about fraction division? One serving (1/2 cup) of broccoli contains 47 mg of calcium. Kids ages 9-18 need to get 1300 mg of calcium daily to build strong bones. How many cups of broccoli would this be? See the Acquisition-Fluency-Generalization scheme for division Two types of division Partitive (fair shares) We want to share 12 cookies equally among 4 kids. How many cookies does each kid get? How would you solve this with objects? The number of groups is known; the number in each group is unknown. Measurement (repeated subtraction) For our bake sale, we have 12 cookies and want to make bags with 2 cookies in each bag. How many bags can we make? How would you solve this with objects? The number in each group is known; the number of groups is unknown. Why is this important? A box of Cheerios contains 3 4 1 12 2 cups. Each serving is cups. How many servings are in a box of Cheerios? How much is left over? Partitive or Measurement division? Write 3 additional problems like these. Dividing a fraction by a whole number We have ½ of a pizza and want to share it equally among 4 people. How much pizza does each person get? 1/2 ÷ 4 Try this with fraction manipulatives. 8/3 ÷ 4 What’s a procedure? Dividing a whole number by a fraction We have a dozen large cookies and want to give ½ cookie to each child. How many children can we serve? 1 ÷ 2 12 (How many times does ½ go into 12?) What’s a procedure? Dividing a fraction by a fraction A serving size is ¼ cup. How many servings are in 5/4 cup? Solve it. Write it as an equation. 5 4 ÷ 1 4 How many ¼’s are in 5/4? This is the same as asking “How many 1’s are in 5?” Procedure: Get common denominators. Dividing a fraction by a fraction Cheerios problem – solved by mental math or a drawing (C-R-A) 1 3 12 ÷ 2 4 Translated to symbols Procedure? Find common denominators... 50 3 ÷ 4 4 then ask “how many 3’s in 50?” Write two more similar problems to solve with common denominators. What about “invert and multiply?” See pp. 29-34 in Operations with Fractions for an instructional approach. Why “invert and multiply” works: 1 12 2 3 4 ÷ = 25 3 ÷ 2 4 = 25 4 × 2 3 ÷ 3 4 × 4 3 The last step is justified by recognizing that if we multiply both numbers in a division problem (i.e. in a fraction) by a constant, we get an equivalent problem. Decimals 2.3 x 1.8 A generalization from multiplying fractions: Multiplying decimals Understanding-Skill-Application C-R-A A-F-G Think-Pair-Share Poster 2.3 x 1.8 See the decimals and percents assessment Why do we “count decimal places?” 2.3 x 1.8 .2 4 1.6 1.3 2 5.1 4 Yellow 2 x 1 Orange 0.3 x 1 Blue 2 x 0.8 Green 0.3 x 0.8 2.3 1.8 Decimals Forever!