PROFESSIONAL LEARNING

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Hector Summers
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1 . 2. PROFESSIONAL LEARNING

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3 ORIGO Education

4 ORIGO Education 2

5 Show 4 in each of these models. For each model, consider the following questions: What attribute is the focus? What is the whole? What does equal-sized mean? What does the fraction indicate? Set Model: Area Model: Length Model: Number Line Model: 0 2 ORIGO Education 3

$What equal- sized means in the model. What the frac7on indicates. Represen7ng Frac7ons et odel Represen7ng Frac7ons rea odel The num er a discrete (counta le) quan7ty.$ $The frac7on indicates the count of o ects in the su set compared to the defined set of o ects. The area a con7nuous (measurea le) quan7ty. The whole is determined y the defined area or region.$ $The same area represents equal- sized parts. The frac7on indicates the area of the part compared to the area of the whole.$

6 Represen7ng Frac7ons Teachers and students need to consider the following: The type of quan7ty that the model is intended to represent (con7nuous or discrete?). How the whole is defined. What equal- sized means in the model. What the frac7on indicates. Represen7ng Frac7ons et odel Represen7ng Frac7ons rea odel The num er a discrete (counta le) quan7ty. The whole is determined y a defined count of a collec7on or set. The same number of items represents equal- sized parts. The frac7on indicates the count of o ects in the su set compared to the defined set of o ects. The area a con7nuous (measurea le) quan7ty. The whole is determined y the defined area or region. The same area represents equal- sized parts. The frac7on indicates the area of the part compared to the area of the whole. Represen7ng Frac7ons ength odel Represen7ng Frac7ons um er ine odel special type of length model. The length a con7nuous (measurea le) quan7ty. The whole is determined y a defined en t. The same en t represents equal- sized parts. The frac7on indicates the length of the part compared to the length of the whole. The length a con7nuous (measurea le) quan7ty. The whole is determined y a unit of stance from to. The same stance represents equal- sized parts. The frac7on indicates the loca7on of a point in rela7on to the distance from with regard to the defined unit. ORIGO Education 4

Represen'ng Frac'ons Type of Model Type of Quan/ty Whole Meaning of Equal- Sized Parts Set model discrete determined by a defined count of a collec'on or set same number of items Area model con'nuous

7 Represen'ng Frac'ons Type of Model Type of Quan/ty Whole Meaning of Equal- Sized Parts Set model discrete determined by a defined count of a collec'on or set same number of items Area model con'nuous determined by a defined area or region same area Length model con'nuous determined by a defined length same length Number line model con'nuous unit of distance from 0 to same distance What the Frac/on Indicates the count of objects in the subset compared to the defined set of objects the area of the indicated part compared to the area of the indicated whole the length of the indicated part compared to the length of the indicated whole the loca'on of a point in rela'on to the distance from 0 with regard to the defined unit ORIGO Education 5

8 ORIGO Education 6

9 Step Up. Complete these to show how you would solve each problem. Draw lines and color parts to show each share. a. Two slices of banana bread are shared equally by 3 people. How much of one whole slice will be in each share? slices shared by 2 = people b. Three pies are shared equally by 8 people. How much of one whole pie will be in each share? pies shared by = people 2. Each large rectangle below is one whole. Complete the sentence, draw lines, and color parts to show each share. a. 3 5 is the same as b. 4 7 is the same as divided by divided by Step Ahead Complete the sentences so that the quotient in the division sentence is the same as the product in the multiplication sentence. Then use the diagram to show why they are the same. 3 4 = 3 = Stepping Stones Student Journal Grade 5/ Module / Lesson ORIGO Education 7

10 Interpreting Fractions Frac%ons as Numerator Denominator Meaning of 3/4 Part of a Whole number of equal- sized parts indicated number of equal- sized parts in the whole 3 parts out of 4 equal- sized parts Numbers or Measures number or count of equal- sized parts (unit frac:ons) number of equal- sized parts (unit frac:ons) needed to create the whole 3 counts (repe::ons) of the unit frac:on /4 /4 + /4 + /4 Quo%ents number of items in the whole number of shares or equal- sized parts 3 4 the result when divided or shared ORIGO Education 8

11 3 Hundredths 3 Hundredths ORIGO Education 9

12 Exploring Equivalent Fractions ORIGO Education 0

13 Exploring Fractions on Number Lines ORIGO Education

14 Fraction Fill Up Two or more players Each player needs a fraction wall. The group needs one cube. Write these fractions on the cube Take turns to roll the cube. Shade that fraction on a single strip if space allows. It can be the fraction shown on the cube or an equivalent fraction. The winner is the first person to exactly fill 2 strips (other than the strips that shows one whole.). 2 0 ORIGO Education 2

15 4 Comparing Fractions Reasoning with Unit Fractions Using Common Denominators Using Common Numerators Using Benchmarks 3 or or or or or or or or or or or or 5 8 ORIGO Education. 5 6 or or or or 4 3 ORIGO Education

17 4 7 2 tenths ones hundredths FOLD FOLD FOLD FOLD FOLD FOLD FOLD FOLD ORIGO Education

19 unit fractions common fractions proper fractions improper fractions mixed numbers decimal fractions ORIGO Education

21 ORIGO Education

23 Common Core State Standards Grades - 5 Fraction Progression Grade Representation & Interpretation.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. Equivalence & Comparison.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. Grade 2 2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. 2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. Grade 3 3.NF.A. Understand a fraction /b as the quantity formed by part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size /b. 3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram. 3.NF.A.2a Represent a fraction /b on a number line diagram by defining the interval from 0 to as the whole and partitioning it into b equal parts. Recognize that each part has size /b and that the endpoint of the part based at 0 locates the number /b on the number line. 3.NF.A.2b Represent a fraction a/b on a number line diagram by marking off a lengths /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. 3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. 3.NF.A.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. 3.NF.A.3b Recognize and generate simple equivalent fractions, e.g., /2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model. 3.NF.A.3c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/; recognize that 6/ = 6; locate 4/4 and at the same point of a number line diagram. 3.NF.A.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Grade 4 4.NF.B.3 Understand a fraction a/b with a > as a sum of fractions /b. 4.NF.B.4a Understand a fraction a/b as a multiple of /b. For example, use a visual fraction model to represent 5/4 as the product 5 (/4), recording the conclusion by the equation 5/4 = 5 (/4). 4.NF.C.5 Express a fraction with denominator 0 as an equivalent fraction with denominator 00, and use this technique to add two fractions with respective denominators 0 and 00.2 For example, express 3/0 as 30/00, and add 3/0 + 4/00 = 34/00. 4.NF.C.6 Use decimal notation for fractions with denominators 0 or 00. For example, rewrite 0.62 as 62/00; describe a length as 0.62 meters; locate 0.62 on a number line diagram. 4.NF.A. Explain why a fraction a/b is equivalent to a fraction (n a)/(n b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as /2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. 4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. ORIGO Education

25 Common Core State Standards Grades - 5 Fraction Progression Understand Operations Computation & Application Grade Grade 2 Grade 3 Grade 4 4.NF.B.3 Understand a fraction a/b with a > as a sum of fractions /b. 4.NF.B.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 4.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 4.NF.B.4b Understand a multiple of a/b as a multiple of /b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 (2/5) as 6 (/5), recognizing this product as 6/5. (In general, n (a/b) = (n a)/b.) 4.NF.B.3 Understand a fraction a/b with a > as a sum of fractions /b. 4.NF.B.3b Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = /8 + /8 + /8 ; 3/8 = /8 + 2/8 ; 2 /8 = + + /8 = 8/8 + 8/8 + /8. 4.NF.B.3c Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. 4.NF.B.3d Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. 4.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 4.NF.B.4c Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? ORIGO Education

27 Common Core State Standards Grades - 5 Fraction Progression Representation & Interpretation 5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (a/b = a b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If people want to share a 50 pound sac of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? Equivalence & Comparison 5.NF.A. 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 + 5/4 = 8/2 + 5/2 = 23/2. (In general, a/b + c/d = (ad + bc)/bd.) Grade 5 ORIGO Education

29 Common Core State Standards Grades - 5 Fraction Progression Grade 5 Understand Operations 5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. 5.NF.B.4a Interpret the product (a/b) q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a q b. For example, use a visual fraction model to show (2/3) 4 = 8/3, and create a story context for this equation. o the same with (2/3) (4/5) = 8/5. (In general, (a/b) (c/d) = ac/bd.) 5.NF.B.5 Interpret multiplication as scaling (resizing), by 5.NF.B.5a Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. 5.NF.B.5b Explaining why multiplying a given number by a fraction greater than results in a product greater than the given number (recognizing multiplication by whole numbers greater than as a familiar case); explaining why multiplying a given number by a fraction less than results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n a)/(n b) to the effect of multiplying a/b by. 5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. 5.NF.B.7a Interpret division of a unit fraction by a non zero whole number, and compute such quotients. For example, create a story context for (/3) 4, and use a visual fraction model to show the quotient. se the relationship between multiplication and division to explain that (/3) 4 = /2 because (/2) 4 = /3. 5.NF.B.7b Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 (/5), and use a visual fraction model to show the quotient. se the relationship between multiplication and division to explain that 4 (/5) = 20 because 20 (/5) = 4. Computation & Application 5.NF.A. 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 + 5/4 = 8/2 + 5/2 = 23/2. (In general, a/b + c/d = (ad + bc)/bd.) 5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + /2 = 3/, by observing that 3/ /2. 5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (a/b = a b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If people want to share a 50 pound sac of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? 5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. 5.NF.B.4b ind the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. ultiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. 5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. 5.NF.B.7c Solve real world problems involving division of unit fractions by non zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share /2 lb of chocolate equally? ow many /3 cup servings are in 2 cups of raisins? ORIGO Education

Fear Not the Fractions

Fear Not the Fractions PROFESSIONAL LEARNING www.origoeducation.com Show 4 in each of these models. For each model, consider the following questions: What is the whole? What does equal-sized mean? What