2.  Honor the challenge in this work and set the tone for teachers as learners  Build conceptual knowledge of fractions, and acknowledge most of us come with procedural  Become proficient with the work in Investigation 1  Know how and where to highlight the standards for students.

3. Investigation 1: Parts of Rectangles – 8 lessons Investigation 2: Ordering Fractions – 7 lessons Investigation 3A: Multiplying Fractions-3 lessons Investigation 4: Working with Decimals 25 lessons – 30 days

4. 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.

5. It’s easy…  Multiply the numerators 5x2 =10  Multiply the denominators 6x3=18  Now you must reduce 10 / 18. divide 10 and 18 by 2 and you get 5 / 9 Do we need to invert? Or is that with division? Draw a representation and write a short story (scenario) to go with it.

6. 5 / 6 + 2 / 3 5 / 6 x 2 / 3 …Solve the problem? …Draw a picture/representation? …Write a word problem?

7. What learning did that take from us? Stolen Opportunity!

8.  What do the Common Core State Standards have to say about HOW students demonstrate their understanding of fractions?  On your standards, highlight the phrase “using (a) visual fraction model(s)” everywhere you see it.

10.  Halves, fourths, and eighths  Thirds and sixths  Fractions of a set In grade 4, expectations are limited to fractions with denominators ….?

11.  At your table… ◦ Create 4 DIFFERENET representations of ¼ of a sandwich. (On the left side of your poster).

12.  How do you know this is ¼?  How could you PROVE it?

13.  How do you know these fourths are equal?

14.  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. “If they don’t look the same, they aren’t equal” COMMON MISCONCEPTION

15.  If the blue is ¼, then what is the white? Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF. 3. a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. b. 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 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

16.  Where is the opportunity to be mindful about standards 4.NF.3 a and b?

17.  At your table… ◦ Create 4 DIFFERENET representations of 1 / 8 of a sandwich.

19.  Using fourths to find eighths

21. Chart: Fractions That are Equal  Look at the bottom of page 34: Discussion- How are Thirds and Sixths Related? Read to the bottom of page 35.  What is the math focus for discussion?  How is the idea of equivalent fractions introduced?

22. Chart: Fractions That are Equal  Look at the bottom of page 34: Discussion- How are Thirds and Sixths Related? Read to the bottom of page 35. If you skipped this discussion, what standard would students miss? 4.NF. 1. 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.

23.  I have a crate of 24 oranges. ¼ go to Mr. Freed. The rest go to Ms. Lee.  What fraction of the oranges will Ms. Lee get?  How many oranges will Mr. Freed get?  How many oranges will Ms. Lee get?

24.  Look at the 3 possible student responses on page 39. Which student best illustrates this standard?  Why?

25. ¼ is greater than ½ ??????

26.  Is 2/3 of the area of a 5 X 12 rectangle more than 2/3 of the area of the 4 X 6 rectangle?  Math Note: p.42

28. How could students prove whether the following equation is true or false? + + + = 1 But I thought students didn’t have to add with unlike denominators! Read teacher note p. 56

29.  Bring some student examples of SAB 14

30.  Students must find common denominators to add fractions.  Students in 4 th grade only add and subtract with common denominators.

31. 1/2

33. Students develop understanding of fraction equivalence and operations with fractions. They recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent fractions. Students extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number.

35. NAES: Smithfield  Monday, Dec. 1------------- Investigation 1 and Fraction Cards  Thursday, January 10-------- Investigation 2 and Multiplying Fractions  Thursday, January 31-------- Decimals  Tuesday, Dec. 18 --------- Investigation 1 and Fraction Cards  Tuesday, January 8-------- Investigation 2 and Multiplying Fractions  Tuesday January 29 ------- Decimals

36. Extend understanding of fraction equivalence and ordering. 4.NF. 1. 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. 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 1/2. 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. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF. 3. Understand a fraction a / b with a > 1 as a sum of fractions 1/ b. a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. b. 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 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. c. 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. d. 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. 4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. a. Understand a fraction a / b as a multiple of 1/ b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4). b. Understand a multiple of a/b as a multiple of 1/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 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) c. 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? Understand decimal notation for fractions, and compare decimal fractions. 4.NF. 5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. 2 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. 4.NF. 6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. 4.NF. 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. _________________  1 Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, 100.  2 Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.