2. By Liza Dallavalle and Michele Ziegler, 2015 Math Resource Teachers for Carroll County Public Schools

6. - partitioning standards in first, second and third grade build the foundation for understanding fractions as numbers.

7. Standard for Mathematical Practice Construct viable arguments and critique the reasoning of others.

15. * Think about your grade level’s Geometry partitioning standard. * In your own words describe what this standard entails.

16. CCSS.Math.Content.1.G.A.3 CCSS.Math.Content.1.G.A.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. CCSS.Math.Content.2.G.A.3 CCSS.Math.Content.2.G.A.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. CCSS.Math.Content.3.G.A.2 CCSS.Math.Content.3.G.A.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.

18. * Read each problem. * Solve and discuss. * Place in order of progression for teaching that builds on students’ understanding of whole number quantities and informal partitioning strategies.

19. First Grade CCSS.Math.Content.1.G.A.3 CCSS.Math.Content.1.G.A.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. 1 st Grade Big Idea 1. Items can be partitioned into equal shares. 2. More shares of the same whole leads to a smaller fraction for each.

20. Using a Familiar Context Read The Doorbell Rang Record how the cookies are shared on a class chart. Talk to a Shoulder Partner: What do you notice? What happens to the shares as more kids come into the kitchen? Standard for Mathematical Practice Look for and express regularity in repeated reasoning.

22. Students are given manipulatives to model (2, 3, 4, 6, and 12) fair shares as they create a class book. Standard for Mathematical Practice Look for and express regularity in repeated reasoning.

23. When creating the book page, the students used wooden blocks to represent the set of items that were equally partitioned. Standard for Mathematical Practice Model with mathematics.

26. Standards for Mathematical Practice Construct viable arguments and critique the reasoning of others. Look for and express regularity in repeated reasoning.

27. Think about the Big Ideas addressed by the learning opportunities we have shared. Talk About It Which three situations best highlight the progression of equal shares in first grade. Be prepared to support your decision.

28. First Grade - All objects MUST be shared and shared equally: 15 grapes shared with 3 children - Relate their understanding of whole numbers to fractions and find them easier to solve than solutions that are less than one: Two children share 5 apples (deal and continue to deal) - 2, 4, 8 still capitalize on halving strategies, just a little harder because it is less than one: Four children and one pie

30. Develop precision of vocabulary related to partitioning by beginning with what they know. Use a variety of manipulatives in familiar contexts to represent equal shares. Connect the representations to clarify language. Standards for Mathematical Practice Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.

31. Connect language to prior knowledge Use models to clarify the language Connect models with prior knowledge

32. Fold shapes to model halves/fourths Create models using different manipulatives Standard for Mathematical Practice Reason abstractly and quantitatively.

33. Standard for Mathematical Practice Model with mathematics.

34. Each group was given a predetermined set of “dishes” (aka cubes). Standard for Mathematical Practice Model with mathematics.

35. 1.G.2. Compose two-dimensional shapes…to create a composite shape… * What if this piece represented half of a cookie? What would the whole cookie look like? * What if this piece represented a quarter of the cookie? What would the whole cookie look like? - Give students a set of different sized pieces and ask them to compose the cookie. - Give students a half/quarter of a circle to trace and the students sketch the other half/quarters. 1.MD.3 Tell and write time in hours and half-hours using analog and digital clocks. * When is the hour hand half way around the clock? - Clock face fold it in half * If an hour is when the hour hand makes a full rotation around the clock, when does the hand on a clock show half of an hour? - Are there other ways to show half an hour?

36. Second Grade CCSS.Math.Content.2.G.A.3 CCSS.Math.Content.2.G.A.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. 2nd Grade Big Ideas 1. Recognize equal shares. 2. Check for identical measures rather than identical shapes. 3. Notice that the size of the pieces relates to the size of the whole.

37. Equal Shares vs Not Equal Shares Sort It Out Standards for Mathematical Practice Attend to precision. Look for and make use of structure.

41. Dividing Cakes “Each group will divide rectangular cakes so that groups of 2 – 10 people will have equal portions.” “Draw lines on each cake to partition it into equal shares for different numbers of people. You may want to fold the paper first. Also be prepared to explain for each of your cakes why you think the shares are equal.” Standards for Mathematical Practice Model with mathematics. Look for and make use of structure.

42. * Sort your cakes into “Easy to Partition” vs “Harder to Partition” - Discuss why. * For each of your cakes, how did you know the shares were equal?

43. * “I’ve bought some apples for you. I am going to put you into groups of three. Then you’ll talk to your group about how you might share the apple equally, so each get the same amount.” * “ After you share your ideas, agree on one plan that you think would work. Then write down the plan. You may want to include drawings and words.” * “When you show me your plan, I’ll follow it and cut your apple.” * “Also I would like you to write how much each person gets.” Each person gets ___. Standard for Mathematical Practice Construct viable arguments and critique the reasoning of others.

44. * What insight in student thinking do these “plans” provide? - What do the students know? - What misconceptions might they have?

45. Think about the Big Ideas addressed by the learning opportunities we have shared (2 nd grade). Talk About It Which two situations best highlight the progression of equal shares in second grade. Be prepared to support your decision.

46. Second Grade - Children have to anticipate how to slice or cut the objects. Many children will attempt to use a repeated halving strategy or trial and error. Moving to 3 will force children to confront their tendencies to use halving strategies.: 3 children share 10 sticks of clay (deal and continue to deal) - Same as above just less than one: 3 children share 1 candy bar

47. Take a look at the remaining two situations. Talk About It Which situation do you think would come first? What makes these situations more challenging than the others presented to first and second grade students?

48. Third Grade - Less than one, but not a unit fraction: Four children share 3 cookies - Less than one, not a unit fraction, with thirds: Three children share 2 pizzas

49. Which brownie do you think is the largest? Would you have more brownie to eat if you selected that one? Which brownie is the smallest? Would you have less brownie to eat if you selected that one? Are any brownies equal in size? Standards for Mathematical Practice Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Brownies

52. Cutting the Cake The baker wants to cut his square cake into fourths a different way every day, how many ways can he cut his cake? Standard for Mathematical Practice Construct viable arguments and critique the reasoning of others.

55. * Mingle. * Compare the shaded portion of your square. * Record your findings. * Mingle and repeat. * Make a group of all the people with the same share of the square shaded. * Prove it! Standard for Mathematical Practice Construct viable arguments and critique the reasoning of others.

58. Ring Around the Halves Standard for Mathematical Practice Construct viable arguments and critique the reasoning of others.

59. Standard for Mathematical Practice Construct viable arguments and critique the reasoning of others.

61. 2.OA.3 Determine whether a group of objects has an even or odd number of members… 2.MD.2 Measure the length of an object twice, using length units of different lengths for the two measurements. Describe how the two measurements relate to the measurement chosen. 2.MD.7 Tell and write time…to the nearest five minutes. 2.MD.8 Solve word problems involving money. 2.MD.9 Show the measurements by making a line plot 2.G.2 Partition a rectangle into rows and columns of same size squares…

62. 2.OA.3 Determine whether a group of objects has an even or odd number of members… -partition a set into two equal groups or into groups of 2 2.MD.2 Measure the length of an object twice, using length units of different lengths for the two measurements. Describe how the two measurements relate to the measurement chosen. - the smaller the unit the more of the units needed - the more you partition the smaller the piece 2.MD.7 Tell and write time…to the nearest five minutes. - partition the clock in half and quarter hours - partition an hour into 60 minutes or 12 groups of 5 minutes 2.MD.8 Solve word problems involving money. - understanding units of pennies, nickels, dimes, and quarters - the more you partition the smaller the amount 2.MD.9 Show the measurements by making a line plot -partitioning the number line into equal units of 1 2.G.2 Partition a rectangle into rows and columns of same size squares… - partitioning

63. * CCSS.Math.Content.3.G.A.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape. CCSS.Math.Content.3.G.A.2 3rd Grade Big Idea 1. Fractions are numbers.

64. Standards for Mathematical Practice Look for and make use of structure. Look for and express regularity in repeated reasoning.

65. 1. Items can be partitioned into equal shares. 2. More shares of the same whole leads to a smaller fraction for each. 3. Checking for identical measures rather than identical shapes. 4. Noticing that the size of the pieces relates to the size of the whole. 5. Fractions as numbers.

67. 3 rd Grade: CCSS.Math.Content.3.NF.1 through 3 * Develop understanding of fractions as numbers. 4 th Grade: CCSS.Math.Content.4.NF.1 through 7 * Extend understanding of fraction equivalence and ordering. * Build fractions from unit fractions by applying and extending previous understandings of operations and whole numbers. * Understand decimal notation for fractions, and compare decimal fractions. 5 th Grade: CCSS.Math.Content.5.NF.1 through 7 * Use equivalent fractions as a strategy to add and subtract fractions. * Apply and extend previous understanding of multiplication and division to multiply and divide fractions.

75. Please feel free to contact us at the following: Michele Ziegler- maziegl@carrollk12.orgmaziegl@carrollk12.org Liza Dallavalle- eadalla@carrollk12.orgeadalla@carrollk12.org

76. * Meghan Engle and Shelley Frock (first grade teachers from Charles Carroll Elementary School), Sandy Johnson and Adelle King (second grade teachers from Linton Springs Elementary School), Shelly Boblits (second grade teacher from Spring Garden Elementary School ), Ingrid Hiltz (first grade teacher from Spring Garden Elementary School) for contributing to this presentation by implementing the lessons, collecting work samples, providing feedback, and submitting photos.