# A Story of Units: A Curriculum Overview for Grades P 5

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4 Sequence of Pre-Kindergarten Modules Aligned with the Standards Module 1: Counting to 5 Module 2: Shapes Module 3: Counting to 10 Module 4: Comparison of Length, Weight, Capacity, and Numbers to 5 Module 5: Addition and Subtraction Stories and Counting to 20 Pre-Kindergarten mathematics focuses on 1. developing an understanding of whole numbers using concrete materials, including concepts of one-to-one correspondence, counting, cardinality, and comparison. 2. describing shapes in their environment. Rationale for Module Sequence in Pre-Kindergarten A Story of Units begins! Beans, pennies, apples, and cubes are some of the units, the objects being counted, that students playfully manipulate during their Pre-Kindergarten year. Lessons are planned from simple to complex so that, by the end of the year, students use their math words to communicate precise observations such as, 3 bananas and 2 bananas make 5 bananas. Notice how this sets the foundation for later mathematics using, for example, place value and fractional units (e.g., 3 hundreds and 2 hundreds make 5 hundreds and 3 fourths and 2 fourths make 5 fourths ). The story kicks off in Module 1 with analyzing, sorting, and making a group sharing an attribute (e.g., things that have wheels). Students count the number of objects in the group in different configurations, match those quantities to numerals, and finally, construct a number stair by placing 1 more cube and verbalizing, 3. One more is 4. Module 1 works exclusively with numbers 1, 2, 3, 4, and 5. In Module 2, students also analyze their world, this time sorting, comparing, re-positioning, identifying, and constructing two-dimensional shapes. They then apply their observations to three-dimensional shapes. This module is carefully positioned between Modules 1 and 3 to allow students time to develop their fluency with numbers to 5 before moving on to the larger numbers of Module 3. 4

5 Module 3 builds the numbers 6 10 in relationship to 5. Students use their fingers to support the 5 + n perspective (e.g., 6 is 5 and 1. When I count to 8 on my fingers, I have 5 fingers here and 3 fingers here ). Thus, numbers 6 10 are 5 and some more, making numbers 6 10 manageable. As in Module 1, students get busy manipulating units into different configurations and then counting them in systematic ways. For example, they learn to place 8 objects in a line and count from left to right. They then learn to arrange those same objects into an array and then into a circle, addressing the complexities that the new configurations bring. After eight objects have been under study for a few lessons, the next number is introduced by putting 1 more object. Notice how this sets the foundation for later mathematics using, for example, place value units and units of 8 (e.g., 236 has 3 tens. 1 more ten is = 246. Five eights is more eight is 48. Six eights is 48 ). In Module 4, students recognize the measureable attributes of length, weight, capacity, and number and learn to use the language of comparison (e.g., shorter than and taller than, heavier than and lighter than, empty and full, more and less). This rich module develops the foundational concept of 2 quantities in relationship to each other. Remember, the second chapter in the Eureka Trilogy is A Story of Ratios. An understanding of ratio begins here at the concrete level as students compare. The critical use of the words more than and less than is embedded in a lovely, playful sequence of exploratory experiences while allowing their skills with counting to 10 develop during fluency practice. In Module 5, as in Grades K 5, comparison work flows directly into addition and subtraction. 5 is more than 3 evolves into 5 is 3 and 2. Students act out math stories with up to 5 objects and learn to decontextualize their stories. For example, a student might use his fingers to represent a story about students and say, 3 children were playing. (Shows 3 fingers.) 2 more children came. (Shows 2 more fingers.) 3 and 2 make 5. Five children are playing now. The student represented the children with his fingers, described the situation with a number sentence ( 3 and 2 make 5 ), and then made a statement returning to the story s context. ( Now there are 5 children playing. ) This ability to decontextualize translates in upper grades into the ability to solve word problems by modeling with math drawings such as tape diagram or arrays. The year culminates with acting out story problems in the Children s Math Theater, a celebration of their learning intended for the whole family. 5

6 Alignment Chart 3 Module 1: Counting to 5 5 (45 days) Standards 4 Addressed in Pre-Kindergarten Modules Know number names and the count sequence. PK.CC.1 Count to 20. PK.CC.2 Represent a number of objects with a written numeral 0 5 (with 0 representing a count of no objects). Count to tell the number of objects. 6 PK.CC.3 PK.CC.4 Understand simple patterns. PK.OA.2 Understand the relationship between numbers and quantities to 10; connect counting to cardinality. a. When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. b. Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. c. Understand that each successive number name refers to a quantity that is one larger. Count to answer how many? questions about as many as 10 things arranged in a line, a rectangular array, or a circle, or as many as 5 things in a scattered configuration; given a number from 1 10, count out that many objects. Duplicate and extend (e.g., What comes next?) simple patterns using concrete objects. 3 When a cluster is referred to in this chart without a footnote, the cluster is addressed in its entirety. 4 The Pre-Kindergarten standards originate from the New York State Common Core Learning Standards. 5 In this module, standards work is limited to within 5. 6 Within 5. 6

7 Standards 4 Addressed in Pre-Kindergarten Modules Sort objects and count the number of objects in each category. 7 PK.MD.2 Sort objects into categories; count the numbers of objects in each category. Module 2: Shapes (15 days) Module 3: Counting to 10 (50 days) Identify and describe shapes (squares, circles, triangles, rectangles). PK.G.1 Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as top, bottom, up, down, in front of, behind, over, under, and next to. PK.G.2 Correctly name shapes regardless of size. Analyze, compare, and sort objects. PK.G.3 Analyze, compare, and sort two- and three-dimensional shapes and objects, in different sizes, using informal language to describe their similarities, differences, and other attributes (e.g., color, size, and shape). PK.G.4 Create and build shapes from components (e.g., sticks and clay balls). Know number names and the count sequence. 8 PK.CC.1 Count to 20. Count to tell the number of objects. PK.CC.3 Understand the relationship between numbers and quantities to 10; connect counting to cardinality. a. When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. b. Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. 7 Within 5. 8 PK.CC.2 is addressed in Modules 1 and 5. 7

8 Standards 4 Addressed in Pre-Kindergarten Modules c. Understand that each successive number name refers to a quantity that is one larger. PK.CC.4 Count to answer how many? questions about as many as 10 things arranged in a line, a rectangular array, or a circle, or as many as 5 things in a scattered configuration; given a number from 1 10, count out that many objects. Module 4: Comparison of Length, Weight, Capacity, and Numbers to 5 (35 days) Module 5: Addition and Subtraction Stories and Counting to 20 (35 days) Compare numbers. PK.CC.5 PK.CC.6 Identify whether the number of objects in one group is more, less, greater than, fewer, and/or equal to the number of objects in another group, e.g., by using matching and counting strategies. Identify first and last related to order or position. Describe and compare measurable attributes. PK.MD.1 Identify measurable attributes of objects, such as length, and weight. Describe them using correct vocabulary (e.g., small, big, short, tall, empty, full, heavy, and light). Know number names and the count sequence. PK.CC.1 Count to 20. PK.CC.2 Represent a number of objects with a written numeral 0 5 (with 0 representing a count of no objects). Understand addition as adding to, and understand subtraction as taking from. PK.OA.1 Understand simple patterns. PK.OA.2 Demonstrate an understanding of addition and subtraction by using objects, fingers, and responding to practical situations (e.g., If we have 3 apples and add two more, how many apples do we have all together?). Duplicate and extend (e.g., What comes next?) simple patterns using concrete objects. 8

11 students start to reorient from 5 toward 10 ones with How much more does 7 need to make ten? These final lessons set the stage for Module 5 wherein 10 ones is the structure on which students build the teen numbers. They are also critical foundation standards for Grade 1. Students must know how much a number needs to make ten in order to use the make ten strategy in Grades 1 and 2, shown to be an important route to place value understanding as they master their sums and differences to 20 by the end of Grade 2. In Module 5, after an extended experience of addition and subtraction with totals up to 10, students progress to investigating numbers For example, thirteen beans are decomposed as 10 beans and 3 beans just as 8 beans are decomposed as 5 beans and 3 beans. Students record their decompositions of the teen numbers as equations, 13 = , and start to think, more is 13. As mentioned at the beginning of the story in Grade 1, the unit one is introduced as students learn to think of the teen numbers as 10 ones and some ones. For the first time, one is not an object but rather a noun! Notice how this sets the stage for expanded form in the upper grades (e.g., 36 = , or 13.6 = ). Module 6 rounds out the year with an exploration of shapes. Students build shapes from components, analyze and compare them, and discover that they can be composed of smaller shapes, just as larger numbers are composed of smaller numbers. Alignment Chart 9 Module 1: Numbers to (43 days) Standards Addressed in Kindergarten Modules Know number names and the count sequence. 11 K.CC.3 Write numbers from 0 to 20. Represent a number of objects with a written numeral 0 20 (with 0 representing a count of no objects). Count to tell the number of objects. 12 K.CC.4 Understand the relationship between numbers and quantities; connect counting to cardinality. a. When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. 9 When a cluster is referred to in this chart without a footnote, the cluster is addressed in its entirety. 10 In this module, standards work is limited to within The balance of this cluster is addressed in Module K.CC.4d is addressed in Module 6. 11

12 Standards Addressed in Kindergarten Modules K.CC.5 b. Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. c. Understand that each successive number name refers to a quantity that is one larger. Count to answer how many? questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1 20, count out that many objects. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from. 13 K.OA.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = and 5 = 4 + 1). Classify objects and count the number of objects in each category. K.MD.3 Classify objects into given categories; count the numbers of objects in each category and sort the categories by count. (Limit category counts to be less than or equal to 10.) Module 2: Two-Dimensional and Three- Dimensional Shapes (12 days) Classify objects and count the number of objects in each category. K.MD.3 Classify objects into given categories; count the numbers of objects in each category and sort the categories by count. (Limit category counts to be less than or equal to 10.) Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres). K.G.1 K.G.2 Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to. Correctly name shapes regardless of their orientations or overall size. 13 The balance of this cluster is addressed in Module 4. 12

13 Standards Addressed in Kindergarten Modules K.G.3 Identify shapes as two-dimensional (lying in a plane, flat ) or three-dimensional ( solid ). Analyze, compare, create, and compose shapes. 14 K.G.4 Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/ corners ) and other attributes (e.g., having sides of equal length). Module 3: Comparison of Length, Weight, Capacity, and Numbers to 10 (38 days) Module 4: Number Pairs, Addition and Subtraction to 10 (47 days) Compare numbers. K.CC.6 Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies. (Include groups with up to ten objects.) K.CC.7 Compare two numbers between 1 and 10 presented as written numerals. Describe and compare measurable attributes. K.MD.1 Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object. K.MD.2 Directly compare two objects with a measurable attribute in common, to see which object has more of / less of the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from. K.OA.1 K.OA.2 Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. (Drawings need not show details, but should show the mathematics in the problem.) Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. 14 The balance of this cluster is addressed in Module 6. 13

14 Standards Addressed in Kindergarten Modules K.OA.3 K.OA.4 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = and 5 = 4 + 1). For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings and record the answer with a drawing or equation. K.OA.5 Fluently add and subtract within Module 5: Numbers and Counting to 100 (30 days) Know number names and the count sequence. K.CC.1 K.CC.2 K.CC.3 Count to 100 by ones and by tens. Count forward beginning from a given number within the known sequence (instead of having to begin at 1). Write numbers from 0 to 20. Represent a number of objects with a written numeral 0 20 (with 0 representing a count of no objects). Count to tell the number of objects. 16 K.CC.4 K.CC.5 Understand the relationship between numbers and quantities; connect counting to cardinality. b. Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. c. Understand that each successive number name refers to a quantity that is one larger. Count to answer how many? questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1 20, count out that many objects. 15 From this point forward, fluency practice is part of students on-going experience. 16 K.CC.4a, K.CC.4b, and K.CC.4c are addressed in Module 1; K.CC.4d is addressed in Module 6. 14

15 Standards Addressed in Kindergarten Modules Work with numbers to gain foundations for place value. K.NBT.1 Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings and record each composition or decomposition by a drawing or equation (such as 18 = ); understand that these numbers are composed of ten ones and one, two three, four, five, six, seven, eight or nine ones. Module 6: Analyzing, Comparing, and Composing Shapes (10 days) Count to tell the number of objects. 17 K.CC.4 Understand the relationship between numbers and quantities: connect counting to cardinality. d. Develop understanding of ordinal numbers (first through tenth) to describe the relative position and magnitude of whole numbers. 18 Analyze, compare, create and compose shapes. 19 K.G.5 K.G.6 Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes. Compose simple shapes to form larger shapes. For example, Can you join these two triangles with full sides touching to make a rectangle? 17 Ordinality is introduced in the context of constructing and manipulating shapes. The balance of this cluster is addressed in Modules 1 and K.CC.4d originates from the New York State Common Core Learning Standards and is not part of the CCSS-M. 19 K.G.4 is addressed in Module 2. 15

23 Standards Addressed in Grade 1 Modules 1.G.2 1.G.3 Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. (Students do not need to learn formal names such as right rectangular prism. ) 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. Module 6: Place Value, Comparison, Addition and Subtraction to 100 (35 days) Represent and solve problems involving addition and subtraction OA.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. (See Standards Glossary, Table 1.) Extend the counting sequence. 1.NBT.1 Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral. Understand place value NBT.2 Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases: a. 10 can be thought of as a bundle of ten ones called a ten. c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones). 32 The balance of this cluster is addressed in Module NBT.2b is addressed in Module 2. 23

26 In Module 2, students learn to measure and estimate using standard units for length and solve measurement problems involving addition and subtraction of length, now encountering the word problem situations involving comparison. A major objective is for students to use measurement tools with the understanding that linear measure involves an iteration of units and that the smaller a unit, the more iterations are necessary to cover a given length. Students work exclusively with metric units (e.g., centimeters and meters) in this module to support upcoming work with place value concepts in Module 3. Units also play a central role in the addition and subtraction algorithms of Modules 4 and 5. An underlying goal for this module is for students to learn the meaning of a unit in a different context, that of length. This understanding serves as the foundation of arithmetic, measurement, and geometry in elementary school. Students also solve word problems involving all addition and subtraction comparison situations, so that by the end of Module 2, they have encountered the full set of situations. All arithmetic algorithms are manipulations of place value units: ones, tens, hundreds, etc. In Module 3, students extend their understanding of baseten notation and apply their understanding of place value to count and compare numbers to 1,000. In Grade 2, the place value units move from a proportional model to a non-proportional number disk model (see the pictures below). The place value table with number disks can be used through Grade 5 for modeling very large numbers and decimals, thus providing students greater facility with, and understanding of, mental math and algorithms. Proportional Model for Place Value Non-Proportional Model for Place Value In Module 4, students apply their work with place value units to add and subtract within 200, moving from concrete to pictorial to abstract. This work deepens their understanding of base ten, place value, and the properties of operations. It also challenges them to apply their knowledge to one-step and two-step word problems. During this module, students also continue to develop one of the required fluencies of the grade: addition and subtraction within

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