# Progressions for the Common Core State Standards in Mathematics (draft)

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1 Progressions for the Common Core State Standards in Mathematics (draft) c The Common Core Standards Writing Team 29 May

2 K, Counting and Cardinality; K 5, Operations and Algebraic Thinking Counting and Cardinality and Operations and Algebraic Thinking are about understanding and using numbers. Counting and Cardinality underlies Operations and Algebraic Thinking as well as Number and Operations in Base Ten. It begins with early counting and telling how many in one group of objects. Addition, subtraction, multiplication, and division grow from these early roots. From its very beginnings, this Progression involves important ideas that are neither trivial nor obvious; these ideas need to be taught, in ways that are interesting and engaging to young students. The Progression in Operations and Algebraic Thinking deals with the basic operations the kinds of quantitative relationships they model and consequently the kinds of problems they can be used to solve as well as their mathematical properties and relationships. Although most of the standards organized under the OA heading involve whole numbers, the importance of the Progression is much more general because it describes concepts, properties, and representations that extend to other number systems, to measures, and to algebra. For example, if the mass of the sun is x kilograms, and the mass of the rest of the solar system is y kilograms, then the mass of the solar system as a whole is the sum x y kilograms. In this example of additive reasoning, it doesn t matter whether x and y are whole numbers, fractions, decimals, or even variables. Likewise, a property such as distributivity holds for all the number systems that students will study in K 12, including complex numbers. The generality of the concepts involved in Operations and Algebraic Thinking means that students work in this area should be designed to help them extend arithmetic beyond whole numbers (see the NF and NBT Progressions) and understand and apply expressions and equations in later grades (see the EE Progression). Addition and subtraction are the first operations studied. Ini- 2

7 7 Table 1: Addition and subtraction situations Result Unknown Change Unknown Start Unknown Add To A bunnies sat on the grass. B more bunnies hopped there. How many bunnies are on the grass now? A B l A bunnies were sitting on the grass. Some more bunnies hopped there. Then there were C bunnies. How many bunnies hopped over to the first A bunnies? A l C Some bunnies were sitting on the grass. B more bunnies hopped there. Then there were C bunnies. How many bunnies were on the grass before? l B C Take From C apples were on the table. I ate B apples. How many apples are on the table now? C B l C apples were on the table. I ate some apples. Then there were A apples. How many apples did I eat? C l A Some apples were on the table. I ate B apples. Then there were A apples. How many apples were on the table before? l B A Total Unknown Both Addends Unknown 1 Addend Unknown 2 Put Together /Take Apart A red apples and B green apples are on the table. How many apples are on the table? A B l Grandma has C flowers. How many can she put in her red vase and how many in her blue vase? C l l C apples are on the table. A are red and the rest are green. How many apples are green? A l C C A l Difference Unknown Bigger Unknown Smaller Unknown Compare How many more? version. Lucy has A apples. Julie has C apples. How many more apples does Julie have than Lucy? How many fewer? version. Lucy has A apples. Julie has C apples. How many fewer apples does Lucy have than Julie? A l C C A l More version suggests operation. Julie has B more apples than Lucy. Lucy has A apples. How many apples does Julie have? Fewer version suggests wrong operation. Lucy has B fewer apples than Julie. Lucy has A apples. How many apples does Julie have? A B l Fewer version suggests operation. Lucy has B fewer apples than Julie. Julie has C apples. How many apples does Lucy have? More suggests wrong operation. Julie has B more apples than Lucy. Julie has C apples. How many apples does Lucy have? C B l l B C In each type (shown as a row), any one of the three quantities in the situation can be unknown, leading to the subtypes shown in each cell of the table. The table also shows some important language variants which, while mathematically the same, require separate attention. Other descriptions of the situations may use somewhat different names. Adapted from CCSS, p. 88, which is based on Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity, National Research Council, 2009, pp This can be used to show all decompositions of a given number, especially important for numbers within 10. Equations with totals on the left help children understand that = does not always mean makes or results in but always means is the same number as. Such problems are not a problem subtype with one unknown, as is the Addend Unknown subtype to the right. These problems are a productive variation with two unknowns that give experience with finding all of the decompositions of a number and reflecting on the patterns involved. 2 Either addend can be unknown; both variations should be included.

11 11 MP1 Understand the approaches of others and identify corre- as groupings, things crossed out, numbers labeling parts or totals, and letters or words labeling aspects of the situation. The symbols,, or may be in the drawing. Students should be encouraged to explain their drawings and discuss how different drawings are the same and different. MP1 spondences Later in the year, students solve addition and subtraction equations for numbers within 5, for example, 2 1 l or 3 1 l, while still connecting these equations to situations verbally or with drawings. Experience with decompositions of numbers and with Add To and Take From situations enables students to begin to fluently add and subtract within 5. K.OA.5 K.OA.5 Fluently add and subtract within 5. Finally, composing and decomposing numbers from 11 to 19 into ten ones and some further ones builds from all this work. K.NBT.1 This is a vital first step kindergarteners must take toward understanding base-ten notation for numbers greater than 9. (See the NBT Progression.) The Kindergarten standards can be stated succinctly, but they represent a great deal of focused and rich interactions in the classroom. This is necessary in order to enable all students to understand all of the numbers and concepts involved. Students who enter Kindergarten without knowledge of small numbers or of counting to ten will require extra teaching time in Kindergarten to meet the standards. Such time and support are vital for enabling all students to master the Grade 1 standards in Grade 1. 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 (e.g., 18 = ); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

17 17 for all of the kinds of numbers students encounter in later grades (multi-digit whole numbers, fractions, decimals, variables). Students can also continue to represent any situation with a situation equation and connect such equations to diagrams. MP1 Such connections can help students to solve the more difficult problem situation subtypes by understanding where the totals and addends are in the equation and rewriting the equation as needed. MP1 By relating equations and diagrams, students work toward this aspect of MP1: Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs.

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