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Stage 1: the Number Sequence 1 stage 1: the number sequence Big Idea: Quantity Quantity is the big idea that describes amounts, or sizes. It is a fundamental idea that refers to quantitative properties; the size of things (magnitude), and the number of things (multitude). Why is Quantity Important? Quantity means that numbers represent amounts. If students do not possess an understanding of Quantity, their knowledge of foundational mathematics will be undermined. Understanding Quantity helps students develop number conceptualization. In or- der for children to understand quantity, they need foundational experiences with counting, identifying numbers, sequencing, and comparing. Counting, and using numerals to quantify collections, form the developmental progression of experiences in Stage 1. Children who understand number concepts know that numbers are used to describe quantities and relationships’ between quantities. For example, the sequence of numbers is determined by each number’s magnitude, a concept that not all children understand. Without this underpinning of understanding, a child may perform rote responses, which will not stand the test of further, rigorous application. The developmental progression of experiences in Stage 1 help students actively grow a strong number knowledge base. Stage 1 Learning Progression Concept Standard Example Description Children complete short sequences of visual displays of quantities beginning with 1. Subsequently, the sequence shows gaps which the students need to fill in. The sequencing 1.1: Sequencing K.CC.2 1, 2, 3, 4, ? tasks ask students to show that they have quantity and number names in order of magnitude and to associate quantities with numerals. 1.2: Identifying Find the Students see the visual tool with a numeral beneath it. Students match the displayed K.CC.3 quantity with its corresponding numeral. Then they see a visual tool with a quantity, and Number number ‘3’ choose the numeral that matches the displayed amount. 1.3: Count The student sees the initial position in a sequence missing, then the last. He progresses K.CC.2 4, 5, 6, 7, ? to visual sequential displays which contain 2 missing amounts, which he fills in further Forward showing his understanding of order. 1.4: Count K.CC.2 9, 8, 7, 6, ? Count backward in the same progression of tasks as 1.3, again using numbers Backward Copyright © 2014 Symphony Learning LLC. Permission is granted to reproduce this sheet for classroom use. Teacher Overview 2 stage 2: more / less / same Big Idea: Quantity Quantity is the big idea that describes amounts, or sizes. It is a fundamental idea that refers to quantitative properties; the size of things (magnitude), and the number of things (multitude). Why is Quantity Important? Quantity means that numbers represent amounts. If students do not possess an understanding of Quantity, their knowledge of foundational mathematics will be undermined. Understanding Quantity helps students develop number conceptualization. In order for children to understand quantity, they need foundational experiences with counting, identifying numbers, sequencing, and comparing. Counting, comparing the magnitude of collections, and using numerals to quantify collections, form the developmental progression of experiences in Stage 2. Children who understand number concepts know that numbers are used to describe quantities and relationships’ between quantities. For example, the sequence of numbers is determined by each number’s magnitude, a concept that not all children understand. Without this underpinning of understanding, a child may perform rote responses, which will not stand the test of further, rigorous application. The developmental progression of experiences in Stage 2 helps students actively grow a strong number knowledge base. Stage 2 Learning Progression Concept Standard Example Description Find the next number without counting . Find ‘One More’ first with a visual representation of an amount, and then with a number name, without counting. Amounts are visually displayed, and then, a single numeral is given. The goal is for a student to build on the con- Find ‘1 more’ 2.1: Find ‘1 More’ K.CC.4 cept of order of magnitude, and to understand each successive number in sequence is ‘1 than 3 more than’ the number before; that ‘the next number after equals one more.’ Four means ‘the number that is 1 more than 3.’ Having ‘number-after’ knowledge asks the student to enter the sequence at any point and show next number without having to start at one. This level continues the idea of sequence magnitude by first showing a visual of an amount, and then a number name-not in any sequence. It tasks the student with finding Find ‘1 less’ 2.2: Find ‘1 Less’ K.CC.4 ‘the number that comes before’ or, ‘1 less.’ Having the ‘number before’ knowledge shows than 8 that the student can enter the sequence at any point and name ‘the number before.’ 2.1 and 2.2 introduce number relationships as the next learning progression after number size. Students apply their small-number recognition, their understanding of order, counting, quantity size, to a comparison of collections within the 0-9 range. The tasks allows students to use counting and ordering knowledge in a practical manner, and develop a Find more 2.3: Find More K.CC.6 concrete understanding of numerical relationships. A number farther along a number line than 3 represents a larger quantity. A collection of dots labeled ‘5’ is more than the collection labeled ‘2.’ A taller bar is more than a shorter bar; longer becomes equated with more in the bar context This level continues the tasks and skills begun in 1.7, as needed in the concept of finding Find less 2.4: Find Less K.CC.7 less, fewer, and shorter. As in 1.7, the different visual environments give breadth to using than 3 different comparative words. Continues a child’s intuitive understanding of relative amounts of more and less, and Find a number 2.5: Find Same K.CC.7 extends it to that of ‘same.’ The words used to describe the relationships move from “the the same as 4 same as” to, ‘equals’, with the use of the equals symbol to accompany number names. Copyright © 2014 Symphony Learning LLC. Permission is granted to reproduce this sheet for classroom use. Teacher Overview stage 3: 3 addition & subtraction to 5 Big Idea: Parts to Whole Parts-to-Whole is the big idea that underlies addition and subtraction. The central concept is that there is a whole that can be partitioned into a certain number of parts. If we combine the parts, they equal the whole. If the whole is 8, the parts could be 6 and 2. Combine the two parts (6 + 2), they equal the whole (8). We can change the order of the parts (2 + 6) and they still equal the whole. We can also find several different ways of making a whole (8) out of two parts, such as 7 + 1 or 3 + 5, or three parts, such as 4 + 3 + 1. A part can be taken away from the whole leaving another part left over. The whole is 8, we take away 5, 3 is left over. A student that has developed in-depth understanding of the Parts-to-Whole big idea can see addition and subtraction as different ways of forming number relationships, often called “fact families,” or, related facts. Why is Parts to Whole Important? Understanding how numbers are related to each other signals that children are ready to experience that each number is more than a distinct character; larger numbers, or wholes, are made up of smaller numbers, or parts. When the student sees the iconic 5 dots on a number card, combined with an additional 2 dots, she can count or add on and know there are 7 dots in total. The 5-length bar with a 2-length bar added on takes on the length that is the same as the 7-bar. Two jumps on the number line past the 5 mark, is the same as 2 numbers past 5, which in turn shows that 7 is two more than 5. Children begin by changing a small collection of dots, or bars, or number line jumps, to a larger amount by virtue of more dots, longer bars, or end points farther along the number line. Stage 3 Learning Progression Concept Standard Example Description 3.1 introduces Parts-to-Whole first using visual tools, and then by bridging to symbols. Equations appear for the first time in the program, as well as an extension of the equal 3.1: sign, first seen in Stage 2.5. Number equations show the meaning of the + symbol, and how the = symbol is used in mathematics. The placement of the = symbol is both at K.OA.1 1 + 2 = ? Addition: Missing the beginning and the “end” of so that students become familiar and flexible with both Result to 5 appearances while combining parts to make a new whole.The missing result comes after combining parts, or appears before parts are combined. Two sides of an equation are equal when the = sign is used. Still keeping the numbers small, students are given the whole, and need to find what parts 3.2: combine with another to make that whole. In 3.2, they see how numbers can be broken apart in a variety of ways. They encounter the idea that order doesn’t matter. And they K.OA.2 1 + ? = 3 Addition: Missing that the quantity remains the same even when parts are broken up differently. Students Change to 5 encounter the same progression of environments offered in 3.1 with the results being reinforcement of part-part-whole and more fluency with the pairings to 5. Copyright © 2014 Symphony Learning LLC. Permission is granted to reproduce this sheet for classroom use.
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