AISJ Mathematics Scope & Sequence Grades PreK /14

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1 The Scope and Sequence document represents an articulation of what students should know and be able to do. The document supports teachers in knowing how to help students achieve the goals of the standards and understanding each standard conceptually. It should be used as a tool to assist in planning and implementing a high quality instructional program. The Sequence of Units provides a snapshot of the recommended pacing of instruction across a year. The unpacking section contains rich information and examples of what the standard means; this section is an essential component to help both teachers and students understand the standards. The progressions provides valuable information for pre assessment as well as information on what follows. Sequence of Units for Pre- Kindergarten Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit Me and My AISJ Community Caring for Living Things Celebrations Imagination How Things Work Integrated Stand Alone CC 1.1 a CC 1.1 b CC 1.1 c MD 2 a MD 2 d CC 1.1 a OA 1.3 a MD 2 a MD 2 d CC 1.1 a CC 1.1 b CC 1.1 c G 3 a G 3 b OA 1.3 a MD 2 d MD 2 a OA 1.3 a MD 2 d MD 2 a 1

2 Sequence of Units for Kindergarten 2

3 Sequence of Units for Grade 1 3

4 Mathematical Standards Standard 1: Number and Algebra: a) Counting and Cardinality: Learners understand that numbers are a naming system. (CC) b) Numbers Base Ten: Learners understand that the base ten place value system is used to represent numbers and number relationships. (NBT) c) Operational Thinking and Algebra: Learners understand that numbers and algebra represent and quantify our world and can be used to solve problems. (OA) d) Number and Operations Fractions: Learners understand that fractions and decimals are ways of representing whole- part relationships. (NF) Standard 2: Measurement and Data: Learners understand that objects and events have attributes that can be measured and compared using appropriate tools. Standard 3: Geometry: Learners understand that geometry models and quantifies structures in our world and can be used to solve problems. Standard 1: Number and Algebra: a) Counting and Cardinality: Learners understand that numbers are a naming system. Benchmarks Know number names and the count sequence. Grade Level Expectations Grade PreK Grade K Grade 1 PK.CC.1 Count verbally to 10 by K.CC.1 Count to 100 by ones and ones. by tens. PK.CC.2 Recognize the concept of just after or just before a given number in the counting sequence up to 10. K.CC.2 Count forward beginning from a given number within the known sequence (instead of having to begin at 1). PK.CC.3 Identify written numerals K.CC.3 Write numbers from 0 to 20. Represent a number of objects 4

5 with a written numeral 0-20 (with 0 representing a count of no objects). Count to tell the number of objects. PK.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 Ability to apply the strategies of touching objects as they are counted and by organizing the objects in a row Knowledge of and ability to apply one- to- one correspondence when counting. b) Recognize that the last number name said tells the number of objects counted. c) Recognize that each successive number name refers to a quantity that is one larger. 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. 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. d) Develop understanding of ordinal numbers (first through tenth) to describe the relative position and magnitude of whole numbers. 5

6 Compare numbers. PK.CC.5 Represent a number (0-5, then to 10 by producing a set of objects with concrete materials, pictures, and/or numerals (with 0 representing a count of no objects). PK.CC.6 Recognize the number of objects in a set without counting (Subitizing 0-5 objects) PK.CC.7 Explore relationships by comparing groups of objects up to 10, to determine greater than/more or less than, and equal to/same 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 (includes groups with up to 5 objects). K.CC.5 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. 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. K.CC.7 Compare two numbers between 1 and 10 presented as written numerals. Standard 1: Number and Algebra: b) Numbers Base Ten: Learners understand that the base ten place value system is used to represent numbers and number relationships. Work with numbers to gain foundations for place value PK.NBT.1 Investigate the relationship between ten ones and ten 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 6

7 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. Extend the counting sequence Understand place value. Use place value understanding and 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. 1.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. b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, s even, eight, or nine ones. 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). 1.NBT.3 Compare two two- digit numbers based on meanings of the 7

8 properties of operations to add and subtract. tens and ones digits, recording the results of comparisons with the symbols >, =, and <. Use place value understanding and properties of operations to add and subtract. 1.NBT.4 Add within 100, including adding a two- digit number and a one- digit number, and adding a two- digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two- digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten. 1.NBT.5 Given a two- digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used. 1.NBT.6 Subtract multiples of 10 in the range from multiples of 10 in the range (positive or 8

9 zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Standard 1: Number and Algebra: c) Operational Thinking and Algebra: Learners understand that numbers and algebra represent and quantify our world and can be used to solve problems. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from. PK.OA.1 Explore addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, or verbal explanations. PK.OA.2 Decompose quantity (less than or equal to 5, then to 10) into pairs in more than one way (e.g., by using objects or drawings). PK.OA.3 For any given quantity from (0 to 5, then to 10) find the quantity that must be added to make 5, then to 10, e.g. by using objects or drawings. K.OA.1 Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. K.OA.2 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. 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 9

10 or equation (e.g., 5 = and 5 = 4 + 1). K.OA.4 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 5. Represent and solve problems involving addition and subtraction. 1.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. 1.OA.2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. 10

11 Understand and apply properties of operations and the relationships between addition and subtraction. 1.OA.3 Apply properties of operations as strategies to add and subtract. Example: if = 11 is known, then =11 is also known. (Commutative property of addition.) To add , the second two numbers can be added to make a ten, so = = 12. (Associative property of addition.) 1.OA.4 Understand subtraction as an unknown- addend problem. For example, subtract 10 8 by finding the number that makes 10 when added to 8. Add and subtract within OA.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). 1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., = = = 14); decomposing a number leading to a ten (e.g., 13 4 = = 10 1 = 9); using the relationship between addition and subtraction (e.g., knowing that = 12, one knows 12 8 = 4); 11

12 and creating equivalent but easier or known sums (e.g., adding by creating the known equivalent = = 13). Work with addition and subtraction equations. 1.OA.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 1, = 2 + 5, = OA.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from. PK.OA.1 Explore addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, or verbal explanations. PK.OA.2 Decompose quantity (less than or equal to 5, then to 10) into pairs in more than one way (e.g., by using objects or drawings). K.OA.1 Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. K.OA.2 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by 12

13 PK.OA.3 For any given quantity from (0 to 5, then to 10) find the quantity that must be added to make 5, then to 10, e.g. by using objects or drawings. using objects or drawings to represent the problem. 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). K.OA.4 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. Represent and solve problems involving addition and subtraction. K.OA.5 Fluently add and subtract within 5. 1.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. 1.OA.2 Solve word problems that call for addition of three whole 13

14 numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Understand and apply properties of operations and the relationships between addition and subtraction. 1.OA.3 Apply properties of operations as strategies to add and subtract. Example: if = 11 is known, then =11 is also known. (Commutative property of addition.) To add , the second two numbers can be added to make a ten, so = = 12. (Associative property of addition.) 1.OA.4 Understand subtraction as an unknown- addend problem. For example, subtract 10 8 by finding the number that makes 10 when added to 8. Add and subtract within OA.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). 1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., = = 14

15 = 14); decomposing a number leading to a ten (e.g., 13 4 = = 10 1 = 9); using the relationship between addition and subtraction (e.g., knowing that = 12, one knows 12 8 = 4); and creating equivalent but easier or known sums (e.g., adding by creating the known equivalent = = 13). Work with addition and subtraction equations. 1.OA.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 1, = 2 + 5, = OA.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation Standard 2: Measurement and Data: Learners understand that objects and events have attributes that can be measured and compared using appropriate tools. 15

16 Describe and compare measurable attributes. PK.MD.1 Describe measurable attributes of objects, such as length or weight. PK.MD.2 Directly compare two objects with a measurable attribute in common, using words such as longer/shorter; heavier/lighter; or taller/shorter. K.MD.1 Describe measureable 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 measureable attribute in common, to see which object has more of / less of the attribute, and describe the difference. For examples, directly compare the heights of two children and describe one child as taller/shorter Measure lengths indirectly and by iterating length units. 1.MD.1 Order three objects by length; compare the lengths of two objects indirectly by using a third object. 1.MD.2 Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same- size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of 16

17 length units with no gaps or overlaps Tell and write time and money. 1.MD.3 Tell and write time in hours and half- hours using analog and digital clocks. Recognize and identify Sth African currency, their names, and their value. Sorts objects and counts the number of objects in each category. PK.MD.3 Sort objects into given categories. PK.MD.4 Compare categories using words such as greater than/more, less than, and equal to/same. K.MD.3 Classify objects into given categories: count the number of objects in each category and sort the categories by count. Represent and interpret data. 1.MD.4 Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another. Standard 3: Geometry: Learners understand that geometry models and quantifies structures in our world and can be used to solve problems. Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, PK.G.1 Match like (congruent and similar) shapes. K.G.1 Describe objects in the environment using names of shapes, and describe the relative positions of these objects using 17

18 cubes, cones, cylinders, and spheres). Analyze, compare, create, and compose shape PK.G.2 Group the shapes by attributes. PK.G.3 Correctly name shapes (regardless of their orientations or overall size). PK.G.4 Match and sort shapes. PK.G.5 Describe three- dimensional objects using attributes. PK.G.6 Compose and describe structures using three- dimensional shapes. Descriptions may include shape attributes, relative position, etc. terms such as above, below, beside, in front of, behind, and next to. K.G.2 Correctly name shapes regardless of their orientations or overall size. K.G.3 Identify shapes as two- dimensional (lying in a plane, flat ) or three- dimensional ( solid ). Analyze, compare, create, and compose shapes. 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). K.G.5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes. K.G.6 Compose simple shapes to form larger shapes. For example, Can you join these two triangles 18

19 with full sides touching to make a rectangle? 19

20 Reason with shapes and their attributes. 1.G.1 Distinguish between defining attributes (e.g., triangles are closed and three- sided) versus non- defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes. 1.G.2 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 shapes. 1 1 Students do not need to learn formal names such as right rectangular prism. 1.G.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 20

21 into more equal shares creates smaller shares. Appendix 1 Table 1 Common addition and subtraction situations Add to Take from Put Together/ Take Apart 3 Compare 4 Result Unknown Change Unknown Start Unknown Two bunnies sat on the grass. Three more Two bunnies were sitting on the grass. Some Some bunnies were sitting on the grass. Three bunnies hopped there. How many bunnies more bunnies hopped there. Then there more bunnies there. Then there were five are on the grass now? were five bunnies. How many bunnies bunnies. How many bunnies were on the grass =? hopped over to the first two 2 +? = 5 before?? + 3 = 5 (Gr 2) (K) (Gr 1) Five apples were on the table, I ate two apples. How many apples are on the table now? 5 2 =? (K) Five apples were on the table, I ate some apples. Then there were three apples. How many apples did I eat? 5 -? = 3 (Gr 1) Some apples were on the table I ate two apples. Then there were three apples. How many apples were on the table before?? 2 = 3 (Gr 2) Total Unknown Added Unknown Both Addends Unknown Three red apples and two green apples are on the table. How many apples are on the table? Five apples are on the table. Three are red and the rest are green. How many apples are green? Grandma has five flowers. How many can she put in her red vase and how many in her blue vase? =? 3 +? = 5, 5 3 =? (K) 5 = 0 + 5, 5 = = 1 + 4, 5 = (K) 5 = 2 + 3, 5 = (Gr 1) Difference Unknown Bigger Unknown Smaller Unknown ( How many more? version): Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy? (Gr 1) (Version with more ): Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have? (Gr 1) (Version with more ): Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have? (Gr 2) ( How many fewer? version): Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Lucy? 2 +? 5, 5 2 =? (Gr 1) (Version with fewer ): Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have? ?,? + 2 -? (Gr 1) (Version with fewer ): Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have? 5 3 =?? + 3 = 5 (Gr 2) 21

22 These take apart situations can be used to show all the decomposition of a given number. The associated equations, which have the total on the left of the equal sign, help children understand that the sign does not always mean makes or results in but always does mean is the same number as. Either addend can be unknown, so there are three variations of these problem situations. Both addends Unknown is a productive extension of this basic situation especially for small numbers less than or equal to 10. For the Bigger Unknown or Smaller Unknown situations, one version directs the correct operation (the version using more for the bigger unknown and using less for the smaller unknown). The other versions are more difficult. Adapted from Box 2 4 of Mathematics Learners in Early Childhood, National Research Council (2009. Pp ). Table 2 Common multiplication and division situation (Grades 3-5) Unknown Product 3 X 6 -? There are 3 bags with 6 plums in each bag. How many plums are there in all? Group Size Unknown ( How many in each group? Division) 3 X? 18, and 18 3 =? If 18 plums are shared equally into 3 bags, then how many plums will be in each bag? Number of Groups Unknown ( How many groups? Division)? x 6 18, and ? If 18 plums are to be packed 6 to a bag, then how many bags are needed? Equal groups Measurement example. You need 3 lengths of string, each 6 inches long. How much string will you need altogether? There are 3 rows of apples with 6 apples in each row. How many apples are there? Measurement example. You have 18 inches of string, which you will cut into 3 equal pieces. How long will each piece of string be? If 18 apples are arranged into 3 equal rows, how many apples will be in each row? Measurement example. You have 18 inches of string, which you will cut into pieces that are 6 inches long. How many pieces of string will you have? If 18 apples are arranged into equal rows of 6 apples, how many rows will there be? Arrays 2 Area 3 Area example. What is the area of a 3cm by 6cm rectangle? A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost? Area example. A rectangle has area 18 square centimeters. If one side is 3cm long, how long is a side next to it? A red hat costs $18 and that is 3 times as much as a blue hat costs. How much does a blue hat cost? Area example. A rectangle has 18 square centimeters. If one side is 6cm long, how long is a side next to it? A red hat costs $18 and a blue hat costs $6. How many times as much does the red hat cost as the blue hat? Compare Measurement example. A rubber band is 6cm long. How long will the rubber band be when it is stretched to be 3 times as long? Measurement Example: A rubber band is stretched to be 18cm long and that is 3 times as long as it was at first. How long was the rubber band at first? Measurement example. A rubber band was 6cm long at first. Now it is stretched to be 18cm long. How many times as long is the rubber band now as it was at first? General a x b =? a x? = p, and p a =?? x b = p, and p b =? 22

23 The language in the array examples shows the easiest form of array problems. A harder form is to use the terms rows and columns. The apples in the grocery window are in 3 rows and 6 columns. How many apples are in there? Both forms are valuable. Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array problems include these especially important measurement situations. The first examples in each cell are examples of discrete things. These are easier for students and should be given before the measurement examples. 23