Avery County & ASU Partnership Project in Mathematics AAPP-Math Grades K-2 Day 1 – 2/18/14 Dr. Tracy Goodson-Espy Dr. Lisa Poling Dr. Art Quickenton

Counting & Cardinality Kindergarten

Counting & Cardinality  Know number names and the count sequence.  Count to tell the number of objects.  Compare numbers.

Big Ideas in This Domain 1. Rote Counting (0-100) vs. Real Counting (1-20) 2. Subitizing 3. Development of Forward Number Word Sequence (FNWS) 4. Comparing Numbers

Real Counting vs. Reciting How do you know when a child can really count a set of objects as opposed to reciting a list of “counting words”?

Vocabulary  Number: An idea or abstraction that represents a quantity  Numeral: A symbol representing numbers  Cardinal number: Used to describe how many elements are in a finite set  Ordinal number: Numbers used to denote order, i.e., “second in line”

Real Counting Requires Understanding 5 Principles Any collection of real or imagined objects can be counted 2. Counting numbers are arranged in a sequence that does not change a) Understand that each successive number name refers to a quantity that is one larger correspondence: One and only one number is used for each item counted, and each item is counted only once

Real Counting Requires Understanding 5 Principles 4. Order-irrelevance: The order in which items are counted is irrelevant 5. Cardinality principle: There is a special significance to the last number counted. It is not only associated with the last item but also represents the total number of items in the set

Subitizing

Perceptual Subitizing Recognizing a number without using other mathematical processes.

Conceptual Subitizing How do people look at an 8-dot domino pattern and recognize it as 8?  Conceptual subitizing plays an advanced organizing role.  People recognize the number pattern as a composite of parts and as a whole.

Development of FNWS Count forward beginning from a given number within the known sequence (instead of having to begin at 1).  Students begin a rote forward counting sequence from a number other than 1. Thus, given the number 4, the student would count, “4, 5, 6, 7 …” This objective does not require recognition of numerals. It is focused on the rote number sequence

Compare Numbers  Students use various counting or matching strategies to compare two quantities and tell which is larger.

Operations and Algebraic Thinking Kindergarten, First Grade, & Second Grade

OA - Kindergarten  Understand addition as putting together and adding to, and;  Understand subtraction as taking apart and taking from.

OA - Kindergarten 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.

OA - Kindergarten 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.

OA - Kindergarten  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).

OA - Kindergarten  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.

OA - Kindergarten K.OA.5 Fluently add and subtract within five.

OA – First Grade 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.

OA – First Grade Represent and solve problems involving addition and subtraction. 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.

OA – First Grade Understand and apply properties of operations and the relationship between addition and subtraction. 1.OA.3 Apply properties of operations as strategies to add and subtract. (Students need not use formal terms for these properties.) Examples: 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.)

OA – First Grade Understand and apply properties of operations and the relationship between addition and subtraction. 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.

OA – First Grade Add and subtract within OA.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

OA – First Grade Add and subtract within 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)

OA – First Grade Add and subtract within OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as:  decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9);  using the relationship between addition and subtraction (e.g., knowing that if, = 12, then one knows 12 – 8 = 4);

OA – First Grade Add and subtract within OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as:  Creating equivalent but easier or known sums (e.g., adding by creating the known equivalent = = 13).

OA – First Grade 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 – First Grade Work with addition and subtraction equations. 1.OA.8 Determine the unknown whole number in an addition or subtraction equation relating to three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11 5 = __ – = __

OA – Second Grade Represent and solve problems involving addition and subtraction. 2.OA.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

OA – Second Grade 2.OA.2 Fluently add and subtract within 20 using mental strategies. (Note: See standard 1.OA.6 for a list of mental strategies). By end of Grade 2, know from memory all sums of two one-digit numbers.

OA – Second Grade Work with equal groups of objects to gain foundations for multiplication. 2.OA.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

OA – Second Grade Work with equal groups of objects to gain foundations for multiplication. 2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Estimation  Estimation encourages children to think about the total quantities and contributes to number sense.  Relationships of numbers to real-world quantities and measures and the use of numbers in simple estimations can help children develop the flexible, intuitive ideas about numbers.

Estimation  Estimation can be used to check the reasonableness of exact answers found by paper/pencil or calculator methods.  Measurements can be approximated using known referents as the unit in the measurement process. Finding how many are in a sub-section and multiplying by the number of sub-sections can estimate a large number of objects in a given area.

Make An Estimate

Estimating Beans from A Collection of Math Lessons by Marilyn Burns  How many scoops of counters will fill the jar?

Reflecting on the Learning  When did you decide or know your estimate or solution was reasonable? Explain your thinking.

Estimation activities that make sense are those that are connected to:  - the children’s experience with counting.  -what the children know about in real life.

Estimation in Primary Grades  Offer children a benchmark or a point of reference when asking them to make an estimate. This leads to a more reasonable estimate and builds number sense.

Do’s and Don’ts of Estimation DoDon’t Have children verbalize the thinking they used to derive their answers. End the estimation activity once the children have arrived at a reasonable estimate. Make children accountable for their estimate. Have them record or share with a partner. Have the children think of an estimate and keep it in their head. Allow children an opportunity to refine their estimate on the basis of new information. Restrict students to their first estimate. Use numbers with children are familiar. Use numbers with which the children have had minimal experience.

Look at the Common Core State Standards  Which standards were embedded in the bean estimation lesson?

Plan an Estimation Task for Young Learners.  1. Form groups of like grade levels.  2. Work together to plan a student-centered estimating activity that involves using an estimation jar and a quantity appropriate for your grade level.  3. Be prepared to share your lesson with the whole group.  Here are some examples and questions to think about…

 How many pony beads are in the jar? How would you begin to estimate these? What if we change to tennis balls or rice?  How many colored tiles are in the estimation jar? Are there more red tiles, more blue tiles, or is the amount of red and blue tiles the same?  Gather up several stacks of classroom books and make a pile in a central meeting area of your room. The pile of books could be placed in a large crate or basket to simulate an estimation jar. Ask the students about how many books they think are in the pile? Do you think there is enough for every student in the room to have one book? Ask for estimates that seem unreasonable, too low, and then reasonable.

Classroom Application  Present the estimation task you developed in this session to your class before the next session.  Reflect on three of these questions:  How did your students respond?  What did you learn about your students’ thinking and estimation skills?  What evidence of number sense did students show?  How does this look different at Grades K, 1 and 2?