## Math - Operations and Algebraic Thinking

**K.OA.1**

**Anchor Standard**

**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.

Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)

**Understand**

__addition__**as**

__putting together__and__adding to__, and understand

__subtraction__**as**

__taking apart__and__taking from.__Students demonstrate the understanding of how objects can be joined (addition) and separated (subtraction) by representing addition and subtraction situations in various ways.

__This objective is focused on understanding the__

__concept of addition and subtraction__, rather than reading and solving addition and subtraction number sentences (equations).

Common Core State Standards for Mathematics states, “Kindergarten students should see addition and subtraction equations, and student writing of equations in kindergarten is encouraged, but it is not required.” Please note that it is not until First Grade when “Understand the meaning of the equal sign” is an expectation (1.OA.7).

Therefore, before introducing symbols (+, -, =) and equations, kindergarteners require numerous experiences using joining (addition) and separating (subtraction) vocabulary in order to attach meaning to the various symbols. For example, when explaining a solution, kindergartens may state, “Three

*and*two

*is the same amount as*5.” While the meaning of the equal sign is not introduced as a standard until First Grade, if equations are going to be modeled and used in Kindergarten, students must connect the symbol (=) with its meaning (is the same amount/quantity as).

***For numbers 0 – 10, Kindergarten students choose, combine, and apply strategies for answering quantitative questions. This includes quickly recognizing the cardinalities of small sets of objects, counting and producing sets of given sizes, counting the number of objects in combined sets, or counting the number of objects that remain in a set after some are taken away. Objects, pictures, actions, and explanations are used to solve problems and represent thinking.

Mathematically proficient students communicate precisely

**about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are:**

__by engaging in discussion__**join, add, separate, subtract, and, same amount as, equal, less, more, total.**

**K.OA.2**

**Anchor Standard**

**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.

**Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.**

Kindergarten students solve four types of problems within 10:

- Result Unknown/Add To
- Result Unknown/Take From
- Total Unknown/Put Together-Take Apart
- Addend Unknown/Put Together-Take Apart

Kindergarteners use counting to solve the four problem types by acting out the situation and/or with objects, fingers, and drawings.

__Add To - Result Unknown:__For example: Three squirrels were in the yard. Three more squirrels joined them. How many squirrels are in the yard now? 3 + 3 = ?

__Take From - Result Unknown__For example: Six cookies were in the bowl. I ate two cookies. How many cookies are in the bowl now? 6 – 2 = ?

__Put Together/Take Apart - Total Unknown__For example: Four yellow bananas and three green bananas are on the desk. How many bananas are on the desk? 4 + 3 = ?

__Put Together/Take Apart - Addend Unknown__For example: Six bananas are on the table. Four are green and the rest are yellow. How many bananas are yellow? 4 + ? = 6 OR 6 – 4 = ?

Another example: Eight jelly beans were in the bowl. I ate three jelly beans. How many jelly beans are in the bowl now? 8 – 3 = ? OR 3 + ? = 8

**K.OA.3**

**Anchor Standard**

**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 = 2 + 3 and 5 = 4 + 1).

**Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.**

Students develop an understanding of part-whole relationships as they recognize that a set of objects (5) can be broken into smaller sub-sets (3 and 2) and still remain the total amount (5). In addition, this objective asks students to realize that a set of objects (5) can be broken in multiple ways (3 and 2; 4 and 1). Thus, when breaking apart a set (decompose), students use the understanding that a smaller set of objects exists within that larger set (inclusion).

Example:

**“Bobby Bear is missing 5 buttons on his jacket. How many different ways can you use blue and red buttons to put on his jacket? Draw a picture of all your ideas.**

Students could draw pictures of:

4 blue and 1 red button

3 blue and 2 red buttons

2 blue and 3 red buttons

1 blue and 4 red buttons

Kindergarten students need ample experiences breaking apart numbers and using the vocabulary “and” & “same amount as” before symbols (+, =) and equations (5= 3 + 2) are introduced

__. If equations are used, a mathematical representation (picture, objects) needs to be present as well.__

**K.OA.4**

**Anchor Standard**

**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.

**Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.**

Students build upon the understanding that a number (less than or equal to 10) can be decomposed into parts (K.OA.3) to find a missing part of 10. Through numerous concrete experiences, kindergarteners model the various sub-parts of ten and find the missing part of 10.

For example: When working with cubes, a student has 6 cubes and has to determine how many more cubes he/she needs in order to make 10. A student determines that 4 more cubes are needed to make a total of 10. In addition, kindergarteners use various materials to solve tasks that involve decomposing and composing 10.

Question: “A full case of juice boxes has 10 boxes. There are only 6 boxes in this case. How many juice boxes are missing?

Answers:

For example

**:**

__Using a Ten-Frame__“I used a ten frame for the case. Then, I put on 6 counters for juice still in the case. There’s no juice in these 4 spaces. So, 4 are missing.”

For example:

__Think Addition__“I counted out 10 counters because I knew there needed to be ten. I pushed these 6 over here because they were in the container. These are left over. So there’s 4 missing.”

For example:

__Fluently add/subtract__“I know that it’s 4 because 6 and 4 is the same amount as 10.”

**K.OA.5**

**Anchor Standard**

**K.OA.5**Fluently add and subtract within 5.

Students are fluent when they display accuracy (correct answer), efficiency (a reasonable amount of steps in about 3-5 seconds* without resorting to counting), and flexibility (using different strategies to solve problems).

Students develop fluency by understanding and internalizing the relationships that exist between and among numbers. Oftentimes, when children think of each “fact” as an individual item that does not relate to any other “fact”, they are attempting to memorize separate bits of information that can be easily forgotten. Instead, in order to fluently add and subtract, children must first be able to see sub-parts within a number (inclusion, K.CC.4.c).

Once they have reached this milestone, children need repeated experiences with many different types of concrete materials (such as cubes, chips, and buttons) over an extended amount of time in order to recognize that there are only particular sub-parts for each number. Therefore, children will realize that if 3 and 2 is a combination of 5, then 3 and 2 cannot be a combination of 6.

For example, after making various arrangements with toothpicks, students learn that only a certain number of sub-parts exist within the number 4: (4 = 0; 1 + 3; 2 + 2; 3 + 1; 0 + 4)

Then, after numerous opportunities to explore, represent and discuss “4”, a student becomes able to fluently answer problems such as, “One bird was on the tree. Three more birds came. How many are on the tree now?” and “There was one bird on the tree. Some more came. There are now 4 birds on the tree. How many birds came?”

Numerous experiences with breaking apart actual sets of objects and developing relationship between numbers help children internalize parts of number and develop efficient strategies for fact retrieval.