Fluently Decompose and Recompose Numbers Additive Composition (parts to wholes) Multiplicative Composition (equal parts) Place Value Understandings ( tenness) Confidently Express and Represent Numbers in flexible forms (notations) expressions, words, symbols Confidently compare numbers with reference to magnitude & density Can flexibly move between one to one and many to one correspondence to interpret numbers, (unitize) & apply that ability to understanding place value and multiplicative growth. Understand equal, the equal sign and inverse relationships. Understand the number properties (associative & commutative, special case for one and zero) Use the above understandings to develop personally meaningful and efficient strategies for solving problems. Expect to make sense of numbers, problems and the solutions they create. If......then and therefore are willing to estimate reasonable results for problems. Kindergarten set the stage with subitizable units as the starting point for decomposing and recomposing numbers to 10. Grade ones must trust 2, 3, 5 without counting so they can use them to express and represent numbers to 10, then to 20. Kindergarteners set the stage for understanding equal by comparing subitizable units to see without counting that they are equal and coming to understand the “cancelling” affect of adding and subtracting an equal amount. If Grade ones need to understand that adding and subtracting an equal amount leaves the total unchanged, but adding more than you subtract or subtracting more than you add create an imbalance or inequality. These understandings form a foundation for equal and inverse relationships. Magnitude and density are illuminated by the physical and visual models. You can see numbers in numbers and therefore how they are related by quantity. Five is in 6, 7, 8, 9, 10. Ten is in the teens. If a student identifies a collection as 9, then 5, 6, 7, 8 must be “in it”. The commutative property is visible in the models. Students can see that 3 + 5 = 5 + 3. You can ”read” the parts in any order in a dot collection, blocks collection. Cuisenaire rods or in the ChunkitZ puzzle.

Kin: Subitize (recognize at a glance) and name familiar arrangements of 1 to 5 objects or dots. [C, CN, ME, V] Gr 1: Subitize (recognize at a glance) and name familiar arrangements of 1 to 10 objects or dots. Tell student: “ I am going to show a collection. You tell me how many are in it.” Show slide for 3 seconds, no longer. Expect immediate response of 5. Kinder Outcomes: N2,N3,N4 Pre cursor to making sense of addition and subtraction is understanding inverses. This task is a basic understanding of equal and ”balancing” out transformations. Equal in, equal out… no change. Same affect as adding zero.. I did 5 +2 -2 = 5. Not appropriate to connect equations. Grade 1 Outcomes: N2, N3,N5, N7,N9 PR4

In response to how many student immediately says 3

In response to how many student immediately says 5

In response to how many student immediately says 2 1N2, 1N3,1N10

In response to how many student immediately says 4 1N2, 1N3,1N10

A key concept in additive reasoning is the idea of inverse relationships. Understanding inverses is fundamental to solving problems like 8 - ? = 3 Understanding inverses begins way before equations. The following task is not about adding and subtracting. It is about equal and maintaining equal.

If you start with a quantity, then add and take away the same number of items, you have just as many as you had before. These two transformations cancel each other out. The number remains unchanged. If Kindergartners have mastered 5 as a subitizable unit they can perform the following task. If Grade ones cannot answer correctly you need to do more work with identifying and trusting 5, then using five to describe collections.

5 apples Tell student: “ I am going to show a collection. You tell me how many are in it.” Show slide for 3 seconds, no longer. Expect immediate response of 5. Kinder Outcomes: N2,N3,N4 Pre cursor to making sense of addition and subtraction is understanding inverses. This task is a basic understanding of equal and ”balancing” out transformations. Equal in, equal out… no change. Same affect as adding zero.. I did 5 +2 -2 = 5. Not appropriate to connect equations. Grade 1 Outcomes: N2, N3,N5, N7,N9 PR4 N2,N3,N4

I put them in this bag I put the apples in this bag. 1N2,1N3

How many apples in the bag? I put the apples in this bag. How many apples are in the bag? Expect immediate response of 5. You put 5 apples in the bag. 1N2,1N3

Dad took 2 Just before lunch Dad took two apples out of my bag. After lunch Mom came home from shopping and she put 2 apples in the bag. How many apples are in the bag now? Expected response 5. 1N2,1N3

Mom put two in After lunch Mom came home from shopping and she put 2 apples into the bag. How many apples are in the bag now? Expected response 5. 1N2,1N3

How many apples in the bag? How many apples are in the bag? Expect immediate response of 5. 1N2,1N3

If you start with a quantity, then add and take away the same number of items, you have just as many as you had before. These two transformations cancel each other out. The number remains unchanged. If Grade ones cannot answer correctly you need to do more work with identifying and trusting 5, then using five to describe collections.

If Kindergartners trust 2, 3 and 5 they are able to identify them in larger collections. Kinder: Represent and describe numbers 2 to 10, concretely and pictorially. [C, CN, ME, R, V] Subitize (recognize at a glance) and name familiar arrangements of 1 to 5 objects or dots. [C, CN, ME, V] Gr one: Subitize (recognize at a glance) and name familiar arrangements of 1 to 10 objects or dots. [C, CN, ME, V] Demonstrate an understanding of counting by: • indicating that the last number said identifies “how many” • showing that any set has only one count • using counting-on • using parts or equal groups to count sets. [C, CN, ME, R, V] Represent and describe numbers to 20, concretely, pictorially and symbolically. [C, CN, V]

Circle sets of 3. Student confidently, efficiently and accurately circles all 3 sets of 3, leaves one out. Grade 1 and 2 confidently responds that I see 3 sets of 3 and 1 left. Oh by the way I also see 2 fives. It’s 10.

Circle sets of 2. Student confidently, efficiently and accurately circles all 4 sets of 2. Grade 1 and 2 recognize that 4 twos is 8. 1N2,

Please circle 5. Student confidently, efficiently and accurately circles 5 with little hesitation.

Circle 5. Student confidently, efficiently and accurately circles a set of 5. 1N2,1N3

We each have a plate. I am going to put some blocks in my plate. Kin Outcomes: Number Subitize (recognize at a glance) and name familiar arrangements of 1 to 5 objects or dots. [C, CN, ME, V] Relate a numeral, 1 to 10, to its respective quantity. [CN, R, V] Represent and describe numbers 2 to 10, concretely and pictorially. [C, CN, ME, R, V] Compare quantities 1 to 10, using one-to-one correspondence. [C, CN, V] And shape and space Use direct comparison to compare two objects based on a single attribute, such as length (height), mass (weight) and volume (capacity). [C, CN, PS, R, V] Quantity is an attribute of these groups. Grade one: (Describe equality as a balance and inequality as an imbalance, concretely and pictorially (0 to 20). [C, CN, R, V] ALL EQUALITIES ARE NOT ABOUT BALANCE. Students need to know more than just “balance”. Record equalities, using the equal symbol. [C, CN, PS, V] Grade One: N2,N3,N5,N7,N8(?),N9 PR4,Pr5 SS1 (quantity is an attribute) me Child’s name

“ I put a set of blocks in my plate “ I put a set of blocks in my plate. Can you put a set that is equal in your plate, please. Alternate prompt: “Make a set that has as many.” Do not add other clues or prompts. If the child asks for help: Repeat your statement: “Please make a set that is equal.” or “A set that has as many.” Expect student to place 2 on his or her plate. Have available a handful of blocks, different colours. Expect no hesitation and no concern with matching colours. Two is the answer. me Child’s name

Clear off both plates. This time put out 3 blocks and ask child to make a set that is equal. A set that has as many. Expect quick response of 3. Child might stop to count 1,2,3. You need to determine if he or she knew 3 and just wanted to count. Perhaps need to ask :”Can you just pick up 3 without counting?” me Child’s name

Leave the blocks out and say to child: “Please , change your set so that it has more than mine.” Expect any number of blocks to be added. If he or she only puts one block in, ask: “Please show me another way to have more in your set.” (Is child willing to put more than one at a time in?) or you may need to simply grab 3 more in each hand and drop into each set. We need to know the child is not caught on adding one at a time side to side. This is about trusting small collection and knowing equal. me Child’s name

Say to child, “Let’s start over Say to child, “Let’s start over. Clear both plates and put 4 back on your plate. Say to child: “Please make a set that is less than mine.” Expect child to put out 0, 1, 2 or 3 blocks. Grade one prompt. “Is there more than one way to have less?” or teacher clears off student response and places a different amount but still less and asks: “Does this show less than mine?” We want to know that child understands 0, 1, 2, 3 are all less than 4. me Child’s name

Start at empty plates. Teacher puts 3 in each plate, at same time… Start at empty plates. Teacher puts 3 in each plate, at same time…. No trickery. Child can clearly see 3 put down together on each. me Child’s name

Teacher says: “We both have 3. Our sets are equal. Do you agree Teacher says: “We both have 3. Our sets are equal. Do you agree?” Expect child to say yes. Teacher says: “ Watch ( in clear sight of the child place 2 more blocks simultaneously on each plate. No colour coding.) me Child’s name

Ask child: “Are the sets still equal. ” Expect child to say yes Ask child: “Are the sets still equal?” Expect child to say yes. No comment about add 2 or 5 is needed. me Child’s name

Clear the plates and start over. me Child’s name 1N2,1N3

In clear view and at same time, in one hand each, place 5 in each plate. Ask child: ”Are my sets equal?” Expected response yes. Child might say 5 in each but immediate response we want is yes to equal. me Child’s name 1N2,1N3

In full view of child, at the same time pick up two from one plate and three from the other. Keep them in full view. Ask: “Are they still equal?” Expected response no. child might say something about now you have 3 and 2 or you have more on one side than the other but the question you want answered is, “Are they equal?” If child does not answer ask again. me Child’s name 1N2,1N3

N2 I have 5 dots. Do you agree? Yes with no hesitation. Kin: N2 Subitize Grade 1: N2 Subitize N3, N4, N7 Conservation N2

Teacher rearranges the dots and asks. How many dots do I have Teacher rearranges the dots and asks. How many dots do I have? Expect immediate response of 5.

Without inverses students are not demonstrating an understanding of addition. Additive reasoning is the skill that will carry them through all levels. The curriculum in Kindergarten does not make mention. These tasks however inform teachers. In Grade one the outcomes that link subtraction to addition (corresponding/ related) Grade one: N9 Demonstrate an understanding of addition of numbers with answers to 20 and their corresponding subtraction facts, N10: Describe and use mental mathematics strategies for basic addition facts and related subtraction facts to 18. [C, CN, ME, PS, R, V] In the outlined box: Understand and apply strategies for addition facts up to and including 9 + 9 and related subtraction facts. Recall addition facts to a sum of 5 and related subtraction facts.

The blue paper is covering some of the dots The blue paper is covering some of the dots? How many of the 5 dots did I cover? Expect to hear 3 with little hesitation. Child might add 2 and 3 is 5.

I have 4 dots. How many more to have 5? Expect to hear 1.

I covered some. How many are covered?

7 5 6 8 Put these in order. Kin Understand and apply strategies for addition facts up to and including 9 + 9 and related subtraction facts. Recall addition facts to a sum of 5 and related subtraction facts.

Put these in order. And explain the order Put these in order. And explain the order. (Expect comments about more and less, possibly one more or one less, possibly plus one, minus one, possibly its in counting order or this is how you count. Any of these is fine.

7 6 5 8 Match (Match and order?) Kin Relate a numeral, 1 to 10, to its respective quantity. [CN, R, V] .

Ask which pair of scissors is longer. Expect immediate response Ask which pair of scissors is longer. Expect immediate response. Record the response. Say to child: please show me how you would prove to someone that pair is longer. Expect to see student line up with a starting point that is clear between them. Use direct comparison to compare two objects based on a single attribute, such as length (height), mass (weight) and volume (capacity). [C, CN, PS, R, V] Kin SSM Use direct comparison to compare two objects based on a single attribute, such as length (height), mass (weight) and volume (capacity).[C, CN, PS, R, V] Gr 1

Trace all the faces. Sort 3-D objects, using a single attribute. [C, CN, PS, R, V] Build and describe 3-D objects. [CN, PS, V]

Why can I put them together in one set? Sort 3-D objects, using a single attribute. [C, CN, PS, R, V] Build and describe 3-D objects. [CN, PS, V] Why can I put them together in one set?