1. 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 = You can ”read” the parts in any order in a dot collection, blocks collection. Cuisenaire rods or in the ChunkitZ puzzle.

2. 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. Not appropriate to connect equations.
Grade 1 Outcomes: N2, N3,N5, N7,N9 PR4

3. In response to how many student immediately says 3

4. In response to how many student immediately says 5

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

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

7. 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.

8. 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.

9. N2,N3,N4 Place 5 apples on the table. Say to student I have 5 apples.
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. Not appropriate to connect equations.
Grade 1 Outcomes: N2, N3,N5, N7,N9 PR4
N2,N3,N4

10. Put the 5 apples in the bag saying as you do, I put my 5 apples in this bag.
1N2,1N3

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

12. I left my bag on the table and my mom came in and took 2 apples out of the bag.
1N2,1N3

13. A little while later, my sister came in from shopping and she put 2 apples into the bag. How many apples are in the bag now? Expected response 5.
1N2,1N3

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

15. 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.

17. 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]

18. 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.

19. 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,

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

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

22. Equal: Match a set that has as many
1N2,1N3

23. At the same time place 3 blocks in each plate. Ask, are the sets equal
At the same time place 3 blocks in each plate. Ask, are the sets equal? Expect to hear yes.
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)

24. Place 5 blocks in one plate and ask the child to make a set that is equal. (if you have not been using the term equal then 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.

25. Leave both sets showing and ask the student to make his/her set be less than yours. Expect student to remove 1,2,3,4,or 5 blocks. When finished ask: Is your set less than mine? Expect yes.

26. Clear off both plates. This time put out 3 blocks and ask student to make a set that has more than yours. Please make me a set that is more. Accept 4 and up. When finished ask student: Does your set have more than mine? Does my set have less than yours?

27. 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?”

28. Place 4 blocks in your set and ask child to show a set that is one less. Expect child to show 3. When finished ask child how do you know your set has one more? Is it automatic or is it a count.

29. Place 3 blocks in your plate and ask child to show the set that is one more. When finished ask: Is your set one more than mine?

30. 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.

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

32. 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 and related subtraction facts. Recall addition facts to a sum of 5 and related subtraction facts.

33. 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.

34. I covered some. How many are covered?

35. 7 5 6 8
Put these in order.
Kin Understand and apply strategies for addition facts up to and including and related subtraction facts. Recall addition facts to a sum of 5 and related subtraction facts.

36. 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.

37. 7 6 5 8 Match (Match and order?)
Kin Relate a numeral, 1 to 10, to its respective quantity. [CN, R, V] .

38. 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

39. 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]

40. 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?