1. Routines for Reasoning
Connecting Representations –
Early Numeracy and Place Value

2. Routine: Connecting Representations Early Representations

3. Today’s Thinking Goals
Think like mathematicians
Look at and make sense number sentences
Connect pictures to numbers

4. Thinking Questions What do I see? What do the numbers mean?
How are the pictures like the numbers?
How many thinking questions should we give kids at one time? Do we develop these as we go?
How is this situation behaving?
What kind of problem is this?
Does this problem remind me of another I’ve solved?
How can I decompose this problem to help me understand it better?

5. I notice __________, which makes me think __________.

6. a) = 15
b) = 15
c) = 15
d) = 15

7. What could 13 look like with a rekenrek?
“I noticed ________, so I __________.”
“They noticed ________, so they __________.”

8. Reflection
“ I can connect a number to a picture by _______________.”

9. Set 2

10. I notice __________, which makes me think __________.

11. 1
a) = 10
b) = 10 2
c) = 10 3
d) 8 +2 = 10

12. What would = 10 look like?
“I noticed ________, so I __________.”
“They noticed ________, so they __________.”

13. Reflection
“ I can connect a number to a picture by _______________.”

14. Set 3

15. I notice __________, which makes me think __________.

16. 1
a) 8 + 5 b) 2
c) 7 + 6 3
d) 9 + 4

17. What would 11 + 2 look like? “I noticed ________, so I __________.”
“They noticed ________, so they __________.”

18. Reflection
“ I can connect a number to a picture by _______________.”

19. Set 3.5

20. I notice __________, which makes me think __________.

21. 1 a) b) 2 c) 3 d)

22. What would 13 – 9 look like? “I noticed ________, so I __________.”
“They noticed ________, so they __________.”

23. Routine: Connecting Representations Base Ten Representations

24. Today’s Thinking Goals
Think like mathematicians
Look at and make sense number sentences
Connect pictures to numbers

25. Thinking Questions What do I see? What do the numbers mean?
How are the pictures like the numbers?
How many thinking questions should we give kids at one time? Do we develop these as we go?
How is this situation behaving?
What kind of problem is this?
Does this problem remind me of another I’ve solved?
How can I decompose this problem to help me understand it better?

26. I notice __________, which makes me think __________.

27. “I noticed ________, so I thought__________.”1
“They noticed ________, so they __________.”
a) 63 2
b) 36
c) 27 3
d) 72

28. What would 27 look like? “I noticed ________, so I __________.”
“They noticed ________, so they __________.”

29. Reflection
“ I can connect a number to a picture by _______________.”

30. Set 2

31. I notice __________, which makes me think __________.

32. “I noticed ________, so I thought__________.”1
“They noticed ________, so they __________.”
a) 231 2
b) 321
c) 213 3
d) 132

33. What would 321 look like? “I noticed ________, so I __________.”
“They noticed ________, so they __________.”

34. Reflection
“ I can connect a number to a picture by _______________.”

35. Set 3

36. Today’s Thinking Goals
Think like mathematicians
Connect pictures to numbers
Think about pairs that make 20

37. Thinking Questions What do I see? What do I know about the number 20?
How are the numbers and the pictures connected?
How many thinking questions should we give kids at one time? Do we develop these as we go?
How is this situation behaving?
What kind of problem is this?
Does this problem remind me of another I’ve solved?
How can I decompose this problem to help me understand it better?

38. I notice __________, which makes me think __________.
How many more blocks do you need to make 20?
I notice __________, which makes me think __________.

39. 1 2 3 How many more do you need to make 20? a) 5 b) 6 c) 7 d) 9
“I noticed ________, so I thought__________.”
“They noticed ________, so they __________.”
a) 5 1
b) 6 2
c) 7 3
d) 9

40. Reflection
“ I can connect a number and a picture to make 20 by _______________.”