1. Thursday, June 29, 2016 Strong Start Math Project
Monday night – remind to bring device with a QR code, possibly charger, headphones. Strategically place partners.
Materials:
CGI books
Interpretation guides
Standards
OA Progressions
Blank problem types/tape diagrams grid (slide 69) – 1 per person
Marbles problem types cards (slide 13) – 2 sets per table
Use figures on pp of CGI book (slide 24) – 1 set per table

2. Agenda Comparing Numbers Learning Trajectory
Symbolic Representations of Word Problem Situations
Written Representations of Word Problem Situations
Lunch
Addition & Subtraction Learning Trajectory
Equality
Equality Learning Trajectory
Individual Project Work Time

3. Symbolic Representations
of Word Problems

4. Symbolic Representations
SITUATION EQUATIONS
Reflect the action in the story in the manner in which happens
SOLUTION EQUATIONS
Reflect the student’s thinking in the order or manner they solved the problem

5. What might an equation look like for each of these problem types?
Add to (Join) Result Unknown: Connie had 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether? = _____ (situation equation) = _____ (solution equation)

6. What might an equation look like for each of these problem types?
Take From (Separate) Result Unknown: Connie had 13 marbles. She gave 5 to Juan. How many marbles does Connie have left? 13 – 5 = _____ (situation equation) 13 – 5 = _____ (solution equations)

7. What might an equation look like for each of these problem types?
Add to (Join) Change Unknown: Connie has 5 marbles. How many more marbles does she need to have 13 marbles altogether? 5 + ____ = 13 (situation equation) ____ = 13 – 5 (possible solution equation)

8. What might an equation look like for each of these problem types?
Take From (Separate) Change Unknown: Connie had 13 marbles. She gave some to Juan. Now she has 5 marbles left. How many marbles did she give to Juan? 13 - _____ = 5 (situation equation) ____ = 13 – 5 (possible solution equation)

9. What might an equation look like for each of these problem types?
Take From (Separate) Start Unknown: Connie had some marbles. She gave 5 to Juan. Now she has 8 marbles left. How many marbles did Connie have to start with? _____ – 5 = 8 (situation equation) _____ = (possible solution equation)

10. What might an equation look like for each of these problem types?
Add to (Join) Start Unknown: Connie had some marbles. Juan gave her 5 marbles. Now she has 13 marbles. How many marbles did she have to start with? _____ + 8 = 13 (situation equation) 8 + ____ = – 8 = 5
(possible solution equations)

11. Learning Intention & Success Criteria
We are learning
the CCSSM expectations around story problem structures.
that students progress through levels of reasoning as they solve story problems.
how we might support students’ representations of their thinking.
Reorienting ourselves to our goals around these standards.

12. Representing
Level 1 and Level 2
Thinking

13. “As children progress to Level 2 strategies they no longer need representations that show each quantity as a group of objects.”
-p. 16 OA Progressions
As children leave behind the need to represent each quantity, what implications does that have for us as teachers?

14. Representing problem situations with equations
Typical Instructional Sequence
Physical
Modeling with Objects ?
What’s missing from this sequence?
Representing problem situations with equations
Symbolic
Visual
Research tells us that as students grapple with new concepts they need to spend time with written representations. During this time, we want ensure we are making connections particularly in the direction of physical  visual  symbolic.

15. Margaret has 4 toy cars. Cassandra has 3 toy cars
Margaret has 4 toy cars. Cassandra has 3 toy cars. How many toy cars do they have altogether?
Physical
Symbolic
Use concrete objects to form two groups and put the two
groups together.
Visual
4 + 3 = 7
Have teachers grab a board and marker.
Split the board in half.
On the left, show a representation.
Have them hold up to show the different ways of representing.
1,2,3
1,2,3 4
1,2,3,4,5,6,7…7 cars

16. Representational Math Drawings
Drawing pictures that represent concrete objects provides a bridge to help young children connect their concrete representations to the abstract world of mathematical symbols.
“Math drawings facilitate reflection and discussion because they remain after the problem has been solved.”
p. 8 OA Progressions
Children need many opportunities to create such drawings.
3 min
So what might these math drawings look like?

17. CCSSM Suggested Math Drawing: Tape Diagram
What is a tape drawing?
A drawing that looks like a segment of tape, used to illustrate number relationships. Also known as strip diagrams, bar model, fraction strip or length model.
--CCSSM Glossary p. 87
Why are they important?
Next level of abstraction
Serves as a model of thinking to support transitioning from Level 1 to Level 2

18. East Meadow School District October, 2013
Tape Diagrams
Tape Diagram?
Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7
Math Drawings (1.OA.1, 1.OA.2)
Math Drawings (2.OA.1, 2.OA.2, 2.MD.5)
Visual Fraction Model (3.NF.3a-d)
Visual Fraction Model (4.NF.3, 4.NF.4, 4.OA.2)
Visual Fraction Model (5.NF.2-4, 6, 7)
Tape Diagrams (6.RP-3) Visual Fraction Model (6.NS-1)
Visual Model for Problem Solving (7RP1-3)
Number Line Diagram (7.NS-1)
Why are they important?
Next level of abstraction

19. “These diagrams are a major step forward because the same diagrams can represent the adding and subtracting situations for all kinds of numbers students encounter in later grades (multi-digit whole numbers, fractions, decimals, variables).”
--p. 17 OA Progressions
Stress that due to the coherence in the CCSS, we need “lasting” models that can be learned and applied to a wide range of problem situations and range of numbers

20. Margaret has 4 toy cars. Cassandra has 3 toy cars
Margaret has 4 toy cars. Cassandra has 3 toy cars. How many toy cars do they have altogether?
Physical
Symbolic
Use concrete objects to form two groups and put the two
groups together.
Visual
4 + 3 = 7
1,2,3
1,2,3 4
1,2,3,4,5,6,7…7 cars

21. Margaret has 4 toy cars. Cassandra has 3 toy cars
Margaret has 4 toy cars. Cassandra has 3 toy cars. How many toy cars do they have altogether? 4
7 cars
5,6,7 4
5, 6, 7
7 cars 4 3
7 cars
Show discrete and then continuous or
Lay out color tiles to show both quantities.

22. Progression of Tape Diagrams
Students begin by using concrete objects and drawing pictorial models
Evolves into using bars to represent quantities
Enables students to become more comfortable using letter symbols to represent quantities later at the secondary level (Algebra) 15 7 ?

23. Foundations for Tape Diagrams in PK-1
Students make use of concrete objects (or picture cut-outs) to make sense of the part-whole and comparison concepts. Then, they progress to drawing rectangular bars as pictorial representations of the models, and use the models to help them solve abstract word problems

24. Tape Diagram
Practice
Part-Part-Whole

25. He has 14 toy cars altogether.
Example A
Connor has toy 6 cars. He gets 8 more toy cars for his birthday. How many toy cars does he have now?
6 + 8 = 14
He has 14 toy cars altogether.

26. East Meadow School District October, 2013
Tape Diagrams
Example B
Laura has a vase with 13 flowers. She puts 7 more flowers in the vase. How many flowers are in the vase? There are 20 flowers in the vase. ?
Number of flowers in the vase 13 7
= 20
“We do” problem
What category? Add to result unknown
ENY G1 M4 L19

27. Example C
174 children went to summer camp. If there were 93 boys, how many girls were there?
93 + ? = 174
174 – 93 = ?
There were 81 girls.

28. Example D
Julie brought 4 pieces of watermelon to a picnic. After Amy brings her some more pieces of watermelon, she has 9 pieces. How many pieces of watermelon did Amy bring Julie?
9 pieces of watermelon 4 ?
4 + ? = 9
9 – 4 = ?
Amy brought 5 pieces
of watermelon to Julie.

29. East Meadow School District October, 2013
Tape Diagrams
Example E
Some yellow beads were on Sally’s bracelet. After she put 14 purple beads on the bracelet, there were 18 beads. How many yellow beads did Sally’s bracelet have at first?
18 beads on the necklace ? 14
Yellow beads
Purple beads
? + 14 = 18
18 – 14 = ?
Sally had 14 yellow beads at first
ENY G1 M4 L20

30. East Meadow School District October, 2013
Tape Diagrams
Example G
Kiana found some shells at the beach. She gave 8 shells to her brother. Now she has 9 shells left. How many shells did Kiana find at the beach?
8 shells
9 shells
? shells
ENY G1 M4 L21

31. East Meadow School District October, 2013
Tape Diagrams
Stop to Reflect
What did you notice about the structure of the problems in Problem Set 1?
They were all addition or subtraction problems, and were all conducive to use of the Part-Whole model.
Ask participants to turn and talk. Gather a few ideas from the group. Then display the summary bullet on the slide

32. Part-Whole Model Addition & Subtraction
Part + Part = Whole Whole – Part = Part

33. Tape Diagram
Practice
Comparison

34. East Meadow School District October, 2013
Tape Diagrams
Example A
Danielle wrote 8 letters. Katrina wrote 12 letters. How many more letters did Katrina write than Danielle?
Number of letters Danielle wrote
Number of letters Katrina wrote 8 12 ?
= 4
Katrina wrote 4 more letters than Rose.
ENY G1 M6 L1

35. East Meadow School District October, 2013
Tape Diagrams
Example B
Nicole saw 14 foxes at the zoo. She saw 5 more monkeys than foxes. How many monkeys did Nicole see?
Number of monkeys
Number of foxes 14 ? 5
= 19
Nicole saw 19 monkeys at the zoo.
ENY G1 M6 L17

36. East Meadow School District October, 2013
Tape Diagrams
Stop to Reflect
What did you notice about the structure of problems in Problem Set 2?
They were all addition or subtraction comparison problems, and were all conducive to use of the Additive Comparison model.

37. The Comparison Model Addition & Subtraction
larger quantity – smaller quantity = difference
smaller quantity + difference = larger quantity

38. Practice Return to the story problems you created.
Create a tape diagram which illustrates each story you and your partner wrote.
Need a better question: How do the different models/representations develop a deeper understanding of the operations of addition and subtraction?

39. Learning Intention & Success Criteria
We are learning
the CCSSM expectations around story problem structures.
that students progress through levels of reasoning as they solve story problems.
how we might support students’ representations of their thinking.
Reorienting ourselves to our goals around these standards.

40. Focus Topics or Standards
Reflection/Summary
Summarize some key points and classroom ideas related to the topics or focus standards in this session.
Focus Topics or Standards
Summary of Key Points
Classroom
Ideas to Try

41. Addition and Subtraction
Learning Trajectory
for
Addition and Subtraction

42. Examining the Trajectory
1st Pass – Look for Problem Types
2nd Pass – Look for Levels of Thinking
Don’t think should puzzle piece together as we’ve done others.

43. Reflecting on the Trajectory
What do you see that confirms what we’ve discussed the past two days?
What surprises you?

44. EGG CHICKEN DINOSAUR HAWK! Rock, Paper, Scissors, SHOOT!
Brain Break!
EGG CHICKEN DINOSAUR HAWK! Rock, Paper, Scissors, SHOOT!

45. Understanding Equality Mequon-Thiensville School District
Relational Thinking
Understanding Equality
Alyssa Murphy
Michelle Painter
Mequon-Thiensville School District

46. Agenda
-Warm up -What does the equal sign mean? -Student work/Levels of equality -Ways to promote understanding of the equal sign -Relational thinking practice -Questions

47. Learning Intention and Success Criteria
Learning Intention: We are learning to examine the concept of equality. Success Criteria: We will be successful when we can… -understand what the equal sign means -look at equations and solve them relationally
Alyssa will do this.

48. Solve the equations below:
= ___ = ___ + 5
Michelle does this.
Keep track of your strategy

49. Sharing
-Think, then turn and talk. -Share your strategy and thoughts with a colleague at your table.
= ___ + 27
8 + 4 = ___ + 5
Alyssa talks.

50. = ___ = ___ + 5
Next discuss… -How do you think your students would solve this problem? -Would their strategy be different than yours? -Why do you think they would answer that way? As you talk through these questions, begin to think about a working definition of the equal sign that might make sense to both you and your students.
Michelle will go through questions.
What does the equal sign mean? Alyssa charts responses.

51. What does the equal sign mean?
Alyssa did this.

52. What is equality? Why is it important?
•Mathematical equality can be loosely defined as the principle that two sides of the equation have the same value and are thus interchangeable. (Kieran, 1981; Wittgenstein, 1961) •Knowledge of the equal sign as an indicator of mathematical equality is foundational to children’s mathematical development and serves as a key link between arithmetic and algebra. (Matthews, 2012)
Michelle does this.
Turn and talk about your thoughts regarding the quotes. What does interchangeable mean when you’re thinking of equality?
Algebra question to prompt discussion:

53. 1.OA.7
Work with addition and subtraction equations. 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, 5+2=2+5, 4+1=5+2
Alyssa does this.
Where does the idea of understanding the meaning of the equal sign come up. Consider students in first grade and the expectation set as they are working to understand this fundamental idea for developing algebraic reasoning.
This comes up one time in the common core standards. Realizing the importance of this skill, we wanted to see what our students understood about the equal sign and then how to teach them about the equal sign as a relationship between both sides. We gave our students a pre assessment on their understanding of the equal sign. Then, we implemented a unit on equality and relational thinking.

54. Student Work
Take a look at the student work samples with your shoulder partner.
Group the samples into 2 groups:
“got it” and “didn’t get it”
Guiding Questions…
How did students arrive at their answers?
What might this work show us about students’ understanding of the equal sign?
Michelle does this.
Wait until they’re done sorting.
Read table 1 on page 320.

55. Matthews Levels of Understanding the Equal Sign
Level One: Rigid Operational Level Two: Flexible Operational Level Three: Basic Relational Level Four: Comparative Relational
Michelle does this.
Turn and talk to your shoulder partner about things that surprised you or things that are new to you.

56. 4th Grade Success: = n + 27

57. 2nd Grade Success: 8 + 4 = ___ + 5

58. Levels of Student Understanding of Equality
Re-sort based on level
Levels of Student Understanding of Equality
Level
Description
Core Equation Structures
Level 1.
Rigid
Operational
Students can solve equations or evaluate true-false statements successfully that only have operations on the left side of the equal sign.
Equations with operations on left: a + b = c
4 + £ = T or F: = 7
3 + 4 = £ T or F: = 8
£ + 4 = T or F: = 8
Level 2.
Flexible Operational
Students can successfully solve equations with operations on the right side of the equal sign or interpret statements that have no operations.
Equations with operations on right: c = a + b
£ = T or F: 8 = 3 + 4
7 = £ T or F: 7 = 3 + 4
Equations with no operations: a = a
7 = £ T or F: 7 = 7
£ = n T or F: n = n
Level 3.
Basic
Relational
Students can successfully solve or evaluate statements with operations on both sides of the equal sign, and explain or give correct relational definitions of the equal sign.
Equations with operations on both sides:
a + b = c + d a + b – c = d + e
5 + 7 = 6 + £ 6 x = £ + 20
7m + 3m + 5 = m + 5 – 9
Level 4. Comparative Relational
Students can successfully use short-cuts (e.g., compensation strategies) and properties of the operations to solve equations or evaluate statements.
Equations that can be most efficiently solved by applying simplify transformations:
Use a shortcut to tell if the equation
“ = ” is true or false?
Figure out the value of m in "3 x 27 = 60 + m" without fully multiplying out 3 x 27.
Michelle. After 8 minutes, one person will get up and chart where they think the letter should go according to the level.
Source: Matthews, P., Rittle-Johnson, B., McEldoon, K., & Taylor, R. (2012). Measure for measure: What combining diverse measures reveals
about children’s understanding of the equal sign as an indicator of mathematical equality. Journal for Research in Mathematics Education, 43(3),

59. Student Work Sample Levels
Level 1: C, D, F, G, H, J Level 2: none Level 3: A, K (E) Level 4: B, E, I, (E)
Alyssa does this.

60. Ways to promote the understanding of =
Lower grades
-Representations (balances, ten frames, rekenreks, counters)
-True/False statements
-Word Problems
Upper grades
-True/False statements
-Solving for unknown variable
-Deep mathematical discourse
Give lesson plans.

61. Questions

62. Strong Start Math Project University of Wisconsin-Milwaukee, 2015-2018
Disclaimer
Strong Start Math Project
University of Wisconsin-Milwaukee,
This material was developed for the Strong Start Math project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors.
This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.