1. Using Student Work to Think About Instruction
Sept. 24, 2015

2. Tasks for the year
Grade 1 & 2 – Equal Sharing Grade 3, 4, & 5 – Multiplicative Thinking Grade 6, 7, 8 – Ratio & Proportional Reasoning

3. It thus differs from assessment designed primarily to serve
“Assessment for learning is any assessment for which the priority in its design and practice is to serve the purpose of promoting students’ learning.
It thus differs from assessment designed primarily to serve
the purposes of accountability, or of ranking, or of
certifying competence.
An assessment activity can help learning if it provides information that teachers and their students can use as feedback in assessing themselves and one another and in modifying the teaching and learning activities in which they are engaged. Such assessment becomes ‘formative assessment’ when the evidence is actually used to adapt the teaching work to meet learning needs.”
ASSESSMENT FOR LEARNING: Working Inside the Black Box: Assessment for Learning in the
Classroom Phi Delta Kappan September (1): 8-21

4. Student Work Analysis Process
Do the task. (at least 2 times)
Work together to complete the top of the SWA form.
Use your work to discuss the mathematics and the connection to the standard.
Describe expectations for student work in only 2 areas “meeting and approaching” the standard.
Write a sticky note for each category.
Sort your papers according to your expectations.

5. Student Work: Identifying Entry Points
Focus on the “Approaching” student work samples
Step #1 - Individually
Select a few papers from your approaching pile. Use the work to identify common misconceptions and common understandings.
Use a post-it to briefly describe the misconception or understanding.
Be prepared to share.
Step #2 – With partner(s)
Place each piece of selected student work in the middle of the table. Take turns sharing the evidence you noted.
Below are the work samples from Hi-Mount students that need to be copied on card stock and cut apart.
Achantia—Grade 4 (Item #1 or Item #2)
Jordan—Grade 4 (Item #1 or Item #2)
Miracle—Grade 5 (Item #1)
Zalaina—Grade 4 (Item #2)
Elijah—Grade 4 (Item #2)

6. Student Work: Peek and Pass
Step #3 Discuss as a table group:
What do you notice about the students’ thinking?
What “wonderings” do you have about these students’ understanding of math in the task?
Below are the work samples from Hi-Mount students that need to be copied on card stock and cut apart.
Achantia—Grade 4 (Item #1 or Item #2)
Jordan—Grade 4 (Item #1 or Item #2)
Miracle—Grade 5 (Item #1)
Zalaina—Grade 4 (Item #2)
Elijah—Grade 4 (Item #2)

7. Next steps Based on the analysis of the student work you brought to
class, write up a 1-2 paragraph synopsis of your findings.
What was the task asking students’ to think about?
What did you learn about your students’ current understanding?
How are you thinking about using the information to identify instructional moves that will strengthen student understanding of the concepts in your pre-assessment task?
What challenges do you anticipate?
How might Number Talks help strengthen student thinking around this math idea?

8. The Road to Proportional Reasoning
Using
Core Mathematics Partnership
Building Mathematical Knowledge and
High-Leverage Instruction for Student Success
Thursday Sept. 30, 2015

9. Learning Intentions & Success Criteria
We are learning to…
Understand the development of ratio and proportional reasoning and the structure of problems that support that development.
We will be successful when….
We can explain the standards progression for the development of multiplicative thinking and align problems to that standards progression.

10. Proportional Reasoning
Proportional reasoning has been referred to as the capstone of the elementary curriculum and the cornerstone of algebra and beyond.
It begins with the ability to understand multiplicative relationships, distinguishing them from relationships that are additive.
Van de Walle,J. (2009). Elementary and middle school teaching developmentally. Boston, MA: Pearson Education.

11. The Big Race
Connie ran 50 meters. Beth ran 200 meters. Use the information above to write an additive (absolute) question & a multiplicative (relative) question.
Use whte boards – split it in half.
Connie ran 50 meters. Melissa ran four times as far as Melissa. How far did Melissa run?
Hold up and read to clarify the two formats.

12. Proportional Reasoning vs Proportions
Proportional reasoning goes well beyond the notion of setting up a proportion to solve a problem—it is a way of reasoning about multiplicative situations.
In fact, proportional reasoning, like equivalence is considered a unifying theme in mathematics.

13. What is a ratio? Read 2 paragraphs from the RP Progression.
How would you define the word ratio?
A pair of non-negative numbers a:b which are not both zero. (CCSSM glossary)
An ordered pair of numbers that express a multiplicative (relative) comparison.
Uses of ratios
Part-to-Part Comparison: number of girls to number of boys
Part-to-Whole Comparison: number of girls to number of
children in the family
Read the progressions p. Ratios, Rates Proportional Relationships 2-3
Make this handout.
Read 2 paragraphs starting at bottom of page 2 and complete the top paragraph on page 3.

14. Jones and King Families
The Jones Family (GBGBB)
The King Family (GBBG)

15. Ratio and Proportion: Parts of a Larger Whole
Ratio and proportion do not develop in isolation. They are part of an individual’s multiplicative conceptual field, which includes other concepts such as multiplication, division, and rational numbers.
Lo, J., & Watanabe, T. (1997). Developing ratio and proportional schemes: A story of a fifth grader. Journal for Research in Mathematics Education, 28,

16. First Steps on the Road: Multiplicative Comparison

17. 4.OA.1, 4.OA.2 & 5.NF.5a Work in a triad. Study one standard.
Divide your slate in half. On one side, rephrase this standard and on the other side, provide an example.
Share with your partner.

18. Standard 4.OA.1
Interpret a multiplication equation as a comparison, e.g., interpret 35 equals 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.
This will give us a glimpse into their understanding of multiplicative comparison problems/.
Focus on the different vocabulary “ 35 is 5 times as many as 7 and 7 times as many 5.” This completes the student’s understanding of x situations. 3OA1 is about groups of objects and 3OA3 is groups, arrays and measurement. May have to refer them back to reading p. 22 and 24.

19. Standard 4.OA.2
Cluster: Use the four operations with whole numbers to solve problem. 4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

20. Standard 5.NF.5a
Cluster: Apply and Extend previous understandings of multiplication and division to multiply and divide fractions.
5.NF.5 Interpret multiplication as scaling (resizing), by:
a: Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

21. Multiplicative Comparison Problems
Read the OA handout about
Multiplicative Comparison
Problems.
Highlight as you read, noting ideas that you understand and those that still confuse you.
Handout in dropbox

22. Sense Making….
Share with your triad a few ideas that struck you as critical to developing a sound understanding of multiplicative comparison problems.
5 minute partner share
Whole group discussion. Give them a situation like:
Blue hat $10
Red hat $30
Larger quantity unknown: The red hat costs $10. The blue hat cost 3 times as much as the red hat. How much does the blue hat cost?
Smaller quantity unknown: The blue hat cost $30. The cost of the blue hat is three times as much as the red hat. How much does the red hat cost?
Compare quantity unknown: The red hat cost $10 and the blue hat costs $30. The blue hat is how many times as expensive as the red hat?

23. A Parking Lot Problem
A fourth grade class counted the number of vehicles that went by the front entrance of the school between 9 o'clock and 10 o'clock. The total number of vehicles counted was 156. There were 3 times as many passenger cars as trucks. How many passenger cars and how many trucks were counted?
Use a tape diagram to illustrate and solve this problem.
Ask teachers to draw the strip diagram before showing them the sample.
Emphasize that the structure of the problem is the same. You still have 3 to 1 comparison
Picture is the same

24. Comparing Heights of Buildings
The Burj Khalifa (Dubai) is about 2 times as tall as the Eiffel Tower (Paris). The Eiffel Tower is about as tall as the Willis Tower (Chicago).
Which of these buildings is the tallest? Which is the shortest? Explain.
Draw pictures to illustrate.

25. Eiffel Tower
Willis Tower
Burj Khaliafa

26. Learning Intentions & Success Criteria
We are learning to…
Understand the development of ratio and proportional reasoning and the structure of problems that support that development.
We will be successful when….
We can explain the standards progression for the development of multiplicative thinking and align problems to that standards progression.

27. Disclaimer Core Mathematics Partnership Project
University of Wisconsin-Milwaukee,
This material was developed for the Core Mathematics Partnership 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.