1. Developing Linear Thinking & Extending to a Ruler
Core Mathematics Partnership
Building Mathematical Knowledge and
High-Leverage Instruction for Student Success
Tuesday July 19, 2016
8:00-10:30

2. Maria’s Crayon
5 minutes
What do students need to understand in order to do this task?

3. [Length] is the basic geometric measurement
[Length] is the basic geometric measurement. Length and unit iteration are critical in understanding and using the number line in Grade 3 and beyond. - K-5, Geometry Measurement Progressions, p. 4

4. Learning Target & Success Criteria
We are learning to…
help children understand the connection between a number line and a ruler.
We will be successful when we can…
use unit length language to connect linear measurements and rulers.
explain the connection between number lines and rulers.

5. Second Grade Length Measurement Standards
1. Read standards 2.MD.1, 2.MD.2, 2.MD.3, 2.MD.4
How do they build from what we learned yesterday?
What is the new expectation for length measurement in 2nd grade?

6. Transitioning towards Standard Units
Take out your K-5, Geometric Measurement Progression
and read the third full paragraph on page 9.

7. 2.MD.5 & 2.MD.6 By the end of 2nd grade…
2.MD.5 Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.
2.MD.6 Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the number 0, 1, 2…and represent whole-number sums and differences within 100 on a number line diagram.

8. Let’s Get Started! Draw a 0-10 number line on your white board.
What were some things you considered as you were drawing your number line?
Turn and share your thinking with your neighbor.

9. What are you attending to?
This is sophisticated thinking!! 1 2 10
Wait a minute…do I have enough room
for the other numbers??? 10 5
So what does this have to do with measurement?
How many hash marks do I need? 4? 3? 5? 1 2 3

10. Lingering Questions…
What do we want children to understand about linear models [number lines and rulers] and how do we go about building that understanding?
Read the excerpt from NCTM, K-2 Geometry and Measurement (blue).
Find the section titled: Using number lines.
Read the first three paragraphs.

11. Where does this work start?
Before using number lines and rulers children need to know what the units and numbers represent.
Van de Walle, Lovin, Karp, & Bay-Williams, , p. 135.
Where does this work start?
What is a tool that will support this goal?

12. Number Path
What do you notice about the number path? Using counters, show 8 on your number path. What statements can you make about 8? What understandings will students need in order to transition from counting to linear measurement?
Looking for: “8 is one more than 7,” “8 is 3 more than 5” understand sequentialist, hierarchical inclusion order

13. This square is where I say eight.
There are eight squares.
This is the distance to 8.
Discrete counting vs. linear counting.

14. What understandings are young students developing as they transition counting to linear models?
Understand that a square represents one count or one unit.
Use their understanding of benchmark numbers to understand that linear measurement is additive.
See a linear representation of quantity.
Where is 5? Where is 10

15. It’s All About the Unit!

16. Defining the Unit Think of our experiences with the number path…
What is the total quantity represented on this linear model? How do you know?
Where is 6? Justify your answer.
Where is 17? Justify your answer.
How does this linear model support an understanding of number relationships?
Where would 0 be?
“6” marks the spot that is the distance from 0.
Make sure “unit(s)” in the language

17. Hash Marks How do the numbers help us define the unit?
How do the numbers help us define the unit?
How do the hash marks support understanding of the unit?
What connections do you see to a ruler?
Give us more definition to the units. The 5 helps us define that that’s the distance of 5 units.
They’re all the same size.

18. A Ruler: Number Line Paired with Units of Measure
This is the total length of 7 units. 9 10 8 7 6 5 4 3 2 1
These are 7 length units.
Attach quote from progressions.

19. Grade 2 Progressions
Read the K-5 Geometric Measurement Progression - Grade 2.
Read starting on p.12 through the first two paragraphs on page 13.
How does the information on these pages connect to your work with kids?
How does the information connect to Math Practice 6: Attend to Precision?
10 minutes

20. Practicing Unit Language
Turn over 2 number cards.
Use a paperclip to plot each on your number line.
Work with your partner to figure out the distance between the two numbers.
Use “unit” language to explain your reasoning.
Example: “I picked the numbers 3 and 7. There are 4 length units between the numbers 3 and 7.”

21. Transitioning to an Unmarked Ruler
Consider the red and white ruler.
Estimate how many units it would take to measure the length of your large white board.
Practice your “length-unit” language to explain the length
of the whiteboard you measured.

22. Scaffolding Experiences to A Ruler
Study the next set of rulers on your table.
What do you notice?
How are they similar to the other rulers?
How are they different from typical rulers found in classrooms?

23. Applying the Measurement Process to Length Measurement
Identify an item.
Talk through the measurement process with your partner.
“My item will be measured in ___________ units.
(inch/centimeter/meter/feet)
The _________ is _________ length units long.
The __________ is _________ (in/cm/ft/m) long.”

24. Maria’s Crayon
5 minutes
What do students need to understand in order to do this task?

25. Analyzing Maria’s Crayon Student Work Gr. 2 and Gr. 3
Begin with Grade 2 work. Distribute copies of Maria’s Crayon among your table group.
Study the work with a table partner.
Note understandings and misconceptions
Record on sticky notes and place on work.
Repeat with Grade 3 student work.
What trends did you note in the student work?
Check the time and we can divide table and one side look at grade 2 and one side look at grade 3.

26. Learning Target & Success Criteria
We are learning to…
help children understand the connection between a number line and a ruler.
We will be successful when we can…
Use length-unit language to connect linear measurements and rulers.
explain the connection between number lines and rulers.

27. “Teachers don’t let kids do it enough!”
A Walk Away Thought…
What is the number one mistake teachers make
when teaching children Measurement?
“Teachers don’t let kids do it enough!”
Rosemary Irons, WMC 2015
Do you agree or disagree with Rosemary’s
statement?

28. PRR: Measurement
Review your notes and the Progressions (if needed) for Grade 1 & 2
What insights did you gain about the measurement process?
How does your learning about the measurement process impact the work you do with students in your grade level? Provide a specific example.

29. Core Mathematics Partnership Project
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.

30. Scaffolding from Number Line to Rulers
Semi-Structured Number Line
Unmarked Unit Measure
Ruler