1. The call to action:

2. Understanding addition and subtraction There is a lot of important work to be done to ensure that students understand addition and subtraction in 1 st and 2 nd grade. The Call to Action is looking for units that connect understanding addition and subtraction to length.

3. Opportunity for coherence Measurement Operations and Algebraic Thinking Connecting the two domains together enhances students’ understanding of both.

4. Linear measurement work in grade 1 and 2 asks students to consider the quantity of units required to represent length Students combine and compare lengths to deepen the understanding of the meaning of addition and subtraction

5. From the Draft K–5 Progression on Measurement and Data (measurement part)Draft K–5 Progression on Measurement and Data (measurement part) Length and unit iteration are critical in understanding and using the number line in Grade 3 and beyond. Length is… one of the most prevalent metaphors for quantity and number, e.g., as the master metaphor for magnitude (e.g., vectors, see the Number and Quantity Progression) https://commoncoretools.files.wordpress.com/2012/07/ccss_progression_gm_k5_2012_07_21.pdf

6. From the Draft K–5 Progression on Measurement and Data (measurement part)Draft K–5 Progression on Measurement and Data (measurement part) To use a number line diagram to represent whole numbers as lengths students need to understand that number lines have specific conventions the use of a single position to represent a whole number and the use of marks to indicate those positions. a number line diagram is like a ruler in that consecutive whole numbers are 1 unit apart, thus they need to consider the distances between positions and segments when identifying missing numbers Students think of a number line diagram as a measurement model and use strategies relating to distance, proximity of numbers, and reference points. https://commoncoretools.files.wordpress.com/2012/07/ccss_progression_gm_k5_2012 _07_21.pdf By connecting measurement to addition and subtraction, students strengthen their understanding of a number line as iterations of equal size and of measurement as it relates to a quantity of units.

7. 1 st grade standards Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

8. Do the math: This task illustrates how using length units in addition and subtraction word problems connect students understanding of iterating units into work with operations. The context of the problem encourages students to use the number line, reinforcing the understanding of representing equal length, non-overlapping units. https://www.illustrativemathematics.org/content-standards/tasks/196

9. 2 nd grade standards Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

10. This task illustrates the connection between addition and subtraction and moving equal units on the number line. The numbers students are working with are kept small to allow for focus on the connection between the operation and units of length. https://www.illustrativemathematics.org/content-standards/tasks/1081

11. Think about this… We use the phrase “out of proportion” in all sorts of situations.

12. Like this:

13. Or this: 15 years in prison for stealing a gumball.

14. Something in each was out of proportion. What do we mean by that?

15. We should be able to describe each situation in the same mathematically precise way.

16. Hint: We need to focus on two things simultaneously, not one thing. Proportionality is based on a relationship between two quantities.

17. The Relevant Standards:

18. Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

21. Important elements:

22. From the Progressions Ratios arise in situations in which two (or more) quantities are related In the Standards, a quantity involves measurement of an attribute http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

23. From the Progressions Some authors distinguish ratios from rates, using the term “ratio” when units are the same and “rate” when units are different; others use ratio to encompass both kinds of situations. The Standards use ratio in the second sense, applying it to situations in which units are the same as well as to situations in which units are different. http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

24. More from the Progressions http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

25. Proportional relationships involve collections of pairs of measurements in equivalent ratios. ratio notation should be distinct from fraction notation A collection of equivalent ratios can be graphed in the coordinate plane. The graph represents a proportional relationship. The unit rate appears in the equation and graph as the slope of the line, and in the coordinate pair with first coordinate 1. More from the Progressions http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

26. Grade 6 (From the Progressions) http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

27. As students generate equivalent ratios and record them in tables, their attention should be drawn to the important role of multiplication and division in how entries are related to each other In other words, when the elapsed time is divided by 2, the distance traveled should also be divided by 2. More generally, if the elapsed time is multiplied (or divided) by N, the distance traveled should also be multiplied (or divided) by N More from the Progressions http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

28. As students become comfortable with fractional and decimal entries in tables of quantities in equivalent ratios, they should learn to appreciate that unit rates are especially useful for finding entries More from the Progressions http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

29. Sidetrack: Sue and Julie were running equally fast around a track. Sue started first. When she had run 9 laps, Julie had run 3 laps. When Julie completed 15 laps, how many laps had Sue run? Cramer, K., Post, T., & Currier, S. (1993). Learning and Teaching Ratio and Proportion: Research Implications. In D. Owens (Ed.), Research Ideas For the Classroom (pp. 159-178) NY: Macmillan Publishing Company. Downloaded 6/9/2011 from http://www.cehd.umn.edu/rationalnumberproject/93_4.html http://www.cehd.umn.edu/rationalnumberproject/93_4.html

30. Grade 7 (Progressions) They work with equations in two variables to represent and analyze proportional relationships. Students recognize that graphs that are not lines through the origin and tables in which there is not a constant ratio in the entries do not represent proportional relationships. they write equations of the form y = cx, where c is a constant of proportionality unit rate as the amount of increase in y as x increases by 1 unit in a ratio table [slope triangle] http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

32. From the EQuIP Call to Action

33. Sample tasks:

34. https://www.illustrativemathematics.org/content-standards/6/RP/A/tasks/496

37. https://www.illustrativemathematics.org/content-standards/6/RP/A/3/tasks/1982

39. https://www.illustrativemathematics.org/content-standards/7/RP/A/2/tasks/181

40. Task A text book has the following definition for two quantities to be directly proportional: We say that y is directly proportional to x if y=kx for some constant k. For homework, students were asked to restate the definition in their own words and to give an example for the concept. Below are some of their answers. Discuss each statement and example. Translate the statements and examples into equations to help you decide if they are correct. https://www.illustrativemathematics.org/content-standards/7/RP/A/2/tasks/1527

41. Marcus: This means that both quantities are the same. When one increases the other increases by the same amount. An example of this would be the amount of air in a balloon and the volume of a balloon. Sadie: Two quantities are proportional if one change is accompanied by a change in the other. For example the radius of a circle is proportional to the area. Ben: When two quantities are directly proportional it means that if one quantity goes up by a certain percentage, the other quantity goes up by the same percentage as well. An example could be as gas prices go up in cost, food prices go up in cost. Jessica: When two quantities are proportional, it means that as one quantity increases the other will also increase and the ratio of the quantities is the same for all values. An example could be the circumference of a circle and its diameter, the ratio of the values would equal π. https://www.illustrativemathematics.org/content-standards/7/RP/A/2/tasks/1527