1. Understanding New Expectations for Rigor in Mathematics
High School Level

2. Turn and talk to the person next to you…name three aspects of rigor
So What is Rigor?
Turn and talk to the person next to you…name three aspects of rigor

3. 3 Aspects Conceptual Understanding Procedural Skill and Fluency
Application:
……which means

4. Rigor includes:
Conceptual Understanding: Students need a conceptual understanding of key concepts, such as place value and ratios. Teachers support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than just a set of mnemonics or discrete procedures.

5. High school rigor includes:
Conceptual Understanding:
Explain and use the relationship between the sine and cosine of complementary angles.
Other words that indicate need for conceptual understanding are interpret, recognize, describe, etc.

6. Rigor includes:
Procedural Skill and Fluency: Students need to have speed and accuracy when performing calculations. Teachers should structure class/homework time for students to practice core functions such as single-digit multiplication so students have access to more complex concepts and procedures.

7. High school rigor includes:
Procedural Skill and Fluency: The ability to see structure in expressions and to use this structure to rewrite expressions is a key skill in everything from advanced factoring (e.g., grouping) to summing series to the rewriting of rational expressions to examine the end behavior of the corresponding rational function.

8. Rigor includes:
Application: Students need to be able to use math flexibly for applications. Teachers should provide opportunities for students to apply math in context. Teachers in content areas outside of math, particularly science, ensure that students are using math to make meaning of and access content.

9. High school rigor includes:
Application: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods

10. And… 8 Mathematical Practices Application to real‐world Modeling
Multiple approaches

11. What is evidence of aspects of rigor?
What do students “say and do” in the classroom?
Turn and talk.

12. Analyzing a Task to Better Understand Rigor
Michelle’s Conjecture

13. Michelle wanted to investigate the effects of the vertex on a graph of f(x) = x2 +6x when f(x) is replaced by f(x + k).
Michelle graphed functions of the form f(x + k) for k = 1, 2, 3 and 4. For each of the functions she graphed, the x-coordinate of the vertex was negative and different for each value of k, the the y-coordinate of the vertex was the same for each value of k. Michelle made three conjectures based on her results.
The x-coordinate of the vertex depends on the value of k.
The x-coordinate of the vertex is negative for all values of k.
The y-coordinate is independent of the value of k.

14. Sub-Claim C: Highlighted Practices MP
Sub-Claim C: Highlighted Practices MP.3,6 with Connections to Content3 (expressing mathematical reasoning)
The student expresses grade/course level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others, and/or attending to precision when making mathematical statements..
CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others.
For High School, Sub-Claim C includes Major, Additional and Supporting Content.
CCSS.Math.Practice.MP6 Attend to precision.

15. Build new functions from existing functions.
CAS: 2.1.e.i (CCSS.Math.Content.HSF.BF.B.3) Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Colorado Academic Standard: 2.1.e.i was aligned to the Common Core State Standard F.BF.B.3

16. Evidence Table Evidence Statement Key Evidence Statement Text
Clarification MP
Calculator
F-BF.3-4
Identify the effect on the graph of a quadratic function of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases using technology.
1) Illustrating an explanation is not assessed here (See sub-claim D)
5, 3, 8
Item Specific

17. How do we know if students are successful?
How are conceptual understanding, procedure fluency, and/or application emphasized in this task?
How could evidence for math content be demonstrated in student work?
How could evidence for the standards of mathematical practice be demonstrated in student work?

18. Does the item address this claim?

19. Does the item address this standard?
Build new functions from existing functions.
CAS: 2.1.e.i (CCSS.Math.Content.HSF.BF.B.3) Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

20. Can we gather this evidence?

21. WARNING
The evidence tables are meant to clarify the details of assessment items. Within a standard there may only be a portion that needs clarification and that is the only part addressed in the evidence tables. The tables are only a supplementary document.

22. Does the item address this standard of mathematical practice?
CCSS.Math.Practice.MP3 Construct viable argument and critique the reasoning of others.