Reviewing Using the Instructional Materials Evaluation Tool: Mathematics

Module 102: Rigor and Balance (AC 1)

www.achievethecore.org

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Essential Questions

• How does the Instructional Materials Evaluation Tool (IMET) reflect the major features of the Standards and the Shifts?

• What understandings support high-quality, accurate application of the IMET metrics?

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Goals

• Understand how aligned materials embody the shifts inherent in the Common Core State Standards

• Understand the precise meaning of each metric of the IMET

• Recognize examples and non-examples related to each metric of Alignment Criterion 1

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Non Negotiable vs. Alignment Criteria

Non- Negotiable Criteria Alignment Criteria

Each metric rated Meets/Does Not Meet

Each metric rated Meets/Partially Meets/Does Not Meet

Every metric must be met to rate the criterion as Meets

Must receive 5 out of 6 points to rate the criterion as Meets

Alignment Criterion 1: Rigor and Balance

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Alignment Criterion 1: Materials must reflect the balances in the Standards and help students meet the Standards’ rigorous expectations.

Rigor

IMET p. 17

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What Is Rigor?

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The Three-Legged Stool

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AC 1 Metric 1A: The materials support the development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content Standards or cluster headings.

Conceptual Understanding

IMET pp. 18-19

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Conceptual Understanding

• Teach more than “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives

• Students are able to see math as more than a set of mnemonics or discrete procedures

• Conceptual understanding is about mathematical ideas, often distinct from context.

• Conceptual understanding supports the other aspects of rigor (procedural skill and fluency, and application)

IMET p. 19

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AC 1A: Is conceptual understanding attended to thoroughly where the Standards set explicit expectations for understanding or interpreting?

4.OA.A Use the four operations with whole numbers to solve problems.

4.OA.A.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 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.

4.OA.B.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.

4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

Handout p. 2

IMET p. 18

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AC 1A: Do the materials feature high-quality conceptual problems and conceptual discussion questions?

[I]n solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Problems are problems because students haven’t yet learned how to solve them; students are learning from solving them.

- K-8 Publishers’ Criteria, Spring 2013, p. 17

IMET p. 18

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AC 1A: Do the materials feature high-quality conceptual problems and conceptual discussion questions?

4.NF.A

Handout p. 3

IMET p. 18

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AC 1A: Do the materials feature high-quality conceptual problems and conceptual discussion questions?

Which number is larger?

1.7 or 17 twelfths

Explain how you can tell without drawing apicture.

4.NF.C

Handout p. 3

IMET p. 18

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AC 1A: Do the materials feature opportunities to identify correspondences across mathematical representations?

Handout p. 4

IMET p. 18

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AC Metric 1A: Review a Full Lesson

- Review the grade 4 lesson on pp. 5-8 of your handout.

- Note evidence for each question associated with AC Metric 1A on p. 19 of your IMET.

- Discuss:- Gut check: Would these materials meet, partially meet, or not meet AC 1A? What is your evidence?

- What other things would you want to look at in these materials to determine your rating?

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AC Metric 1B: The materials are designed so that students attain the fluencies and procedural skills required by the Standards.

Procedural Skill and Fluency

IMET pp. 20-21

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Procedural Skill and Fluency

• The Standards require speed and accuracy in calculation.

• Materials structure class time and/or homework time for students to practice core functions such as single-digit multiplication so that they are more able to understand and manipulate more complex concepts

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Required Fluencies in K-6

GradeStandard

Required Fluency

K K.OA.5 Add/subtract within 5

1 1.OA.6 Add/subtract within 10

2 2.OA.22.NBT.5

Add/subtract within 20 (know single-digit sums from memory)Add/subtract within 100

3 3.OA.73.NBT.2

Multiply/divide within 100 (know single-digit products from memory)Add/subtract within 1000

4 4.NBT.4 Add/subtract within 1,000,000

5 5.NBT.5 Multi-digit multiplication

6 6.NS.2,3 Multi-digit divisionMulti-digit decimal operations

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IMET p. 21

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AC 1B: Is progress toward fluency and procedural skill interwoven with students’ developing conceptual understanding of the operations in question?

Handout p. 9

IMET p. 20

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AC 1B: Do the materials in grades K–6 provide repeated practice toward attainment of fluency Standards?

IMET p. 20

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AC Metric 1B: Review a Full Lesson

• Review the grade 4 lesson on pp. 10-13 of your handout.

• Note evidence for each question associated with AC Metric 1B on p. 21 of your IMET.

• Discuss:- Gut check: Would these materials meet, partially meet, or not meet AC 1B? What is your evidence?

- What other things would you want to look at in these materials to determine your rating?

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AC Metric 1C: The materials are designed so that teachers and students spend sufficient time working with applications, without losing focus on the Major Work of each grade.

Application

IMET pp. 22-23

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Application

• Students can use appropriate concepts and procedures for application even when not prompted to do so.

• Teachers provide opportunities at all grade levels for students to apply math concepts in “real world” situations, recognizing this means different things in K-2, 3-5, and 6-8

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IMET p. 23

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AC 1C: Are there single- and multi-step contextual problems that develop the mathematics of the grade, afford opportunities for practice, and engage students in problem solving?

A grower packs 4,568 peaches. He packs the most peaches possible, dividing them equally into 9 boxes, and then gives away the remaining peaches.

• How many of the remaining peaches does he give away?

• If he sells 7 boxes, how many peaches does he have left?

Handout p. 14

IMET p. 22

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AC 1C: Do application problems particularly stress applying the Major Work of the grade?

1) A clown needs 275 balloons for a party he is going to, but the balloons only come in packs of 8. How many packs of balloons does he need to buy?

2) A florist splits 878 flowers into vases by putting 9 flowers in each vase. She needs more flowers to fill the last vase. How many more flowers does she need so that the last vase also has 9?

Handout p. 15

IMET p. 22

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AC 1C: Does modeling build slowly across K–8, with applications that are relatively simple in earlier grades and when students are encountering new content? In grades 6–8, do the problems begin to provide opportunities for students to make their own assumptions or simplifications in order to model a situation mathematically?2.OA.A.1A pencil costs 59 cents, and a sticker costs 20 cents less. How much do a pencil

and a sticker cost together?

7.EE.B.4Shelbi is running in a half marathon today. About ½ hour before the race, she parks her car in a garage about ¼ mile from the starting line. The prices for parking are shown in the following table:

As she is running, she remembers that she only has $13. What is the slowest pace Shelbi can run, in minutes per mile, and still pay less than $13 for parking? Explain your answer. State any assumptions you made. Handout p. 16

IMET p. 22

Less than 1 hour $8

1 hour to less than 2 hours $12

2 hours to less than 3 hours $15

3 hours or more $18

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AC Metric 1C: Review a Full Lesson

• Review the grade 4 lesson on pp. 17-20 of your handout.

• Note evidence for each question associated with AC Metric 1C on p. 23 of your IMET.

• Discuss:- Gut check: Would these materials meet, partially meet, or not meet AC 1C? What is your evidence?

- What other things would you want to look at in these materials to determine your rating?

IMET p. 24

Wrapping Up

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Alignment Criterion 1 and Metrics

AC #1: Balance and Rigor• 1A: Conceptual Understanding• 1B: Procedural Skill and Fluency• 1C: Application

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Essential Questions -- Discuss

• How does the Instructional Materials Evaluation Tool (IMET) reflect the major features of the Standards and the Shifts?

• What understandings support high-quality, accurate application of the IMET metrics?