Assessment in this unit
Key Ideas, Expectations, and Opportunities
Using Assessment to Meet Individual Needs
The explicit expectations and assessment tasks in this unit describe what it means to “get it.” Providing feedback on these expectations helps identify students who need to access the content another way, need further practice opportunities, or are ready to extend or deepen their understanding of a concept. Instructional opportunities that help support the varied needs of students also need to be identified. These opportunities provide models that can be replicated or used multiple times, and can be used in a variety of settings (e.g., home, transitions, support classroom, as a center).
The Assessment Program serves the
following purposes:
- It provides information to teachers about what students know and can do. This information is used to guide instruction. An activity may help teachers answer questions about whole-class instruction: What do I do next? In the next minute? Next lesson? Next class? Next unit? Other assessments may help teachers decide how to support individual students, including those who struggle with a concept and those who are ready to be challenged.
- It communicates the goals of instruction to parents and students. What teachers choose to assess communicates to the class what they value. For example, if teachers want students to work hard at communicating problem-solving strategies, then it is important to assess mathematical communication.
- It provides feedback to students and parents about student progress. This includes teacher evaluation of student progress as well as students' assessment of their own progress.
Key Mathematical Ideas
The mathematical content in Math Trailblazers is organized around a set of Key Ideas. These Key Ideas are based on the National Council of Teachers of Mathematics (NCTM) Standards for the grade band as well as current thinking in the mathematics education community, e.g., Charles (2005), NCTM (2000), Van de Walle (2006). There is a set of Key Ideas for each content strand: Number, Algebra, Geometry, Measurement, and Data. They are based on “big ideas” in mathematics and describe what students should be able to do within each strand. The Key Ideas are shown in the table in Figure 1.
| 1. Number Sense: Understand the base-ten number system, recognize relationships among quantities and numbers, and represent numbers in multiple ways. |
2. Operations: Understand the meaning of numerical operations and their application for solving problems. |
3. Computation and Estimation: Use efficient and flexible procedures to compute accurately and make reasonable estimates. |
|
| 1. Identifying Patterns: Identify and describe patterns and relationships, including how a change in one variable relates to a change in a second variable. |
2. Tables and Graphs: Represent patterns and relationships with graphs, tables, and diagrams. |
3. Symbols: Represent patterns and relationships with symbols (includes using variables in formulas and as unknowns in equations). |
4. Using Patterns: Apply relationships, properties, and patterns to solve problems, develop generalizations, or make predictions. |
| 1. Shapes: Identify, describe, classify, and analyze 2- and 3-dimensional shapes based on their properties. |
2. Orientation and Location: Use coordinate systems to specify locations and describe spatial relationships. |
3. Motion: Apply transformations (slides, flips, and turns) and use symmetry to analyze mathematical situations. |
4. Geometric Reasoning: Use visualization, spatial reasoning, and geometric modeling to solve problems. |
| 1. Measurement Concepts: Understand measurable attributes of objects or situations (length, area, mass, volume, size, time) and the units, systems, and processes of measurement. |
2. Measurement Skills: Use measurement tools, appropriate techniques, and formulas to determine measurements. |
| 1. Data Collection: Select, collect, and organize data to answer questions, solve problems, and make predictions. |
2. Data Representation: Select and create appropriate representations, including tables and graphs, for organizing, displaying, and analyzing data. |
3. Data Description: Describe a data set by interpreting graphs, identifying patterns, and using statistical measures, e.g., average and range. |
4. Using Data: Apply relationships and patterns in data to solve problems, develop generalizations, and make predictions. |
Figure 1: Key Ideas for Math Trailblazers (Key Ideas addressed in Unit 8 are shaded.)
Expectations
To monitor students' growth across and within grades, there are a set of Expectations that describe what students are “expected” to do within each content strand. Expectations show the growth of the mathematical content within the Key Ideas for each strand.
| EXPECTATIONS |
|
Use this list of Expectations to assess students on the key concepts and skills in this unit. |
| E1.* |
Represent fractions using area models (circle pieces, fraction strips, drawings) and number lines. |
| E2.* |
Use words and numbers to name fractions. |
| E3.* |
Recognize that the same fractional parts of different-sized unit wholes are not equal. |
| E4.* |
Identify the unit whole when given a fractional part of a whole. |
| E5.* |
Name and represent fractions greater than one as mixed numbers and improper fractions using models (fraction strips, circle pieces, number lines). |
| E6.* |
Write number sentences from area models of fractions (e.g., 1/2 = 3/6, 1/3 + 1/3 = 2/3, 1/3 + 1/3 + 1/3 = 1/3 × 3). |
| E7.* |
Make connections among representations of fractions including symbols, words, area models, and number lines. |
| E8.* |
Find equivalent fractions using area models (circle pieces, fraction strips, drawings) and multiplication and division strategies. |
| E9.* |
Compare and order fractions using area models, number lines, and one-half as a benchmark. |
| E10.* |
Add and subtract fractions with like denominators using area models. |
| E11. |
Multiply fractions by a whole number (e.g., 1/3 × 3 = 1, 2/3 × 6 = 1/3 × 6 × 2). |
| E12.* |
Demonstrate fluency with the division facts for the 9s. |
| E13.* |
Determine the unknown number in a multiplication or division sentence relating three whole numbers for the 9s facts. |
Targeted Practice
This unit provides opportunities for additional targeted practice for some of the Expectations. See the chart in Figure 3 and the descriptions that follow. These opportunities connect directly to assessment tasks so the practice can be tailored to the current level of student progress.
- For students who are struggling with the Expectation, practice is targeted toward the foundational concepts and skills involved and often provides a different way to access the content.
- For students who are making significant progress toward the Expectation, practice is designed to help move toward proficiency and autonomy.
- For students who are already meeting the Expectation, opportunities are provided to deepen or extend understanding.
| Expectation |
Opportunities for Targeted Practice |
| E1. |
Represent fractions using area models (circle pieces, fraction strips, drawings) and number lines. |
|
| E2. |
Use words and numbers to name fractions. |
|
| E3. |
Recognize that the same fractional parts of different-sized unit wholes are not equal. |
|
| E4. |
Identify the unit whole when given a fractional part of a whole. |
|
| E5. |
Name and represent fractions greater than one as mixed numbers and improper fractions using models (fraction strips, circle pieces, number lines). |
- Play Show Me from Lesson 2
- Order Fraction Cards on the Benchmark Number Line from Lesson 9 and 11
|
| E6. |
Write number sentences from area models of fraction (e.g., 1/2 − 3/6, 1/3 + 1/3 = 2/3, 1/3 + 1/3 + 1/3 = 1/3 × 3). |
|
| E7. |
Make connections among representations of fractions including symbols, words, area models, and number lines. |
|
| E8. |
Find equivalent fractions using area models (circle pieces, fraction strips, drawings) and multiplication and division strategies. |
|
| E9. |
Compare and order fractions using area models, number lines, and one-half as a benchmark. |
|
| E12. |
Demonstrate fluency with the division facts for the 9s. |
- See the Letter Home for ways to use the Triangle Flash Cards
|
| E13. |
Determine the unknown number in a multiplication or division sentence relating three whole numbers for the 9s facts. |
- See the Letter Home for ways to use the Triangle Flash Cards
|
Figure 3: Expectations for Unit 8 with opportunities for targeted practice
Workshop
Much of the targeted practice is in the Lesson 7 Workshop: Many Ways to Show a Fraction and the Lesson 11 Workshop: More Than, Less Than, or Equal To which each provide menus of activities that revisit key concepts and skills developed earlier in the unit. Based on students' self-assessment of their confidence with Content Expectations, students select activities from a Workshop Menu (Figure 4). Teacher guidance can help students find the appropriate level of practice based on evidence from earlier assessment tasks.
Figure 4: Workshop Menu from Lesson 7
Practice Menu
The Teacher Guide also has an optional Practice Menu in Lesson 8 for the Equivalent Fractions pages in the Student Guide. Teachers may offer the Practice Menu for students to self-assess and choose practice based on the strategies they use to find equivalent fractions [E8].
Games
There are several games in this unit that can be used to provide targeted practice. These games can be played in centers, as part of class transitions, in another setting, or at home.
Activities
There are two activities in this unit that can also be a vehicle for providing targeted practice. Once introduced in the unit, these activities can be tailored to meet the needs of students. Like games, these activities can be used during class transitions or used in other settings or at home.
- Circle Pieces Review Master from Lesson 6
- Order Fraction Cards on the Benchmark Number Line from Lessons 9 and 11
) on the Unit 8 Key Assessment Opportunities Chart. See Figure 2. In this unit, students will also get better acquainted with the Math Practices Expectations by discussing them in the context of a specific problem, receiving feedback, reviewing a peer's work, and revising their work.