Assessment in this unit

The following components of the assessment program provide teachers with the tools to plan instruction and communicate student progress on the mathematical content in Grade 5.

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Resources for Teachers

  • Charles, R. “Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics.” Journal of Mathematics Education Leadership, 7 (3), pp. 9–24, 2005.
  • Dacey, L., and J.B. Lynch. Math for All: Differentiating Instruction Grades 3–5. Math Solutions Publications, CA, 2007.
  • Guskey, T.R., and J.M. Bailey. Developing Grading and Reporting Systems for Student Learning. Corwin Press, Inc., CA, 2001.
  • Jorgensen, M. “Assessing Habits of Mind: Performance-Based Assessment in Science and Mathematics.” Eric Clearinghouse for Science, Mathematics, and Environmental Education, Columbus, OH, 1994.
  • Krulik, S., and J.A. Rudnick. Assessing Reasoning and Problem Solving: A Sourcebook for Elementary School Teachers. Allyn and Bacon, Pearson, NJ, 1998.
  • Mathematics Assessment: A Practical Handbook for Grades 3–5. J.K. Stenmark and W.S. Bush, ed. National Council of Teachers of Mathematics, Reston, VA, 2001.
  • Murray, M., with J. Jorgensen. The Differentiated Classroom: A Guide for Teachers, K–8. Heinemann, Portsmouth, NH, 2007.
  • Pellegrino, J.W., and S.R. Goldman. “Beyond Rhetoric: Realities and Complexities of Integrating Assessment into Classroom Teaching and Learning.” In The Future of Assessment: Shaping Teaching and Learning. C. Dwyer ed. Erlbaum, Mahwah, NJ, 2007.
  • Principles and Standards for School Mathematics. National Council of Teachers of Mathematics, Reston, VA, 2000.
  • Tomlinson, C.A. How to Differentiate Instruction in Mixed-Ability Classrooms. 2nd Ed. Association for Supervision and Curriculum Development, Alexandria, VA, 2001.
  • Van de Walle, J.A., and L. Lovin. Teaching Student-Centered Mathematics: Grades 3–5. Pearson Education, Boston, 2006.
Key Ideas Tools for Feedback Individual Needs Assessment

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).

Tools for Feedback and Monitoring

Monitoring Progress
Progress can be monitored and reported over time using the Unit Assessment Record and Unit Individual Assessment Record and by collecting reviewed student work samples in a portfolio.

Feedback Boxes

The Assessment Program serves the
following purposes:

  1. 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.
  2. 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.
  3. 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.
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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.

Number
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.
Algebra 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.
Geometry 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.
Measurement 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.
Data 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 10 are shaded.)
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Expectations

To monitor students' growth across and within grades, there is 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* Identify and find equivalent fractions using tools (e.g., area models, number lines) and multiplication and division strategies.
E2* Represent and identify the simplest form of a fraction using tools (e.g., area models) and multiplication and division strategies.
E3* Represent addition, subtraction, multiplication, and division of fractions with area models, number lines, number sentences, drawings, and stories.
E4* Multiply and divide fractions using area models, drawings, and number lines.
E5 Solve word problems involving addition, subtraction, and multiplication of fractions.
E6 Explain the effects of factors less than and greater than 1 on the product of fractions (e.g., is the product of 1/2 × 3 larger or smaller than 3).
E7 Choose appropriately from among estimation and computation strategies.
E8* Add and subtract fractions including those with unlike denominators using area models and paper-and-pencil methods.
E9 Estimate sums, differences, products, and quotients of fractions using benchmarks and mental math strategies.
E10 Find common denominators and use them to add, subtract, and compare fractions.
* Denotes Benchmark Expectation
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Unit Assessment Record

Use this vehicle to record student progress on each of the unit's Content Expectations. The Key Assessment Opportunities Chart identifies activities within each lesson that are appropriate for assessing each Expectation in the unit.

Unit 10 Assessment Record

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Unit Individual Assessment Record

Upon completion of each unit, transfer information from the Unit Assessment Record to Individual Assessment Records. This latter tool becomes a compilation of the progress for each child across the units.

Individual Assessment Record

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Feedback Boxes

Feedback Boxes are provided with several activities to report progress toward these Expectations. Look for the box ( ) on the Unit 10 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 own work.

Turkey and Potatoes
Feedback Box		 Expectation Check
In		 Comments
Represent the simplest form of a fraction. [Q# 3, 4B] E2
Represent multiplication of fractions with area models, number lines, number sentences, and drawings. [Q# 2] E3
Multiply fractions using area models, drawings, and number lines. [Q# 1–4] E4
Solve word problems involving multiplication of fractions. [Q# 3–4] E5
  Yes… Yes, but… No, but… No…
MPE2. Find a strategy. I choose good tools and an efficient strategy for solving the problem [Q# 1–4]
MPE4. Check my calculations. If I make mistakes, I correct them.
MPE5. Show my work. I show or tell how I arrived at my answer so someone else can understand my thinking.
Figure 2: Sample Feedback Box from Lesson 7
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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.

Expectation Opportunities for Targeted Practice
E1.* Identify and find equivalent fractions using tools (e.g., area models, number lines) and multiplication and division strategies.
E3.* Represent addition, subtraction, multiplication, and division of fractions with area models, number lines, number sentences, drawings, and stories.
E4.* Multiply and divide fractions using area models, drawings, and number lines.
E5. Solve word problems involving addition, subtraction, and multiplication of fractions.
E7. Choose appropriately from among estimation and computation strategies.
E8.* Add and subtract fractions including those with unlike denominators using area models and paper-and-pencil methods.
E9. Estimate sums and differences of fractions using benchmarks and mental math strategies.
E10. Find common denominators and use them to add, subtract, and compare fractions.

*Denotes Benchmark Expectation

Figure 3: Expectations for Unit 10 with opportunities for targeted practice
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Workshop

Much of the targeted practice is in the Lesson 6 Workshop: Add and Subtract Fractions and the Lesson 11 Workshop: Multiply and Divide Fractions. These Workshops 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 for Lesson 11
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Games

There are two 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.

  • Circle Duets from Lesson 3
  • Closest To from Lesson 5