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.
  • Murray, M., with J. Jorgensen. The Differentiated Classroom: A Guide for Teachers, K–8. Heinemann, Portsmouth, NH, 2007.
  • Mathematics Assessment: A Practical Handbook for Grades 3–5. J.K. Stenmark and W.S. Bush, eds. National Council of Teachers of Mathematics, Reston, VA, 2001.
  • 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.
  • National Council of Teachers of Mathematics. 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, Unit Individual Assessment Record, Math Facts Class 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 7 are shaded.)
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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* Demonstrate understanding of division of multidigit numbers by one- and two-digit numbers using models.
E2* Show connections between models and strategies for multidigit division.
E3 Interpret remainders from division of multidigit numbers.
E4 Follow the order of operations (e.g., using parentheses).
E5* Estimate quotients for division of multidigit numbers by one- and two-digit numbers.
E6* Divide numbers that are multiples of ten.
E7* Divide multidigit numbers by one- and two-digit divisors using paper and pencil.
E8 Use ratios to solve problems.
E9 Estimate the area of shapes with curved sides.
E10 Choose appropriate units to measure variables (e.g., length, area, volume, mass).
E11 Represent variables and procedures.
E12 Make a point graph and draw a best-fit line.
E13 Find the mean and median of a data set.
E14 Make predictions and generalizations using data tables, graphs, and averages.
E15* Demonstrate fluency with the multiplication and division facts for the last six facts
(4 × 6, 4 × 7, 4 × 8, 6 × 7, 6 × 8, 7 × 8).
* 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 7 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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Math Facts Class Record

Use this tool to track the progress of students' fluency with the fact groups that are practiced and assessed in the Daily Practice and Problems.

Math Facts Class 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 7 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.

How Many Drops
Feedback Box		 Expectation Check
In		 Comments
Use ratios to solve problems. E8
Make predictions and generalizations using data tables, graphs, and averages. E14
  Yes… Yes, but… No, but… No…
MPE1. Know the problem. I read the problem carefully. I know the questions to answer and what information is important.
MPE2. Find a strategy. I choose good tools and an efficient strategy for solving the problem.
MPE5. Show my work. I show or tell how I arrived at my answer so someone else can understand my thinking.
MPE6. Use labels. I use labels to show what numbers mean.
Figure 2: Sample Feedback Box from Lesson 8
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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.* Demonstrate understanding of division of multidigit numbers by one- and two-digit numbers using models.
E2.* Show connections between models and strategies for multidigit division.
E3. Interpret remainders from division of multidigit numbers.
E5.* Estimate quotients for division of multidigit numbers by one- and two-digit numbers.
E6.* Divide numbers that are multiples of ten.
E7.* Divide multidigit numbers by one- and two-digit divisors using paper and pencil.
E15.* Demonstrate fluency with the multiplication and division facts for the last six facts (4 × 6, 4 × 7, 4 × 8, 6 × 7, 6 × 8, 7 × 8).

*Denotes Benchmark Expectation

Figure 3: Expectations for Unit 7 with opportunities for targeted practice
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Workshop

Much of the targeted practice is in Lesson 4 Workshop: Division Strategies. This workshop provides a menu 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 4
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Games

There are two games that can be used to provide targeted practice for the concepts developed in this unit. These games can be played in centers, as part of class transitions, in another setting, or at home.

  • Division Digits Game from Lesson 3
  • Quotient Quest from Lesson 4