Framework for Intervention

Instructional Materials


The revised 2005 Mathematics Framework sets the stage for addressing the intervention needs of students. Appendix E: Mathematics Intervention and Algebra Readiness Instructional Materials describes a Mathematics Intervention Program for grades 4-7. It delineates six “volumes” to represent the subset of standards that should be addressed in an intervention program serving strategic and intensive students.

Appendix E serves as a foundation for the identification and alignment of intervention resources, although instructional materials designed specifically for intervention will not be available until the 2007 mathematics instructional materials adoption. Until that time, schools and districts will need to conduct an alignment process using their existing state-adopted materials. This process involves matching key content standards to the intervention volumes, then determining which grade-level instructional materials would be most appropriate for intervention instruction. With this information, schools can construct an effective mathematics intervention program using their existing grade-level instructional materials.

Here is an example of how this alignment process might be undertaken by a school and/or district collaborative team designing intervention for fifth-grade students.

Example: Aligning Intervention to Grade-Level Instruction

Student scores from the mathematics portion of the California Standards Test identified ordered pairs, linear functions, and equations as areas requiring intervention.

Step 1—The planning team reviewed Appendix E of the 2005 Mathematics Framework, which aligns six intervention volumes (number skills, fractions and decimals, ratios and percentages, mathematical processes, functions and equations, and measurement) to key grade-level standards.

Volume 5 of the intervention guide deals with Functions and Equations, which correlates to the areas identified for intervention. It suggests that the following three content standards be considered in a fifth-grade intervention program addressing this topic area: Algebra and Functions 1.5; Statistics, Data Analysis, and Probability 1.4; and Statistics, Data Analysis, and Probability 1.5. One additional standard—Algebra and Functions 1.4—was identified as foundational for algebra readiness.

The chart below lists these standards and their references in the intervention guide. The number of CST questions indicates the relative importance of each standard. The team also looked at the standards in relationship to the California High School Exit Exam (CAHSEE) and found that three of the standards were key for that assessment as well.

Fifth Grade Intervention Alignment Example

California Standard


Algebra Readiness

Number of CST questions


Algebra and Functions 1.4: Identify and graph ordered pairs in the four quadrants of the coordinate plane


Topic 7



Algebra and Functions 1.5: Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs of integers on a grid

Volume 5




Statistics 1.4: Identify ordered pairs of data from a graph and interpret the meaning of the data in terms of the situation depicted by the graph

Volume 5




Statistics 1.5: Know how to write ordered pairs correctly; for example (x,y)

Volume 5




Step 2—Once the planning team determined the alignment of key grade-level standards to the intervention topic, the next step was to explore the released test items from the California Standards Tests related to those standards. This gave the planning team an opportunity to specifically identify what students are being asked to know and be able to do. In this case, the team identified seven Grade 5 released test items from CDE’s 2005 Released Test Questions, <> and matched each question to the standard or standards tested.

Step 3—Next, the planning team identified the academic vocabulary required for learning these four standards. Specifying the academic vocabulary required, and developing strategies for teaching that vocabulary, is a key part of successful mathematics intervention for English Learners.

  • Target Vocabulary: coordinate plane, coordinates, quadrant, Quadrant I, Quadrant II, Quadrant III, Quadrant IV
  • Review Vocabulary: axis, axes, coordinate grid, graph, grid, horizontal, label, location, ordered pair, origin, plot, point, vertical, x-axis, y-axis

Step 4—The planning team then located the targeted standards in their adopted instructional materials. Questions that the team considered when undertaking this materials review and planning for intervention instruction included:

  • How are the assessments in the instructional materials similar to or different from the CST released items?
  • What additional language or instructional strategies should be implemented to serve strategic and intensive students so that they can learn efficiently from basic grade-level instructional materials?

Below is a listing of which resources the team identified for addressing students’ intervention needs using adopted mathematics textbooks for Grade 5.

Harcourt Math

  • Topics: Graph Relationships and Integers on the Coordinate Plane; Using an Equation to Graph
  • Screening: Assessment Guide, Inventory Test, pages 149-151; Chapter 24 pre-test, page 437
  • Teach: Student Textbook, pages 438-449; Practice, Reteach, Challenge Masters, sections 24.1, 24.2, and 24.3; Spiral Review, page 583, items 13-16
  • Remediation: Student Textbook, Trouble-Shooting, page 437, items 5-19
  • Progress monitoring: Teacher Edition, pages 439, 441, and 445
  • Summative assessment: Assessment Guide, pages 149-150 and 151-152; Chapter 24 test, 18 items; Cumulative Review, page 451

Houghton Mifflin Math

  • Topics: Ordered Pairs and Graphing the Equation
  • Screening: Assessment Guide, Inventory Test, pages 5-6, no items for identified standards; Chapter 12 pre-test; Assessment Guide, page 92, items 1-11
  • Teach: Student Textbook, pages 574-6, 577, 584-91; Practice, Reteach, Challenge Masters, 12-2, 12-5, 12-6, 12-7, 12-8; Spiral Review, Teacher’s Edition, page 583, items 13-16
  • Remediation: Fourth Grade, Chapter 11
  • Progress monitoring: Student Textbook, Quick Check, page 592, items 1-7, and page 593, items 1-3; Teacher’s Edition, Explain Your Thinking, Lesson 2; Teacher’s Edition, Assess and Close, Lessons 2, 3, 5, 6, 7, and 8; Teacher’s Edition, Error Analysis, Lessons 2, 5, and 7
  • Summative assessment: Assessment Guide, Chapter 12 tests, Form A items 5-21 and 23, Form B items 5-21and 23; Student Textbook, Chapter 12 test, items 8-17, 19-20; Assessment Guide, Write About It Quarterly Test 4, items 22-25; Assessment Guide, end-of-year test, item 25
  • Reteach: Refresher Master, pages 45-46; Student Lessons, Mixed Review, pages 1-3, 5, 7-8; Student Textbook, Cumulative Review, page 583, items 2-3, page 593, items 1-2; Student Textbook, Extra Practice, sets B, C, D, and E; Student Textbook, Chapter Review, pages 598-599, items 8-13, 17-24; Student Textbook, Spiral Review, pages 67-72; Student Textbook, Another Look, page 602; Student Textbook, Enrichment, page 603

Macmillan/McGraw-Hill Math

  • Topics: Ordered Pairs and Graphing the Equation
  • Screening: Assessment Guide, pre-test forms A and B, pages 145-148, items 13-16; Bridge the Gap Transition Handbook, Prerequisite Skills Test, Chapter 10, page 127
  • Teach: Student Textbook, pages 446-449, 450-451 ,452, 472, 476-477, 478; Practice/Reteach, sections 10-6, 10-7
  • Remediation: Skill Builders, pages 128-129
  • Progress monitoring: Assessment Guide, Monitoring Student Progress, pages 157-188, no items for identified standards
  • Summative assessments: Student Textbook, Performance Assessment, page 478, items 23-25; Chapter 10, test items 23-25, Assessment Guide, post-test forms A and B, pages149-152, items 13-16