GPAEA Appendix 13B: Diagnostic Process and Intervention Recommendations

“What math does my student need for intervention?”

~and~

“What should I use for intervention?”

What math topic(s) does my student need for intervention?

• Work across the rectangles on Slide 5 left to right.
• The larger the rectangle, the more priority it should be given.
• For some rectangles, clarifying questions have been added.
• The higher the grade level of the student, the more important it is to intervene in the all the largest rectangles instead of working in the smaller rectangles.
• Basic Fact Fluency is a long-term effort, 5 minutes a day, everyday.
• Don’t choose an intervention that has not been taught yet in universal instruction.
• For older students, you should at least consider what content will provide the most access to current and upcoming topics in universal instruction
• Educators should diagnose before they provide an intervention

​

​

Do you want to…

Diagnose to know what math concepts the student needs to learn? Take me there!

Intervene with a high quality math intervention? Take me there!

​

• OR -

• OR -

Progress Monitor

Take me there!

​

Number Ideas to 20

Place Value

​

​

Grades 2 and up

Meaning of Operations

Add/Subtr

Early Algebraic Thinking

Basic Fact Fluency Grade 1 and up

Can they count?

Can they solve word problems?

Do they understand the equal sign?

Fractions

​

​

Grades 4 and up

Multi-Digit Computation

Add/Subtr

Only after place value work:

Expressions and Equations

Ratios and Proportions

Grade 6 and up

Can they compute with integers?

** Each rectangle links to diagnostic questions.

Meaning of Operations

Mult/Div

Multi-Digit Computation

Mult/Div

Back to choice slide

Number Ideas to 20

Prepare to Diagnose:

Print for each student being assessed:

• Organizer to record information

​

Materials needed:

• Unifix cubes
• Dot cards (can print from Subitizing Dots)
• Numeral Cards

​

You want to know:

• Does the student have a stable count sequence? One-to-one correspondence? Cardinality? Order irrelevance? Abstraction? See this short video for explanations.
• Does the student subitize to 5 with dots? With ten frames?
• Does the student compare numbers less than 10?

Back to flowchart

Early Algebraic Thinking

Prepare to Diagnose:

Print one for each student.

• Early Algebraic Thinking Data Organizer - Make a Copy
• Meaning of the Equals Sign (Grade 1+)

​

​

You need to know:

• Does the student understand the equal sign?

​

​

Back to flowchart

Place Value

Prepare to Diagnose:

Print for each student being assessed:

• Data organizer for the below bullets
• Task prompts
• Diagnose Place Value Student Sheet

​

Materials needed: Unifix cubes; Place Value blocks optional

​

You need to know:

1. Does the student see a two-digit number both as a collection of all ones and as tens and ones?
2. Does the student understand what each digit in a two-digit number represents?
3. Given any two-digit number, does the student know how many tens can be made? (Unitizing a ten)
4. Can the student operate on tens and ones separately for two-digit addition and subtraction?

Back to flowchart

Meaning of Operations

Add and Subtract

Prepare to Diagnose:

Print for each student.

• Meaning of Operation Student Task Sheet - Add/Subtract
• Data Organizer: Meaning of Operations - Addition and Subtraction

​

Materials:

• 25 counters

​

You need to know:

• Can the student draw a diagram and annotate it to show what is happening in an addition or subtraction problem?
• If not, can the student use manipulatives to act out the context for addition/subtraction?
• Can the student choose equations to match an addition or subtraction context?

​

Back to flowchart

Meaning of Operations

Multiply and Divide

Prepare to Diagnose:

• Data Organizer: Meaning of Operations - Multiplication and Division
• Meaning of Operation Student Task Sheet - Multiply/Divide

​

Materials Needed:

• Counters
• Grid paper
• One colored paper plate and many white plates of the same size

​

You need to know:

• Can the student draw a diagram and annotate it to show the actions in the context for multiplication/division?
• If not, can the student use manipulatives to act out the context for multiplication/division?
• Does the student understand multiplication as equal groups?
• Does the student understand multiplication as a comparative multiplication (scalar)?

​

Back to flowchart

Basic Fact Fluency

Prepare to Diagnose:

Print 1 form from the Boxes for the fact set you need:

• Assessing within 10 Addition and Subtraction with Boxes
• Assessing within 20 Addition and Subtraction with Boxes
• Assessing Multiplication with Boxes

Organize the data for an individual or a classroom::

• Add and subtract within 10: spreadsheet
• Add and subtract within 20: spreadsheet
• Multiplication: spreadsheet

Back to flowchart

You need to know:

• Which foundational fact groups does the student know?
• Addition/Subtraction:
• +/- 1, 2; +/- 0, 10; combinations of ten; and Doubles
• Multiplication/Division
• Times by 2, 5, 10; times by 1 and 0; Perfect squares and Doubling
• Does the student use strategies for some facts, such as near-doubles?
• Is the student counting on or counting all to figure out facts?
• Does the student use the distributive property for some facts?

​

Fractions

Prepare to Diagnose:

Print one for each student.

• Organizer for recording data
• Pre-Assessment for Positive Integers
• Understanding a Fraction as a Number Mini-Assessment Student Pages
• Additional Assessment Items for Fraction Models (use a separate color of paper)

​

You need to know:

• Does the student understand the structure of the number line for whole numbers?
• For both the area model and the number line model, can the student identify the unit? Identify a fraction? Represent a fraction? Express a non-unit fraction as a composition of unit fractions?
• Can the student explain the role of the numerator and denominator?
• Can the student compare fractions?
• Can the student generate equivalent fractions?

​

​

Back to flowchart

Multi-Digit Computation Addition and Subtraction

​

You need to know (assuming place value has been addressed as discussed above):

• For addition and subtraction:
• Can the student operate using a strategy based on separating place value or properties of operations?
• Does the student know what the regrouping numbers mean?
• For example, how does the student explain the 1 regrouped to the tens column when adding 34 + 27? Does the student know this is one ten and not a one?

​

​

​

Back to flowchart

Prepare to Diagnose:

Print one for each student:

• Multi-Digit Addition and Subtraction Computation Student Sheet
• Data Organizer

Multi-Digit Computation Multiplication and Division

You need to know (assuming place value has been addressed as discussed above):

• For multiplication:
• Can the student operate using partial products, area model, or another sense-making strategy?
• Does the student know what the regrouping numbers mean?
• For example, how does the student explain the 2 regrouped to the tens column when adding 14 × 5? Does the student know this is 2 tens and not 2 ones?
• For division:
• Can the student operate using partial quotients or another sense-making strategy?

​

​

​

Back to flowchart

Prepare to Diagnose:

Print one for each student:

• Multi-Digit Multiplication and Division Computation Student Sheet
• Data Organizer

Ratios and Proportions

Prepare to Diagnose:

Print one for each student:

• Ratios and Proportional Reasoning Data Organizer
• Student Tak Prompts (Answer keys are available at the links for each task.)

​

Consider investigating:

• Can the student solve problems using multiplicative comparisons?
• Can the student write a ratio represented by various models?
• Double number line?
• Discrete model?
• Ratio table?
• Does the student distinguish between part-part ratios and part-whole ratios?
• Can the student solve problems using unit rates?

​

​

Back to flowchart

Expressions and Equations

Prepare to Diagnose:

Print one for each student:

• Algebraic Expressions and Equations Data Organizer - Make a Copy
• Algebraic Expressions and Equations Assessment Student Sheet

​

You need to know:

• Does the student use order of operations correctly?
• Can the student describe the difference between an expression and an equation?
• Can the student represent a situation as an expression?
• Can the student operate with positive and negative integers? (Grade 7 and beyond)
• Can the student solve one-step equations? Two-step equations?
• Can the student interpret graphs of linear equations? Graphs of linear systems? (Grade 8 and beyond)

​

​

​

​

​

Back to flowchart

Number Ideas to 20

Place Value

​

​

Grades 2 and up

Meaning of Operations

Add/Subtr

Early Algebraic Thinking

Basic Fact Fluency Grade 1 and up

Can they count?

Can they solve word problems?

Do they understand the equal sign?

Fractions

​

​

Grades 4 and up

Multi-Digit Computation

Add/Subtr

Only after place value work:

Expressions and Equations

Ratios and Proportions

Grade 6 and up

Can they compute with integers?

** Each rectangle links to recommended interventions

Meaning of Operations

Mult/Div

Multi-Digit Computation

Mult/Div

Back to choice slide

Number Ideas to 20

Recommended Intervention:

• Number Sense Intervention
• Number Sense Interventions Information and how to purchase
• Appropriate for grades K-2

​

• Kathy Richardson - Developing Number Concepts
• Kathy Richardson Materials order information
• Appropriate for grades K-3
• Specifically Number Concepts: Book 1

​

• Bridges
• Bridges Math Intervention - One-pager and purchase info

​

• Learning Trajectories
• Make a free account
• Kickstart
• ​

Back to flowchart

Early Algebraic Thinking

Recommended Intervention:

• Equal sign
• True/False Equations - professional learning by https://tedd.org/
• True/False Equations - Slides created and ready to go!

Equal sign and properties of operations

• Leap One-Pager and purchase information

​

​

​

Back to flowchart

Back to flowchart

Place Value

Recommended Intervention:

• Bridges Math Intervention - One-pager
• Administer Volume 1 Placement Assessment Part 1

​

• Kathy Richardson
• Specifically Developing Number Concepts: Book 3
• Usually recommended as a bridge between Number Sense Intervention and SOAR
• See your GPAEA math consultant for Checks for Understanding

​

• SOAR Numbers Base Ten- See your GPAEA math consultant
• Specifically Numbers in Base Ten Binder

​

• MathLinks Essentials (for Middle School and above)
• Mathlinks Essentials Order Information

​

Back to flowchart

Meaning of Operations

Add and Subtract

Recommended Intervention:

• Pirate Math Equation Quest Information One-Pager
• Free

​

• Bridges Math Intervention - One-pager
• Administer Volume 4 Placement Assessment Part 1

​

• Kathy Richardson
• Specifically: Developing Number Concepts: Book 2
• Gap Closing - Intermediate, Junior
• Module 6 Relating Situations to Mathematical Operations
• Free

Back to flowchart

Meaning of Operations

Multiply and Divide

​

Recommended Intervention:

• Pirate Math Equation Quest Information One-Pager
• Free

​

• Bridges Math Intervention - One-pager
• Administer Volume 7 Placement Assessment Part 1

​

• Kathy Richardson
• Specifically Developing Number Concepts: Book 3
• Gap Closing - Intermediate, Junior
• Module 6 Relating Situations to Mathematical Operations
• Free

Back to flowchart

Basic Fact Fluency

Recommended Intervention:

• Bridges Math Intervention - One-pager
• For Basic Fact Fluency (addition and subtraction within 5, 10 and 20) administer Volume 2 Placement Assessment Part 1

​

• https://sites.google.com/gpaea.org/gpaeamath/basic-fact-fluency
• Assess to identify which strategies need instruction

​

• Bridges Math Intervention - One-pager
• For Basic Fact Fluency (multiplication and division) administer Volume 5 Placement Assessment Part 1

Back to flowchart

Fractions

Recommended Intervention:

• SOAR - See your GPAEA math consultant
• Number Operations: Fractions
• Bridges Math Intervention - One-pager
• Administer Volume 8 Placement Assessment Part 1

​

• Learning through Mathematical Representations
• Learning through Mathematical Representations (LMR) One-Pager
• Provides explicit instruction on the structure of number line also
• AIM
• https://meadowscenter.org/resource/project-aim-algebra-modules/
• AIM Intervention Modules - One Pager
• Free
• Gap Closing
• Free
• MathLinks Essentials (for Middle School and above)
• Mathlinks Essentials Order Information

Back to flowchart

Multi-Digit Computation Addition and Subtraction

​

​

​

​

​

Recommended Intervention:

• Bridges Math Intervention - One-pager
• Administer Volume 3 Placement Assessment Part 1

​

• SOAR - See your GPAEA math consultant
• Specifically Numbers in Base Ten Binder: Folder D

Back to flowchart

Multi-Digit Computation Multiplication and Division

Recommended Intervention:

• SOAR - See your GPAEA math consultant
• Specifically Numbers in Base Ten Binder: Folder D

​

• Bridges Math Intervention - One-pager
• Administer Volume 6 Placement Assessment Part 1

​

• MathLinks Essentials (for Middle School and above)
• Mathlinks Essentials Order Information

Back to flowchart

Ratios and Proportions

Recommended Intervention:

• SOAR - See your GPAEA math consultant
• Specifically Ratios and Proportions Binder
• AIM
• https://meadowscenter.org/resource/project-aim-algebra-modules/
• AIM Intervention Modules - One Pager
• Free
• Gap Closing
• Free
• MathLinks Essentials (for Middle School and above)
• Mathlinks Essentials Order Information

Back to flowchart

Expressions and Equations

Recommended Intervention:

• SOAR - See your GPAEA math consultant
• Specifically Expressions and Equations Binder
• AIM
• https://meadowscenter.org/resource/project-aim-algebra-modules/
• AIM Intervention Modules - One Pager
• Free
• Concept of Variable Math Talks
• Free slide decks with instructions
• Gap Closing
• Free
• MathLinks Essentials (for Middle School and above)
• Mathlinks Essentials Order Information

​

​

​

​

​

Back to flowchart

Number Ideas to 20

Place Value

​

​

Grades 2 and up

Meaning of Operations

Add/Subtr

Early Algebraic Thinking

Basic Fact Fluency Grade 1 and up

Can they count?

Can they solve word problems?

Do they understand the equal sign?

Fractions

​

​

Grades 4 and up

Multi-Digit Computation

Add/Subtr

Only after place value work:

Expressions and Equations

Ratios and Proportions

Grade 6 and up

Can they compute with integers?

** Each rectangle links to recommended interventions

Meaning of Operations

Mult/Div

Multi-Digit Computation

Mult/Div

Back to choice slide

Number Ideas to 20

Progress Monitoring Measures:

• Acadience -Free
• Fastbridge Early Math progress monitoring measures
• ​

Back to flowchart

Early Algebraic Thinking

Progress Monitoring Measures:

• Complex Quantity Discrimination - CQD (Grades 2, 3, or 4) - Contact your Great Prairie math consultant, free
• ​

​

​

​

Back to flowchart

Back to flowchart

Place Value

Progress Monitoring Measures:

• Complex Quantity Discrimination - CQD (Grades 2, 3, or 4) - Contact your Great Prairie math consultant

​

Back to flowchart

Meaning of Operations

Add and Subtract

Progress Monitoring Measures:

• ​

​

Back to flowchart

Meaning of Operations

Multiply and Divide

​

Progress Monitoring Measures:

• ​

Back to flowchart

Basic Fact Fluency

Progress Monitoring Measures:

​

• https://kcm.nku.edu/mathfactfluency/tools.php
• Progress Monitoring Tool for Addition
• Progress Monitoring Tool for Multiplication

Back to flowchart

Fractions

Progress Monitoring Measures:

​

• Complex Quantity Discrimination - CQD 4 - Contact your Great Prairie math consultant
• Estimation Measures - Contact your Great Prairie math consultant

Back to flowchart

Multi-Digit Computation Addition and Subtraction

​

​

​

​

​

Progress Monitoring Measures:

​

• Complex Quantity Discrimination - CQD 4 - Contact your Great Prairie math consultant
• Estimation Measures - Contact your Great Prairie math consultant

Back to flowchart

Multi-Digit Computation Multiplication and Division

Progress Monitoring Measures:

​

• Complex Quantity Discrimination - CQD 4 - Contact your Great Prairie math consultant
• Estimation Measures - Contact your Great Prairie math consultant

Back to flowchart

Ratios and Proportions

Progress Monitoring Measures:

​

• AAIMS - Contact your Great Prairie Math Consultant

​

​

Back to flowchart

Expressions and Equations

Recommended Intervention:

• Progress Monitoring Measures:

​

• AAIMS - Contact your Great Prairie Math Consultant

​

​

​

​

​

Back to flowchart