Math Intervention Suite

Stuck on Counting Pips?

Apply blue tape across the face of the 2-tiles so individual squares are invisible. Force the child to treat the bar as a solid block.

Loses Count When Screened?

Use partial screening: leave the last tile half exposed. Tap on the cloth rhythmically to reinforce spatial location of the hidden units.

Confuses Tiles vs. Dots?

Ask: "Hold up tiles in your left hand, count dots with your right finger." Anchor vocabulary: tiles (things) vs. dots (pips).

10-Second Exit Check: Hide 3 tiles (6). Place 1 tile outside. Ask: "6 dots are hiding. How many dots in all?"

Intervention Triage Two Tile Leap

Two Tile Leap • AVMR II Intervention Cue Ring Set Print on cardstock, laminate, and attach with binder ring

CARD 12 Grand Challenge

A cloth hides 18 dots.

Every piece is a 2-tile. How many tiles are hiding?

Track by twos: 2, 4, 6, 8, 10, 12, 14, 16, 18!

Number of Tiles:

Two Tile Leap • AVMR II Student Task Cards Page 2 of 2 • Cards 7–12

Target: Part Size vs Part Count

Compare Pair A (Same Num): \(\frac{3}{4}\) vs \(\frac{3}{8}\) • "Which is larger? Why?"

Compare Pair B (Same Den): \(\frac{2}{6}\) vs \(\frac{5}{6}\) • "Which is larger? Why?"

Compare Pair C (Benchmark): \(\frac{5}{4}\) vs \(\frac{7}{8}\) • "Which is greater than 1 whole?"

Look-Fors: • Same num: recognizes 4ths are larger pieces than 8ths. • Benchmarks \(\frac{5}{4}\) as exceeding 1.

Fraction Foundations • Clinical Progress Monitoring Protocol Page 1 of 3

Student Assessment Probe Form A • Fraction Concepts

Name:  

Date:  

Fraction Thinking Check-In

1. Partitioning the Whole Strip Show equal shares

a. Partition this entire bar into 4 equal shares. Then shade \(\frac{3}{4}\).

Write the unit fraction for 1 share:

2. Iterating the Unit Fraction Building fractions greater than 1

This small strip is \(\frac{1}{3}\) of a candy bar:

\(\frac{1}{3}\)

← This is 1 unit share.

Draw a candy bar that is \(\frac{5}{3}\) long by repeating this unit piece:

3. Fractions on the Number Line Locating 0 to 2

Partition the number line and plot these three fractions: \(\frac{2}{4}\), \(\frac{4}{4}\), and \(\frac{6}{4}\).

0

1

2

4. Show Equivalence

Split the bottom bar to prove that \(\frac{1}{2} = \frac{2}{4}\):

\(\frac{1}{2}\)

5. Compare Using >, <, =

\(\frac{3}{4}\)

\(\frac{3}{8}\)

\(\frac{5}{6}\)

\(\frac{2}{6}\)

\(\frac{5}{4}\)

1

Fraction Foundations Assessment Probe Page 2 of 3

Intervention Roster & Construct Mastery Matrix Multi-Tiered Support System

Small Group Progress Tracker

Track individual student scheme progression across Baseline (B), Mid-Point (M), and Post-Intervention (P).

Intervention Group Roster (5 Students)

Coding: 0 = Pre-Fractional • 1 = Part-Whole • 2 = Iterative • 3 = Relational

Fractional Construct Hierarchy & Decision Rules

Level 0 Pre-Fractional

Counts shaded pieces as whole objects ("3 things"). Ignores fair shares or unequal sizes.

Move: Fair-sharing tasks with paper folding & dough.

Level 1 Part-Whole Only

Successfully partitions & shades \(\frac{3}{4}\), but cannot conceptualize fractions > 1 ("You can't have 5 thirds").

Move: Cuisenaire rods & iterating unit strips past 1.

Level 2 Iterative Scheme

Treats \(\frac{1}{b}\) as a measuring unit. Iterates 5 times to make \(\frac{5}{3}\). Places fractions on 0–2 number lines.

Move: Multiplicative splitting for equivalence (\(\frac{2}{4} = \frac{1}{2}\)).

Level 3 Relational Scheme

Reasons flexibly about inverse denominator size (\(\frac{3}{4} > \frac{3}{8}\)), generates equivalent forms, benchmarks to 1.

Move: Ready for addition/subtraction with like units.

MTSS Tier Guidance & Decision Framework Post-Assessment Action Plan

Tier 3 (Intensive 1:1) Students at Level 0. Daily 15-min intervention with physical paper strips and equal sharing.

Tier 2 (Targeted Small Group) Students at Level 1. 3× weekly 25-min intervention focusing on unit fractions as iterable lengths on number lines.

Tier 1 (Core Alignment) Students at Level 2+. Solid on iterating; ready for classroom core instruction and equivalence tasks.

Fraction Foundations Progress Monitoring Protocol • Steffe & Olive MTSS Page 3 of 3

Teacher Instructional Flow Core Script & Facilitation Protocol

The Three Concrete-to-Linear Phases

Lead students sequentially from physical folding to mental iteration, and finally to linear measurement.

PHASE 1

Equipartitioning & Disembedding (The Reference Whole)

8 Minutes • Paper Folding

Teacher Action:

Give each student two identical paper strips. Keep one strip untouched as the "Master Whole." Have students fold the second strip in half, then half again. Crease sharply to produce 4 equal segments. Cut along folds to produce four separate unit pieces.

Verbatim Script:

Say: "Hold up one cut piece. How many of these exact pieces did it take to make our Master Whole?" [Wait: 4 pieces]

Say: "Because it takes 4 equal pieces to recreate the whole, we call this one piece one-fourth, written as \(\frac{1}{4}\)."

Crucial Disembedding Probe: "Lay 3 cut pieces next to the Master Whole. Show me \(\frac{3}{4}\). Is the Master Whole still one whole? Did our pieces lose their name?"

Student Target: Recognizes that \(\frac{3}{4}\) is formed by 3 pieces of size \(\frac{1}{4}\), and the whole remains invariant.

PHASE 2

Iterating the Unit Fraction (Breaking the 1-Whole Ceiling)

10 Minutes • Measuring Unit

Teacher Action:

Have students use a single \(\frac{1}{4}\) strip as a measuring ruler. Instruct them to iterate (repeat) the strip along a line, chanting the fraction count as they lay each piece down: "\(\frac{1}{4}\), \(\frac{2}{4}\), \(\frac{3}{4}\), \(\frac{4}{4}\)... and one more makes \(\frac{5}{4}\)!"

Verbatim Script:

Say: "We count whole numbers: 1, 2, 3, 4. We can count fractions the exact same way by counting units of \(\frac{1}{4}\)!"

Chant & Touch: "\(\frac{1}{4}\)... \(\frac{2}{4}\)... \(\frac{3}{4}\)... \(\frac{4}{4}\)... What comes next?"

Address Misconception: If student hesitates at \(\frac{4}{4}\), ask: "Do we stop counting inches when we reach a foot? No! We keep counting: 5 fourths (\(\frac{5}{4}\)), 6 fourths (\(\frac{6}{4}\))!"

Key Probe: "Compare your \(\frac{5}{4}\) train to our Master Whole strip. Is \(\frac{5}{4}\) longer, shorter, or the same? Why?"

PHASE 3

Transfer to the Number Line (Lengths from 0)

12 Minutes • Linear Representation

Teacher Action:

Unroll the 0-to-2 number tape. Align the Master Whole strip from 0 to 1 to establish the unit interval. Then use the \(\frac{1}{4}\) unit piece to physically hop along the line, drawing tick marks and recording values.

Verbatim Script:

Say: "The number line starts at 0. A fraction is the distance from 0."

Prompt: "Place your \(\frac{1}{4}\) piece at 0. Make a mark at the end. That mark is \(\frac{1}{4}\). Hop again. That's \(\frac{2}{4}\). Hop again: \(\frac{3}{4}\). Hop again: \(\frac{4}{4}\)."

Ask: "Look where \(\frac{4}{4}\) landed! What whole number is right there? Why do \(\frac{4}{4}\) and 1 live at the exact same address?"

Challenge: "Keep hopping past 1! Plot \(\frac{5}{4}\), \(\frac{6}{4}\), and \(\frac{8}{4}\). Where does \(\frac{8}{4}\) land?" [At 2]

Fraction Builders • Step-by-Step Teacher Intervention Guide Page 2 of 3

Equivalence & Relational Reasoning Deepening Fractional Magnitude

Equivalence & Magnitude Comparisons

Guiding students to reason about the size and quantity of shares without relying on mechanical cross-multiplication tricks.

Equivalence Protocol: Multiplicative Splitting

3.NF.3 Scheme

The Action Routine:

Fold a strip in half (\(\frac{1}{2}\)). Shade one half. Now fold the strip in half again without changing the shaded region.

\(\frac{1}{2}\) becomes \(\frac{2}{4}\)

The Reasoning Questions:

Ask: "Did the shaded space get bigger, smaller, or stay the exact same?"

Ask: "Why are there now 2 pieces shaded instead of 1? What happened to the size of each piece?"

Target Anchor Rule: "Each piece split into 2 smaller pieces. We have twice as many pieces, but each piece is half as big! That's why \(\frac{1}{2} = \frac{2}{4}\)."

Reasoning-Based Comparison Matrix (3 Scenarios)

Same Numerator \(\frac{3}{4}\) vs \(\frac{3}{8}\)

Compare 3 fourths to 3 eighths.

Reasoning: We have the same number of pieces (3), but fourths are cut into fewer shares, so fourths are much bigger pieces than eighths! Therefore \(\frac{3}{4} > \frac{3}{8}\).

Same Denominator \(\frac{2}{6}\) vs \(\frac{5}{6}\)

Compare 2 sixths to 5 sixths.

Reasoning: The pieces are the exact same size (sixths). 5 pieces is clearly more pieces than 2 pieces! Therefore \(\frac{5}{6} > \frac{2}{6}\).

Benchmark to 1 \(\frac{5}{4}\) vs \(\frac{7}{8}\)

Compare a fraction > 1 to < 1.

Reasoning: \(\frac{4}{4}\) is 1 whole, so \(\frac{5}{4}\) is greater than 1. But \(\frac{7}{8}\) needs one more eighth to reach 1. So \(\frac{5}{4} > \frac{7}{8}\).

Common Misconceptions & Instant Teacher Pivots

Whole-Number Bias in Denominators: Student says \(\frac{1}{8} > \frac{1}{4}\) because 8 is bigger than 4. Pivot: "Would you rather share a pizza with 4 kids or 8 kids? More shares means smaller pieces!"

Counting Tick Marks Instead of Spaces: Student counts the tick lines on the number line instead of unit interval distances. Pivot: "A frog doesn't count lines—the frog counts hops! Trace the jump from 0 to 1."

1-Minute Exit Transfer Check

The Mystery Fraction on 0–2 Line

Teacher places a clip past 1 at \(\frac{6}{4}\). Ask: "How many fourth-hops did it take to get here from 0? Name this fraction!"

Mastery Evidence States "\(\frac{6}{4}\)" and notes it is past 1

Fraction Builders • Step-by-Step Teacher Intervention Guide Page 3 of 3

\(\frac{3}{4}\)

\(\frac{3}{8}\)

Same Numerator

My Reasoning:

\(\frac{5}{6}\)

\(\frac{2}{6}\)

Same Denominator

My Reasoning:

\(\frac{5}{4}\)

\(\frac{7}{8}\)

Benchmark to 1

My Reasoning:

Fraction Foundations Assessment Suite • Student Independent Probe Page 2 of 2

Exemplar Justification: "\(\frac{4}{4}\) lands right on 1 because 4 fourths makes 1 whole distance. \(\frac{6}{4}\) is 2 more fourths past 1."

Fraction Foundations Assessment Guide • Tasks 1 to 3 Page 1 of 2

Scoring Rubrics: Tasks 4 & 5 • Diagnostic Action Profile Evaluation Page 2

TASK 4

Equivalence & Multiplicative Splitting (Max 4 Pts)

Score: [     ] / 4

3–4 Pts: Full Understanding

Accurately splits Model B into 4 equal parts, shades 2 parts (\(\frac{2}{4}\)). Explains that cutting each half into 2 smaller pieces creates twice as many pieces that are each half as big.

1–2 Pts: Partial / Emerging

Draws \(\frac{2}{4}\) correctly on the visual model, but written justification is superficial ("they look like the same size") without multiplicative reasoning.

0 Pts: Minimal / Pre-Fractional

Claims \(\frac{2}{4}\) is greater than \(\frac{1}{2}\) because 2 and 4 are larger numbers than 1 and 2. Fails to partition Model B into fourths.

Exemplar Justification: "When you split each half into 2 pieces, you get 4 pieces in the whole. The shaded part now has 2 pieces. It covers the same amount of space because each piece is half as big."

TASK 5

Reasoning-Based Comparison Matrix (Max 4 Pts)

Score: [     ] / 4

3–4 Pts: Full Understanding

Correctly places all 3 signs (>, >, >). Written explanations articulate piece size (same numerator), piece count (same denominator), and benchmarking to 1 whole.

1–2 Pts: Partial / Emerging

Gets 2 of 3 signs correct, but reasoning relies on memory tricks or drawing imprecise circles rather than analytical size/count properties.

0 Pts: Minimal / Pre-Fractional

Exhibits whole-number bias: states \(\frac{3}{8} > \frac{3}{4}\) because 8 > 4. Cannot explain comparisons conceptually.

Exemplar Justification: "\(\frac{3}{4} > \frac{3}{8}\) because both have 3 pieces, but fourths are bigger than eighths. \(\frac{5}{4} > \frac{7}{8}\) because \(\frac{5}{4}\) is greater than 1 whole, while \(\frac{7}{8}\) is less than 1."

Composite Score Conversion & MTSS Intervention Profile

17–20 Points Level 3: Relational Mastery

Full iterative scheme. Flexibly compares, proves equivalence, and reasons on number lines.

Tier 1 Core Alignment

12–16 Points Level 2: Iterative Scheme

Iterates unit fractions past 1 and maps number lines. Needs targeted scaffolding for equivalence.

Tier 2 Targeted Group

7–11 Points Level 1: Constrained Part-Whole

Partitions and shades within 1 whole. Cannot conceptualize fractions > 1; weak number line sense.

Tier 2/3 Intensive Group

0–6 Points Level 0: Pre-Fractional

Persistent whole-number bias. Ignores fair shares. Needs concrete folding and equal sharing tasks.

Tier 3 1:1 Intervention

Clinical Interventionist Qualitative Notes & Action Plan:

Fraction Foundations Assessment Guide • Tasks 4 & 5 • Diagnostic Action Profile Page 2 of 2

8 Minutes • Splitting Scheme

Teacher Action:

Show a strip partitioned into fourths with 3 parts shaded (\(\frac{3}{4}\)). Slice each fourth vertically down the center into 2 equal parts.

Verbatim Script:

Say: "Here is \(\frac{3}{4}\). Watch closely: I am going to split every fourth in half."

Ask: "How many total pieces make the whole strip now?" [Wait: 8 pieces]

Ask: "How many shaded pieces do we have now?" [Wait: 6 pieces]

Anchor Rule: "Did the shaded space change? No! We have twice as many pieces, but each piece is half as big. That's why \(\frac{3}{4} = \frac{6}{8}\)!"

PHASE 2

Number Line Navigation & The \(\frac{1}{2}\) Benchmark

10 Minutes • Relational Comparison

Teacher Action:

Unroll the 0-to-3 number tape. Mark \(\frac{1}{2}\), \(1\), \(1\frac{1}{2}\), \(2\), and \(2\frac{1}{2}\) with bold flags. Ask students to compare fractions using \(\frac{1}{2}\) as a benchmark.

Verbatim Script:

Present: \(\frac{3}{8}\) and \(\frac{4}{6}\). Ask: "Which is greater without finding common denominators?"

Benchmark Reasoning: "Look at \(\frac{3}{8}\). What is half of 8? (4). So \(\frac{4}{8}\) is \(\frac{1}{2}\). Is \(\frac{3}{8}\) less than or more than \(\frac{1}{2}\)?" [Less than]

Prompt: "Now look at \(\frac{4}{6}\). What is half of 6? (3). So \(\frac{4}{6}\) is more than \(\frac{1}{2}\)! Therefore, \(\frac{4}{6} > \frac{3}{8}\)!"

Plot on Line: Have student physically place clips at \(\frac{3}{8}\) (before \(\frac{1}{2}\)) and \(\frac{4}{6}\) (after \(\frac{1}{2}\)).

PHASE 3

Adding Mixed Numbers with Recomposition (\(1\frac{3}{4} + 2\frac{2}{4}\))

12 Minutes • Distributing & Joining

Teacher Action:

Use the regrouping mat. Separate wholes into the Wholes Box and fourths into the Fractions Box. Join wholes (\(1+2=3\)) and join fourths (\(\frac{3}{4} + \frac{2}{4} = \frac{5}{4}\)).

Verbatim Script:

Say: "We distribute the addition: wholes add with wholes, fourths add with fourths!"

Ask: "We have \(3\) wholes and \(\frac{5}{4}\). What do you notice about \(\frac{5}{4}\)?"

Recompose: "Four fourths (\(\frac{4}{4}\)) makes another whole! So we trade \(\frac{4}{4}\) for \(1\) whole block. What is our new total?" [\(3 + 1 = 4\) wholes, with \(\frac{1}{4}\) left over \(\rightarrow 4\frac{1}{4}\)]

Verify on Number Tape: Start at \(1\frac{3}{4}\), hop \(2\) wholes to \(3\frac{3}{4}\), then hop \(\frac{2}{4}\) to land on \(4\frac{1}{4}\).

Mixed Numbers Intervention Protocol • Grade 4 Schemes Page 2 of 3

Disembedding in Subtraction Regrouping & Assessment Matrix

Subtracting Mixed Numbers & Regrouping

Mastering the disembedding routine for unbundling wholes without standard whole-number subtraction confusion.

Phase 4: Regrouping Protocol for \(3\frac{1}{4} - 1\frac{3}{4}\)

Core G4 Target

The Problem Setup:

We need to subtract \(\frac{3}{4}\), but we only have \(\frac{1}{4}\) in the fraction box.

\(3\frac{1}{4} = 2 + \frac{4}{4} + \frac{1}{4} = 2\frac{5}{4}\)

The Teacher Questions:

Ask: "Can we take 3 fourths away from 1 fourth? No! Where can we get more fourths?"

Prompt: "Unbundle 1 whole block into 4 fourths. Move them to the fraction box."

Execute: "Now we have \(2\) wholes and \(\frac{5}{4}\). Now subtract \(1\frac{3}{4}\): wholes: \(2-1=1\); fourths: \(\frac{5}{4} - \frac{3}{4} = \frac{2}{4}\). Final answer: \(1\frac{2}{4}\) (or \(1\frac{1}{2}\))!"

Diagnostic Observation Matrix (G4 Construct Progression)

Observed Behavior

Construct Stage

In-the-Moment Teacher Pivot

Subtracts smaller fraction from larger: \(3\frac{1}{4} - 1\frac{3}{4} = 2\frac{2}{4}\).

Whole Number Trap Ignores fraction order.

Physical barrier: Lay down 3 full strips and 1 fourth. Ask: "Physically pick up and remove 1 whole and 3 fourths. Show me how you get the 3 fourths."

Converts to improper fractions mechanically (\(\frac{13}{4} - \frac{7}{4}\)) but makes counting errors.

Algorithmic Reliant Lacks distributing sense.

Regrouping Mat: Anchor students in decomposing only 1 whole: "\(3\frac{1}{4} = 2\frac{5}{4}\)". Keeps numbers manageable.

Decomposes flexibly; reasons with \(\frac{1}{2}\) benchmark; navigates 0–3 number line fluently.

Relational & Distributive Full G4 Mastery.

Extend to unlike units: "If we have fourths and eighths, how can our splitting scheme help us add them?"

1-Minute Mastery Check

The Regrouping Mystery

Solve: \(4\frac{1}{5} - 2\frac{3}{5}\). Show how you unbundle 1 whole into fifths!

Target Evidence Rewrites as \(3\frac{6}{5} \rightarrow 1\frac{3}{5}\)

Mixed Numbers Intervention Protocol • Grade 4 Schemes Page 3 of 3

0

1

2

3

Plot: \(\frac{5}{4}\)

Plot: \(\frac{7}{4}\)

Plot: \(\frac{10}{4}\)

3

Comparison Arena: Compare & Explain

Write >, <, or = and your mathematical reasoning

\(\frac{3}{8}\)

\(\frac{4}{6}\)

Reasoning:

\(\frac{5}{10}\)

\(\frac{4}{8}\)

Reasoning:

4

Mixed Number Operations

Show complete work and final simplified answers

Problem A: Addition \(1\frac{3}{4} + 1\frac{3}{4}\)

Show your work:

Final Answer:

Problem B: Subtraction \(3\frac{1}{4} - 1\frac{3}{4}\)

Show your work:

Final Answer:

Mixed Numbers Intervention Mat • Grade 4 Advanced Schemes Page 2 of 2 • Independent Practice (Without Hints)

Composite Score Conversion & MTSS Intervention Profile

17–20 Points Level 3: Distributive Mastery

Mastered G4 standards. Regroups fluently and reasons with benchmarks.

Tier 1 Core Ready

12–16 Points Level 2: Emergent Regrouping

Navigates number lines well. Needs support unbundling wholes in subtraction.

Tier 2 Targeted Group

7–11 Points Level 1: Constrained Part-Whole

Struggles with fractions > 1. Lacks splitting schemes for equivalence.

Tier 2/3 Intensive Group

0–6 Points Level 0: Whole-Number Bias

Adds denominators across; reverses subtraction order. Needs concrete bars.

Tier 3 1:1 Intervention

Clinical Interventionist Observations & Next Steps:

Mixed Numbers Assessment Suite • Scoring Guide & MTSS Rubric Page 3 of 3 (Examiner Rubric)

Goal: Decompose & Recompose Fraction Navigators

○

CARD 4 Decomposing Scheme

Subtracting in Friendly Steps (\(3\frac{1}{4} - 1\frac{3}{4}\))

Teacher Strategy

Decompose the subtrahend \(1\frac{3}{4}\) into: \(1\) whole, \(\frac{1}{4}\), and \(\frac{2}{4}\).

Verbatim Prompts

Step 1 (Take 1 away): "Take 1 whole away from \(3\frac{1}{4}\) first: \(3\frac{1}{4} - 1 = 2\frac{1}{4}\)."

Step 2 (Decompose \(\frac{3}{4}\)): "We still need to subtract \(\frac{3}{4}\). Split \(\frac{3}{4}\) into friendly parts: \(\frac{1}{4}\) and \(\frac{2}{4}\)."

Step 3 (Jump to friendly 2): "Take \(\frac{1}{4}\) away first: \(2\frac{1}{4} - \frac{1}{4} = 2\)."

Step 4 (Take remaining \(\frac{2}{4}\)): "Now take \(\frac{2}{4}\) away from 2: \(2 - \frac{2}{4} = 1\frac{2}{4}\) (or \(1\frac{1}{2}\))!"

Why This Works: "No messy unbundling rules! Students count back through the friendly whole number 2."

Goal: Decompose & Jump Back Fraction Navigators

Mixed Numbers Intervention Cue Ring Set • Grade 4 Advanced Schemes Print on cardstock, laminate, and attach with binder ring