Why Modeling & Reasoning Matters Here
Our DoDEA CCRS Summative data show that Modeling and Reasoning items carry the most available points on the math assessment — and represent our students' greatest area of growth. This 3-session PD series gives every GES teacher concrete, classroom-ready tools to shift that trend.
What Are We Actually Measuring?
Unpack Type II and Type III items. Do the tasks as students. Surface what makes these hard.
Building It Into Daily Instruction
Move from assessment awareness to daily practice. Design a math talk. Use the lesson template.
Looking at Student Work Together
Score real student responses. Identify gap patterns. Build a classroom action plan.
What's in This PD Package
All resources available in the tabs above
Classroom Strategy Reference Card
Six research-backed strategies with at-a-glance descriptions, grade-band notes, and a "try it this week" prompt for each. Designed to live in a teacher's plan book.
Sample Type II & III Tasks
Grade-banded sample tasks (K–1, 2–3, 4–5) modeled on actual summative item types. Includes facilitator scoring notes and common student errors.
Student-Facing Sentence Frames
Reasoning and modeling language frames differentiated by grade band. Ready to post in classrooms, print as bookmarks, or embed in task sheets.
Lesson Planning Template
A fillable planning template that prompts teachers to embed a Type II or III task, identify sentence frames to use, and plan for discourse. Editable in-browser.
Session 1: What Are We Actually Measuring?
60 minutes · Whole-group with grade-band breakouts
FACILITATION AGENDA
Why We're Here: The Data
Display the school's summative sub-score breakdown. Highlight Modeling and Reasoning. Keep it brief — one slide, one number. Ask: "What do you notice about where our students are leaving points on the table?"
🎯 Do the Task — Be the Student
Facilitator reads aloud one Type II and one Type III sample task (use the Sample Tasks tab, appropriate grade). Teachers solve individually, then compare answers with a partner.
- After Type II: "What did the task ask you to do beyond calculate?"
- After Type III: "What real-world decisions did you have to make before you could do any math?"
📌 Facilitator Notes
Use a Grade 3–5 task even if K–2 teachers are present — the experience of grappling is the point, not whether everyone solves it. It's OK to sit with productive struggle. Resist giving hints for at least 5 minutes. Normalize discomfort: "This is what our students feel. Let's pay attention to that."
📊 Anatomy of a Reasoning Item vs. a Modeling Item
Direct instruction — 8 minutes max. Cover:
- Reasoning (Type II / Sub-claim C): Students construct written arguments, justify solutions, or critique reasoning. Tied to MP.3 and MP.6. Always includes a written response part. Worth 3–4 points each.
- Modeling (Type III / Sub-claim D): Multi-part, real-world context. Students must identify what math is needed, set it up, solve, and interpret. Worth 3–6 points each. Content can come from any prior grade.
- Why students struggle: They know the math but can't explain it. They don't know how to enter a messy real-world problem. They stop when they hit unfamiliar contexts.
🗂 Sort, Score, and Notice
Groups of 3–4 by grade band (K–1, 2–3, 4–5). Each group receives 4–5 printed student work samples (facilitator prepares these from DoDEA practice test responses or fabricated samples). Groups use the summative rubric to score each response, then discuss:
- What did the student do well?
- What is missing that cost them points?
- What classroom instruction would have helped this student?
📌 Facilitator Notes
Fabricate 3 student samples per task: a 0–1 point response (no explanation), a partial response (right answer, weak justification), and a full-credit response. Teachers need to see all three to calibrate. You don't need real student data for Session 1 — use the sample tasks in this guide.
💡 What Does This Mean for Monday?
Each grade band shares one "aha" from the breakout. Facilitator charts responses on a whiteboard under two columns: What students need and What we can do. Introduce the Strategy Reference Card — teachers read it, circle one strategy to try before Session 2.
📬 Between-Session Commitment
Each teacher names one strategy they will try and brings back a brief note on what happened — a student quote, a sample of work, or a reflection sentence. This feeds Session 2.
Materials for Session 1
Prepare and gather before the session
Preparation Checklist
- Print 1 Type II and 1 Type III task per teacher (use Sample Tasks tab, grade 3–5 version)
- Fabricate or source 4–5 student work samples per grade-band group (at varying score levels)
- Print Strategy Reference Card — one per teacher
- Display school's sub-score data on screen (1 slide)
- Post anchor chart with two columns: "What students need" / "What we can do"
- Timer visible to all — hold agenda pacing
Session 2: Building It Into Daily Instruction
60 minutes · Whole-group with grade-band breakouts
FACILITATION AGENDA
Report Back: What Did You Try?
Pairs share their between-session attempt in 60 seconds each. Facilitator listens for: Where did students get stuck? What surprised you? What worked? Chart 3–4 responses publicly. Validate the attempt regardless of outcome — the point is building the habit of trying.
🗣 Math Talk: Learn It, Then Lead It
Facilitator runs a 5-minute Math Talk with the whole group (use a number string or "What do you notice / wonder?" about a visual quantity). Debrief the experience:
- What moves did the facilitator make?
- How did it connect to Reasoning (MP.3)?
- What would this look like in your classroom, at your grade?
Then teachers in pairs plan a 5-minute Math Talk for their own class, using the Lesson Template. Share one with the group.
📌 Facilitator Notes
A Math Talk isn't a test-prep activity — it's the daily habit that builds the language and reasoning muscles students need for Type II items. Emphasize: you are building discourse norms over time, not teaching to a question type. Good Math Talk prompts: "Which is greater, and how do you know?" / "Is this always true?" / "Can you say that a different way?"
📐 The CRA Progression & Why It Matters for Modeling
Brief direct instruction — 8 minutes. Cover the Concrete → Representational → Abstract progression with a live example (e.g., fraction division K–2 equivalent: sharing a cookie). Key message:
- Modeling tasks require students to build their own representation — if they've only ever practiced at the abstract level, they don't know how to enter an unfamiliar real-world scenario.
- CRA isn't just for young learners — a Grade 5 student modeling a budget problem benefits from drawing or using a table before writing equations.
- Anchor charts, diagrams, and manipulatives are not "scaffolds for struggling students" — they're the pathway to mathematical modeling for all students.
📝 Design a Lesson — Use the Template
Grade-band groups use the Lesson Planning Template to plan one lesson that embeds a Type II or Type III task. Teachers must identify:
- The Math Talk that opens the lesson
- The task (from Sample Tasks tab or their curriculum)
- Which Sentence Frames will be posted/provided
- How they will respond to students who are stuck
- How they will collect evidence of reasoning (exit ticket, partner share, written response)
📌 Facilitator Notes
Circulate during the breakout. The most common sticking point: teachers want to scaffold the task into steps before students even try it. Redirect: "What if you let them struggle for 5 minutes first? What would you listen for?" The goal is low-floor, high-ceiling tasks where every student can enter, not tasks with the modeling already done for them.
🔄 Share + Strengthen
One representative from each grade band shares their planned lesson opening (Math Talk) and task. Group gives one warm comment and one "wonder" question. Teachers revise their template based on feedback.
📬 Between-Session Commitment
Each teacher teaches the lesson they planned. They bring 3 samples of student work (one low, one mid, one high response) to Session 3 for calibration. No prep needed beyond collecting the work — keep it simple.
Materials for Session 2
Prepare and gather before the session
Preparation Checklist
- Prepare a Math Talk prompt for the whole-group demonstration (visual quantity, number image, or number string)
- Print Lesson Planning Template — one per teacher
- Print Sentence Frames — one set per grade band
- Have Sample Tasks tab open on screen or printed for reference
- Concrete manipulatives available for CRA demonstration (base-ten blocks, fraction tiles, etc.)
- Anchor chart ready: "CRA — Three Ways to Enter a Problem"
Session 3: Looking at Student Work Together
60 minutes · Whole-group with grade-band breakouts
FACILITATION AGENDA
Quick Win Share
Popcorn style — teachers each give one sentence: "One thing I noticed when I taught the lesson was…" Facilitator charts patterns without judgment. This validates effort and sets the norm that we learn from all outcomes.
📋 Calibration: What Does a Full-Credit Response Look Like?
Facilitator displays one student response on screen (fabricated or from practice test). Group scores it independently using the summative rubric language. Then share out — surface any disagreements and discuss what language in the rubric resolves them. Key anchors to establish:
- Reasoning: A justification must explain the why, not just restate the what.
- Modeling: Setting up the problem correctly earns partial credit even if the final answer has a computation error.
- Sentence frames on the response page can coach the writing, but the reasoning must be the student's own.
🔍 Collaborative Student Work Protocol
Each teacher brings 3 samples of student work. Groups use the following structured protocol:
- Read silently (2 min): Each person reads one sample without discussion.
- Score independently (1 min): Assign a score using rubric language.
- Share scores (1 min): Say your score — no explanation yet.
- Discuss discrepancies (3 min): What in the student work led to different scores?
- Name the gap (1 min): What skill or habit is missing?
- Name the instruction (1 min): What would help this student before the next task?
Rotate through 3–4 samples per group. Groups track the most common error pattern they saw.
📌 Facilitator Notes
The protocol feels slow at first — hold the structure. The goal is calibration and naming, not speed. If a group finishes early, ask: "If you saw this pattern in every student in your class, what one instructional move would you make?" Anchor the debrief to instruction, not to assessment scores.
📌 Pattern Share + Root Cause
Each grade band shares their most common error pattern. Facilitator charts across all groups. Ask the group to look across all patterns: "What do these errors have in common? What are students NOT yet able to do?" Expected patterns to listen for:
- Students compute correctly but write no explanation
- Students use informal language without mathematical precision
- Students can't enter the problem — leave it blank
- Students answer Part A but abandon multi-part items
🎯 My 3-Action Classroom Plan
Teachers independently complete their classroom action plan using the bottom section of the Lesson Template. They commit to three specific, ongoing actions:
- Daily: A Math Talk routine (e.g., "Every Monday and Wednesday, I will open with a 'Which One Doesn't Belong?' or number string")
- Weekly: A task that requires written justification or real-world modeling (e.g., "Every Friday task will include a 'Show your thinking' prompt")
- Monthly: Collect and look at student work on a Type II or III item with a colleague
🐻 GES Closing
Each teacher reads their Daily commitment aloud (30 seconds each). Facilitator closes: "The students who leave points on Modeling and Reasoning aren't students who don't know the math. They're students who haven't yet been asked to explain, model, and justify — consistently, over time. That changes here."
Materials for Session 3
Preparation Checklist
- Teachers bring 3 student work samples each (reminded at Session 2 close)
- Print scoring rubric — one per teacher (from DoDEA summative resources)
- Anchor chart pre-made: "Most Common Error Patterns" (fill in live)
- Printed Lesson Template with Action Plan section for each teacher
- Optional: Post-it notes for pattern charting during debrief
Classroom Strategy Reference Card
Six high-leverage moves for Modeling & Reasoning — K–5
1. Math Talks (Daily Habit)
Open every math lesson with a 5–8 minute structured discussion around a visual, number string, or open question. Pose it, give think time (60 sec), then invite responses without evaluating. Chart multiple strategies. Ask: "Did anyone think about it differently?"
Why it works: Builds the oral reasoning language students need to write justifications on Type II items. Repeated daily exposure builds discourse norms.
🎯 Try it: "Which is greater — ⅓ or ¼ — and how do you know?"2. "Show Your Thinking" Prompts
Add a written justification requirement to at least one problem per lesson. Replace "Solve:" with "Solve and explain how you know your answer is correct." Accept drawings, equations, and words — all count as mathematical reasoning at the K–5 level.
Why it works: Type II items require written justification. Students who have only practiced computation are unprepared. Regular writing about math builds the habit.
🎯 Try it: Swap one practice problem per day with a "Solve + Explain" version.3. Real-World Problem Contexts (Weekly)
Once per week, present a math task embedded in a real scenario before telling students what operation to use. Let them read and identify: What information matters? What are we trying to find? What math is needed? This is the entry skill for Type III modeling tasks.
Why it works: Modeling items always appear in real-world contexts. Students who only practice decontextualized computation can't identify what math to apply when the context is new.
🎯 Try it: "The school store has 48 pencils and 6 classrooms. How many pencils can each class get? Is that fair?"4. Concrete–Representational–Abstract (CRA)
Introduce new concepts with physical objects first (concrete), then drawings or diagrams (representational), then symbols and equations (abstract). Never skip to abstract. For modeling tasks, explicitly ask students to draw a model before they write an equation.
Why it works: Students who can't build or sketch a representation of a real-world problem can't complete a modeling task. The diagram is not a crutch — it IS the mathematical model.
🎯 Try it: Before solving any word problem, require a quick sketch or diagram. No equation before a picture.5. Error Analysis
Periodically show students a worked example with an error (real or fabricated). Ask: "What did this student do? Where did they go wrong? How would you explain the mistake?" This is directly aligned to MP.3 — critiquing the reasoning of others.
Why it works: Explaining someone else's error requires the same language as justifying your own answer. Error analysis is low-threat and high-engagement for students who won't volunteer their own reasoning.
🎯 Try it: "Alex says 5 × 0 = 5. Look at Alex's work. What did Alex misunderstand? How would you help Alex?"6. Sentence Frames for Mathematical Language
Post and teach specific sentence frames for reasoning and modeling. Model using them yourself. Require students to use at least one frame in written justifications. Gradually fade the frames as students internalize the language.
Why it works: Many students know the math but can't produce the academic language the rubric requires. Sentence frames are language scaffolds, not answer scaffolds — the mathematical reasoning still belongs to the student.
🎯 Try it: Post the grade-band frames from this PD on your math focus wall this week.Sample Type II & III Tasks by Grade Band
Modeled on DoDEA CCRS Summative item types — for PD and classroom use
Kindergarten / Grade 1 · Numbers and Operations
Part B: Draw a picture to show how you know. Then use words to explain what happened to the apples.
Reasoning demand: Students must represent the situation (draw) AND explain in words — not just state an answer. Full credit requires both a correct representation and a sentence that explains the relationship (e.g., "He gave 3 apples because 7 minus 3 equals 4"). Common error: Drawing the correct picture but writing no explanation, or writing "3" with no connection to the context.
Kindergarten / Grade 1 · Operations and Algebraic Thinking
There are 5 children at the snack table. The teacher has 12 crackers to share.
Part A: Circle the picture that shows one way to share the crackers so every child gets the same amount. (Provide 3 options: 2 each with 2 left over, 2 each equally, 3 each with 3 left over.)
Part B: How many crackers will be left over? Show how you know using pictures or numbers.
Part C: Is it possible for every child to get exactly the same number of crackers with no leftovers? Circle: Yes / No. Explain your thinking.
Modeling demand: Students must interpret the real-world context, represent it, solve, AND evaluate whether the scenario is possible. Part C is the modeling heart — it requires reasoning about whether a situation is mathematically possible. Note: Content can include K concepts for Grade 1 students; this is a normal summative feature.
Grades 2–3 · Number and Operations in Base Ten
Part B: Explain your reasoning. Give at least two examples that support your answer. Use words and numbers in your explanation.
Reasoning demand: This item directly assesses MP.3 — students must construct a viable argument. Full credit requires a correct claim AND two valid examples AND a written explanation that connects the examples to the claim (not just listing examples). Common error: Students pick the correct answer but only write "because it always works" — this earns 0 points for justification.
Grades 2–3 · Measurement and Data / Operations
The GES Grizzlies are planting a rectangular classroom garden. They have 24 feet of fence to go around the entire garden.
Part A: Draw a rectangle that could be their garden. Label the length and width. Make sure your fence measurement is correct.
Part B: What is the area of your garden? Show your work.
Part C: Your friend says: "A garden that is 10 feet long and 2 feet wide would use exactly 24 feet of fence." Do you agree? Explain how you know.
Modeling demand: Students must select their own dimensions (there are multiple valid answers — this is intentional), calculate perimeter, find area, and evaluate a claim. Part C builds in the reasoning sub-skill. Part A has no single correct answer — assessing whether students can accept mathematical ambiguity is part of the modeling standard.
Grades 4–5 · Number and Operations — Fractions
Part B: Explain the error in Maya's reasoning. Then show the correct way to add ½ + ⅓ and explain each step.
Reasoning demand: This is an error analysis format aligned to MP.3. Full credit requires: (1) correctly identifying Maya as wrong, (2) explaining why the approach is mathematically incorrect (denominators must be equal before adding), (3) correct steps to find ½ + ⅓ = 5/6, and (4) a written explanation that narrates each step — not just numbers. Common error: Students show the correct calculation but never address what Maya did wrong.
Grades 4–5 · Operations and Algebraic Thinking / Fractions
The GES school store sells pencils for $0.25 each and erasers for $0.40 each. Marcus has $3.00 to spend.
Part A: Marcus wants to buy exactly 6 pencils and spend the rest on erasers. How many erasers can he buy? Show all your work.
Part B: Marcus changes his mind. He wants to buy the same number of pencils and erasers and spend as close to $3.00 as possible without going over. How many of each can he buy? Show how you figured this out.
Part C: Marcus's friend Emma says she can spend exactly $3.00 with no money left over buying only pencils and erasers. Is that possible? Explain your reasoning.
Modeling demand: Part A is procedural setup. Part B requires students to develop their own strategy (trial and refinement, equations, tables) — there is no single procedure given. Part C is the high-value reasoning component: students must reason about whether $3.00 = 0.25a + 0.40b has whole-number solutions, without necessarily using algebra. This is where the 6-point version of a modeling item earns its weight.
Student-Facing Sentence Frames
Mathematical language scaffolds for Reasoning and Modeling — K–5
🐣 Kindergarten – Grade 1
🌱 Grades 2 – 3
🌳 Grades 4 – 5
Modeling & Reasoning Lesson Planning Template
GES Mathematics · Complete before teaching · Bring to Session 3