Lead the Journey
A 90-minute introduction followed by two 60-minute workshops.
Audience: PK–8 faculty and Algebra I educators. Participants do not need advanced mathematics credentials. Facilitation should foreground reasoning, models, and classroom transfer rather than speed or procedural recall.
Preparation
Prepare fraction strips or paper rectangles, graph paper, pencils, rulers, and a projected version of this website. Pair teachers from different grade bands where possible. Ask participants to identify the whole and the unit whenever a fraction is used.
A shared lens from Fraction Fix
Watch for six learning breakpoints: fractions not understood as one number; an unstable referent whole; equivalence reduced to a trick; persistent whole-number bias; procedures before meaning; and knowledge bound to one model. These are hypotheses to test through student work, not assumptions that a learner or curriculum is deficient. Use context → model → student language → equation → generalization → efficient procedure; revisit earlier representations when evidence calls for them.
Each chapter now offers a completed solution, contrasting approaches, incorrect-work analysis, a partially completed problem, and independent transfer. Study and explain first, then fade support. Keep the revealed reasoning closed until participants have attempted each task.
Session 1 · Experience the Story (90 minutes)
0–10 min — The opening dilemma. Present two differently sized loaves, each divided in half. Ask whether the resulting halves have equal amounts. Introduce the Whole.
10–30 min — Chapters I–III. Explore equal sharing, number lines, and equivalence. Ask teachers to construct 1/2 = 2/4, then explain 1/2 + 1/4 by common units.
30–55 min — Chapters IV–VI. Model 2/3 of 1/2 with an area model, then transition to part-to-part ratios and equivalent ratio tables. Discuss how multiplicative comparison differs from additive comparison.
55–70 min — Chapter VII. Compare y=3x with y=3x+2. Connect constant ratios to slopes and distinguish proportional from merely linear relationships.
70–90 min — Return to the map. Teachers identify their grade-level chapter, one prerequisite, one later use, and a student misconception they want to anticipate.
Facilitation within Session 1: During 10–30, compare the Chapter 2 strip and number-line explanations and diagnose the Chapter 3 claim 1/2 + 1/3 = 2/5. During 30–55, compare area and scaling methods in Chapter 4, then distinguish recipe ratios and proportional cost. During 55–70, compare the two correct equation-solving methods in Chapter 7. During 70–90, use “A Glimpse Ahead” to name a prerequisite and later use; record which breakpoint the evidence supports. These prompts fit within the existing time blocks.
Session 2 · Understand Your Chapter (60 minutes)
0–10: Revisit the whole journey. 10–25: Grade teams map their actual standards and curriculum lessons onto the story. 25–40: Study student work or anonymous sample responses for misconceptions. 40–55: Cross-grade partners describe what they need students to understand before the next grade. 55–60: Commit to one instructional adjustment.
Facilitation within Session 2: During 10–25, select the developmentally appropriate core task; label previews explicitly. During 25–40, annotate a completed solution before comparing methods, then identify the first invalid step in the chapter’s incorrect response. Complete its faded problem without opening the key. During 40–55, change representation or context and attempt the independent transfer task; compare explanations only after individual work. Use the closing five minutes to name a breakpoint and an instructional response supported by evidence.
Session 3 · Teach the Story (60 minutes)
0–10: Share a successful representation or persistent question. 10–35: Redesign one existing fraction lesson using a narrative problem, a visual model, explicit mathematical language, and targeted practice. 35–50: Peer feedback across grades. 50–60: Choose evidence to collect from students and a date to review it.
Facilitation within Session 3: During 10–35, build a short sequence into an existing lesson: explain a correct worked example, compare a second method, repair a plausible error, complete missing reasoning, then solve an independent transfer problem. Include one future connection from “A Glimpse Ahead.” During 35–50, partners check units, equal wholes, method validity, developmental fit, and whether the scaffolds actually fade. During 50–60, choose an immediate exit task and a delayed check 1–2 weeks later, including a changed context or representation.
Teacher field guide · Planning template
For any grade, complete these seven prompts:
- Our chapter: Which mathematical idea are students developing?
- What came before: What must they already understand about wholes, units, or equivalence?
- What comes next: Where will this idea reappear?
- Show it: Which concrete or visual model makes the meaning visible?
- Evidence: What explanation or transfer task demonstrates understanding beyond a correct answer?
- Fade support: Which completed example and contrasting method will learners explain? Which plausible error will they repair, and which reasoning will they complete?
- Transfer and retention: What independent new-context problem and later check will test whether learning travels beyond the original model?
Facilitator cautions
Keep the story subordinate to mathematical precision. Avoid suggesting that fractions always represent parts of shapes: they are numbers, quotients, operators, measures, and comparisons in different contexts. Clarify part-to-part versus part-to-whole ratios. Treat the grade labels as a conceptual map rather than a strict standards list. Combine conceptual models with explicit instruction, fluency, and cumulative practice.
Use the classroom practice
Each chapter ends with a printable student worksheet and a separate teacher key. Chapters 5–7 include an optional final page, “Think Like a Mathematician,” with nonroutine problems and complete extension solutions. Use one challenge for sustained reasoning or discussion after the core practice. In Session 2, examine the grade-band guidance before choosing tasks. In Session 3, use selected tasks with the redesigned lesson and retain one independent item as evidence. Keep keys separate from student copies; use the collection’s models and partial reasoning to support learning, then check a new related task later.
Use a consistent instructional lens
Before adding another model, ask whether the current one can show the next relationship. Keep the whole, unit, quantities, and labels consistent while translating from objects to pictures to equations. Then change the context to test transfer.
This resource uses a Tang-informed design lens: purposeful models, number relationships, connected representations, and accurate, efficient, flexible calculation. It complements our AoPS-inspired emphasis on justification and nonroutine problems. This is our synthesis, not an official Tang Math program or endorsement.
Tang Math instructional approach ↗
Use the chapter decision cues
Read “Listen for,” “Reduce support when,” and “Return to a model when” alongside each worksheet. Ask students to name the unit, estimate where useful, connect the model to symbols, and try a related problem independently. Change one feature at a time in the short comparison sequences.
Make worked examples active
Explain a completed example; compare a second method; diagnose the visible incorrect claim before opening its interpretation; complete missing reasoning; then try an independent task. A demonstration is not independent evidence.
Collect a delayed sample
Choose the matching page from the delayed-check booklet 1–2 weeks after instruction. Keep teacher answers separate. Record accuracy, conceptual explanation, and independent transfer; note prompts or models supplied. Compare immediate and delayed evidence to choose the next teaching step. This classroom routine is separate from the Fraction Audit.
Evidence of impact
Collect a brief pre/post teacher explanation of equivalence, a revised lesson from each grade team, and a student exit task focused on representation and explanation. Compare the quality of reasoning and the persistence of misconceptions, not only teacher satisfaction. Track computation accuracy, conceptual explanation, and problem-solving/transfer separately. Record the whole, unit, representation, suspected breakpoint, and next instructional step; compare immediate performance with the delayed check before inferring retention. Successful imitation of a worked example alone is insufficient evidence of independent understanding.