Lead the Multiplicative Journey
A 90-minute introduction followed by two 60-minute workshops.
Preparation
Bring counters, cups, bundles of ten, paper strips, graph paper, and the chapter worksheets. Pair teachers across grade bands. Ask “What does one mean here?” and “What quantity does the answer count?” Use anonymous student work where available.
Session 1 · Experience the Story (90 minutes)
0–10: Arrange five counters in two ways. Discuss structure without requiring speed.
10–30: Chapters 1–3: exchange a ten for ten ones; compare two division stories for 24 ÷ 4. Draw an array and identify rows, columns, and units.
30–55: Chapters 4–6: predict the effect of factors below and above one; compare recipe scaling with adding equally; test a fixed-fee cost model for proportionality.
55–70: Chapter 7: compare coordinate ratio with slope using two points. Connect a table, slope triangle, and equation.
70–90: Return to the journey map. Identify one prerequisite, one breakpoint, and one future connection from “A Glimpse Ahead.” Record evidence that would distinguish an error in computation from an error in the unit.
Session 2 · Understand Your Chapter (60 minutes)
0–10: Revisit the six breakpoint hypotheses on the overview.
10–25: Grade teams map the chapter to local standards and choose appropriate worksheet tasks. Study a correct solution and explain each unit.
25–40: Compare a second method; diagnose the first invalid step in incorrect work; complete the faded reasoning before opening its key.
40–55: Solve a transfer task independently, then compare approaches across grades. Upper-grade teams may try a “Think Like a Mathematician” challenge.
55–60: Select one adjustment supported by the evidence.
Session 3 · Teach the Story (60 minutes)
0–10: Share a persistent question or student explanation.
10–35: Adapt an existing lesson: context → model → language → equation → generalization → procedure. Include an explained example, contrasting method, plausible error, partial solution, and independent task. Fade support as learners demonstrate understanding.
35–50: Peer review for units, valid methods, developmental fit, model variety, and workspace. Keep optional challenges optional.
50–60: Choose an immediate check and a delayed transfer check 1–2 weeks later.
Teacher field guide
- What is the unit, and what does each number count?
- What earlier understanding does this require?
- Does the context involve an additive change or a multiplicative comparison?
- Which model and second method reveal the structure?
- Which incorrect inference will we investigate?
- Which support will be faded before independent work?
- What new context and later check will test transfer and retention?
Evidence and cautions
Track computation accuracy, conceptual explanation, and independent transfer separately. A correct product does not by itself establish understanding of the quantities. A repeated addition model suits whole-number groups but does not exhaust multiplication; scaling extends the meaning. A constant additive change does not guarantee proportionality. Do not require formal symbolic multiplication in PK or treat chapter bands as readiness tests.
Collect an annotated teacher explanation, one adapted lesson, student work with units, and a delayed check. Use findings to plan support, not to assume a curriculum or learner is deficient.