Number Talks: Multiplication
A free, research-backed teacher tool for collaborative exploration of mental multiplication strategies. Designed for grades 2-5.
What is a Number Talk?
A short (5-15 minute) whole-class routine where the teacher poses one mental-math problem, students solve it without paper, and several students share how they thought about it while the teacher records the strategies on the board.
Stay neutral and curious — accept every strategy on its own terms before connecting or comparing, so students learn that reasoning is the goal and the answer is just where the reasoning lands.
What a Number Talk Looks Like
Mr. Kim: Here's today's problem: 9 times 7. Solve it in your head. When you have an answer, thumb up. If you find a second way, put up a second finger.
Mr. Kim: Who got something different from their neighbor? Let's see. Declan?
Declan: I got 61. I did 9 times 8 is 72, and then I took away 9. 72 minus 9 is... 61.
Mr. Kim: Walk me through that subtraction. 72 minus 9 — how did you do that part?
Declan: I went 72 minus 10 is 62, then add 1 back... wait, that's 63, not 61.
Mr. Kim: Say more about that.
This tool helps teachers prepare for Number Talks by previewing which strategies a problem might invite, generating word problems, and providing coaching commentary on facilitation techniques.
Features
- Strategy analysis — 10 mental math strategies scored per problem: skip-count, break apart, compensation, doubling, near square, and more
- Word problems — 1,200+ templates across 12 domains, matched to factor ranges, with cognitive features (extraneous info, implied factors)
- Problem of the Day — daily auto-advancing problem with fullscreen projection mode for the classroom
- PDF export — student blanks and multi-strategy teacher copy with no single "best" strategy
- Classroom transcripts — 25 fictional Number Talk scenes with line-by-line coaching commentary
- Projection mode — clean fullscreen display with participation prompt, sentence starters, and strategy vocabulary
- Grade-appropriate — factor ranges, strategy scoping, and word problem selection respect grade 2-5 expectations
Strategy Analysis
The tool scores 10 mental multiplication strategies per problem, each with a fit rating and stepwise breakdown. The strategies, their mathematical basis, and their research lineage:
Skip-count
Works for any problem, but it really sings when the chosen step is 2, 5, or 10 and the hop count stays modest. It can still be reasonable for 3s and 4s; by 6s, 7s, and 8s it is usually just a fallback unless there are very few hops. The most concrete multiplication move — a direct link to repeated addition. Relies on: definition of multiplication as repeated addition. Source: Parrish (2014) treats it as an entry-level Number Talks move; CGI (Carpenter et al.) codes it as a Counting Strategy.
Break apart (distributive property)
The universal move — split one factor into friendlier parts and distribute. Strongest for clean 5-splits (6–9), clean 10-splits (11–19), tiny leftover chunks such as 301 = 300 + 1, and ordinary place-value splits. Extends naturally to multi-digit multiplication. Relies on: a × (m + n) = a × m + a × n. Source: Bay-Williams & SanGiovanni (2021) call this "Break Apart to Multiply." Fosnot draws it on an "open array."
Compensation (over/under)
Anchor to a nearby friendly landmark, do the easy version, then adjust. Canonical cases are 9 → 10, 11 → 10, 24 → 25, 49 → 50, and 99 → 100. In the implementation, offsets of 1 or 2 from powers of 10 and major landmarks like 25 or 50 qualify, while other round tens and hundreds only qualify at ±1. Relies on: n × b = (n+k) × b − k × b. Source: Van de Walle names the 9-specific version "nifty nines." Pam Harris calls the general move "The Over Strategy."
Doubling chain
Lights up when one factor is a power of 2 in the doubling-chain range. The short chains 2, 4, and 8 are strongest; 16 is still viable but longer. Chain from 1 × (other) by repeated doubling — since ×2 is foundational, the whole ×4 and ×8 rows largely come for free. Relies on: associativity of multiplication. Source: Pam Harris uses doubling chains as her canonical "constructible" problem string; Baroody (2006) lists doubling among Phase 2 reasoning strategies.
Fives / halving tens
Best when one factor is 5, 25, or 50. Compute ×10 or ×100 first, then halve or quarter the result. ×5 and ×50 are the cleanest cases; ×25 is also strong, especially when the other factor is even so the quartering stays clean. Relies on: the fact that 5 is half of 10. Source: Pam Harris's "Five is Half of Ten."
Doubling and halving
Halve one factor and double the other — the product stays the same. It sings when the transformed pair lands on a power of 10 or another major landmark such as 25, 50, or 100, whether that friendliness comes from the doubled side or the halved side. Requires at least one even factor. Relies on: a × b = (a/2) × (2b). Source: Harris treats this as one of the highest-leverage mental-math moves; Bay-Williams & SanGiovanni (2021) list it as an advanced derived-fact strategy.
Associative regroup
Spot a composite factor (like 12 = 4 × 3) and regroup so a useful chunk joins the other factor: 12 × 5 = (4 × 3) × 5 = 4 × 15 = 60. The implementation also treats small prime-power families such as 9 × 27 = 3² × 3³ as especially strong regrouping territory. Relies on: (a × b) × c = a × (b × c). Source: Pam Harris calls this "Flexible Factoring." Fosnot uses the ratio table as the visual model.
Near square
Lights up when the factors are close enough to a square for their size. For small facts that usually means within 1 or 2; for larger factors the allowable gap grows slowly. Anchor to a nearby square (6 × 6 = 36) and add or subtract strips. Squares are disproportionately memorable (Siegler fan effect). Relies on: distributivity plus the empirical observation that squares are easier to recall. Source: Bay-Williams & SanGiovanni (2021) call this "Use a Square."
Place-value shift (×10, ×100)
Fires when one factor is exactly a power of 10. The move is to "shift the digits" — multiplying by 10 shifts every digit one column to the left. Scales past memorized facts: 42 × 10 = 420 is as trivial as 7 × 10 = 70. Relies on: base-10 place value. Source: Bay-Williams & SanGiovanni (2021) classify ×10 as foundational. Note: "add a zero" is linguistically shaky once decimals enter the picture.
Direct retrieval
The student just knows it — Phase 3 mastery (Baroody, 2006). The strongest retrieval cases in the implementation are foundational pairs and small perfect squares; ordinary 1×1 facts are still plausible retrieval territory, but that depends heavily on the individual student. Bay-Williams & SanGiovanni (2021) are emphatic that retrieval is the result of good Phase 2 reasoning work, not the goal of flashcards.
Turnaround (commutative property)
Pure commutativity — flip the factors to pick the easier orientation. Doesn't reduce the work, but it halves the fact set the student has to memorize. Relies on: a × b = b × a. Source: Russell, Schifter & Bastable (2011) frame commutativity as a claim kids investigate and prove for themselves.
Research Basis & Bibliography
Strategy analysis is opinionated heuristic, not published ground truth. The following sources informed the tool's design:
- Bay-Williams, J. M. & SanGiovanni, J. J. (2021). Figuring Out Fluency in Multiplication and Division. Corwin. The most comprehensive contemporary source for the foundational-vs-derived framing and for each named multiplication strategy.
- Parrish, S. (2014). Number Talks: Helping Children Build Mental Math and Computation Strategies. Math Solutions. The canonical classroom-routine source.
- Harris, P. Math is Figureoutable (blog + podcast). Problem Strings methodology, the "Over" strategy, "Five is Half of Ten," and "Flexible Factoring."
- Baroody, A. J. (2006). "Why children have difficulties mastering the basic number combinations". Teaching Children Mathematics. The three-phase model (counting → reasoning → mastery).
- Carpenter, T. P., et al. Children's Mathematics: Cognitively Guided Instruction. Heinemann. CGI's four-tier strategy coding (direct modeling → counting → derived fact → recall).
- Fosnot, C. T. & Dolk, M. (2001). Young Mathematicians at Work: Constructing Multiplication and Division. Heinemann. The "landscape of learning" approach and the open array as a thinking model.
- Russell, S. J., Schifter, D. & Bastable, V. (2011). Connecting Arithmetic to Algebra. Heinemann. Kids investigate commutativity, associativity, distributivity as conjectures.
- Van de Walle, J. A., Karp, K. S. & Bay-Williams, J. M. Elementary and Middle School Mathematics: Teaching Developmentally. Pearson.
- Boaler, J. (2015). "Fluency Without Fear". youcubed at Stanford. The affective case for why multi-strategy flexibility matters more than memorized speed.
- Hickendorff, M., Torbeyns, J. & Verschaffel, L. (2019). "Multi-digit Addition, Subtraction, Multiplication, and Division Strategies". Springer International Handbook of Mathematical Learning Difficulties.
- Hickendorff, M. (2017). "Dutch sixth graders' use of shortcut strategies". European Journal of Psychology of Education. Empirical counterweight: even students taught strategies via RME curricula use them infrequently on unprompted work.
- Threlfall, J. (2002). "Flexible mental calculation". Educational Studies in Mathematics. Theoretical critique of strategy-choice models.
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