Multiplication Facts: Strategies Beyond Memorization
Disclosure: This post contains affiliate links. If you make a purchase through these links, I may earn a commission at no additional cost to you. As an Amazon Associate, I earn from qualifying purchases.
This week’s focus is 3.OA.B.5: Apply properties of operations as strategies to multiply and divide. The key word in this standard is apply. Students use properties to develop strategies for learning multiplication facts; they are not expected to memorize or name the formal properties.
Learning multiplication facts involves more than memorizing products. In this week’s
Math Standard Monday video, a 6-by-7 array shows how students can use facts they already know to determine one they do not yet recall. The array makes the relationships among the facts visible and sets the stage for the strategies explored below.
Let’s extend that thinking and look more closely at:
what multiplication facts are and how fact recall develops
why students use properties as strategies rather than memorize property names
how arrays make the distributive relationship visible
how flexible multiplication strategies support fluency and eventual recall
What Are Multiplication Facts?
Multiplication facts describe relationships between two factors and their product. For example, 6 × 7 = 42 represents the relationship among 6, 7, and 42. In Grade 3, students develop fluency with multiplication and division within 100 and, by the end of the grade, are expected to know from memory all products of two one-digit numbers.
Knowing basic multiplication facts from memory is an important goal, but it is not the starting point—or the whole story. Facts are not isolated. For example, 6 × 7 is connected to 6 × 5, 6 × 2, 6 × 6, and 5 × 7. Students use these relationships to determine facts they do not yet recall, and those connections support increasingly fluent recall over time.
Basic facts also become building blocks for multiplication beyond the facts students are expected to memorize. A product less than 100 does not automatically make an expression a basic multiplication fact. Although 6 × 16 = 96, 6 × 16 is not a product of two one-digit numbers. Students can use known facts with place-value understanding and the distributive property:
6 × 16 = (6 × 10) + (6 × 6)
= 60 + 36
= 96
Notation Note: *The parentheses visually separate the two partial products; they are not mathematically required. Students could also write 6 × 10 + 6 × 6 = 96. The focus is on using known facts, place-value understanding, and the distributive property—not teaching a procedure for evaluating expressions with grouping symbols.
Use Properties as Strategies—Not Vocabulary Words
Students use properties of operations as multiplication strategies. They are not expected to memorize the formal names of the properties or identify a property from a list.
This work builds on the understanding developed in last week’s post, Equal Groups Multiplication: Building Conceptual Understanding: students must first interpret what each factor represents and connect the expression to a meaningful model or context.
A student may use the commutative property by using 7 × 6 to determine 6 × 7. A student may use the distributive property by decomposing 6 × 7 into 6 × 5 and 6 × 2. What matters is that the student understands and can explain the relationship—not that the student says “commutative property” or “distributive property.”
Instead of asking, “Which property did you use?” try asking:
Which multiplication fact did you already know?
How did that fact help you?
Where can you see the smaller facts in your model?
How does your expression or equation show your thinking?
Use a Known Fact to Determine an Unknown Fact
When a multiplication fact is not yet known, students can use facts they already know. For 6 × 7, one useful decomposition is 6 × 5 and 6 × 2:
6 × 7 = (6 × 5) + (6 × 2)
= 30 + 12
= 42
This strategy works because 7 can be decomposed into 5 and 2. Six groups of 7 contain six groups of 5 and six groups of 2, so the original quantity has not changed; it has simply been partitioned into friendlier facts. Here, 5 and 2 are useful because multiplication facts involving 5 and 2 are commonly among the first facts students know.
The Array Makes the Distributive Relationship Visible
The equations record the reasoning, but the array reveals why it works. When the seven columns are separated into five yellow columns and two blue columns, students can see both partial products inside the original array.
Color coding can help students coordinate the array and the equations. The five yellow columns align with 6 × 5 = 30 and the yellow term (6 × 5), while the two blue columns align with 6 × 2 = 12 and the blue term (6 × 2). Using the same colors reduces the visual matching students must do, making it easier to focus on how the partial products combine. Color is a temporary scaffold—not the mathematics itself—but it can make the distributive relationship visible and keep the decomposition grounded in the model rather than a disconnected procedure.
Students are not being taught to break apart numbers because a rule tells them to. They are learning that an array can be partitioned and that the products represented by the smaller arrays combine to equal the product represented by the whole array.
One Multiplication Fact, More
Than One Useful Strategy
Decomposing 7 into 5 and 2 is not the only option. Seven could also be decomposed into 3 and 4 or 6 and 1. A student who knows 6 × 6 might reason:
6 × 7 = (6 × 6) + (6 × 1)
= 36 + 6
= 42
Students can also decompose the other factor. A student who knows 5 × 7 might add one more group of 7:
6 × 7 = (5 × 7) + (1 × 7)
= 35 + 7
= 42
These decompositions are mathematically equivalent, but they may not be equally useful to every student. A flexible multiplication strategy begins with the facts the student actually knows. Our goal is not to require one preferred decomposition; it is to help students choose an efficient relationship and explain why it works.
Students may orient or partition an array differently. A horizontal or vertical partition can be mathematically valid, but the partition must match the factor being decomposed in the equation. In the arrays shown, 6 rows of 7 means that a vertical partition decomposes the 7, while a horizontal partition decomposes the 6. If a student uses the opposite orientation, ask what the rows and columns represent and check that the model, explanation, and equation remain consistent.
Strategies Build Toward Recall
Using a strategy is not the opposite of knowing multiplication facts. Purposeful strategy use is part of the path toward fluent recall.
Students may move from using representations or skip-counting, to deriving an unknown fact from known facts, to selecting increasingly efficient strategies, and eventually to recalling the fact from memory. This progression does not happen at the same rate for every student—or for every fact. A student may immediately recall 5 × 8, derive 6 × 8 from it, and still need a model to determine 7 × 8.
This developmental view of fluency—and the use of assessments such as fact sorts to determine appropriate next steps—is explored in Jennifer Bay-Williams and Gina Kling’s Math Fact Fluency: 60+ Games and Assessment Tools to Support Learning and Retention (Paid Link).
Fact sorts can help teachers identify where students are along this path. Students might sort facts into categories such as:
facts I am still learning (representing or skip-counting)
facts I can figure out using another fact (strategy)
facts I know right away (automatic)
The purpose is not simply to sort the cards. The sort provides evidence teachers can use to plan instruction. Which facts are already secure? Which relationships does the student recognize? Is the student relying on counting when a more efficient strategy is within reach? Which known fact could help the student determine the next set of facts?
Teachers then help students move forward by connecting models to equations, highlighting useful relationships, comparing strategies, and providing meaningful practice with a manageable set of facts. As connections become more familiar, students should be encouraged to use more efficient reasoning and eventually recall facts they no longer need to reconstruct.
Strategies should therefore be temporary supports, not permanent procedures students must perform for every fact. When a student repeatedly uses 6 × 5 and 6 × 2 to determine 6 × 7, the relationship among 30, 12, and 42 becomes increasingly familiar. Eventually, the student may recall 6 × 7 = 42 without decomposing it. The strategy has done its job.
Multiplication Facts Beyond Memorization
Students are expected to know from memory all products of two one-digit numbers by the end of Grade 3 (3.OA.C.7). Recall matters, but it is one outcome of fluency—not the entire learning process.
Multiplication fact fluency includes accuracy, efficiency, flexibility, and appropriate strategy selection. Recall is one expression of fluency; accurate, efficient derivation from a known fact is another stage that teachers can develop toward recall.
Timed practice may reveal how quickly a student produces answers, but it does not show which facts are secure, which facts the student can derive, or which relationships the student understands. Teachers need more specific evidence so they can respond to what each student is ready to learn next.
When students develop multiplication facts through models, properties, and connected reasoning—and teachers deliberately help them progress from less efficient methods toward increasingly efficient strategies and recall—they gain more than a collection of memorized products. They develop a connected structure they can use with division, multi-digit multiplication, fractions, and algebra.
Try This Activity: Which One Doesn’t Belong?
The activity slide presents four multiplication expressions and invites students to decide which one does not belong. Students must support their choice with mathematical reasoning, and more than one response may be valid.
Access the free 3.OA.B.5 Which One Doesn’t Belong? Activity Slide.
A Which One Doesn’t Belong routine does not have one predetermined correct answer. The value comes from noticing relationships, comparing the structure of the expressions, and defending a choice with mathematical evidence. Avoid telling students what to notice or confirming a response too quickly. After students justify an initial choice, invite them to find another possible answer.
If the routine is new to you, read Christopher Danielson’s reflections on the purpose and origins of Which One Doesn’t Belong?
As students work, listen for whether they compare the mathematical structure of the expressions rather than rely only on surface-level differences. Their explanations can reveal which facts they recognize, how they interpret the factors, and whether they understand how known facts can be used to determine an unfamiliar fact.
Pay attention to the connections students make across the four expressions. If a student’s reasoning is unclear, ask one or more of these questions: “Which known facts do you see?” “What relationships do you notice among the expressions?” “Could a different expression be the one that does not belong?”
Help Students Move from Skip-Counting to Strategy to Recall
Want to help students develop multiplication fact recall through understanding rather than memorization alone? Join me this November in Toms River for Fluency Beyond Fast: 3–5, a full-day workshop designed for Grades 3–5 educators and also helpful for Grade 6. Register here.
Comments