See the connection
7 × 6 = (5 × 6) + (2 × 6) = 42
Split an array of 7 rows of 6 into 5 rows and 2 rows. There are 30 dots and 12 dots: 42 altogether.
Quick-reference strategies
Choose a relationship the learner understands. These are useful options, not a required sequence or a guarantee of instant recall.
| Fact or set | Try this | Example & explanation |
|---|---|---|
| ×0 and ×1 | Use equal groups | 6 × 0 = 0; 6 × 1 = 6 Six groups of zero have no items; six groups of one have six items. |
| ×2 | Double | 2 × 7 = 7 + 7 = 14 Two groups contain two equal quantities. |
| ×3 | Double, then add one group | 3 × 6 = 12 + 6 = 18 Three groups are two groups plus one group. |
| ×4 | Double twice | 4 × 6: 6 → 12 → 24 Doubling a quantity twice makes four copies. |
| ×5 | Halve a tens product | 5 × 8 = 80 ÷ 2 = 40 Five equal groups are half of ten equal groups. |
| ×6 and ×7 | Use five groups plus the rest | 6 × 8 = 40 + 8 = 48; 7 × 8 = 40 + 16 = 56 Six is five plus one; seven is five plus two. |
| ×8 | Double three times | 8 × 6: 6 → 12 → 24 → 48 Each doubling doubles the number of copies: 2, 4, then 8. |
| ×9 | Ten groups minus one group | 9 × 7 = 70 − 7 = 63 Nine equal groups are one group short of ten. |
| ×10 | Use tens and place value | 10 × 7 = 7 tens = 70 Ten copies of seven equal seven copies of ten. |
| ×11 and ×12 | Ten groups plus one or two groups | 11 × 6 = 60 + 6 = 66; 12 × 6 = 60 + 12 = 72 Keep the group size the same while splitting the number of groups. |
1. Build equal groups and arrays
Connect the number of groups, the size of each group, and the total to the equation. Arrays make the structure visible and prepare children to split products into parts.
Try it together
Make 3 rows of 4 counters. Label 3 × 4 = 12. Rotate the array and compare 4 × 3: the same total, a different grouping.
2. Establish useful anchor facts
Facts involving 2, 5, and 10 often provide useful starting points. Connect them to doubling, familiar five-structures, and place value, rather than treating them as isolated lists.
Try it together
For 4 × 6, double 2 × 6. Show 12 + 12 = 24 with two matching parts of an array.
3. Break apart an unfamiliar product
Use known products and show why the parts recombine. Draw a dividing line in an array so the written equation matches the visual model.
Try it together
Draw 7 rows of 6. Separate 5 rows from 2 rows and use the worked example above. Ask why 30 + 12 counts every dot exactly once.
4. Mix facts once the strategy is secure
Predictable table recitation can hide difficulty retrieving individual facts. After teaching a relationship, include individual questions in a varied order and revisit them after a gap.
Try it together
Mix 5 × 6, 7 × 6, and a few established facts. If recall stalls, return to the array and strategy, then retry later in the session.
Try it: think, then check
Work it out before opening the answer. Use counters or a drawing if helpful. The explanation matters as much as the result.
Double twice to find 4 × 7.
Show answer & explanation
28
Double 7 to get 14, then double 14 to get 28.
Use 10 × 8 to solve 9 × 8.
Show answer & explanation
72
10 × 8 = 80. Remove one group of 8: 80 − 8 = 72.
Use five groups plus two groups to find 7 × 6.
Show answer & explanation
42
5 × 6 = 30 and 2 × 6 = 12. Add them: 42.
A simple routine to try
Adapt the amount and pace to the learner. This is a suggested routine, not a fixed prescription.
- Show one array and explain its groups.
- Derive a new product from an anchor.
- Retrieve a small varied set with feedback.
- Revisit it later alongside previously learned facts.
What to watch for
Skip counting can support meaning, but a child who must restart a long sequence for every question may benefit from a derived fact. Avoid introducing several new shortcuts at once.
When to take the next step
Increase the range when the learner can represent equal groups and use anchors accurately across sessions. Extend by asking for two different partitions of the same array.
Common questions
Which times table should we teach first?
There is no single required sequence for every learner. Meaningful facts involving 2, 5, and 10 are often useful anchors; choose the next relationships based on what the child can already explain and recall.
What is the trick for the 8 times table?
One useful strategy is doubling three times. For 8 × 7, double 7 to 14, double 14 to 28, and double 28 to 56. Use this when doubling is familiar; an array or a nearby known product may be easier for some learners.
Research behind the guidance
These sources support the teaching principles. The examples and routines are FluentFacts’ practical applications, not individually tested interventions or a promise of outcomes.
Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades (2021)
What Works Clearinghouse · Institute of Education SciencesSupports systematic instruction, mathematical language, representations, number lines, and carefully used fluency activities in elementary intervention.
Put it into practice: Teach one relationship explicitly, connect a model to its equation, and give specific feedback before asking for independent recall.
Organizing Instruction and Study to Improve Student Learning (2007)
Institute of Education SciencesRecommends spacing learning over time, connecting concrete and abstract representations, and using retrieval to strengthen learning.
Put it into practice: Revisit a small fact set across days. Ask the learner to produce an answer before checking, then correct mistakes with a useful strategy.
Strengthening the Student Toolbox (2013)
John Dunlosky · American EducatorReviews practice testing and distributed practice across learning contexts. It does not establish one ideal schedule for every child.
Put it into practice: Use short recall attempts with feedback rather than only rereading a fact list. Spread those attempts across sessions instead of one long sitting.
Adding It Up: Helping Children Learn Mathematics (2001)
National Research CouncilDescribes mathematical proficiency as interwoven strands, including conceptual understanding, procedural fluency, strategic competence, reasoning, and productive disposition.
Put it into practice: Look for accurate answers, flexible strategies, and an explanation of why the fact works. Record progress beyond speed alone.
Put the idea into practice.
Choose an operation and a suitable fact range in FluentFacts. Begin with facts the learner understands.
Choose your practice Or create a worksheet →