Content description: recall and demonstrate proficiency with multiplication facts up to 10 x 10 and related division facts; extend and apply facts to develop efficient mental strategies for computation with larger numbers without a calculator
Elaboration 1
E1: using arrays on grid paper or created with blocks or counters to develop, represent and explain patterns in the 10 × 10 multiplication facts; using the arrays to explain the related division facts
Teach and do: Build 4 rows of 7 with counters. Record 4 × 7 = 28 and 7 × 4 = 28, then 28 ÷ 4 = 7 and 28 ÷ 7 = 4. Add a fifth row: the total grows by 7 to 35. Students build a different array, label rows/columns/whole and explain one pattern and both divisions. Look for equal rows and a connection between whole and factors.
Elaboration 2
E2: using materials or diagrams to develop and record multiplication strategies such as doubling, halving, commutativity, and adding one more or subtracting from a group to reach a known fact; for example, creating multiples of 3 on grid paper and doubling to find multiples of 6; recording and explaining the connections to the ×3 and ×6 multiplication facts: 3, 6, 9, … doubled is 6, 12, 18, …
Worked connections: Use 3 × 8 = 24 and double 24 to obtain 6 × 8 = 48; use 10 × 6 = 60 and halve 60 to obtain 5 × 6 = 30; turn 3 × 8 into 8 × 3; add one group of 6 to 6 × 6 = 36 to get 7 × 6 = 42; remove one group of 6 from 10 × 6 = 60 to get 9 × 6 = 54. Students show a chosen connection with an actual diagram and explain what changes.
Elaboration 3
E3: using known multiplication facts for 2, 3, 5 and 10 to establish multiplication facts for 4, 6, 7, 8 and 9 in different ways; for example, using multiples of 10 to establish the multiples of 9 as “to multiply a number by 9 you multiply by 10 then take the number away”; 9 × 4 = 10 × 4 – 4, so 9 × 4 is 40 – 4 = 36; using multiple of 3 as “to multiply a number by 9 you multiply by 3, and then multiply the result by 3 again”
Strategy menu: Build fours by doubling twos; sixes by doubling threes; sevens by joining fives and twos; eights by doubling fours; nines by subtracting one group from tens. For 9 × 5, either 50 − 5 = 45 or 3 × 5 = 15 then 3 × 15 = 45. In the second method there are three lots of three groups of 5, making nine groups. Ask students to compare two strategies for one fact, rather than memorise a rule without meaning.
Elaboration 4
E4: using arrays and known multiplication facts for twos and fives to develop the multiplication facts for sevens, applying the distributive property of multiplication; for example, when finding 6 × 7, knowing that 7 is made up of 2 and 5, and using an array to show that 6 × 7 is the same as 6 × 2 + 6 × 5 = 12 + 30 which is 42
Worked array: In 6 rows of 7, colour 2 columns blue and 5 gold. The parts give 6 × 2 = 12 and 6 × 5 = 30, so 6 × 7 = 42. Students create a sevens array of their own, show the 2+5 column split and write both partial products and the sum. If a learner writes 6 × 2 + 5, ask where the other five rows of gold counters went.
Elaboration 5
E5: using known multiplication facts up to 10 × 10 and the inverse relationship of multiplication and division to establish corresponding division facts
Worked inverse: From 7 × 8 = 56, there are 8 groups of 7 in 56 and 7 groups of 8 in 56. Thus 56 ÷ 7 = 8 and 56 ÷ 8 = 7. Distinguish “how many groups?” from “how many in each group?” With square facts the two division statements coincide: 6 × 6 = 36 gives 36 ÷ 6 = 6.
Elaboration 6
E6: designing, creating and playing instructive card games that involve the recall, recognition and explanation of the 10 × 10 multiplication facts and related division facts
Create and play: Use three multiplication facts with different products and factors 1–10. Make four cards per family: a multiplication, its product and two related divisions. Write rules: match a product, explain a mental strategy, then check a related division. Play three turns with a partner and keep a written record of the cards, strategy and division check. For 6 × 8, match 48, explain 3 × 8 = 24, so 6 × 8 = 24 + 24 = 48, then say 48 ÷ 6 = 8. The adult checks the actual 12 cards, rules and recorded play; varied correct games are welcome.