The number 72 will eventually appear 12 times and the number 48 will appear 10 times. If the table were continued further, other larger numbers such as 72 and 48 would appear many more than 4 times.

The numbers which appear 4 times have two distinct factorizations into single digit numbers (up to the order of the factors): for example, $24 = 4 imes 6 = 3 imes 8$. There are some numbers, such as 9, 16, and 36, which appear 3 times. Several numbers appear 4 times in the table: 6, 8, 12, 18, and 24.

If we extended the table to include products of all numbers, then every number $n$ (including all perfect squares) would appear at least twice as $n imes1$ and $1 imes n$. Some products on the diagonal, such as 2 $ imes$ 2 = 4, do also appear elsewhere because they have a different factorization (4 $ imes$ 1 and 1 $ imes$ 4 in this case). This is the commutative property of multiplication and several of these squares are shaded below:įor a number on the diagonal like $49 = 7 imes 7$, the two factors are the same and so when we switch their order, we don't get a different entry in the table. This makes sense because a number such as 48 = 6 $ imes$ 8, not on the diagonal, also appears as 48 = 8 $ imes$ 6. These squares are each shaded in different colors in the picture below: These are all along the diagonal of the table. The numbers which only appear once in the table are 1, 25, 49, 64, and 81. MP3, Construct Viable Arguments and Critique the Reasoning of Others. If students work on this task in groups and share their insights, then it is also a good opportunity to engage in MP7, Look For and Make Use of Structure, since the goal of the task is to help students understand the multiplication table from the point of view of factorizations. The main standard for mathematical practice which aligns with this task is The first non-prime number not appearing in the table is $22 = 2 imes 11$, the product of the smallest prime number with the smallest prime number bigger than 9. This makes sense in the context because a prime number cannot be written as a product of two smaller whole numbers. The last part of this question gives an opportunity for the teacher to discuss prime numbers since the list of numbers the students produce will all be primes. Working through the table to see where different numbers appear, the students will have a good opportunity to observe the symmetry of the table which comes from the commutative property of multiplication: $a imes b = b imes a$. For example, 24 appears 4 times in the table:Ģ4 = 3 imes 8 = 8 imes 3 = 6 imes 4 = 4 imes 6.įor the 9 by 9 multiplication table shown, only the numbers 1 through 9 appear with all of their factorizations. The table shows some, but not necessarily all, factorizations of different numbers. The goal of this task is to encourage students to study the multiplication table, a familiar object, from a novel point of view.

3 times table up to 100