Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Arithmetic is full of patterns, and third graders are asked to notice them and, more importantly, to explain why they happen. Look at the fours column of a multiplication table: 4, 8, 12, 16, 20. Every product is even. The reason is that 4 times any number can be split into two equal parts, each equal to 2 times that number, and anything that splits into two equal parts is even.
Other patterns worth exploring include the diagonal of square numbers, the way the nines column has digits that add to 9 (up to 9 × 9), the symmetry of the multiplication table across its diagonal because of the commutative property, and the rule that an odd number plus an odd number always makes an even number. In an addition table, numbers along each diagonal stay the same because one addend goes up as the other goes down. Students describe what they see, test it with more examples, then give a reason that uses equal groups, arrays or properties of operations. That move from noticing to explaining is the heart of mathematical reasoning.
Saying "they are all even" is only half the task. Ask why: students should connect 4 × n to n + n + n + n or to two equal groups of 2 × n.
A student who notices 3 × 2 and 3 × 4 are even may claim all multiples of 3 are even. Encourage testing more cases, such as 3 × 5 = 15.
Some children think an odd number times an odd number is even because odd plus odd is even. Build 3 × 5 with counters and pair them up to see one is left over.
Patterns down a column and across a row are the same in a multiplication table, but students sometimes compare a row to a column of a different number. Highlight one row or column at a time.
Find 4 × 7 and explain why every product in the fours column is even.
Answer: 4 × 7 = 28. Products of 4 are always even because 4 times a number equals two equal addends, each 2 times the number.
Give students a blank multiplication chart and colored pencils and ask them to shade all even products. The striking pattern sparks good questions. Then focus on one pattern at a time, asking students to write an explanation that a classmate could check with counters or an array.
Test items often show a table or sequence and ask students to choose the rule or the true statement about it, sometimes with a written explanation. Practise both multiple-choice identification and short written reasoning.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: A) They are all even.
6 times a number is 3 times the number plus 3 times the number, two equal parts, so every product of 6 is even (6, 12, 18, ...).
Answer: 35
The pattern counts by 7s (the 7s column), so the next number is 28 + 7 = 35.
Answer: C) Odd, because 35 cannot be split into two equal whole-number groups
5 × 7 = 35. An odd number of groups of an odd number is always odd: when you pair things up, one is left over.
Answer: 45
Count on by 9: 36 + 9 = 45, and 4 + 5 = 9, so the digit-sum pattern continues.
Answer: A) Because 4 × 6 = 24 and 6 × 4 = 24
Order does not change a product, so 4 × 6 and 6 × 4 are both 24. This is why the table is symmetric.
Answer: D) One addend increases by 1 and the other decreases by 1
Down adds 1 to one addend and left takes 1 from the other, so the total does not change, for example 5 + 3 = 6 + 2.
Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.
A full lesson with slides, activities and an exit ticket on arithmetic patterns and properties, pitched to grade 3 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 3.OA.D.9 with an answer key, ready in about a minute.
Make a worksheet →Turn arithmetic patterns and properties into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Yes. The standard asks students to identify patterns and explain them using properties of operations, so a short written or spoken reason is expected, not just the pattern.
Even and odd products, the symmetry of the multiplication table, columns that count by a fixed number, and diagonals in the addition table are all accessible and lead to strong explanations.