Thought for the day: Allow conversations to flow smoothly between noticing and wondering. Observations lead to questions, which lead to more observations and more questions, and so on!
Concepts: addition facts, addition strategies, counting by threes (or other multiples), properties of addition (possibly simple multiplication and division concepts and strategies)
I notice three addition problems.
I notice that three numbers in a row are being added.
I notice that the answers are 18, 21, and 24, and they get 3 larger each time.
I wonder why the answers keep going up by 3 every time.
I wonder if there is a fast way to add three numbers in a row.
I notice that the first number plus the last number is the same as the middle number plus itself.
I notice that I can find the sum by adding the middle number three times.
I wonder if (or why) the answers to these kinds of problems will always be "counting by 3" numbers.
I wonder if I can draw pictures that show how these addition problems work.
I wonder what will happen if I add numbers that jump by two (like 1 + 3 + 5 or 2 + 4 + 6).
I wonder what will happen if I add four whole numbers in a row. (Will the answer always be a “counting by 4 number?)
I wonder if I can do the problems backwards (find three numbers in a row that equal some number).
As they notice and wonder, students may create their own addition problems with patterns. Many of their ideas will flow from things that they have wondered about. For example, they may
Create new addition strategies.
Create pictures that illustrate addition patterns and lead to new strategies.
Create addition expressions that equal some number chosen in advance.
Create new kinds of addition expressions with new patterns.
Reflecting and Extending
I notice that when I add three consecutive whole number, the answer is always a “counting by 3” number (a multiple of 3).
I notice that any three consecutive numbers always contain exactly one multiple of 3.
I notice that a multiple of 3 plus a multiple of 3 is always a multiple of 3.
I notice that I can always make all three numbers the same by shifting 1 from the largest number to the smallest number.
I wonder if I can use what I discover to add long strings of consecutive numbers quickly.
I wonder if there would be patterns like this if I multiplied consecutive numbers.
When you add 1 to each of the three numbers you are adding, you add 1 + 1 + 1 = 3 to the sum, which explains why the answers keep increasing by 3.
Whenever you add three consecutive counting numbers, the answer is a multiple of 3 (a "counting by 3 number'). If fact, the answer is always 3 times the middle number. You can see this by shifting 1 from the largest number to the smallest number (subtracting 1 from the largest number and adding it to the smallest number). For example, if you begin with 5 + 6 + 7 and shift 1 from the 7 to the 5, you get 6 + 6 + 6.
In order to find three consecutive numbers whose sum is some chosen whole number, students may begin by estimating, testing, and revising. Some may eventually discover that they can decompose the total into three equal parts (divide by 3) in order to get the middle number. This can be a great opportunity for students to explore the concept of division before receiving direct instruction on it.
If the sum is not a multiple of 3, it will not be possible to find three consecutive whole numbers whose sum is the number. Some students who encounter this situation may end up thinking about mixed numbers! For example 16 = 4 1/3 + 5 1/3 + 6 1/3.
Students who are able to explore sums of longer strings of numbers may discover all sorts of related patterns involving multiples of other numbers such as 4 and 5.