Thought for the day: Patterned lists of equations help students discover meanings and make connections.
Samples of noticing and wondering
I notice that the answer is always on the left.
I notice that the first number on the right (minuend) stays the same.
I notice that the number being subtracted (the subtrahend) goes down by 1 each time.
I notice that the answer (difference) goes up by 1 each time.
I notice that the 1 and the 3 trade places in two of the equations.
I wonder why the difference goes up by 1 when the subtrahend goes down by 1.
I wonder what happens to the difference when the minuend goes up by 1.
I wonder if the pattern keeps going if you put more equations into the list.
I wonder if the difference always goes up by the same amount that the subtrahend goes down.
I wonder if you can fit equations with fractions in between the equations in the list.
You can use this type of prompt to help your students understand operations more deeply. In this case, recognizing how the difference is affected when you change the minuend and/or subtrahend is an important part of making sense of the concept of subtraction. Deeper understanding can also lead to new strategies. For example, students may discover that increasing both the minuend and subtrahend by the same amount leaves the difference unchanged. They could use this idea to make certain subtraction calculations easier: for example, by changing 17 – 9 into 18 – 10.
Some students may wonder what happens if you continue the list downward (or upward). This will lead to a discussion of negative numbers. The ideas of "how much more" and "counting up" are great ways to think about subtraction with negative numbers (probably more helpful than the "taking away" idea), especially when you visualize the equations on a number line.