The Investigations curriculum and Jessica Shumway’s book, Number Sense Routines contain so many wonderful math routines. Routines designed to give students access to the mathematics and elicit many ways of thinking about the same problem. One of the more open routines, is Today’s Number. In Today’s Number, a number is posed to the class and the teacher can ask students for questions about that number, expressions that equal that number, or anything they know about that number. I love this routine, and while it is more commonly used in the primary grades, I used it often in my 5th grade classroom. While I would capture so much amazing student thinking, I always felt like all of that great thinking was left hanging out there. I could see some students were using what they knew about operations and properties to generate new expressions for the given number, however I wondered how many saw each expression as individual, unconnected ideas.
After I read Connecting Arithmetic to Algebra, I had a different ending to Today’s Number, an ending that pushed students to look explicitly at relationships between expressions. I tried it out the other day in a 3rd grade classroom.
I asked students for expressions that equaled Today’s Number, 48. I was getting a lot of addition, subtraction and multiplication expressions with two numbers, so I asked students if they could think of some that involved division or more than two numbers. I ran out of room so I moved to a new page and recorded their ideas.
After their thinking was recorded, I asked the students which expressions they saw a connection between. This is where my recording could improve tremendously, but I drew arrows between the two expressions as students explained the connection.
In case the mess is hard to see, these are some connecting ideas that arose:
Commutative Property: 3 x 16 and 16 x 3, 6 x 8 and 8 x 6, 12 x 4 and 4 x 12
Fraction and Fraction addition: 48/1 and 24/1 + 24/1 and 24 + 24
Subtracting from 100 and 1000: 100-52 and 1000-952
Multiplication and Repeated Addition: 4 x 12 and 12+12+12+12
Adjusting Addends in similar ways: 38+10 and 18 + 30, 40 + 8 and 48+0
Other ideas that I don’t particularly know how to categorize:
10 x 4 + 8 = 10 x 4 + 4 x 2
58 x 1 – 10 = 58 – 10
The second page got even more interesting:
“Groups of” and Decomposition: 7 x 4 + 2 + 18 = 14 + 14 + 2 + 10 + 8 . This student saw the two 14s as two groups of 7 and then the 18 decomposed into 10 + 8.
Halving and Halving the Dividend and Divisor: 192÷4 = 96÷2. This student actually used the 192 to get the expression with 96.
Another variation of the one above was 200 ÷ 4 – 2 = 100 ÷ 2 – 2.
Other cool connection:
96 ÷2 = (48 + 48) ÷ 2; This student saw the 96 in both expressions since they were both dividing by 2.
I think asking students to look for these connections pushes them to think about mathematical relationships so expressions don’t feel like such individual ideas. I can imagine the more this is done routinely with students, the more creative they get with their expressions and connections. I saw a difference in the ways students were using one expression to get another after just pushing them try to think of some with more than 2 numbers and some division.