Rename the Number is an incredibly flexible math routine that develops number sense. It appears in many curriculum materials under different names (e.g. Today’s Number, Number of the Day), encourages creativity and flexibility with numbers and operations, and can fuel rich discussions about patterns in our number system and the behavior of operations. Last year, two 5th grade teachers at Young Leaders Elementary School in the Bronx, came up with a brilliant modification of this routine that helps students write and interpret numerical expressions.
Here’s a CCSS standard that you may find difficult to make sense of and teach:
Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. CC.5.OA.2
The teachers did not initially aim to address this standard. Instead, they wanted to create a routine, with many access points, that would help students operate with increasingly large numbers. So, they gave their students 3 target numbers (e.g. 87, 870, and 8700) that were related by powers of 10, and asked students to Rename the Numbers. They gave students about 5-7 minutes to write expressions in their math notebooks for any or all of the target numbers and then asked students to share some of their expressions with the class.
Here is where the brilliance comes in. After a student shared an expression, the teacher recorded it under the appropriate target number. For example:
87 870 8,700
(4 x 20) + 7
Then she asked the class to create a similar expression for each of the other target numbers. For example:
87 870 8,700
(4 x 20) + 7 (4 x 200) + 70 (4 x 2,000) + 700
Creating similar expressions pushed students to grapple with the relationships among factors and products, addends and sums, subtrahends, minuends, and differences and dividends, divisors, and quotients. What do you have to do to factors, addends, dividends, divisors, subtrahends and/or minuends to create an expression that is 10 times greater or smaller than your original expression?
As they grappled and shared ideas, students noticed and discussed patterns, made conjectures, tested them and formed generalizations (e.g. “Multiplying one factor by 10 increases the product by 10”). These are all important components of Mathematical Practice 3: Create viable arguments and critique the reasoning of others. The teachers posted the generalizations and both teachers and students referred to them on a regular basis.
By doing this routine regularly, students ALSO learned to, “interpret numerical expressions without evaluating them”. They learned to create and recognize expressions that were 10 or 100 times greater or smaller than other expressions without calculating the answers! Eventually, the teachers even included DECIMALS in the target numbers.
Here are some examples from their students’ notebooks:
What do you notice? What do you think YOUR students could learn, and what could YOU learn about your students from this routine?
What does the work below reveal about what this student DOES and DOES NOT yet understand about creating similar expressions?
Oh yeah, this also makes a great homework assignment!
My thanks to Bonnie Rosario and Kelly Jo Lillian at PS 369X, Young Leaders Elementary School.