Building Number Sense

How do we teach grade level standards for students that need support with their number sense? When students are learning new content, using mental math, visuals, and manipulatives provide opportunities to deepen their understanding of numbers by thinking through a new context. Take these examples of using double number lines to work with percent increase and decrease from Illustrative Mathematics Grade 7 Unit 4 Lesson 7.

Students have to first label the tick mark above 100% with 12, which is covered in previous lessons. As a second step, students need to see that 12 divided into two equal intervals. How do we support students that may struggle with determining the first tick mark should be 6? First, we have to acknowledge that there are several ways to arrive at the answer, listed below in order of a predictable progression of learning about division as an operation:

1. Skip counting
2. Multiplication facts (knowing that 6 times 2 is 12)
3. Division

Students that are skip counting, may start by using guess and check, and while a calculator or multiplication chart may support the calculation, it may not support the building of number sense. Instead, a number line with every integer between 0 and 12 or 12 counters, can help students see the idea of equality inherent in the partitioning of the double number line. To strengthen number sense, relating back to the fundamental understanding of the operation is paramount. Sharing these strategies for labeling the 6 (and eventually the 18) in the order above allows for students to see connections between their strategy and the next strategy on the progression of learning. Teachers can draw connections between skip counting and multiplication to provide a bridge so skip counting students can try a new strategy in subsequent problems, and similarly draw connections to the process of division. There's also a potential to connect to a calculation with fractions in:

Question 2:

For the second question, students may again try to find a number to skip count or add by to reach 15,000. After some trial and error, they'll likely land correctly on 3,000. Some students may leverage the structure (SMP 7) to think about a number that 5 times will give 15,000. This offers the opportunity to again revisit division as an idea of equal groups or equal sized jumps on a number line, as well as discussions about how powers of ten, in this case thousands, can be as easy to skip count by as their single digit counterparts. 

One of the learning targets is "Interpret a description of a situation to identify the original amount, the new amount, the change, and corresponding percentages. Label these on a double number line diagram". Some may view calculation difficulties as a barrier to achieving this outcome, but if we frame these difficulties as an opportunity to build number sense, the accumulation of these opportunities over the course of a year can have an outsized impact on student's future abilities to work flexibly with numbers.