by Leah Silver, Math Coordinator, JK-4

How would you solve 72 – 49?

Perhaps you begin by stacking these numbers on top of each other and crossing numbers out in your head. Maybe you’re borrowing from another number, and writing new numbers on top of old numbers. And when you get your answer, you seem to have solved this problem using a strategy totally disconnected from the numbers themselves.

I’d like to invite you to consider other strategies—strategies that may be more efficient and utilize more of your number sense. Can you think of a way of counting up from 49 (not only by ones!) until you get to 72? Or perhaps shift the problem to 73 – 50, a simpler problem to solve that would land us on the same answer?

At Grace, our goal is for our students to efficiently pull apart and put together numbers using strategies grounded in number sense and place value. The open number line, shown below, empowers our students to solve problems strategically. 

In a recent visit to the third grade classes, I was so impressed with our students’ comfort with using the open number line as a tool for solving subtraction problems. Here are some strategies they used to solve 72 – 49. 

How do our students build this comfort with the number line? Our number line work actually begins with our youngest learners, in JK. If you walk into our JK classrooms, you’ll see our students use a number path, pictured below. Under the guidance of Ms. Moller and Ms. Lasfargeas, students will guess a hidden number, put number cards and pictures in numerical order, and make hops forward and backward. Through these activities, students make the connection between quantities and their numerical representations, practice number sequence and cardinality and start to link addition and subtraction.

Our Kindergarten students spend a lot of time building on their number path work from JK. Each month they focus on a different range of numbers within 50: they begin with work from one to twenty, ten to thirty, twenty to forty and so on. Throughout the year, our Kindergarten students use their number path to practice counting forwards and backwards; identify numbers that come before, after and between numbers; use inequalities (greater than and less than); and begin to understand our number system as a base ten system. Hap, the friendly grasshopper, accompanies our students throughout the year to help them hop forwards and backwards.

This work continues into first grade, and by the time our students get to second grade, they are ready to use open number lines to solve problems. Students first create a train of Unifix cubes and then write a number line above it; this helps them connect the concrete and the abstract.

By the end of second grade, our students begin to use the open number line to solve problems like 72 – 49, shown above. Familiar with the tool of the number line, students are able to deploy it to solve more complex problems. 

But it doesn’t end there! Our third graders also use the number line to build their concept of multiplication. In the game Frog Jump Multiplication, students use the number line to make jumps of different groups of numbers, and represent their jumps with a multiplication equation.

And later in third grade and in early fourth grade, students may be presented with an open number line puzzle like this one.

Teaching our students to solve complex addition, subtraction, multiplication and division problems is a team effort that begins in our JK classrooms. When students are comfortable with these visual models as math tools, they can actually use these tools as a way to keep track of and demonstrate their thinking as they solve challenging problems.

Please enjoy some pictures from our students in different grades using number paths and number lines to build their math thinking!