Problem Solving: Working Backwards

Strategy:  Make a Path

 Sometimes students must work backwards to solve a problem.  This method also works as an introduction to some algebraic problems in which the starting number is unknown.  This method and the accompanying problem sets were developed to introduce students to one method of recording these problems by creating the forward path, then reversing the operations to create a backward path students may use to work backwards and identify the starting number.

 Once students learn this method, they become quite proficient at using the paths to identify the starting number.   The paths are easily generated on blank paper or on whiteboards so that these problems are a great do-now for the beginning or end of class periods.

Suggestions for modeling this strategy:

• Read the problem through, then model creating and labeling the forward path step-by-step, using an overhead or whiteboard.  
• Draw arrows and circles to create a backward path that is matched arrow to arrow and circle to circle.
• Label the backward path by using the opposite operation since students are working backwards (undoing).
• Start at the end number and work backward along the new backward path, writing the correct number in each circle until students reach the beginning number.
• Check work by using the beginning number along the forward path to make sure that it creates the ending number.

 Suggestions for Guided Practice:

• Read a problem slowly, pausing at each step to allow students to create a forward path and correctly label it.
• Read the problem  once again so that students may check their forward path.
• Challenge students to create the backward path with labels then work backwards to identify the starting number.
• Encourage students to check their work by using the starting number across the forward path to make sure that it works.
• Distribute copies of the Making a Path problem sets for students to work on independently or in pairs.

Enrichment:

• Challenge students to create their own original problems, complete with a solution and answer.  Use them as do-nows for the class and/or place these problems in the classroom math center.

Materials:

• Making Paths:  Grade 3 includes directions, 16 student problems and answer key.
• Making Paths:  Grade 4 includes directions, 16 student problems and answer key.
• Make A Path Templates that may be inserted in clear plastic sheet protector and used with dry erase pens for quick use