Monday, June 22, 2009

Walk the Line

Concept To Teach: Adding and Subtracting with Positive and Negative Integers

Standards Addressed: Numbers and Operations, Algebra
7.1.1 Develop, analyze, and apply models (including everyday contexts), strategies, and procedures to compute with integers, with an emphasis on negative integers.

General Goals: Build mathematical models, expand common arithmetic skills.

Specific Objectives: Students will learn to add and subtract using both positive and negative integers by building a physical model of each operation.

Required Materials: Masking tape and dark marker (for number lines), index cards for problems

Anticipatory Set (Lead-In): Show a small-scale model of an airplane, car, or rocket and ask students about the importance of building a model: "Why would the engineer designing a [plane, car, or rocket] spend so much time building a model first?" Students' answers can lead into a discussion of creating models in math as a way to develop a deeper understanding of the concept, and to develop quick procedures for computation.

Step-By-Step Procedures:

  1. Before the lesson, use masking tape and a dark marker to make a number line from -10 to +10 on the floor in a large open area of the classroom or hall. Spacing should be about a foot apart.
  2. Invite students to gather with you around the number line; they should bring their math journal and a pencil to record examples.
  3. Ask for a volunteer to "walk the line" to show how they would get the answer to 5 + 3. You might ask questions to draw out the student's thought process: "Where did you start?" "Where did you end up?" "How did you get from one number to the next?" Have students use terms such as "forward" and "backward" consistently in their discussions. Remind students to record each problem in their math journals.
  4. Ask another volunteer to "walk the line" to show 5 - 3. Ask the same "thinking" questions as in Step 3, but this time ask if the class observers can think of another way of getting the same answers. This is where you will begin to guide students to build the model: the student could walk forward for 5 steps then backward for 3, or they could walk forward for 5 steps then turn around and walk forward 3 steps; both should arrive at the same answer. Each of these, however, represents a different arithmetical expression -- the first is 5 + (-3) and the second is 5 - 3. Students should come up with methods to show walking off positive and negative numbers as well as the operations of addition and subtraction.
  5. The finished model could look like this:
    1. Walk off a positive number by walking forward.
    2. Walk off a negative number by "moonwalking" backward.
    3. Adding faces students in the positive direction.
    4. Subtraction faces students in the negative direction.
  6. Ask another volunteer to "walk the line" to show -5 + 3. Assess whether students understand the components of the model, as well as their execution. Discuss the "thinking" questions of Step 3 with this volunteer as well.
  7. At this point, the teacher should determine whether the class would benefit more from practicing in small groups or doing more whole-class demonstrations.
    1. Other examples to "walk the line:" -5 - 3, -5 - (-3)
    2. Invite students to come up with their own single-digit problems for classmates to practice.
    3. Subtracting negative numbers is a difficult concept for many students to visualize; the beauty of this model is that it provides a way for students to see how "subtracting a negative means adding a positive" really works. Be sure to point this out in a closure discussion.

Plan For Independent Practice: Tiered assignment consisting of problems that students must draw out their physical steps and movements on graph paper to reinforce concept, then several drill problems to increase proficiency and fluency.

Assessment Based On Objectives: Assess student movement and accuracy of answers.

Adaptations (For Students With Learning Disabilities): Use easier problems to start, then build up to tougher ones; use graph paper for those who cannot easily move on the physical number line.

Extensions (For Gifted Students): Use larger numbers; have students quickly estimate answers; ask students to create a different model for addition and subtraction of integers.

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