Lesson #1
Objective: Students will be able to add whole numbers using rods, cubes and number lines.
Timing: 1 - 50 minute period
Materials: Cuisenaire rods and cubes, graph paper, rulers.
Opening:
Using an overhead projector or board in the front of the room, display the following number line.
Ask the students to discuss with their group mates, how long this line is? Place answers from the class on the board/screen. (Possible answers: 5, 4, 1, etc…) Have students prove their answers. Offer them graph paper, rulers, calculators, anything they need to prove their hypothesis.
Go over these proofs with the class. We want students to justify their answers with the idea that these are all equal parts, so whatever unit they are counting by (d), the total length would be the iterations (n) times the unit 1/d. In other words, 1x4 or (1/4) x4 or
Symbolically (n)1/d. Discuss with the students how this breaks down. Gives lots of examples. It would be prudent to keep this in the positive integer, whole number realm for this opening lesson.
This opening will help students ground themselves in a unit to work with, that the iterations of this unit are the same and that each unit has value, (shown through length on a number line).
Student Experimentation/Documentation:
Using rods, cubes and the number line students will practice showing addition problems and then use those models to sum. This practice will be a way to help stop the tendency our students have of counting marks on a line or a grid. Moving them towards seeing number lines and grids as forms of linear measurement is our goal.
For each of the activities involving number lines, we will begin with the physical manipulation of rods and cubes. Students will need to begin by measuring the length of a rod with cubes in order to establish the concept of length in their own minds. Using Cuisenaire rods students will then create trains, end to end, to represent addition. After establishing this process with rods alone students should add cubes to begin working with 2 digit numbers that are not strictly powers of ten. See figure below.
These rods will translate easily to working on a number line. Given a problem such as 13 + 11, students should first attempt to draw this on a number line by bracketing 13 and 11 and finding the sum. It will be necessary to stop students that are not representing unit
values with the same amount of space. In other words, have them get out the ruler or use graph paper so the idea of the iteration of a unit, whatever it is, is represented by the same amount of space.
Once this is established moving into representing addition problems in expanded form will help students in many ways. The number line in figure d. shows the problem 13 + 11 expanded into 10 + 3 + 10 +1. This is a process we will continue with throughout this unit. For this lesson continue to have the students expand ever larger numbers. To differentiate, have students work with multiplication and/or powers of 10 when representing a deconstructed number.
Students should document their work with both the rods and cubes as well as number lines in either a math journal or on a separate sheet of paper. This will allow teachers a good anecdotal assessment opportunity. Be on the look out for students representing the same unit with different lengths.
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