How to Apply the Properties of Arithmetic to Rational Numbers

Math Lesson: Computation with Rational Numbers / Grade Level: 7
Lesson Summary:
The teacher reviews properties of whole numbers and the leads students through building a number line with positive and negative integers and rational numbers. Students next use a number line to add and subtract rational numbers. Then students create examples of each property using rational numbers. Advanced students investigate the term integer and compare it to rational number. Struggling students engage in creating examples of each property using integers.
Lesson Objectives:.
The students will know…

the differences between fractions, decimals, and positive and negative integers.
how to apply the properties of arithmetic to rational numbers.

The students will be able to…

add and subtract rational numbers using a number line.
identify the additive inverse of numbers.
use properties of arithmetic to calculate with fractions, decimals, and integers.

Learning Styles Targeted:
Visual / Auditory / Kinesthetic/Tactile
Pre-Assessment:
1)Write the following equations on the board, and then ask students to explain the properties of addition or multiplication that each represents:

3 + 2 = 2 + 3 and 3 × 2 = 2 × 3 [Commutative property: the order in which two numbers are added or multiplied does not matter.]
(3 + 2) + 4 = 3 + (2 + 4) and (3 × 2) × 4 = 3 × (2× 4) [Associative property: When three or more numbers are added or multiplied, the answer is the same regardless of the grouping of the factors.]
(12 ÷ 4) = (8 ÷ 4) + (4 ÷ 4) [Distributive property over addition: The factors in addition, subtraction, multiplication, and division can be decomposed and operated on individually and then added together.]
3 – 2 = 1 and 2 + 1 = 3 and 6 ÷ 2 = 3 and 3 × 2 = 6 [Inverse operations of addition and subtraction and multiplication and division.]
3 × 1 = 3 and 3 ÷ 1 = 3 [Identity property of multiplication: Any number multiplied or divided by 1 is itself.]
3 + 0 = 0 and 3 – 0 = 3 [Zero property of addition and subtraction/Identity property of addition]
3 × 0 = 0 [Zero property of multiplication]

2)Note students who appear to have little awareness of the properties of arithmetic.
Whole-Class Instruction
Materials Needed:Number Line*, Properties of Operations*
Procedure:
Presentation
1)Draw a long horizontal number line on the board and put arrows on either end. Place a 0 in the middle of the number line and 5 on the positive end before the arrow and -5 before the negative end arrow.
2)Have student volunteers place whole positive and negative integers 1 to 5 and –1 to –5 on the number line, spacing them as equally as possible. Applaud any efforts to use a measure to make equal divisions between numbers.
3)Ask what comes halfway between 1 and 2. As students provide the answer, write above the number line and 1.5 below the number line.
4)Then ask what comes halfway between 1 and 1.5. As the answer is given, write above the number line and 1.25 below the number line. Then fill in the halves and quarters between each positive integer on the number line.
5)Ask whether the opposites of these numbers would be the same distance from zero on a number line as the positive numbers,and once students agree that they would, include the same negative rational numbers as positive rational numbers.
6)Write the words: positive integer, negative integer, and rational number on the board and discuss the definition of each as a whole number, a negative whole number, or a number between two integers that can be expressed as a fraction, decimal, or ratio.
7)Using the number line, ask students to show you how to add 2 to 2.5 or . [By hopping to 3.5 then 4.5 while counting +1, +2.]
8)Using the number line, ask students to show you how to subtract 4 from . [By hopping to , , , and thenwhile counting -1, -2, -3, -4.]
9)Using the number line, have students tell you how to add 2.25 to -4.5. [First, add 2 to -4.5 by hopping to -3.5 then -2.5 while counting +1, +2. Then, add 0.25 to -2.5 by hopping to -2.25 while counting +0.25.]
10)Have students explain why -3.5 + 4.25 = 0.75.
Guided Practice
11)Distribute the Number Lineto students. Have them plot 0 and then plot the positive and negative integers.
12)Next have them label positive and negative halves with fractions above the number line and decimals below.
13)Have them plot positive and negative 1/4s and 1/8s in decimals and fractions.
14)Now play a round of follow the leader by calling out a series of addition and subtraction equations and having students add and subtract to see if they can all follow the sequence and end up on the same place on the number line.
15)Tell everyone to start at 2 and then follow this sequence +2, -1.5, +1, , +. [Students should wind up at .] Then repeat the sequence starting at the additive inverse of 2, which is -2. [Students should wind up at .] Explain the difference in the result by comparing the difference in the additive inverses.
16)Discuss why they might or might not have arrived at the same point on the number line. Continue with new sequences that you record on the board until all students are able to follow the sequence.
Independent Practice
17)Divide the class into pairs. Hand out the Properties with Operationsworksheet, and have students create one example of each property using either decimals or fractions. After ten minutes have students present their findings and discuss their reasoning.
Closing Activity
18)Have students present and compare their examples of each property and confirm that the same properties used in calculating whole numbers are used when calculating rational numbers.
Advanced Learner
What Is an Integer?
Materials Needed:Internet access or dictionary
Procedure:
1)Have students investigate the etymology of the word integer and how it is used in mathematics. Have them compare the words integer and rational number to distinguish between the two.
2)Have them present their findings to the class.
Struggling Learner
Properties of Whole Numbers
Materials Needed:Properties of Operations*master
Procedure:
1)Divide students into groups of two and give them the Properties of Operations.
2)Have each group create an example of each property using whole numbers.
3)Review and compare student results and have them explain their reasoning.

*see supplemental resources