 Unit 2

Integers and Rational Numbers

Unit Description:

Students continue to build an understanding of the number line in Unit 2 from their work in Grade 6. They develop a unified understanding of numbers, recognizing fractions, decimals (that have a finite or repeating decimal representation), and percents as different representations of rational numbers.  Students should apply and extend their understanding of addition, subtraction, multiplication, and division to add, subtract, multiply and divide within the entire set of rational numbers. Students should begin this unit representing addition and subtraction on a horizontal or vertical number line diagram and finish the unit being able to apply properties of operations as strategies to add, subtract, multiply and divide rational numbers. They should also apply their understanding of positive and negative numbers to establish the rules for multiplying signed numbers. Additionally, students should understand that integers can be divided, provided that the divisor is not zero, and develop an understanding that the quotient of integers (with non-zero divisors) is a rational number.  Students should leave this unit with a deeper conceptual understanding of positive and negative rational numbers and be able to use them to solve real-world and mathematical problems including real-world problems where the sum is zero. Although procedural skill and fluency should be improved through this unit, conceptual understanding should be the basis for discovery and instruction.  Unit 2 includes rational numbers as they appear in expressions and equations—work that is continued in Unit 3.  A focus on equivalent expressions is important as student prepare for work with equations and inequalities in Unit 3.

Standards for Mathematical Practice

MP.1 Make sense of problems and persevere in solving them.

MP.2 Reason abstractly and quantitatively.

MP.3 Construct viable arguments and critique the reasoning of others.

MP.4 Model with mathematics.

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

MP.8 Look for and express regularity in repeated reasoning.

Louisiana Student Standards for Mathematics (LSSM)

 NS:  The Number System A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. a. Describe situations in which opposite quantities combine to make 0.  For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative.  Show that a number and its opposite have a sum of 0 (are additive inverses).  Interpret sums of rational numbers by describing real‐world contexts. c. Understand subtraction of rational numbers as adding the additive inverse, . Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real‐world contexts. d. Apply properties of operations as strategies to add and subtract rational numbers. *I can describe situations in which opposite quantities combine to make 0. *I can apply the principal of subtracting rational numbers in real-world contexts. 7.NS.A.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)( –1) = 1 and the rules for multiplying signed numbers.  Interpret products of rational numbers by describing real‐world contexts. b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non‐zero divisor) is a rational number.  If and are integers, then . Interpret quotients of rational numbers by describing real‐world contexts. c. Apply properties of operations as strategies to multiply and divide rational numbers. d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. *I can recognize that the process for multiplying fractions can be used to multiply rational numbers including integers. *I can recognize and describe the rules when multiplying and dividing signed numbers. *I can apply the properties of operations, particularly distributive property, to multiply rational numbers. *I can interpret the products of rational numbers by describing real-world contexts. *I can explain why integers can be divided except when the divisor is 0. *I can recognize that . *I can interpret the quotient of rational numbers by describing real-world contexts. *I can convert a rational number to a decimal using long division. *I can explain the decimal form of a rational number terminates (stops) in zeroes or repeats (non-terminating). 7.NS.A.3 Solve real‐world and mathematical problems involving the four operations with rational numbers. *I can add rational numbers. *I can subtract rational numbers. *I can multiply rational numbers. *I can divide rational numbers. *I can solve real-world mathematical problems by adding, subtracting, multiplying, and dividing rational numbers, including complex fractions. EE:  Expressions and Equations A. Use properties of operations to generate equivalent expressions 7.EE.A.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.” *I can write equivalent expressions with fractions, decimals, and integers. *I can rewrite an expression in an equivalent form in order to provide insight about how quantities are related in a problem context. B. Solve real-life and mathematical problems using numerical and algebraic expressions and equations. 7.EE.B.4a Use variables to represent quantities in a real‐world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.    a. Solve word problems leading to equations of the form and ,where , , and , are specific rational numbers.  Solve equations of these forms fluently.  Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  For example, the perimeter of a rectangle is 54 cm.  Its length is 6 cm.  What is its width? This can be answered algebraically by using only the formula for perimeter to isolate w or by finding an arithmetic solution by substituting values into the formula. *I can identify the sequence of operations used to solve an algebraic equation of the form and . *I can fluently solve equations of the form and with speed and accuracy. *I can solve word problems leading to equations of the form and . *I can compare an algebraic solution to an arithmetic solution by identifying the sequence of the operations used in each approach.

Enduring Understandings:

·         Rational numbers use the same properties as whole numbers.

·         Rational numbers can be used to represent and solve real-life situation problems.

·         Rational numbers can be represented with visuals (including distance models), language, and real-life contexts.

·         A number line model can be used to represent the unique placement of any number in relation to other numbers.

·         There are precise terms and sequences to describe operations with rational numbers.

Essential Questions:

·         Why do I need mathematical operations?

·         What is the relationship between properties of operations and types of numbers?

·         How do I know which mathematical operation (+, -, x, ÷, exponents, etc.) to use?

·         How do I know which computational method (mental math, estimation, paper and pencil, and calculator) to use?

·         How do you add rational numbers?

·         How do you subtract rational numbers?

·         How do you multiply rational numbers?

·         How do you divide rational numbers?

·         How is computation with rational numbers similar to and different from whole number computation?

·         How are rational numbers used and applied in real-life and mathematical situations?

·         How does the ongoing use of fractions and decimals apply to real-life situations?