Unit 2

Integers and Rational Numbers

 

Grade 7 Math 

Unit Length and Description:

 

30 days

 

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:

 

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 realworld contexts.

c.    Understand subtraction of rational numbers as adding the additive inverse, pq = p + (–q).  Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.

d.   Apply properties of operations as strategies to add and subtract rational numbers.

 

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 realworld contexts.

b.   Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number.  If p and q are integers, then –(p/q) = (–p)/q = p/(–q).  Interpret quotients of rational numbers by describing realworld 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.

 

7.NS.A.3  Solve realworld and mathematical problems involving the four operations with rational numbers.

o   Standards Clarification:  Computations with rational numbers extend the rules for manipulating fractions to complex fractions.

 

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.”

o   Standards Clarification:  In this unit, this standard is applied to expressions with rational numbers in them.

 

Focus Standards of 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.

 

Instructional Outcomes:

Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:

 

7.NS.A.1a

·        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.1b

·        I can represent and explain how a number and its opposite have a sum of 0 and are additive inverses.

·        I can demonstrate and explain how adding two numbers, p + q, if q is positive, the sum of p and q will be lql spaces to the right of p on the number line.

·        I can demonstrate and explain how adding two numbers, p + q, if q is negative, the sum of p and q will be lql spaces to the left of p on the number line.

·        I can interpret sums of rational numbers by describing real-world contexts.

·        I can explain and justify why the sum of p + q is located a distance of lql in the positive or negative direction from p on a number line.

·        I can represent the distance between two rational numbers on a number line is the absolute vale of their difference and apply this principal in real-world contexts.

·        I can apply properties of operations as strategies to add and subtract rational numbers.

 

7.NS.A.1c

·        I can identify subtraction of rational numbers as adding the additive inverse property to subtract rational numbers, p-q= p+ (-q).

·        I can apply and extend previous understanding to represent addition and subtraction problems of rational numbers with a horizontal or vertical number line.

·        I can apply properties of operations as strategies to add and subtract rational numbers.

 

7.NS.A.1d

·        I can identify properties of addition and subtraction when adding and subtracting rational numbers.

·        I can apply properties of operations as strategies to add and subtract rational numbers.

 

7.NS.A.2a

·        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 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.

 

7.NS.A.2b

·        I can explain why integers can be divided except when the divisor is 0.

·        I can describe why the quotient is always a rational number.

·        I can comprehend and describe the rules when dividing signed numbers, integers.

·        I can recognize that –(p/q) = -p/q = p/-q.

·        I can interpret the quotient of rational numbers by describing real-world contexts.

 

7.NS.A.2c

·        I can identify how properties of operations can be used to multiply and divide rational numbers (such as distributive property, multiplicative inverse property, multiplicative identity, commutative property for multiplication, associative property for multiplication, etc.).

·        I can apply properties of operations as strategies to multiply and divide rational numbers.

 

7.NS.A.2d

·        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.

 

7.NS.A.3

·        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.

 

7.EE.A.2

·        I can write equivalent expressions with fractions, decimals, and integers.

·        I can rewrite an expression in a equivalent form in order to provide insight about how quantities are related in a problem context.

 

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?