Colorado Academic Standards Online
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clear Content Area: Mathematics - 2019 // Grade Level: Seventh Grade // Standard Category: 1. Number and Quantity
Mathematics - 2019
Seventh Grade, Standard 1. Number and Quantity
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Prepared Graduates:
- MP1. Make sense of problems and persevere in solving them.
- MP2. Reason abstractly and quantitatively.
- MP8. Look for and express regularity in repeated reasoning.
Grade Level Expectation:
7.RP.A. Ratios & Proportional Relationships: Analyze proportional relationships and use them to solve real-world and mathematical problems.
Evidence Outcomes:
Students Can:
- Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. For example, if a person walks \(\frac{1}{2}\) mile in each \(\frac{1}{4}\) hour, compute the unit rate as the complex fraction \(\frac{\frac{1}{2}}{\frac{1}{4}}\) miles per hour, equivalently \(2\) miles per hour. (CCSS: 7.RP.A.1)
- Identify and represent proportional relationships between quantities. (CCSS: 7.RP.A.2)
- Determine whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. (CCSS: 7.RP.A.2.a)
- Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. (CCSS: 7.RP.A.2.b)
- Represent proportional relationships by equations. For example, if total cost \(t\) is proportional to the number \(n\) of items purchased at a constant price \(p\), the relationship between the total cost and the number of items can be expressed as \(t = pn\). (CCSS: 7.RP.A.2.c)
- Explain what a point \(\left(x,y\right)\) on the graph of a proportional relationship means in terms of the situation, with special attention to the points \(\left(0, 0\right)\) and \(\left(1, r\right)\) where \(r\) is the unit rate. (CCSS: 7.RP.A.2.d)
- Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. (CCSS: 7.RP.A.3)
Academic Contexts and Connections:
Colorado Essential Skills and Mathematical Practices:
- Recognize when proportional relationships occur and apply these relationships to personal experiences. (Entrepreneurial Skills: Inquiry/Analysis)
- Recognize, identify, and solve problems that involve proportional relationships to make predictions and describe associations among variables. (MP1)
- Reason quantitatively with rates and their units in proportional relationships. (MP2)
- Use repeated reasoning to test for equivalent ratios, such as reasoning that walking \(\frac{1}{2}\) mile in \(\frac{1}{4}\) hour is equivalent to walking \(1\) mile in \(\frac{1}{2}\) hour and equivalent to walking \(2\) miles in \(1\) hour, the unit rate. (MP8)
Inquiry Questions:
- How are proportional relationships related to unit rates?
- How can proportional relationships be expressed using tables, equations, and graphs?
- What are properties of all proportional relationships when graphed on the coordinate plane?
Coherence Connections:
- This expectation represents major work of the grade.
- In Grade 6, students understand ratio concepts and use ratio reasoning to solve problems.
- This expectation connects with several others in Grade 7: (a) solving real-life and mathematical problems using numerical and algebraic expressions and equations, (b) investigating chance processes and developing, using, and evaluating probability models, and (c) drawing, constructing, and describing geometrical figures and describing the relationships between them.
- In Grade 8, students (a) understand the connections between proportional relationships, lines, and linear equations, (b) define, evaluate, and compare functions, and (c) use functions to model relationships between quantities. In high school, students use proportional relationships to define trigonometric ratios, solve problems involving right triangles, and find arc lengths and areas of sectors of circles.
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Prepared Graduates:
- MP2. Reason abstractly and quantitatively.
- MP3. Construct viable arguments and critique the reasoning of others.
- MP7. Look for and make use of structure.
Grade Level Expectation:
7.NS.A. The Number System: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
Evidence Outcomes:
Students Can:
- 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. (CCSS: 7.NS.A.1)
- 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. (CCSS: 7.NS.A.1.a)
- Demonstrate \(p+q\) as the number located a distance \(\left|q\right|\) 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. (CCSS: 7.NS.A.1.b)
- Demonstrate subtraction of rational numbers as adding the additive inverse, \(p-q=p+\left(-q\right)\). 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. (CCSS: 7.NS.A.1.c)
- Apply properties of operations as strategies to add and subtract rational numbers. (CCSS: 7.NS.A.1.d)
- Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. (CCSS: 7.NS.A.2)
- 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 \(\left(-1\right)\left(-1\right) = 1\) and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. (CCSS: 7.NS.A.2.a)
- 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 \(p\) and \(q\) are integers, then \(-\left(\frac{p}{q}\right) = \frac{-p}{q} = \frac{p}{-q}\). Interpret quotients of rational numbers by describing real-world contexts. (CCSS: 7.NS.A.2.b)
- Apply properties of operations as strategies to multiply and divide rational numbers. (CCSS: 7.NS.A.2.c)
- Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in \(0\)s or eventually repeats. (CCSS: 7.NS.A.2.d)
- Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.) (CCSS: 7.NS.A.3)
Academic Contexts and Connections:
Colorado Essential Skills and Mathematical Practices:
- Solve problems with rational numbers using all four operations. (Entrepreneurial Skills: Critical Thinking/Problem Solving)
- Compute with rational numbers abstractly and interpret quantities in context. (MP2)
- Justify understanding and computational accuracy of operations with rational numbers. (MP3)
- Use additive inverses, absolute value, the distributive property, and properties of operations to reason with and operate on rational numbers. (MP7)
Inquiry Questions:
- How do operations with integers compare to and contrast with operations with whole numbers?
- How can operations with negative integers be modeled visually?
- How can it be determined if the decimal form of a rational number terminates or repeats?
Coherence Connections:
- This expectation represents major work of the grade.
- In previous grades, students use the four operations with whole numbers and fractions to solve problems.
- In Grade 7, this expectation connects with solving real-life and mathematical problems using numerical and algebraic expressions and equations. This expectation begins the formal study of rational numbers (a number expressible in the form \(\frac{a}{b}\) or \(-\frac{a}{b}\) for some fraction \(\frac{a}{b}\); the rational numbers include the integers) as extended from their study of fractions, which in these standards always refers to non-negative numbers.
- In Grade 8, students extend their study of the real number system to include irrational numbers, radical expressions, and integer exponents. In high school, students work with rational exponents and complex numbers.
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