Colorado Academic Standards Online

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Mathematics - 2019

Sixth Grade, Standard 1. Number and Quantity

| Read the Colorado Essential Skills | Expand Set of GLEs Below

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:

6.RP.A. Ratios & Proportional Relationships: Understand ratio concepts and use ratio reasoning to solve problems.

Evidence Outcomes:

Students Can:

Apply the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings to beaks in the bird house at the zoo was $2:1$, because for every $2$ wings there was $1$ beak." "For every vote Candidate $A$ received, Candidate $C$ received nearly three votes." (CCSS: 6.RP.A.1)
Apply the concept of a unit rate $\frac{a}{b}$ associated with a ratio $a:b$ with $b \neq 0$, and use rate language in the context of a ratio relationship. For example, "This recipe has a ratio of $3$ cups of flour to $4$ cups of sugar, so there is $\frac{3}{4}$ cup of flour for each cup of sugar." "We paid $75 for $15$ hamburgers, which is a rate of $5 per hamburger." (Expectations for unit rates in this grade are limited to non-complex fractions.) (CCSS: 6.RP.A.2)
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. (CCSS: 6.RP.A.3)
1. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. (CCSS: 6.RP.A.3.a)
2. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took $7$ hours to mow $4$ lawns, then at that rate, how many lawns could be mowed in $35$ hours? At what rate were lawns being mowed? (CCSS: 6.RP.A.3.b)
3. Find a percent of a quantity as a rate per $100$ (e.g., $30\%$ of a quantity means $\frac{30}{100}$ times the quantity); solve problems involving finding the whole, given a part and the percent. (CCSS: 6.RP.A.3.c)
4. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. (CCSS: 6.RP.A.3.d)

Academic Contexts and Connections:

Colorado Essential Skills and Mathematical Practices:

Use ratio tables to test solutions and determine equivalent ratios. (Entrepreneurial Skills: Critical Thinking/Problem Solving)
Analyze and use appropriate quantities and pay attention to units in problems that require reasoning with ratios. (MP2)
Construct arguments that compare quantities using ratios or rates. (MP3)
Use tables, tape diagrams, and double number line diagrams to provide a structure for seeing equivalency between ratios. (MP7)

Inquiry Questions:

How are ratios different from fractions?
What is the difference between a quantity and a number?
How is a percent also a ratio?
How is a rate similar to and also different from a unit rate?

Coherence Connections:

This expectation represents major work of the grade.
In prior grades, students work with multiplication, division, and measurement. Prior knowledge with the structure of the multiplication table is an important connection for students in creating and verifying equivalent ratios written in symbolic form or in ratio tables (multiplicative comparison vs. additive comparison).
In Grade 6, this expectation connects with one-variable equations, inequalities, and representing and analyzing quantitative relationships between dependent and independent variables.
In Grade 7, students analyze proportional relationships and use them to solve real-world and mathematical problems. In high school, students generalize rates of change to linear and nonlinear functions and use them to describe real-world scenarios.

Prepared Graduates:

MP2. Reason abstractly and quantitatively.
MP4. Model with mathematics.

Grade Level Expectation:

6.NS.A. The Number System: Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Evidence Outcomes:

Students Can:

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for $\frac{2}{3} \div \frac{3}{4}$ and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that $\frac{2}{3} \div \frac{3}{4} = \frac{8}{9}$ because $\frac{3}{4}$ of $\frac{8}{9}$ is $\frac{2}{3}$. (In general, $\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}$.) How much chocolate will each person get if $3$ people share $\frac{1}{2}$ lb of chocolate equally? How many $\frac{3}{4}$-cup servings are in $\frac{2}{3}$ of a cup of yogurt? How wide is a rectangular strip of land with length $\frac{3}{4}$ mi and area $\frac{1}{2}$ square mi? (CCSS: 6.NS.A.1)

Academic Contexts and Connections:

Colorado Essential Skills and Mathematical Practices:

Create and solve word problems using division of fractions, understanding the relationship of the arithmetic to the problem being solved. (Entrepreneurial Skills: Critical Thinking/Problem Solving)
Reason about the contextualized meaning of numbers in word problems involving division of fractions, and decontextualize those numbers to perform efficient calculations. (MP2)
Model real-world situations involving scaling by non-whole numbers using multiplication and division by fractions. (MP4)

Inquiry Questions:

When dividing, is the quotient always going to be a smaller number than the dividend? Why or why not?
What kinds of real-world situations require the division of fractions?
How can the division of fractions be modeled visually?

Coherence Connections:

This expectation represents major work of the grade.
In Grade 5, students apply and extend previous understandings of multiplication and division to divide whole numbers by unit fractions and unit fractions by whole numbers.
In Grade 6, this expectation connects with solving one-step, one-variable equations and inequalities.
In Grade 7, students apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Prepared Graduates:

MP6. Attend to precision.
MP7. Look for and make use of structure.

Grade Level Expectation:

6.NS.B. The Number System: Compute fluently with multi-digit numbers and find common factors and multiples.

Evidence Outcomes:

Students Can:

Fluently divide multi-digit numbers using the standard algorithm. (CCSS: 6.NS.B.2)
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. (CCSS: 6.NS.B.3)
Find the greatest common factor of two whole numbers less than or equal to $100$ and the least common multiple of two whole numbers less than or equal to $12$. Use the distributive property to express a sum of two whole numbers $1$–$100$ with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express $36 + 8$ as $4 \left(9 + 2\right)$. (CCSS: 6.NS.B.4)

Academic Contexts and Connections:

Colorado Essential Skills and Mathematical Practices:

Accurately add, subtract, multiply, and divide with decimals. (MP6)
Recognize the structures of factors and multiples when identifying the greatest common factor and least common multiple of two whole numbers. Use the greatest common factor to rewrite an expression using the distributive property. (MP7)

Inquiry Questions:

How do operations with decimals compare and contrast to operations with whole numbers?
How does rewriting the sum of two whole numbers using the distributive property yield new understanding and insights on the sum?

Coherence Connections:

This expectation is in addition to the major work of the grade.
In Grade 5, students divide whole numbers with two-digit divisors and perform operations with decimals.
In Grade 6, this expectation connects with applying and extending previous understandings of arithmetic to algebraic expressions.
In Grade 7, students apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. Students apply the concept of greatest common factor to factor linear expressions, and extending properties of whole numbers to variable expressions.

Prepared Graduates:

MP2. Reason abstractly and quantitatively.
MP3. Construct viable arguments and critique the reasoning of others.
MP5. Use appropriate tools strategically.

Grade Level Expectation:

6.NS.C. The Number System: Apply and extend previous understandings of numbers to the system of rational numbers.

Evidence Outcomes:

Students Can:

Explain why positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of $0$ in each situation. (CCSS: 6.NS.C.5)
Describe a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. (CCSS: 6.NS.C.6)
1. Use opposite signs of numbers as indicating locations on opposite sides of $0$ on the number line; identify that the opposite of the opposite of a number is the number itself, e.g., $-(-3) = 3$, and that $0$ is its own opposite. (CCSS: 6.NS.C.6.a)
2. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; explain that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. (CCSS: 6.NS.C.6.b)
3. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. (CCSS: 6.NS.C.6.c)
Order and find absolute value of rational numbers. (CCSS: 6.NS.C.7)
1. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret $-3 \gt -7$ as a statement that $-3$ is located to the right of $-7$ on a number line oriented from left to right. (CCSS: 6.NS.C.7.a)
2. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write $-3^\circ\mbox{C} \gt -7^\circ\mbox{C}$ to express the fact that $-3^\circ\mbox{C}$ is warmer than $-7^\circ\mbox{C}$. (CCSS: 6.NS.C.7.b)
3. Define the absolute value of a rational number as its distance from $0$ on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of $-30$ dollars, write $\left|-30\right| = 30$ to describe the size of the debt in dollars. (CCSS: 6.NS.C.7.c)
4. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than $-30$ dollars represents a debt greater than $30$ dollars. (CCSS: 6.NS.C.7.d)
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. (CCSS: 6.NS.C.8)

Academic Contexts and Connections:

Colorado Essential Skills and Mathematical Practices:

Investigate integers to form hypotheses, make observations and draw conclusions. (Entrepreneurial Skills: Inquiry/Analysis)
Understand the relationship among negative numbers, positive numbers, and absolute value. (MP2)
Explain the order of rational numbers using their location on the number line. (MP3)
Demonstrate how to plot points on a number line and plot ordered pairs on a coordinate plane. (MP5)

Inquiry Questions:

Why do we have negative numbers?
What relationships exist among positive and negative numbers on the number line?
How does the opposite of a number differ from the absolute value of that same number?
How does an ordered pair correspond to its given point on a coordinate plane?

Coherence Connections:

This expectation represents major work of the grade.
In previous grades, students develop understanding of fractions as numbers and graph points on the coordinate plane (limited to the first quadrant) to solve real-world and mathematical problems.
In Grade 6, this expectation connects with reasoning about and solving one-step, one-variable equations and inequalities.
In Grade 7, students apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. In Grade 8, students investigate patterns of association in bivariate data.

Need Help? Submit questions or requests for assistance to bruno_j@cde.state.co.us