Grade 6 Mathematics, Quarter 1, Unit 1.1 Understanding...

Cumberland, Lincoln, and Woonsocket Public Schools, with process support from the Charles A. Dana Center at the University of Texas at Austin

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Grade 6 Mathematics, Quarter 1, Unit 1.1

Understanding Ratios and Rates

Overview Number of instructional days: 10 (1 day = 45–60 minutes)

Content to be learned Mathematical practices to be integrated • Understand the concept of a ratio.

• Use ratio language to describe a ratio relationship between two quantities.

• Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0.

• Use rate language in the context of a ratio relationship.

Make sense of problems and persevere in solving them.

• Analyze given information to develop possible strategies for solving the problem.

• Check answers using a different method, and continually ask, “Is my answer reasonable?”

Look for and express regularity in repeated reasoning.

• Look for mathematically sound shortcuts.

• Use repeated applications to generalize properties.

Essential questions • What is a ratio?

• Using ratio language, how would you describe a ratio relationship between two quantities?

• What is a unit rate?

• Using rate language, how would you describe a ratio relationship between two quantities?

• How are a ratio and rate similar/different?

Grade 6 Mathematics, Quarter 1, Unit 1.1 Understanding Ratios and Rates (10 days)


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Written Curriculum

Common Core State Standards for Mathematical Content

Ratios and Proportional Relationships 6.RP

Understand ratio concepts and use ratio reasoning to solve problems.

6.RP.1 Understand 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.”

6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 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 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”1 1 Expectations for unit rates in this grade are limited to non-complex fractions.

Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

8 Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.



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Clarifying the Standards

Prior Learning

In grade 5, students learned to add and subtract fractions with unlike denominators by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. The concept of equivalent fractions connects to the concepts of ratios and rates/unit rates in grade 6.

Current Learning

In grade 6, students understand 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 two wings, there was one beak” and “For every vote Candidate A received, Candidate C received nearly three votes.”

Students understand the concept of a unit rate a/b associated with a ratio a:b with b not equal to 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 3/4 cup of flour for each cup of sugar” and “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”

Future Learning

In grade 7, students will compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. Students will recognize and represent proportional relationships between quantities. Students will use proportional relationships to solve multistep ratio and percent problems.

Additional Findings

According to the PARCC Ratios and Proportional Relationships (Grades 6–7) progression document, “An understanding of ratio is essential in the sciences to make sense of quantities that involve derived attributes such as speed, acceleration, density, surface tension, electric or magnetic field strength, and to understand percentages and ratios used in describing chemical solutions. Ratios and percentages are also useful in many situations in daily life such as in cooking and in calculating tips, miles per gallon, taxes, and discounts.”

According to Principles and Standards for School Mathematics, “by solving problems that require multiplicative comparisons, students will gain extensive experience with ratios, rates, and percents, which helps form a solid foundation for their understanding of, and their facility with, proportionality. The study of rational numbers in the middle grades should build on students’ prior knowledge of whole-number concepts and skills and their encounters with fractions, decimals, and percents in lower grades and in everyday life … Students’ facility with rational numbers and proportionality can be developed in concert with their study of many topics in the middle-grades curriculum” (p. 215).



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Reasoning Proportionally with Percents


Content to be learned Mathematical practices to be integrated • Use ratio reasoning to solve real-world and

mathematical problems.

• Use rate reasoning to solve real-world and mathematical problems.

• Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity).

• Given a part and the percent, solve problems involving finding the whole.


• Identify and execute appropriate strategies to solve the problem.

• Check answers using a different method, and continually ask, “Is this answer reasonable?”

Construct viable arguments and critique the meaning of others.

• Understand and use prior learning in constructing arguments.

• Seek to understand alternative approaches suggested by others and as a result, adopt better approaches.

• Compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and, if there is a flaw in the argument, explain what it is.

Attend to precision.

• Use clear definitions and state the meaning of the symbols they choose, including the use of the equal sign consistently and appropriately.

• Strive for accuracy.

Essential questions • How can you use ratio reasoning to solve real-

world and mathematical problems?

• How can you use rate reasoning to solve real-world and mathematical problems?

• How do you find a percent of a given quantity as a rate per 100?

• When you’re given a part and the percent, how can you find the whole?

Grade 6 Mathematics, Quarter 1, Unit 1.2 Reasoning Proportionally with Percents (10 days)


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Written Curriculum




6.RP.3 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.

c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

Common Core Standards for Mathematical Practice

1 Make sense of problems and persevere in solving them.


3 Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.



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6 Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.


Prior Learning

There is no prior learning connected to the grade 6 standards in this unit.

Current Learning

By the end of unit 4.1, Multidigit Computation and Finding Common Factors and Multiples, students need to be fluently dividing multidigit whole numbers using the standard algorithm. They must also be fluent in adding, subtracting, multiplying, and dividing multidigit decimals using the standard algorithm for each operation. Although the fluency is not assessed until the end of unit 4.1, these skills must be embedded and practiced throughout the first three quarters (as it fits within this unit) to ensure that students are fluent by the end of unit 4.1.

In grade 6, students 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). Specifically, students find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity). Students also solve problems involving finding the whole, given a part and the percent.

Future Learning

In grade 7, students will compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. They will also recognize and represent proportional relationships between quantities. Students will use proportional relationships to solve multistep ratio and percent problems (simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, and percent error).

Additional Findings




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Applications of Ratios and Rates


Content to be learned Mathematical practices to be integrated • Create ratio and rate tables, and double number

lines and equations to represent real-world problems.

• Find missing values in ratio and rate tables.

• Use tables to create graphical representations of rates and ratios on the coordinate plane.

• Use tables to compare ratios.

• Solve unit rate problems involving unit pricing and constant speed problems that include decimal numbers.

• Make tables of equivalent ratios relating quantities with whole-number measurements.

• Use ratio reasoning to convert measurement units and manipulate and transform units appropriately when multiplying or dividing quantities.


• Identify and execute appropriate strategies to solve a problem.

• Analyze given information to develop possible strategies for solving a problem.

Reason abstractly and quantitatively.

• Make sense of quantities and their relationships in problem situations.

• Attend to the meaning of quantities, considering the units involved.

Model with mathematics.

• Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.

• Identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas.

Use appropriate tools strategically.

• Use tools (screwdrivers and hammers and maybe a drill and some bits) when solving a mathematical problem to deepen their understanding of concepts.

• Make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.

Grade 6 Mathematics, Quarter 1, Unit 1.3 Applications of Ratios and Rates (12 days)


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Essential questions • Why is it helpful to make tables of equivalent

ratios relating quantities with whole-number measurements?

• What do you notice when you plot the pairs of values in an equivalent ratio table on the coordinate plane?

• Given a problem involving unit pricing or constant speed, how would you solve the problem using unit rates?

• How can you use ratios to convert measurement units?

• What should you do if your units of measurement need to be changed or are not appropriate for the problem?

• What are the advantages to representing ratio relationships in tables, tape diagrams, double number lines, and equations?

• How do you use ratio and rate reasoning to solve real-world and mathematical problems?

• How do you use ratio reasoning to convert measurements units?

• How do you solve unit rate problems that include decimal numbers including unit pricing and constant speed?

Written Curriculum




6.RP.3 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.

a. 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.

b. 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?

d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.



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2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.



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5 Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.


Prior Learning

In grade 5, students learned to add and subtract fractions with unlike denominators by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. The concept of equivalent fractions connects to the concepts of ratios and rates/unit rates in grade 6.

Current Learning

By the end of unit 4.1, Multidigit Computation and Finding Common Factors and Multiples, students need to be fluently dividing multidigit whole numbers using the standard algorithm. They must also be fluent in adding, subtracting, multiplying, and dividing multidigit decimals using the standard algorithm for each operation. Although the fluency is not assessed until the end of unit 4.1, these skills must be embedded and practiced throughout the first three quarters (as it fits within this unit) to ensure that students are fluent by the end of unit 4.1.

In grade 6, students use ratio and rate reasoning to solve real-world and mathematical problems (by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations). They 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. They also use tables to compare ratios. Students solve unit rate problems including those involving unit pricing and constant speed. For example, “If it takes 7 hours to mow 4 lawns, then at that rate, how many lawns can be mowed in 35 hours? At what rate are lawns being mowed?

Finally, students in grade 6 use ratio reasoning to convert measurement units and manipulate and transform units appropriately when multiplying or dividing quantities.



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Future Learning

In grade 7, students will compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. They will also recognize and represent proportional relationships between quantities. Students will use proportional relationships to solve multistep ratio and percent problems (simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, and percent error).

Additional Findings





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Division of Fractions Using Models


Content to be learned Mathematical practices to be integrated • Interpret and compute quotients of fractions

(fractions divided by fractions) using visual fraction models and equations to represent the problems.

• Solve word problems involving the division of fractions by fractions using visual fraction models and equations to represent the problems.

• Connect the actions performed in the division model to a written method.

Make sense of problems and persevere in solving them. • Identify and execute appropriate strategies to

solve a problem. • Check answers using a different method, and

continually ask, “Does this make sense?”

Reason abstractly and quantitatively.

• Make sense of quantities and their relationships in problem situations.

• Abstract a given situation and represent it symbolically.

Model with mathematics. • Apply the mathematics they know to solve

problems arising in everyday life. • Analyze mathematical relationships to draw

conclusions. Look for and make use of structure. • Look for, develop, generalize and describe a

pattern orally, symbolically, graphically, and in written form.

Look for and express regularity in repeated reasoning.

• Look for mathematically sound shortcuts.

• Use repeated applications to generalize properties.

Essential questions • How do you use a model to demonstrate the

division of a fraction by a fraction? • How does the process of dividing whole

numbers help you to model the quotient of two fractions?

• What quantities do you know? What quantities are you trying to find?

Grade 6 Mathematics, Quarter 1, Unit 1.4 Division of Fractions Using Models (8 days)


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• Describe how to recognize when division is needed in a real-world scenario. Give an example to explain response.

• Division can result from two different actions—sharing (as in sharing equally) or grouping (forming equal-sized groups). Explain what type of division action is involved in this scenario.

• When given a division problem, such as 3/5 ÷ 1/3, what is the question we are trying to answer?

• When given a division problem such as 3/5 ÷ 1/3, what is the quantity 3/5 referring to? (i.e., identify the whole); What is the quantity 1/3 referring to? What will the resulting quotient refer to? (i.e., identify the whole the quotient is describing). Use a model to justify your response.

• When will the quotient be greater than the dividend? When will the quotient be less than the dividend? Justify your reasoning.

• How would you predict that the quotient of 3/5 ÷ 1/3 will compare to the quotient of 3/5 ÷ 2/3? Use a model to verify your prediction.

Written Curriculum


The Number System 6.NS

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

6.NS.1 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 (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?



Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students



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check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.



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8 Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.


Prior Learning

In kindergarten through grade 2, the emphasis was on adding and subtracting whole numbers. In grade 3, students have developed an understanding of fractions as numbers. They also developed an understanding of division of whole numbers within 100 as a relationship to multiplication. In grade 4, students applied and extended their previous understandings of multiplication to multiply a fraction by a whole number. In grade 5, students applied and extended previous understanding of division to divide unit fractions by whole numbers and whole numbers by unit fractions. Students have interpreted division of a unit fraction by a non-zero whole number, interpreted division of a whole number by a unit fraction, and solved real-world problems by using visual fraction models and equations to represent the problem. Refer to page 37 of the CCSS for examples.

Current Learning

A critical area in grade 6 is completing student understanding of division of fractions. Students need to understand and be able to explain why the procedures used to divide fractions make sense. In grade 6, students interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions using visual fraction models and equations to represent the problem. Grade 6 completes the extension of operations with fractions. By the close of sixth grade, students should be fluent with all operations involving whole numbers, fractions, and decimals (i.e., all positive rational numbers). See the CCSS and PARCC progression documents for examples.

Future Learning

In grade 7, students will apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. By the end of grade 7, students will have mastered this concept.



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Additional Findings

According to Principles and Standards for School Mathematics, “the division of fractions has traditionally been quite vexing for students. Although ‘invert and multiply’ has been a staple of mathematics instruction and although it seems to be a simple way to remember how to divide fractions, students have for a long time had difficulty doing so. Some students forget which number is to be inverted, and others are confused about when it is appropriate to apply the procedure. This process can seem very remote and mysterious to many students … An alternative approach involves helping students understand the division of fractions by building on what they know about the division of whole numbers. If students understand the meaning of division as repeated subtraction, they can recognize that 24/6 can be interpreted as ‘how many sets of 6 are there in a set of 24?’ This view of division can be applied to fractions” (p. 219).



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