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

Cranston 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

Multidigit Computation With and Without Decimals

Overview Number of instructional days: 12 (1 day = 45 minutes)

Content to be learned Mathematical practices to be integrated • Fluently divide multidigit numbers using the

standard algorithm.

• Fluently add and subtract multidigit decimals using the standard algorithm.

• Fluently multiply multidigit decimals using the standard algorithms.

• Fluently divide multidigit decimals using the standard algorithms.

Attend to precision.

• State the meaning of the symbols they choose.

• Use the equal sign consistently and appropriately.

• Specify carefully units of measure.

• Calculate accurately and efficiently.

• Give carefully formulated explanations to each other.

Look for and express regularity in repeated reasoning.

• Notice if calculations are repeated.

• Look for both general methods and shortcuts.

• Maintain oversight of the process while attending to the details.

• Evaluate the reasonableness of their immediate results.

Essential questions • How is the standard algorithm for dividing

multidigit whole numbers related to a model of dividing multidigit numbers?

• How are the standard algorithms for adding, subtracting, multiplying, and dividing multidigit decimals related to models of adding, subtracting, multiplying, and dividing multidigit decimals?

• How do you decide where to place the decimal point in your answer? Why does that make sense?

• When will the product be less than the multiplicand? When will the product be greater than the multiplicand? Why does this make sense?

Grade 6 Mathematics, Quarter 1, Unit 1.1 Multidigit Computation With and Without Decimals (12 days)


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• How does the algorithm used to add/subtract decimals compare to the strategies used to add/subtract whole numbers and fractions? (all require adding/subtracting like units)

• How does your algorithm keep track of place value?

• Where did you use the properties of operations (including composing/decomposing) in your algorithm?

• What steps are repeated in a division algorithm?

• How is the process of adding and subtracting decimals different from the process of multiplying decimals?

• How is the algorithm used to divide whole numbers similar to the algorithm used to divide decimal numbers? How is it different?

• When can you expect the quotient to be less than the dividend? When can you expect the quotient to be greater than the dividend? Why does this make sense?

• When dividing, what are your options for dealing with a remainder?

• When you have a decimal in your divisor, what steps do you take to divide? What is the mathematics behind “moving the decimal point” that explains why this action makes sense?

• In a real-world problem situation, how do you decide which operation makes sense for that problem situation?

• How can you verify your computation is reasonable?

Written Curriculum

Common Core State Standards for Mathematical Content

The Number System 6.NS

Compute fluently with multi-digit numbers and find common factors and multiples.

6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.

6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

Common Core Standards for Mathematical Practice

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.



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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.

Clarifying the Standards

Prior Learning

In grade 4, students compared two decimals to the hundredths place and fluently added and subtracted multidigit whole numbers. In grade 5, students achieved fluency in the standard algorithm for multiplying multidigit whole numbers. They also reasoned about dividing whole numbers with two-digit divisors. In addition, students reasoned about adding, subtracting, multiplying, and dividing decimals to the hundredths and illustrated their reasoning using models. They used models to represent and reason about the operations, but fluency with the standard algorithm was not the expectation of the standards at this level.

In grade 5, division strategies with whole numbers entailed decomposing the dividend into like base-10 units and applying properties of operations (Distributive Property) to find the quotient place by place. See the Progressions document, K–5 Number and Operations in Base Ten, for more insights and connections to how fifth graders reasoned about division with whole numbers as well as operations with decimals and their associated models.

Current Learning

In grade 6, students divide multidigit whole numbers with fluency as well as fluently use the standard algorithm to add, subtract, multiply, and divide multidigit decimals. While students may occasionally encounter a real-world problem that goes beyond the thousandths place, the standards do not specify mastery with such numbers.

Future Learning

In grade 6, students are introduced to negative numbers. By grade 7, students will be expected to fluently operate (+, –, ×, /) with all forms of rational numbers (positive and negative fractions, decimals, and integers).



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Additional Findings

According to Curriculum Focal Points, students need to develop an understanding of and fluency with multiplication of fractions and decimals.

According to A Research Companion to Principles and Standards for School Mathematics, the development of algorithms is an essential component of mathematics. In the history of mathematics, methods for solving applied problems have been developed into efficient, fixed procedures.

“The algorithms for multiplication and division depend heavily upon fluency with multiplication and division, with multidigit subtraction. The difficulties that many students have in subtraction noticeably affect division.” (p. 87)


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Greatest Common Factor and Least Common Multiple


Content to be learned Mathematical practices to be integrated • Find the greatest common factor of two whole

numbers less than or equal to 100.

• Find 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 with a common factor as a multiple of a sum with no common factors [e.g., 36 + 8 expressed as 4(9 + 2)].

Reason abstractly and quantitatively.

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

Look for and make use of structure.

• Look for, develop, generalize, and describe a pattern orally, symbolically, graphically, and in written form.

Essential questions • What is your strategy for finding the greatest

common factor? When would you need to use this strategy?

• What is your strategy for finding the least common multiple? When would you need to use this strategy?

• How can you use the Distributive Property to express the sum of two whole numbers? Does your sum have any more factors in common?

• How can you use a model or some other method to prove your two expressions are equivalent?

• How does the Distributive Property make use of factors and multiples?

• How are common multiples and common factors different?

• What is the difference between prime and composite numbers?

Grade 6 Mathematics, Quarter 1, Unit 1.2 Greatest Common Factor and Least Common Multiple (4 days)


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Written Curriculum



Compute fluently with multi-digit numbers and find common factors and multiples.

6.NS.4 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 (9 + 2).


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.

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.


Prior Learning

In grade 4, students gained familiarity with factors and multiples. They found all factor pairs for a whole number in the range 1–100 and determined if a number was a multiple of a one-digit number. Students also determined if numbers 1–100 were prime or composite.



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In grade 5, students wrote simple expressions to record calculations with numbers, such as 2 × (8 + 7); they interpreted this quantity as having a value two times greater than the sum of 8 + 7.

As early as grade 3, students informally used the Distributive Property in working with multiplication and division. For example, third graders decomposed a number to reason about a product:

15 × 10 = (10 + 5) × 10 = 10 groups of 10 + 10 groups of 5 = (10 × 10) + (5 × 10)

Similarly, students used area models to reason about multiplication and division using the Distributive Property. See the Progressions document, K–5 Number in Base Ten, for more information on how the standards connected understanding operations to the Distributive Property.

Current Learning

Students compute 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. This is a supporting standard as students use this skill in recognizing common factors to rewrite expressions as stated in 6.EE.3.

At the end of grade 6, students are at the drill-and-practice instructional level for the greatest common factor and least common multiple. Students are at a level of mastery where they are applying their understanding through problem solving.

Future Learning

In grades 6 and beyond, students will continue to apply properties of operations, including the Distributive Property, to generate equivalent expressions, including algebraic expressions containing variables. Students will use the words sum, term, product, factor, quotient, and coefficient to describe parts of the expression. They will also view one or more parts of the expression as a single entity [e.g., 2(8 + 7) is a product of two factors and notice the structure (8 + 7) as a single entity].

Additional Findings

According to Principles and Standards for School Mathematics, “Students should recognize that different types of numbers have particular characteristics; for example, square numbers have an odd number of factors and prime numbers have only two factors.” (p. 151)

“Tasks, such as the following, involving factors, multiples, prime numbers, and divisibility, can afford opportunities for problem solving and reasoning.

1. Explain why the sum of the digits of any multiple of 3 is itself divisible by 3.

2. A number of the form abcabc always has several prime-number factors. Which prime numbers are always factors of a number of this form? Why?” (p. 217)



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Fractions


Content to be learned Mathematical practices to be integrated • Define reciprocals.

• Use models to represent division of fractions by fractions.

• Divide fractions by fractions (including mixed numbers) using standard algorithm and models.

• Solve word problems involving division of fractions by fractions.

Make sense of problems and persevere in solving them.

• Explain the meaning of a problem and restate it in their own words.

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

• Identify and execute appropriate strategies to solve the problem.

• Evaluate progress toward the solution and make revisions if necessary.

• Check their answers using a different method, and continually ask, “Does this make sense?”

Model with mathematics.

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

• Make assumptions and approximations to simplify a complicated situation, realizing that they may need revision later.

Use appropriate tools strategically.

• Use tools when solving a mathematical problem and 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. Detect possible errors by strategically using estimation and other mathematical knowledge.

Grade 6 Mathematics, Quarter 1, Unit 1.3 Fractions (5 days)


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Essential questions • What is a reciprocal? When do you use it?

• What strategies would you use to divide a fraction by a fraction?

• How would you represent fractional division using models?

• What are the steps for solving word problems involving division with fractions?

Written Curriculum



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?


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.



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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.

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 3, students developed an understanding of fractions as numbers. In grades 4 and 5, students extended an understanding of equivalent fractions and ordering fractions. They built from unit fractions by applying and extending previous understanding of operations of whole numbers. Students understood decimal notation for fractions and compared decimal and fractions. They divided and multiplied fractions.

Current Learning

Sixth graders apply and extend previous understanding of multiplication and division and apply it to dividing fractions. Students compute fluently with multidigit numbers and find the greatest common factor. They apply and extend previous understandings of numbers to the system of rational numbers.

Future Learning

In grade 7, students will apply and extend previous understanding of operations with fractions to multiply and divide rational numbers. They will understand that multiplication is extended from fractions to rational numbers.



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Additional Findings According to Principles and Standards for School Mathematics,

• “Fewer than one-third of the thirteen year old U.S. students tested in the NAEP in 1988 correctly chose the largest number from 3/4, 9/16, 5/8, and 2/3. (Kouba, Carpenter, and Swafford, 1989)” (p. 216)

• “Visual images of fractions as fraction strips should help many students think flexibly in comparing, interpreting, or computing fractions.” (p. 216)

• “From experiences with whole numbers, many students appear to develop a belief that multiplication makes bigger, and division makes smaller. When solving problems… This belief has negative consequences that have been well researched. (Greer, 1992)” (p. 218)


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Ratios


Content to be learned Mathematical practices to be integrated • Understand ratios.

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

• Understand unit rates.

• Use ratios to solve real-world problems.

• Solve unit rate problems, including unit pricing and constant speed.

• Find percent of a quantity as a rate per 100.

• Solve problems involving finding the whole, given a part and a percent.

• Make tables of equivalent ratios and find missing values.

• Plot pairs of values on the coordinate plane.

Make sense of problems and persevere in solving them.

• Explain the meaning of a problem and restate it in their own words.

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

• Identify and execute appropriate strategies to solve the problem.

• Evaluate progress toward the solution and make revisions if necessary.

• Check their answers using a different method and continually ask, “Does this make sense?”

Look for and express regularity in repeated reasoning.

• Use repeated applications to generalize properties.

• Look for mathematically sound shortcuts.

Essential questions • What is a unit rate?

• What does a ratio represent?

• What is unit pricing? How does it relate to ratios?

• How can you use a table to show equivalent ratios and find missing values within the table?

• What steps do you take to plot ordered pairs on a coordinate plane?

Grade 6 Mathematics, Quarter 1, Unit 1.4 Ratios (18 days)


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Written Curriculum


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.

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?

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.


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.



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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 grade 4, students extended understanding of fraction equivalence and ordering. In grade 5, students solved real-world problems using visual fraction models or equations to represent the problem. They graphed points on the coordinate plane to solve real-world and mathematical problems.

Current Learning

In grade 6, students understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. They understand the concept of a unit rate associated with a ratio. (For example, 30% of a quantity means 30/100 times the quantity.) Students use ratio and rate reasoning to solve real-world problems. They use ratio reasoning to convert measurement units and manipulate and transform units appropriately when multiplying and dividing quantities. Students make tables of equivalent ratios relating quantities with whole number measurements.

Future Learning

Seventh graders will analyze proportional relationships and use them to solve real-world and mathematical problems. Students will recognize and represent proportional relationships between quantities. They will use proportional relationships to solve multistep ratio and percent problems. Students will identify the constant of proportionality in tables.

Additional Findings

According to A Research Companion to Principles and Standards for School Mathematics, “Because probabilities are often expressed as ratios, an understanding of ratios provides the cornerstone for an understanding of probability.” (p. 217)

According to Principles and Standards for School Mathematics, “Problems that involve constructing or interpreting scale drawings offer students opportunities to use and increase their knowledge of similarity, ratio, and proportionality. Such problems can be created from many sources such as maps, bulletins, science, and even literature.” (pp. 245–246)

According to NCTM Curriculum and Evaluation Standards, “Proportional reasoning is of such importance that it merits whatever time and effort that must be expended to assure its careful development.” (p. 82)



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