Unit 3: Multi-Digit and Fraction Computation
Students extend their understanding of multiplication and division to divide fractions by fractions, and develop fluency with whole number and decimal operations.
In Unit 3, sixth grade students focus on the number system, extending their understanding of multiplication and division to include fraction division, and developing fluency with whole number and decimal operations. Throughout the unit, students work toward developing and understanding efficient algorithms. By examining the structure of concrete models and patterns that emerge from these structures, students make sense of concepts such as multiplying by a reciprocal of a fraction when dividing or using long division as a shorthand to partial quotients (MP.8). With these efficient computation algorithms, students solve and interpret real-world problems, including rate applications from Unit 2. Throughout this unit, students will develop, practice, and demonstrate fluency with decimal operations; however, practice and demonstration opportunities should continue throughout the year with the goal of fluency by the end of the year. Several opportunities are already built into future units, such as the unit on Expressions and the unit on Equations, but additional opportunities need to be planned for and included. See our Procedural Skill and Fluency Guide for additional information and strategy and activity suggestions.
Throughout elementary grades, students developed their understanding of the base-ten system. They found sums and products and quotients by using concrete models, place value, properties of operations, and the relationships between operations. Intentionally, students did not learn a standard algorithm until they had the conceptual understanding to back it up. Some of these strategies are revisited in this unit in order to ensure that students firmly understand the reasoning behind an algorithm, rather than using it without understanding.
Once students have mastered the positive number system of fractions, decimals, and whole numbers, sixth-grade students will investigate the numbers to the left of 0 on the number line, or negative rational numbers, in Unit 4. In seventh grade, students will learn how to compute with all rational numbers, including negatives, and in eighth grade and high school, students learn about irrational numbers, rounding out their study of the real number system.
Pacing: 20 instructional days (17 lessons, 2 flex days, 1 assessment day)
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The following assessments accompany Unit 3.
Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.
Pre-Unit Student Self-Assessment
Have students complete the Mid-Unit Assessment.
Use the resources below to assess student mastery of the unit content and action plan for future units.
Post-Unit Assessment Answer Key
Post-Unit Student Self-Assessment
Use student data to drive your planning with an expanded suite of unit assessments to help gauge students’ facility with foundational skills and concepts, as well as their progress with unit content.
Suggestions for how to prepare to teach this unit
Prepare to teach this unit by immersing yourself in the standards, big ideas, and connections to prior and future content. Unit Launches include a series of short videos, targeted readings, and opportunities for action planning.
Multiply 12.6 and 4.8 using partial products.
Divide 67,764 by 12 using partial quotients.
Use a Venn diagram to find the GCF of 12 and 18.
The central mathematical concepts that students will come to understand in this unit
Terms and notation that students learn or use in the unit
greatest common factor (gcf)
least common multiple (lcm)
long division/ standard algorithm for division
To see all the vocabulary for Unit 3, view our 6th Grade Vocabulary Glossary.
The materials, representations, and tools teachers and students will need for this unit
To see all the materials needed for this course, view our 6th Grade Course Material Overview.
Topic A: Dividing with Fractions
Interpret division problems as the number of items in each group or the number of groups of a given number of items. Write corresponding multiplication and division problems.
Divide a fraction by a whole number using visual models and related multiplication problems.
Divide a whole number by a fraction using visual models.
Use visual models and patterns to develop a general rule to divide with fractions.
Solve and write story problems involving division with fractions.
Solve problems involving division with fractions.
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Topic B: Computing with Decimals
Add and subtract decimals using the standard algorithm.
Multiply decimals using strategies, and develop an understanding of the standard algorithm.
Multiply decimals using the standard algorithm.
Divide multi-digit whole numbers using the standard algorithm.
Divide numbers with decimal quotients. Divide decimals by whole numbers.
Divide decimals by decimals using the standard algorithm.
Solve problems involving decimals using all four operations.
Topic C: Applying the Greatest Common Factor and the Least Common Multiple
Use prime factorization to represent numbers as products of prime factors.
Find the greatest common factor of two numbers. Solve application problems using the greatest common factor.
Find the least common multiple of two numbers. Solve application problems using the least common multiple.
Solve mathematical and real-world problems using the greatest common factor and least common multiple.
The content standards covered in this unit
— 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?
— Fluently divide multi-digit numbers using the standard algorithm.
— Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
— 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).
Standards covered in previous units or grades that are important background for the current unit
— Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
— Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
— Fluently multiply multi-digit whole numbers using the standard algorithm.
— Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
— Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
— Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
— Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade
— Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
— Find all factor pairs for a whole number in the range 1—100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1—100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1—100 is prime or composite.
Standards in future grades or units that connect to the content in this unit
— 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.
— Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
— 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.
— Make sense of problems and persevere in solving them.
— Reason abstractly and quantitatively.
— Construct viable arguments and critique the reasoning of others.
— Model with mathematics.
— Use appropriate tools strategically.
— Attend to precision.
— Look for and make use of structure.
— Look for and express regularity in repeated reasoning.
Unit Rates and Percent