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Examples
The quotient of 23 and 0.42 can be estimated as a little bigger than 46 because 0.42 is less than one-half and 23 times 2 is 46.Clarifications
Clarification 1: Estimating quotients builds the foundation for division using a standard algorithm.Clarification 2: Instruction includes the use of models based on place value and the properties of operations.
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Equation
- Expression
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
The purpose of this benchmark is for students to explore multiplication and division of multi- digit numbers with decimals using estimation, rounding, place value, and exploring the relationship between multiplication and division. This benchmark connects to the work students did in grade 4 with addition and subtraction of decimals (MA.4.NSO.2.7). Students achieve procedural fluency with multiplying and dividing multi-digit numbers with decimals in grade 6 (MA.6.NSO.2.1)- Instruction of this benchmark focuses on number sense to help students develop procedural reliability while multiplying and dividing multi-digit numbers with decimals.
- During instruction, students should explore how the products and quotients of whole numbers relate to decimals.
- For example, if students know the product of 8 × 7 and the quotient of 56 ÷ 4, then they can reason through 0.08 × 7 or 5.6 ÷ 0.4 through place value relationships. Classroom discussions should allow students to explore these patterns and use them to estimate products and quotients (MTR.4.1, MTR.6.1).
- Teachers should connect what students know about place value and fractions. o For example, because students know that multiplying a number by one-fourth will result in a product that is smaller, multiplying a number by 0.25 (its decimal equivalence) will also result in a smaller product. In division, dividing a number by one-fourth and 0.25 will result in a larger quotient. Continued work in this benchmark will help students to generalize patterns in multiplication and division of whole numbers and fractions (MTR.5.1).
- Models that help students explore the multiplication and division of multi-digit numbers with decimals include base ten representations (e.g., blocks) and place value mats.
Common Misconceptions or Errors
- Students may not understand the reasoning behind the placement of the decimal point in the product. Modeling and exploring the relationships between place value will help students gain understanding.
- Students can confuse that multiplication always results in a larger product, and that division always results in a smaller quotient. Through classroom discussion, estimation and modeling, classroom work should address this misconception.
Strategies to Support Tiered Instruction
- Instruction includes opportunities to explore place value of decimals with concrete models and objects.
- For example, students use place value understanding and a place value chart to compare 0.14 and 0.2. The teacher explains that when comparing decimals, we start with the digit to the far left because we want to compare the greatest place values first. Both values have a 0 in the ones place, so we will move to the tenths place. One-tenth is less than two-tenths, so 0.14 < 0.2.
- For example, students compare 0.3 and 0.03 using decimal grids and represent each value and explain that 0.3 covers a greater are of the decimal grid than 0.03, so 0.3 is greater than 0.03.
- Instruction includes opportunities to predict and explain the relative size of the product of two decimals. Students use models to check their prediction and solve. The teacher guides students to connect that multiplying a given number by a number less than one will result in a smaller number, and that multiplying a given number by a number greater than one will result in a larger number.
- For example, students solve the following problem 0.2 × 0.5. Students should reason about the size of the decimals and connect it back to their fraction understanding and think about the multiplication sign signaling “groups of.” This expression could be interpreted as 0.2 “of” 0.5. This will help with the misconception of multiplying equals a larger product. The picture below illustrates the product of 0.2 and 0.5. If the entire square is 1 unit, the gray region represents 0.5 units, and the red region represents 0.2 units. The overlap in purple contains 10 small squares, each of which represents 0.01 units. Therefore, the overlap portion contains 10 × 0.01 = 0.10 units. The overlap portions show a 0.2 by 0.5 rectangle, so the number of units it contains is the product 0.2 and 0.5.
Instructional Tasks
Instructional Task 1 (MTR.4.1)
Instructional Task 2 (MTR.4.1)
What is the same about the quotients of these expressions? What is different? Explain.
Instructional Task 3 (MTR.5.1)
How can you use 2 × 12 = 24 to help you find the product of 2 × 1.2? Explain.
Instructional Items
Instructional Item 1
Raul reasons that the product of 82 × 0.56 will be greater than 41 and less than 82. Explain whether or not his conclusion is reasonable.*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
Related Courses
Related Access Points
Related Resources
Educational Game
Lesson Plans
Original Student Tutorial
Perspectives Video: Expert
Perspectives Video: Teaching Ideas
Problem-Solving Tasks
STEM Lessons - Model Eliciting Activity
This Model Eliciting Activity (MEA) is written at a 4th grade level.
This activity allows students to think critically using information provided. Students will write a procedure on how they determined which painting company would be suitable for the client.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
In this MEA, students will create a procedure to rank five mini-refrigerators to determine which one should be purchased for the school by the PTA based on size, type, features, energy usage, and cost. In the process, students will solve real-world problems involving the multiplication of multi-digit numbers with decimals to the hundredths, including using money. Students will also determine the volume of a rectangular prism using a formula.
In this Model Eliciting Activity, MEA, the students will be asked to assist a discount shoe store owner, who is planning a one day sale promotion, to choose a famous brand shoe to feature for the one day sale. Students will determine which one will bring in more customers, as well as provide the most profit. Students will need to read a data table, calculate the total profit margin per pair, and the total sales potential profit margin determined by the number of sneakers in stock. Students will also need to consider comfort, durability, and specific details about each brand. A twist is added to the problem when additional stock items are added, plus one of the brands is removed and two new brands are added.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx
In this Model Eliciting Activity, MEA, students will work in collaborative groups to solve multistep problems with whole numbers and decimals by using different mathematical operations such as addition, subtraction, multiplication, and division. The students will be asked to assist a businessman who is planning a party for his employees. They will need to read several ads and decide which company offers the best deal in renting tables, chairs, and tablecloths for the client. They will need to take into consideration the number of guests attending the party and the budget allowed. A twist is added to the problem when the students are asked to consider an additional ad and the fact that the guest list is now slightly larger.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx
This Model Eliciting Activity (MEA) asks students to develop a procedure for choosing a reptile or amphibian to place in a school reception area. Students will need to consider safety, price of animal, cost by week to feed animal, size and cost of the enclosure, and the life span of the animals they are considering. In the second portion of the problem statement, the students will need to prepare a budget and cost analysis for the year to consider if they have still made the correct choices while adding three more animals for consideration. The culminating activity for this MEA will have the student write a proposal for the Principal to state their choice of animal, give a year's budget for cost and care for the animal.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx
Students work in teams to plan the contents of a covered wagon for a family relocating from Missouri to Oregon. Students must calculate the weight and cost of the wagon by adding, subtracting, and multiplying with decimals.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx
In this MEA, students are challenged to choose the snacks that will be in a vending machine in a school. Students will need to divide whole numbers and decimal numbers by whole numbers. Students will work in groups to solve the problem and write a letter to the client explaining their thinking.
Students will create a rating system for workout DVD's according to weight loss, muscle toning, and increased physical condition.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
Original Student Tutorials Mathematics - Grades K-5
Help your town build a dog park by multiplying whole numbers by decimals to the tenths place in this interactive tutorial.
Note: this is an introductory tutorial on multiplying whole numbers by decimals before students move on to multiplying decimals by decimals.
Student Resources
Original Student Tutorial
Help your town build a dog park by multiplying whole numbers by decimals to the tenths place in this interactive tutorial.
Note: this is an introductory tutorial on multiplying whole numbers by decimals before students move on to multiplying decimals by decimals.
Type: Original Student Tutorial
Educational Game
This fun and interactive game helps practice estimation skills, using various operations of choice, including addition, subtraction, multiplication, division, using decimals, fractions, and percents.
Various levels of difficulty make this game appropriate for multiple age and ability levels.
Addition/Subtraction: The addition and subtraction of whole numbers, the addition and subtraction of decimals.
Multiplication/Division: The multiplication and addition of whole numbers.
Percentages: Identify the percentage of a whole number.
Fractions: Multiply and divide a whole number by a fraction, as well as apply properties of operations.
Type: Educational Game
Problem-Solving Tasks
Students are asked to reason about and explain the placement of decimals in quotients.
Type: Problem-Solving Task
When a division problem involving whole numbers does not result in a whole number quotient, it is important for students to be able to decide whether the context requires the result to be reported as a whole number with remainder (as with Part (b)) or a mixed number/decimal (as with Part (c)). Part (a) presents two variations on a context that require these two different responses to highlight the distinction between them.
Type: Problem-Solving Task
Parent Resources
Problem-Solving Tasks
Students are asked to reason about and explain the placement of decimals in quotients.
Type: Problem-Solving Task
When a division problem involving whole numbers does not result in a whole number quotient, it is important for students to be able to decide whether the context requires the result to be reported as a whole number with remainder (as with Part (b)) or a mixed number/decimal (as with Part (c)). Part (a) presents two variations on a context that require these two different responses to highlight the distinction between them.
Type: Problem-Solving Task
