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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 add and subtract multi-digit numbers with decimals to the thousandths with procedural fluency. In grade 4 (MA.4.NSO.2.7), students explored the addition and subtraction of multi-digit numbers with decimals to hundredths using money and manipulatives. In grade 6, students add and subtract positive fractions with procedural fluency.- To demonstrate procedural fluency, students may choose a standard algorithm that works best for them and demonstrates their procedural fluency. A standard algorithm is a method that is efficient and accurate (MTR.3.1).
- When students use a standard algorithm, they should be able to justify why it works conceptually. Teachers can expect students to demonstrate how their algorithm works, for example, by comparing it to another method for addition and subtraction (MTR.6.1).
- Along with using a standard algorithm, students should estimate reasonable solutions before solving. Estimation helps students anticipate possible answers and evaluate whether their solutions make sense after solving.
Common Misconceptions or Errors
- Students can make computational errors while using standard algorithms when they cannot reason why their algorithms work. In addition, they can struggle to determine where or why that computational mistake occurred because they did not estimate reasonable values for intermediate outcomes as well as for the final outcome. During instruction, teachers should expect students to justify their work while using their chosen algorithms and engage in error analysis activities to connect their understanding to the algorithm.
Strategies to Support Tiered Instruction
- Instruction includes estimating reasonable values for sums and differences when adding and subtracting decimals to the hundredths.
- For example, students make reasonable estimates for the sum of 6.32 + 2.84. Instruction includes stating, “Before using an algorithm, we will estimate the sum to make sure that we are using the algorithm correctly and our answer is reasonable. I will use my understanding of rounding decimals to estimate my sum. The addend of 6.32 rounds to 6 when rounded to the nearest whole number and reasonable estimate for my sum would be 9 because 6 + 3 = 9.”
- For example, students make reasonable estimates for the difference of 7.9 − 4.25. Instruction includes stating, “Before using an algorithm, we will estimate the difference to make sure that we are using the algorithm correctly and our answer is reasonable. I will use my understanding of rounding decimals to estimate my difference. The minuend of 7.9 rounds to 8 when rounded to the nearest whole number and the subtrahend 4.25 rounds to 4 when rounded to the nearest whole number. A reasonable estimate for my difference would be 4 because 8 − 4 = 4.” the addend 2.84 rounds to 3 when rounded to the nearest whole number.
- Instruction includes explaining and justifying mathematical reasoning while using an algorithm to add and subtract decimals to the hundredths. Instruction also includes determining if an algorithm was used correctly by analyzing any errors made and reviewing the reasonableness of solutions.
- For example, students use a standard algorithm to determine 6.32 + 2.84 and explain their thinking using a place value understanding. Instruction includes stating, “Begin by lining up the decimal points and place values for each addend. Next, add in hundredths place. 2 hundredths plus 4 hundredths are 6 hundredths. Because the total number of hundredths is less than 10 hundredths it is not necessary to regroup. Next, add in the tenths place. 3 tenths plus 8 tenths are 11 tenths. Because I have more than 10 tenths it is necessary to regroup the 10 tenths to make one whole. After composing a group of 10 tenths there is 1 tenth remaining. Finally, add 6 ones plus 2 ones and the 1 whole that was regrouped from the tenths place. The sum is 9.16. Our sum of 9.16 is close to our estimate of 9, this helps us determine that our answer is reasonable.”
- For example, students use a standard algorithm to determine 7.9 − 4.25 and explain their thinking using place value understanding. The teacher reminds students that 7.9 is equivalent to 7.90 and uses a decimal grid to show the equivalency of 0.9 and 0.90 if needed. Instruction includes stating, “Begin by lining up the decimal points and place values. Next, subtract 4.25 starting in the hundredths place. There are not enough hundredths to subtract 5 hundredths from 0 hundredths. It is necessary to decompose one tenth into 10 hundredths. Now there are 10 hundredths, and there is enough to subtract 5 hundredths. 10 hundredths − 5 hundredths = 5 hundredths. Then, subtract the tenths: 8 tenths − 2 tenths = 6 tenths. Finally, subtract the ones: 7 ones − 4 ones = 3 ones. The difference is 3.65. Our difference of 3.65 is close to our estimate of 4, this helps us determine that our answer is reasonable.”
- For example, students use a standard algorithm to determine 1.9 + 2.3 and explain their thinking using a place value understanding. Instruction includes stating, “Begin by lining up the decimal points and place values for each addend. Next, add in tenths place. 9 tenths plus 3 tenths are 12 tenths. Because I have more than 10 tenths it is necessary to regroup the 10 tenths to make one whole. After composing a group of 10 tenths there are 2 tenths remaining. Finally, add 1 one plus 2 ones and the 1 whole that was regrouped from the tenths place. The sum is 4.2. Our sum of 4.2 is close to our estimate of 4, this helps us determine that our answer is reasonable.”
- For example, students use a standard algorithm to determine 5.2 − 3.8 and explain their thinking using place value disks and their understanding of place value. Instruction includes stating, “Begin by lining up the decimal points and place values. Next, subtract 3.8 starting in the tenths place. There are not enough tenths to subtract 8 tenths from 2 tenths. It is necessary to decompose one whole into 10 tenths. Now there are a total of 12 tenths, and there are enough to subtract 8 tenths. 12 tenths – 8 tenths = 4 tenths. Finally, subtract the ones: 4 ones – 3 ones = 1 one. The difference is 1.4. Our difference of 1.4 is close to our estimate of 1, this helps us determine that our answer is reasonable.”
- Instruction includes the use of place value columns to support place value understanding when using an algorithm to add and subtract decimals.
Instructional Tasks
Instructional Task 1 (MTR.3.1)
Use a standard algorithm to find the difference of eight hundred two and forty - Six thousandths and three hundred and nine tenths. Explain how you use your algorithm to subtract.
Instructional Items
Instructional Item 1
Find the sum and difference of 8.72 and 3.032.*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 Games
Formative Assessments
Lesson Plans
Original Student Tutorials
Perspectives Video: Experts
Tutorial
STEM Lessons - Model Eliciting Activity
On this MEA activity, students will create a procedure to rank five lunch bags as to which one is the best in keeping food and drinks at a safe temperature and appealing to the taste, while keeping design and price on target.
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 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.
MFAS Formative Assessments
Students are asked to solve a word problem that involves adding two decimals by using a strategy based on place value.
Students are asked to solve a word problem that involves subtracting two decimals by using a strategy based on place value.
Original Student Tutorials Mathematics - Grades 6-8
Learn to add decimals to the thousandths using a standard algorithm at the ice cream shop in this interactive tutorial.
Sail through subtracting decimals to the thousandths place using a standard algorithm in this interactive tutorial.
Student Resources
Original Student Tutorials
Sail through subtracting decimals to the thousandths place using a standard algorithm in this interactive tutorial.
Type: Original Student Tutorial
Learn to add decimals to the thousandths using a standard algorithm at the ice cream shop in this interactive tutorial.
Type: Original Student Tutorial
Educational Games
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
This interactive applet gives students practice in making change in U.S. dollars and in four other currencies. Students are presented with a purchase amount and the amount paid, and they must enter the quantity of each denomination that make up the correct change. Students are rewarded for correct answers and are shown the correct change if they err. There are four levels of difficulty, ranging from amounts less than a dollar to amounts over $100.
Type: Educational Game
Tutorial
This tutorial for student audiences will assist learners with a further understanding of the rules for adding and subtracting with decimals. Students will be able to navigate the teaching portion of the tutorial at their own pace and test their understanding after each step of the lesson with a "Try This" section. The "Try This" section will monitor students answers and self-check by a right answer turning orange and a wrong answer dissolving.
Type: Tutorial
Parent Resources
Tutorial
This tutorial for student audiences will assist learners with a further understanding of the rules for adding and subtracting with decimals. Students will be able to navigate the teaching portion of the tutorial at their own pace and test their understanding after each step of the lesson with a "Try This" section. The "Try This" section will monitor students answers and self-check by a right answer turning orange and a wrong answer dissolving.
Type: Tutorial
