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Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Equal sign
- Expression
- Equation
- Whole Number
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
The purpose of this benchmark is for students to add and subtract multi-digit whole numbers with procedural fluency. Students use skills from the procedural reliability stage in Grade 2 to become fluent with efficient and accurate procedures, including standard algorithms for addition and subtraction. In Grade 2, students added and subtracted multi-digit whole numbers up to 1,000. In Grade 3, the magnitude of numbers increases.- A standard algorithm is defined as any efficient and accurate procedure that allows students to add and subtract whole numbers. Students’ choices of standard algorithms for addition and subtraction do not need to be the same (MTR.5.1).
- Students should be able to justify their use of a standard algorithm for adding and subtracting by explaining the steps mathematically. Each student should be able to explain if and when regrouping is needed, and how regrouping is computed using their chosen algorithm. During instruction, teachers and students should work together to relate place value understanding to algorithms (MTR.3.1, MTR.4.1, MTR.5.1).
- Problems should include both vertical and horizontal forms, including opportunities for students to apply the commutative and associative properties.
- Instruction of this benchmark should be taught with MA.3.NSO.1.4. Students should use rounding as a means to estimate reasonable solutions of sums and differences before calculating (MTR.6.1).
Common Misconceptions or Errors
- Students who learn a standard algorithm without being able to explain why it works using place value understanding often make computational errors and/or cannot determine if their solutions are reasonable. To assist students with this misconception, students should justify the algorithm they choose by checking for reasonableness.
- Students who cannot explain their steps mathematically often have difficulty understanding regrouping. Many computational errors are a result of students not discovering this conceptual understanding while they practiced adding and subtracting with procedural reliability in Grade 2 (MA.2.NSO.2.3 and MA.2.NSO.2.4). Instruction that focuses on learning standard algorithms for addition and subtraction as a series of steps without checking for conceptual understanding will contribute to regrouping errors.
Strategies to Support Tiered Instruction
- Instruction includes guiding students through the process of estimating reasonable values for sums and differences using an understanding of place value, addition, and subtraction.
- For example, students make reasonable estimates for the sum of 174 + 253. Instruction includes a prompt such as “Before using an algorithm, we will estimate the sum to make sure that we are using the algorithm correctly and our answer is reasonable. The first addend of 174 is close to the benchmark number 200 and the second addend of 253 is close to the benchmark number 250. So, we can use 200 + 250 = 450 to estimate that our sum should be close to 450.”
- Instruction includes guiding students through the process of explaining and justifying the chosen algorithm and determining if an algorithm was used correctly by reviewing the reasonableness of solutions.
- For example, students use a standard algorithm to solve 174 + 253 and explain their thinking using a place value visual representation. Instruction includes a prompt such as “Begin by adding in the one's place. 4 ones plus 3 ones is 7 ones. Because the total number of ones is less than 10, it is not necessary to regroup. Next, add in the tens place. 7 tens plus 5 tens is 12 tens. Because I have more than 10 tens it is necessary to regroup the 10 tens to make one hundred. After composing a group of 10 tens there are 2 tens remaining. Finally, add 1 hundred plus 2 hundreds. Add the 1 hundred that was regrouped from the tens place. The sum is 427. Our sum of 427 is close to our estimate of 450, this helps us determine that our answer is reasonable”
- For example, students use a standard algorithm to solve 327 − 174 and explain their thinking using a place value visual representation. Instruction includes prompts such as “ Begin subtracting 174 starting in the one's place. 7 ones minus 4 ones are 3 ones. There are not enough tens to subtract 7 tens from 2 tens. It is necessary to decompose one hundred into 10 tens. Now there are 12 tens, and there is enough to subtract 7 tens. 12 tens minus 7 tens equals 5 tens. Finally, subtract the hundreds: 3 hundreds minus 1 hundred equals 2 hundreds. The difference is 253.”
- For example, students use a standard algorithm and base-ten blocks to solve 62 − 37 and explain their thinking using a place value visual representation. Instruction includes a prompt such as “Begin subtracting 37 starting in the one's place. There are not enough ones to subtract 7 ones from 2 ones. It is necessary to decompose one ten into 10 ones. Now there are 12 ones and there is enough to subtract 7 ones. 12 ones take away 7 ones equals 5 ones. Finally, subtract the tens: 5 tens minus 3 tens is 2 tens. The difference is 25.”
- Teacher provides guidance on using strategies based on place value to add and subtract.
- For example, students use strategies based on place value to solve 174 + 253. Students can decompose each number into expanded form, then add each place value separately (add the ones together, the tens together, and the hundreds together). Then, students can add together the sums of the ones, tens and hundreds to compute the sum.
Instructional Tasks
Instructional Task 1 (MTR.3.1, MTR.7.1)
- Miranda finds 492 seashells during her vacation. She now has 1,045 seashells in her collection. How many seashells did she have in her collection before vacation?
- Part A. Solve using a standard algorithm.
- Part B. Indicate one step where you needed to regroup while solving and show how you did it using words or a pictorial model.
- Mount Pleasant Elementary School had a penny fundraiser last week to raise money for families that were affected by the recent storm. The chart below shows how many pennies they raised each day.
- Part A. How many more pennies were collected on Monday and Tuesday than on Thursday?
- Part B. Mrs. William’s class collected 1,627 of Friday’s total pennies. How many pennies did the rest of the school collect on Friday?
- Part C. The principal said the school collected about 6,000 pennies on Monday, Tuesday and Wednesday. Is this a reasonable estimate? Explain how you know.
- Part A. Write a word problem using the equation shown above.
- Part B. Solve the word problem you created in two different ways. Explain your thinking.
- Maya downloaded 856 songs to her music library last month. This month, she downloaded 726 more songs. Then, she deleted 119 songs that she no longer liked.
- Part A. How many songs does Maya have in her music library now?
- Part B. How many more songs will Maya need to download before she has 3,000 songs in her music library?
- Part C. What are two different ways you could solve this problem? Show your thinking.
- Shay wants to find the sum of 2,417 and 3,568. Explain the steps for finding the sum of 2,417 and 3,568. Be sure to include the words thousands, hundreds, tens, ones, and sum in your explanation.
Instructional Items
Instructional Item 1
- What is the sum of 1,432 and 2,981?
Instructional Item 2
- What is the difference of 8,000 and 1,432?
Instructional Item 3
- Cameron set a goal of collecting 7,000 stickers this year. In January, he collected 895 stickers, in February, he collected 472 stickers, and in March he collected 927 stickers. How many more stickers does he need to collect to meet his goal for the year?
- Use rounding to estimate the sum of 4,587 and 926. Then, find the actual sum of the two numbers.
Related Courses
Related Access Points
Related Resources
Educational Games
Educational Software / Tool
Formative Assessments
Lesson Plans
Original Student Tutorials
Perspectives Video: Experts
Problem-Solving Tasks
Teaching Idea
Tutorials
Worksheets
STEM Lessons - Model Eliciting Activity
In this Model Eliciting Activity, MEA, students will choose their top choices of field day activities given the area required for event, safety concerns, clean up required, number of students that can play at a time, and peer comments about the activity. Students will need to make trade-offs in cost when the "twist" provides budget restrictions. Students will count unit squares to calculate area, multiply one-digit numbers by multiples of ten, and add multi-digit whole numbers.
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
Kites for Education is a Modeling Eliciting Activity which presents students with an engineering challenge in which they must analyze data sets and develop a procedure for ranking different kite models. The product ranked as best by the students will hypothetically be sold to customers and the profit used to purchase school textbooks and supplies for school age children impacted by Haiti's devastating earthquake.
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 processes. 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 MEAs visit: https://www.cpalms.org/cpalms/mea.aspx
In this Model Eliciting Activity, MEA, students will create a procedure for ranking pool construction companies based on the number of years in business, customer satisfaction, and available pool dimensions. In a “twist,” students will be given information about discounts available by each company. Students will evaluate their procedure for ranking and change it if necessary.
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 MEA asks the students to compare information provided on various Science Space Camps to be attended by a student during the summer. They will take into account past attendees' reviews of the camps which should create interesting student discussions. They will use knowledge of operations to determine the difference in camp costs.
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 Shady Oak Treehouse Club is doing a makeover and needs help choosing flooring. Students will be asked to figure area, calculate cost, and add installation fees to cost. The students will then rank the flooring and choose the best one for the makeover. The data provided is: a model of the treehouse (in square yards), flooring price per square yard, and ratings for ease of cleaning, comfort, and color choices. In the twist, student will be provided with an installation fee for each flooring material and must decide if, and how, to change their procedure with the new information.
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
MFAS Formative Assessments
Students are asked to complete addition and subtraction problems that can be done more easily by using properties.
Students add four two-digit numbers without the context of a word problem.
Students are asked to add four two digit numbers using place value.
Students are asked to add four two digit numbers by considering how another student added "friendly" numbers first.
Students are asked to solve two addition problems using the standard algorithm.
Students are asked to add four different pairs of numbers (within 1000) using strategies of their own choosing.
Students are asked to complete subtraction problems using the standard algorithm.
Students are asked to find the error in a partially completed subtraction problem using the standard algorithm and to solve the problem correctly.
Students are asked to solve two subtraction problems using the standard algorithm.
Students are asked to complete four subtraction problems (within 1000) using strategies of their own choosing.
The student is encouraged to use compensation to efficiently add multi-digit numbers.
Original Student Tutorials Mathematics - Grades K-5
Learn to add multi-digit numbers using a standard algorithm in this interactive tutorial.
Help Speedy Sam add and subtract as quickly as possible by using the properties of addition and subtraction in this interactive tutorial.
STEM Civics Tutorials and Videos - Grades K-12
United States citizens have a responsibility to vote. In this integrated civics and math tutorial, a class collects voting data to display in a table showing the students' and teachers' choices for a new school project.
Learn how to use a bar graph to summarize voting results at school in this interactive tutorial.
This is part 2 in a two-part series. Click HERE to open Part 1.
Student Resources
Original Student Tutorials
Learn how to use a bar graph to summarize voting results at school in this interactive tutorial.
This is part 2 in a two-part series. Click HERE to open Part 1.
Type: Original Student Tutorial
United States citizens have a responsibility to vote. In this integrated civics and math tutorial, a class collects voting data to display in a table showing the students' and teachers' choices for a new school project.
Type: Original Student Tutorial
Learn to add multi-digit numbers using a standard algorithm in this interactive tutorial.
Type: Original Student Tutorial
Help Speedy Sam add and subtract as quickly as possible by using the properties of addition and subtraction 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
In this activity, students are quizzed on their ability to estimate sums, products, and percentages. The student can adjust the difficulty of the problems and how close they have to be to the actual answer. This activity allows students to practice estimating addition, multiplication, or percentages of large numbers. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Educational Game
Educational Software / Tool
In this activity, students solve arithmetic problems involving whole numbers, integers, addition, subtraction, multiplication, and division. This activity allows students to track their progress in learning how to perform arithmetic on whole numbers and integers. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Educational Software / Tool
Problem-Solving Tasks
This task presents an incomplete problem and asks students to choose numbers to subtract (subtrahends) so that the resulting problem requires different types of regrouping. This way students have to recognize the pattern and not just follow a memorized algorithm--in other words, they have to think about what happens in the subtraction process when we regroup. This task is appropriate to use after students have learned the standard US algorithm.
Type: Problem-Solving Task
This task is intended to assess adding of four numbers as given in the standard while still being placed in a problem-solving context. As written this task is instructional; due to the random aspect regarding when the correct route is found, it is not appropriate for assessment. This puzzle works well as a physical re-enactment, with paper plates marking the islands and strings with papers attached for the tolls.
Type: Problem-Solving Task
Tutorials
In this video tutorial from Khan Academy, learn how to subtract in situations that require regrouping twice using the expanded forms of numbers, as well as the standard algorithm.
Type: Tutorial
In this Khan Academy video tutorial, consider an alternate algorithm for subtracting multi-digit numbers mentally. This video is best for students that are already comfortable with using regrouping to subtract using the standard algorithm.
Type: Tutorial
Worksheet
Parent Resources
Perspectives Video: Experts
Explore ways students may solve addition problems within 100 and how to support their understanding in this video.
FCR-STEM’s Count Us In! initiative is designed to support out-of-school providers and parents in fostering math success and enjoyment among K-5 children.
Type: Perspectives Video: Expert
Explore strategies students may use to solve subtraction problems within 100 and ways to support their understanding by watching this video.
FCR-STEM’s Count Us In! initiative is designed to support out-of-school providers and parents in fostering math success and enjoyment among K-5 children.
Type: Perspectives Video: Expert
Problem-Solving Tasks
This task presents an incomplete problem and asks students to choose numbers to subtract (subtrahends) so that the resulting problem requires different types of regrouping. This way students have to recognize the pattern and not just follow a memorized algorithm--in other words, they have to think about what happens in the subtraction process when we regroup. This task is appropriate to use after students have learned the standard US algorithm.
Type: Problem-Solving Task
This task is intended to assess adding of four numbers as given in the standard while still being placed in a problem-solving context. As written this task is instructional; due to the random aspect regarding when the correct route is found, it is not appropriate for assessment. This puzzle works well as a physical re-enactment, with paper plates marking the islands and strings with papers attached for the tolls.
Type: Problem-Solving Task
Teaching Idea
This guide offers five practical strategies to help after-school providers support meaningful, low-pressure math experiences that lead to deeper understanding and better long-term outcomes for K-5 students.
Type: Teaching Idea
Tutorials
In this video tutorial from Khan Academy, learn how to subtract in situations that require regrouping twice using the expanded forms of numbers, as well as the standard algorithm.
Type: Tutorial
In this Khan Academy video tutorial, consider an alternate algorithm for subtracting multi-digit numbers mentally. This video is best for students that are already comfortable with using regrouping to subtract using the standard algorithm.
Type: Tutorial
