problem solving lesson 4 4

Teaching Problem Solving in Math

• Freebies , Math , Planning

Every year my students can be fantastic at math…until they start to see math with words. For some reason, once math gets translated into reading, even my best readers start to panic. There is just something about word problems, or problem-solving, that causes children to think they don’t know how to complete them.

Every year in math, I start off by teaching my students problem-solving skills and strategies. Every year they moan and groan that they know them. Every year – paragraph one above. It was a vicious cycle. I needed something new.

I put together a problem-solving unit that would focus a bit more on strategies and steps in hopes that that would create problem-solving stars.

The Problem Solving Strategies

First, I wanted to make sure my students all learned the different strategies to solve problems, such as guess-and-check, using visuals (draw a picture, act it out, and modeling it), working backward, and organizational methods (tables, charts, and lists). In the past, I had used worksheet pages that would introduce one and provide the students with plenty of problems practicing that one strategy. I did like that because students could focus more on practicing the strategy itself, but I also wanted students to know when to use it, too, so I made sure they had both to practice.

I provided students with plenty of practice of the strategies, such as in this guess-and-check game.

There’s also this visuals strategy wheel practice.

I also provided them with paper dolls and a variety of clothing to create an organized list to determine just how many outfits their “friend” would have.

Then, as I said above, we practiced in a variety of ways to make sure we knew exactly when to use them. I really wanted to make sure they had this down!

Anyway, after I knew they had down the various strategies and when to use them, then we went into the actual problem-solving steps.

The Problem Solving Steps

I wanted students to understand that when they see a story problem, it isn’t scary. Really, it’s just the equation written out in words in a real-life situation. Then, I provided them with the “keys to success.”

S tep 1 – Understand the Problem.   To help students understand the problem, I provided them with sample problems, and together we did five important things:

• read the problem carefully
• restated the problem in our own words
• crossed out unimportant information
• circled any important information
• stated the goal or question to be solved

We did this over and over with example problems.

Once I felt the students had it down, we practiced it in a game of problem-solving relay. Students raced one another to see how quickly they could get down to the nitty-gritty of the word problems. We weren’t solving the problems – yet.

Then, we were on to Step 2 – Make a Plan . We talked about how this was where we were going to choose which strategy we were going to use. We also discussed how this was where we were going to figure out what operation to use. I taught the students Sheila Melton’s operation concept map.

We talked about how if you know the total and know if it is equal or not, that will determine what operation you are doing. So, we took an example problem, such as:

Sheldon wants to make a cupcake for each of his 28 classmates. He can make 7 cupcakes with one box of cupcake mix. How many boxes will he need to buy?

We started off by asking ourselves, “Do we know the total?” We know there are a total of 28 classmates. So, yes, we are separating. Then, we ask, “Is it equal?” Yes, he wants to make a cupcake for EACH of his classmates. So, we are dividing: 28 divided by 7 = 4. He will need to buy 4 boxes. (I actually went ahead and solved it here – which is the next step, too.)

Step 3 – Solving the problem . We talked about how solving the problem involves the following:

• taking our time
• working the problem out
• showing all our work
• estimating the answer
• using thinking strategies

We talked specifically about thinking strategies. Just like in reading, there are thinking strategies in math. I wanted students to be aware that sometimes when we are working on a problem, a particular strategy may not be working, and we may need to switch strategies. We also discussed that sometimes we may need to rethink the problem, to think of related content, or to even start over. We discussed these thinking strategies:

• switch strategies or try a different one
• rethink the problem
• think of related content
• decide if you need to make changes
• check your work
• but most important…don’t give up!

To make sure they were getting in practice utilizing these thinking strategies, I gave each group chart paper with a letter from a fellow “student” (not a real student), and they had to give advice on how to help them solve their problem using the thinking strategies above.

Finally, Step 4 – Check It.   This is the step that students often miss. I wanted to emphasize just how important it is! I went over it with them, discussing that when they check their problems, they should always look for these things:

• compare your answer to your estimate
• check for reasonableness
• check your calculations
• add the units
• restate the question in the answer
• explain how you solved the problem

Then, I gave students practice cards. I provided them with example cards of “students” who had completed their assignments already, and I wanted them to be the teacher. They needed to check the work and make sure it was completed correctly. If it wasn’t, then they needed to tell what they missed and correct it.

To demonstrate their understanding of the entire unit, we completed an adorable lap book (my first time ever putting together one or even creating one – I was surprised how well it turned out, actually). It was a great way to put everything we discussed in there.

Once we were all done, students were officially Problem Solving S.T.A.R.S. I just reminded students frequently of this acronym.

Stop – Don’t rush with any solution; just take your time and look everything over.

Think – Take your time to think about the problem and solution.

Act  – Act on a strategy and try it out.

Review – Look it over and see if you got all the parts.

Wow, you are a true trooper sticking it out in this lengthy post! To sum up the majority of what I have written here, I have some problem-solving bookmarks FREE to help you remember and to help your students!

You can grab these problem-solving bookmarks for FREE by clicking here .

You can do any of these ideas without having to purchase anything. However, if you are looking to save some time and energy, then they are all found in my Math Workshop Problem Solving Unit . The unit is for grade three, but it  may work for other grade levels. The practice problems are all for the early third-grade level.

• freebie , Math Workshop , Problem Solving

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Curriculum  /  Math  /  4th Grade  /  Unit 1: Place Value, Rounding, Addition, and Subtraction  /  Lesson 4

Place Value, Rounding, Addition, and Subtraction

Lesson 4 of 19

Criteria for Success

Tips for teachers, anchor tasks.

Problem Set

Target Task

Additional practice.

Build numbers to 1,000,000 and write numbers to that place value in standard and unit form.

Common Core Standards

Core standards.

The core standards covered in this lesson

Number and Operations in Base Ten

4.NBT.A.1 — Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

4.NBT.A.2 — Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

Foundational Standards

The foundational standards covered in this lesson

2.NBT.A.1 — Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

2.NBT.A.2 — Count within 1000; skip-count by 5s, 10s, and 100s.

3.NBT.A.3 — Multiply one-digit whole numbers by multiples of 10 in the range 10—90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Extend the place value system to the millions place (or higher), recognizing the patterns within place value (i.e., ones, tens, and hundreds repeat within triples of units, thousands, millions, etc.) (MP.7).
• Visualize the magnitude of 1 million. 
• Appropriately place commas within numbers up to 1 million when they are presented in standard and unit forms.
• Convert between unit and standard form (i.e., 24,078 = 2 ten thousands 4 thousands 7 tens 8 ones).
• Read numbers in word form (e.g., 24,078 as “twenty-four thousand, seventy-eight). (Note: Students will not yet write numbers in word form.)

Suggestions for teachers to help them teach this lesson

Before the Problem Set or at any point to give students more practice with reading numbers, you could have students play "Digit Ski" from  Building Conceptual Understanding and Fluency Through Games  by the Public Schools of North Carolina.

Lesson Materials

• Paper Hundreds Flats (At least 25 per class period) — See Anchor Task 1 for how to prepare this material before the lesson.
• Tape (1 or more per teacher) — Either tape or a stapler will work here
• Optional : Base ten blocks (1 thousand, 1 hundred, 1 ten, and 1 one per teacher) — You could use the image included in Anchor Task 2.
• Stapler (1 or more per teacher) — Either tape or a stapler will work here

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Tasks designed to teach criteria for success of the lesson, and guidance to help draw out student understanding

Look at your paper base ten blocks. The ones piece is the smallest square. Then tens piece is a 10 × 1 strip. The hundreds piece is the larger 10 × 10 square.

a.   Use the paper base ten blocks to construct 1,000. Use tape as needed.

b.   Use the paper base ten blocks to construct 10,000. Use tape as needed.

c.   What comes next? How much bigger will it be?

Guiding Questions

Van de Walle, John A.  Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (Volume II) . Pearson, 2nd edition, 2013.

a.   Look at the ones, tens, hundreds, and thousands base ten blocks. 

• What would you expect a ten thousands base ten block to look like? 
• What would you expect a hundred thousands base ten block to look like? 
• What comes next? What would you expect its base ten block to look like?

b.   What pattern do you notice in the shapes of the base ten blocks? What pattern do you notice in the names of the place values? ​​​​​​

When we write really large numbers, writing commas in between groups of them can help make it easier to write them. We place commas every three places from the right, starting with the ones place. For example, we write 5,678 with the comma placed three places from the right.

a.   Write 430325 in standard form with the correct placement of commas. Then read the number name out loud. 

b.   Write 3 hundred thousands 2 thousands 4 hundreds 5 tens 7 ones in standard form with the correct placement of commas. Then read the number name out loud.

c.   Read 50,038 out loud. Then write it in unit form.

Unlock the answer keys for this lesson's problem set and extra practice problems to save time and support student learning.

Discussion of Problem Set

• How do you know where to place commas in a number? How is this related to three-dimensional representations of numbers? How is it related to the place value names?
• How is the placement of commas related to how we read numbers?
• What are the similarities and differences between standard and unit form? When might we use one over the other?
• Look at #4. Explain how you knew the place value names without ever having hear them before. 

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Write the following numbers in standard form. Include commas where appropriate. 

a.   6 thousands 4 tens 9 ones

b.   7 ten thousands 2 thousands 1 hundred 5 tens 3 ones

c.   8 hundred thousands 4 ten thousands 7 hundreds 9 tens 2 ones

Student Response

Write the following numbers in unit form.

a.   23,091

b.   8,530

c.   360,467

The Extra Practice Problems can be used as additional practice for homework, during an intervention block, etc. Daily Word Problems and Fluency Activities are aligned to the content of the unit but not necessarily to the lesson objective, therefore feel free to use them anytime during your school day.

Extra Practice Problems

Answer keys for Problem Sets and Extra Practice Problems are available with a Fishtank Plus subscription.

Word Problems and Fluency Activities

Help students strengthen their application and fluency skills with daily word problem practice and content-aligned fluency activities.

Topic A: Place Value of Multi-Digit Whole Numbers

Review place value understanding for numbers within 1,000.

4.NBT.A.1 4.NBT.A.2 4.NBT.A.3 4.NBT.B.4

Build numbers to 10,000 and write numbers to that place value in standard, unit, and expanded form. 

4.NBT.A.1 4.NBT.A.2

Model and write numbers to 10,000 with more than 9 of any unit in standard and unit form.

4.NBT.A.1 4.NBT.A.2 4.NBT.B.4

Multiply and divide single units by 10. Recognize that a digit represents 10 times the value of what it represents in the place to its right.

Multiply and divide multiple units by 10. Apply the property of “ten times greater” in the context of problems.

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Topic B: Reading, Writing, and Comparing Multi-Digit Whole Numbers

Read and write multi-digit numbers using base-ten numerals, number names, and expanded form.

Locate multi-digit numbers on a number line and explain their placement.

4.NBT.A.2 4.NBT.A.3

Compare numbers based on the meanings of the digits using >, <, or = to record the comparison.

Topic C: Rounding Multi-Digit Whole Numbers

Round multi-digit numbers to their largest place.

Round multi-digit numbers to any place.

Round multi-digit numbers to any place in more complex cases, including those involving real-world contexts and/or assessing the reasonableness of that estimate. 

Topic D: Multi-Digit Whole-Number Addition and Subtraction

Fluently add multi-digit whole numbers using the standard algorithm involving up to two compositions. Solve one-step word problems involving addition.

Fluently add multi-digit whole numbers using the standard algorithm involving multiple compositions. Solve one-step word problems involving addition.

Solve multi-step word problems involving addition, using rounding to assess the reasonableness of answers.

4.NBT.B.4 4.OA.A.3

Fluently subtract multi-digit whole numbers using the standard algorithm involving up to two decompositions. Solve one-step word problems involving subtraction.

Fluently subtract multi-digit whole numbers using the standard algorithm involving multiple decompositions. Solve one-step word problems involving subtraction.

Solve multi-step word problems involving subtraction, using rounding to assess the reasonableness of answers.

Solve multi-step word problems involving addition and subtraction, using rounding to assess the reasonableness of answers.

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Elementary and Middle School Mathematics: Teaching Developmentally

John van de walle, karen karp, jennifer bay-williams, planning in the problem-based classroom - all with video answers.

Chapter Questions

All the following are recommended teacher moves in a three-phase lesson plan format excepf: A. Sequencing student presentations of their strategies in an intentional manner B. Posing a simpler problem as a way to elicit prior know ledge C. Illustrating how to solve a problem to ensure that students are realy to practice D. Observing students and asking probing questions

Identify the teacher action that would happen in the befor lesson phase. A. Establish clear expectations. B. Listen actively without evaluation. C. Notice students' mathematical thinking. D. Provide worthwhile extensions.

The dmrang phase of a problem-solving lesson is a great opportunity for the teacher to find out what students lenow. Three of the prompes below would help the teacher notice exreft one. Can you identify which one? A. Can I help you ....? B. Why did you ...? C. How did you solve it? D. How does your drawing connect to the equation?

Identify the major goal of the $a f t e r$ phase of a problem-solving lesson. A. Promote a mathematical community of learners. B. Sharing, justifying, and challenging various solutions. C. Listen without evaluation. D. Formalize the main ideas of the lesson and make connections between strategies and mathematical ideas.

Which one of the following statements is trime? A. Design assessments after a lesson is planned in order to assess what was planned for that lesson. B. Start with a worthwhile task, and then see which standards it addresses and plan the lesson around the task. C. Phan the phases of the lesson sequentially, starting with the befwe phase and ending with the after phase. D. Begin with mathematical goals, consider student needs, and use these factors to create the threephase lesson.

All of the following questions with the exception of one would be helpful as a teacher is considering the needs of their students. Identify the one that belongs in an earlier step. A. What mathematical practices and processes will be developed? B. What visual models might support student understanding? C. What vocabulary support might be needed? D. What context would be engaging for a range of learners?

In this chapter the authors have illustrated the decision-making process with a lesson on fixed area. Which one of the statements below would be part of the planning for the during phase of the lesson? A. Find visuals or models to support student understanding B. Options available for presenting the task C. Complete the task ahead to anticipate student challenges D. A checklist for observing student products

If a lesson is well-aligned and worthwhile, it will have the following components exrpt which one? A. Model and practice the right problem-solving method B. Multiple entry and exit points C. Frygages student in mathematical reasoning D. Increases in challenge

What is the purpose of differentiation? A. Provide a variety of tasks to enrich the affer discussion B. Provide chos?ces for students C. Provide more challenging tasks for more capable students D. Provide tasks that are accessible and challenging for all students

Learning center lessons may include all of the following except which? A. The lessons may require more time in the doring phase with more tasks to do at this time. B. The lessons will entail careful observation in the durimg phase to see which stations or which concepts will be the focus of the after phase. C. The lessons may involve additional time in the offer phase to share all solutions from each station. D. The lessons may require more time in the before phase to ensure that the stations are understood.

Which method below would be particularly useful in taking low cognitive demand tasks and turning them into high cognitive demand tasks? A. Tiered lessons B. Number tallos C. Leaming centers D. Open questions

Identify the method below which involves students working on different tasiks all focused on the same learning goal. This method is based on choice and helps stwdents become more self-directed. A. Flexible grouping B. Parallel tasks C. Open questions D. Tiered lessons

Three of the following are listed as impertant items to share with families related to mathematics. Identify the one that would not be considered important. A. Why an inquiry (reasoning, critical thinking, and problem solving) is important in learning mathematics B. The role of technology in learning mathernatics C. The amount of time their chibd should be drilling at home to ensure that they remember thearr slcills D. What a worthwhile task loolss like

What would be a way of addressing parents' questions about the mathematics curriculum and teaching practices? A. Have Math Orientation Worleshoop B. Have a welsite for mathematic homework questions C. Encourage the use of calculators D. Send home NCTM Prinaple to Action document

Below are three methods that will support children in learning mathematies at school and at home. Which method would parents be more familiar, based on their own school mathematics learning? A. Inquiry-based learning B. Cooperative groups C. Problem solving D. Procedural strategies

Homework should be a positive experience for students and their Emilies. Identify the activity below that could be engaging for both the parents and their child(ren). A. Practice page from the texthook B. Activity sheets with basie fact recall C. Review of concepts for weekly assessment D. Data collection of home tasks

4th Grade Problem Solving Strategies

Nys common core grade 4, module 7, lesson 5.

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FREE K-12 standards-aligned STEM

curriculum for educators everywhere!

Find more at TeachEngineering.org .

• TeachEngineering
• Problem Solving

Lesson Problem Solving

Grade Level: 8 (6-8)

(two 40-minute class periods)

Lesson Dependency: The Energy Problem

Subject Areas: Physical Science, Science and Technology

• Print lesson and its associated curriculum

Curriculum in this Unit Units serve as guides to a particular content or subject area. Nested under units are lessons (in purple) and hands-on activities (in blue). Note that not all lessons and activities will exist under a unit, and instead may exist as "standalone" curriculum.

• Energy Forms and States Demonstrations
• Energy Conversions
• Watt Meters to Measure Energy Consumption
• Household Energy Audit
• Light vs. Heat Bulbs
• Efficiency of an Electromechanical System
• Efficiency of a Water Heating System
• Solving Energy Problems
• Energy Projects

TE Newsletter

Engineering connection, learning objectives, worksheets and attachments, more curriculum like this, introduction/motivation, associated activities, user comments & tips.

Scientists, engineers and ordinary people use problem solving each day to work out solutions to various problems. Using a systematic and iterative procedure to solve a problem is efficient and provides a logical flow of knowledge and progress.

• Students demonstrate an understanding of the Technological Method of Problem Solving.
• Students are able to apply the Technological Method of Problem Solving to a real-life problem.

Educational Standards Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards. All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN) , a project of D2L (www.achievementstandards.org). In the ASN, standards are hierarchically structured: first by source; e.g. , by state; within source by type; e.g. , science or mathematics; within type by subtype, then by grade, etc .

Ngss: next generation science standards - science.

View aligned curriculum

Do you agree with this alignment? Thanks for your feedback!

International Technology and Engineering Educators Association - Technology

State standards, national science education standards - science.

Scientists, engineers, and ordinary people use problem solving each day to work out solutions to various problems. Using a systematic and iterative procedure to solve a problem is efficient and provides a logical flow of knowledge and progress.

In this unit, we use what is called "The Technological Method of Problem Solving." This is a seven-step procedure that is highly iterative—you may go back and forth among the listed steps, and may not always follow them in order. Remember that in most engineering projects, more than one good answer exists. The goal is to get to the best solution for a given problem. Following the lesson conduct the associated activities Egg Drop and Solving Energy Problems for students to employ problem solving methods and techniques. 

Lesson Background and Concepts for Teachers

The overall concept that is important in this lesson is: Using a standard method or procedure to solve problems makes the process easier and more effective.

The specific process of problem solving used in this unit was adapted from an eighth-grade technology textbook written for New York State standard technology curriculum. The process is shown in Figure 1, with details included below. The spiral shape shows that this is an iterative, not linear, process. The process can skip ahead (for example, build a model early in the process to test a proof of concept) and go backwards (learn more about the problem or potential solutions if early ideas do not work well).

This process provides a reference that can be reiterated throughout the unit as students learn new material or ideas that are relevant to the completion of their unit projects.

Brainstorming about what we know about a problem or project and what we need to find out to move forward in a project is often a good starting point when faced with a new problem. This type of questioning provides a basis and relevance that is useful in other energy science and technology units. In this unit, the general problem that is addressed is the fact that Americans use a lot of energy, with the consequences that we have a dwindling supply of fossil fuels, and we are emitting a lot of carbon dioxide and other air pollutants. The specific project that students are assigned to address is an aspect of this problem that requires them to identify an action they can take in their own live to reduce their overall energy (or fossil fuel) consumption.

The Seven Steps of Problem Solving

1.  Identify the problem

Clearly state the problem. (Short, sweet and to the point. This is the "big picture" problem, not the specific project you have been assigned.)

2.  Establish what you want to achieve

• Completion of a specific project that will help to solve the overall problem.
• In one sentence answer the following question: How will I know I've completed this project?
• List criteria and constraints: Criteria are things you want the solution to have. Constraints are limitations, sometimes called specifications, or restrictions that should be part of the solution. They could be the type of materials, the size or weight the solution must meet, the specific tools or machines you have available, time you have to complete the task and cost of construction or materials.

3.  Gather information and research

• Research is sometimes needed both to better understand the problem itself as well as possible solutions.
• Don't reinvent the wheel – looking at other solutions can lead to better solutions.
• Use past experiences.

4.  Brainstorm possible solutions

List and/or sketch (as appropriate) as many solutions as you can think of.

5.  Choose the best solution

Evaluate solution by: 1) Comparing possible solution against constraints and criteria 2) Making trade-offs to identify "best."

6.  Implement the solution

• Develop plans that include (as required): drawings with measurements, details of construction, construction procedure.
• Define tasks and resources necessary for implementation.
• Implement actual plan as appropriate for your particular project.

7.  Test and evaluate the solution

• Compare the solution against the criteria and constraints.
• Define how you might modify the solution for different or better results.
• Egg Drop - Use this demonstration or activity to introduce and use the problem solving method. Encourages creative design.
• Solving Energy Problems - Unit project is assigned and students begin with problem solving techniques to begin to address project. Mostly they learn that they do not know enough yet to solve the problem.
• Energy Projects - Students use what they learned about energy systems to create a project related to identifying and carrying out a personal change to reduce energy consumption.

The results of the problem solving activity provide a basis for the entire semester project. Collect and review the worksheets to make sure that students are started on the right track.

Learn the basics of the analysis of forces engineers perform at the truss joints to calculate the strength of a truss bridge known as the “method of joints.” Find the tensions and compressions to solve systems of linear equations where the size depends on the number of elements and nodes in the trus...

Through role playing and problem solving, this lesson sets the stage for a friendly competition between groups to design and build a shielding device to protect humans traveling in space. The instructor asks students—how might we design radiation shielding for space travel?

A process for technical problem solving is introduced and applied to a fun demonstration. Given the success with the demo, the iterative nature of the process can be illustrated.

The culminating energy project is introduced and the technical problem solving process is applied to get students started on the project. By the end of the class, students should have a good perspective on what they have already learned and what they still need to learn to complete the project.

Hacker, M, Barden B., Living with Technology , 2nd edition. Albany NY: Delmar Publishers, 1993.

Other Related Information

This lesson was originally published by the Clarkson University K-12 Project Based Learning Partnership Program and may be accessed at http://internal.clarkson.edu/highschool/k12/project/energysystems.html.

Contributors

Supporting program, acknowledgements.

This lesson was developed under National Science Foundation grants no. DUE 0428127 and DGE 0338216. However, these contents do not necessarily represent the policies of the National Science Foundation, and you should not assume endorsement by the federal government.

Last modified: August 16, 2023

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Problem Solving

Fourth graders complete a worksheet identifying unnecessary information regarding 8 different math problems. In this problem solving instructional activity, 4th graders discuss the meaning of colors on a spotlight in order to develop the skill of how to distinguish unnecessary information. Students then are introduced to the meaning behind each color and according to color sentence strips participate in an activity where they underline, answer, and write specific information.

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Course: Class 6   >   Unit 5

• Subtracting a negative = adding a positive
• Subtracting integers: find the missing value
• Understand subtraction as adding the opposite
• Subtracting negative numbers

Addition and subtraction of integers

• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍