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## 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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## 6 Tips for Teaching Math Problem-Solving Skills

Solving word problems is tougher than computing with numbers, but elementary teachers can guide students to do the deep thinking involved.

A growing concern with students is the ability to problem-solve, especially with complex, multistep problems. Data shows that students struggle more when solving word problems than they do with computation , and so problem-solving should be considered separately from computation. Why?

Consider this. When we’re on the way to a new destination and we plug in our location to a map on our phone, it tells us what lane to be in and takes us around any detours or collisions, sometimes even buzzing our watch to remind us to turn. When I experience this as a driver, I don’t have to do the thinking. I can think about what I’m going to cook for dinner, not paying much attention to my surroundings other than to follow those directions. If I were to be asked to go there again, I wouldn’t be able to remember, and I would again seek help.

If we can switch to giving students strategies that require them to think instead of giving them too much support throughout the journey to the answer, we may be able to give them the ability to learn the skills to read a map and have several ways to get there.

Here are six ways we can start letting students do this thinking so that they can go through rigorous problem-solving again and again, paving their own way to the solution.

## 1. Link problem-solving to reading

When we can remind students that they already have many comprehension skills and strategies they can easily use in math problem-solving, it can ease the anxiety surrounding the math problem. For example, providing them with strategies to practice, such as visualizing, acting out the problem with math tools like counters or base 10 blocks, drawing a quick sketch of the problem, retelling the story in their own words, etc., can really help them to utilize the skills they already have to make the task less daunting.

We can break these skills into specific short lessons so students have a bank of strategies to try on their own. Here's an example of an anchor chart that they can use for visualizing . Breaking up comprehension into specific skills can increase student independence and help teachers to be much more targeted in their problem-solving instruction. This allows students to build confidence and break down the barriers between reading and math to see they already have so many strengths that are transferable to all problems.

## 2. Avoid boxing students into choosing a specific operation

It can be so tempting to tell students to look for certain words that might mean a certain operation. This might even be thoroughly successful in kindergarten and first grade, but just like when our map tells us where to go, that limits students from becoming deep thinkers. It also expires once they get into the upper grades, where those words could be in a problem multiple times, creating more confusion when students are trying to follow a rule that may not exist in every problem.

We can encourage a variety of ways to solve problems instead of choosing the operation first. In first grade, a problem might say, “Joceline has 13 stuffed animals and Jordan has 17. How many more does Jordan have?” Some students might choose to subtract, but a lot of students might just count to find the amount in between. If we tell them that “how many more” means to subtract, we’re taking the thinking out of the problem altogether, allowing them to go on autopilot without truly solving the problem or using their comprehension skills to visualize it.

## 3. Revisit ‘representation’

The word “representation” can be misleading. It seems like something to do after the process of solving. When students think they have to go straight to solving, they may not realize that they need a step in between to be able to support their understanding of what’s actually happening in the problem first.

Using an anchor chart like one of these ( lower grade , upper grade ) can help students to choose a representation that most closely matches what they’re visualizing in their mind. Once they sketch it out, it can give them a clearer picture of different ways they could solve the problem.

Think about this problem: “Varush went on a trip with his family to his grandmother’s house. It was 710 miles away. On the way there, three people took turns driving. His mom drove 214 miles. His dad drove 358 miles. His older sister drove the rest. How many miles did his sister drive?”

If we were to show this student the anchor chart, they would probably choose a number line or a strip diagram to help them understand what’s happening.

If we tell students they must always draw base 10 blocks in a place value chart, that doesn’t necessarily match the concept of this problem. When we ask students to match our way of thinking, we rob them of critical thinking practice and sometimes confuse them in the process.

## 4. Give time to process

Sometimes as educators, we can feel rushed to get to everyone and everything that’s required. When solving a complex problem, students need time to just sit with a problem and wrestle with it, maybe even leaving it and coming back to it after a period of time.

This might mean we need to give them fewer problems but go deeper with those problems we give them. We can also speed up processing time when we allow for collaboration and talk time with peers on problem-solving tasks.

## 5. Ask questions that let Students do the thinking

Questions or prompts during problem-solving should be very open-ended to promote thinking. Telling a student to reread the problem or to think about what tools or resources would help them solve it is a way to get them to try something new but not take over their thinking.

These skills are also transferable across content, and students will be reminded, “Good readers and mathematicians reread.”

## 6. Spiral concepts so students frequently use problem-solving skills

When students don’t have to switch gears in between concepts, they’re not truly using deep problem-solving skills. They already kind of know what operation it might be or that it’s something they have at the forefront of their mind from recent learning. Being intentional within their learning stations and assessments about having a variety of rigorous problem-solving skills will refine their critical thinking abilities while building more and more resilience throughout the school year as they retain content learning in the process.

Problem-solving skills are so abstract, and it can be tough to pinpoint exactly what students need. Sometimes we have to go slow to go fast. Slowing down and helping students have tools when they get stuck and enabling them to be critical thinkers will prepare them for life and allow them multiple ways to get to their own destination.

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

## 5 Teaching Mathematics Through Problem Solving

Janet Stramel

In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)

What is a problem in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.

According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.

There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.

Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.

Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.

Teaching through problem solving focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.

Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):

- The problem has important, useful mathematics embedded in it.
- The problem requires high-level thinking and problem solving.
- The problem contributes to the conceptual development of students.
- The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
- The problem can be approached by students in multiple ways using different solution strategies.
- The problem has various solutions or allows different decisions or positions to be taken and defended.
- The problem encourages student engagement and discourse.
- The problem connects to other important mathematical ideas.
- The problem promotes the skillful use of mathematics.
- The problem provides an opportunity to practice important skills.

Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.

Key features of a good mathematics problem includes:

- It must begin where the students are mathematically.
- The feature of the problem must be the mathematics that students are to learn.
- It must require justifications and explanations for both answers and methods of solving.

Problem solving is not a neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.

But look at the b ack.

It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.

When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!

## Mathematics Tasks and Activities that Promote Teaching through Problem Solving

## Choosing the Right Task

Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:

- Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
- What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
- Can the activity accomplish your learning objective/goals?

## Low Floor High Ceiling Tasks

By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].

The strengths of using Low Floor High Ceiling Tasks:

- Allows students to show what they can do, not what they can’t.
- Provides differentiation to all students.
- Promotes a positive classroom environment.
- Advances a growth mindset in students
- Aligns with the Standards for Mathematical Practice

Examples of some Low Floor High Ceiling Tasks can be found at the following sites:

- YouCubed – under grades choose Low Floor High Ceiling
- NRICH Creating a Low Threshold High Ceiling Classroom
- Inside Mathematics Problems of the Month

## Math in 3-Acts

Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:

Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.

In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.

Act Three is the “reveal.” Students share their thinking as well as their solutions.

“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:

- Dan Meyer’s Three-Act Math Tasks
- Graham Fletcher3-Act Tasks ]
- Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete

## Number Talks

Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:

- The teacher presents a problem for students to solve mentally.
- Provide adequate “ wait time .”
- The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
- For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
- Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.

“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:

- Inside Mathematics Number Talks
- Number Talks Build Numerical Reasoning

## Saying “This is Easy”

“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.

When the teacher says, “this is easy,” students may think,

- “Everyone else understands and I don’t. I can’t do this!”
- Students may just give up and surrender the mathematics to their classmates.
- Students may shut down.

Instead, you and your students could say the following:

- “I think I can do this.”
- “I have an idea I want to try.”
- “I’ve seen this kind of problem before.”

Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.

## Using “Worksheets”

Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?

What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.

Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.

One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”

You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can

- Provide your students a bridge between the concrete and abstract
- Serve as models that support students’ thinking
- Provide another representation
- Support student engagement
- Give students ownership of their own learning.

Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.

any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method

should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning

involves teaching a skill so that a student can later solve a story problem

when we teach students how to problem solve

teaching mathematics content through real contexts, problems, situations, and models

a mathematical activity where everyone in the group can begin and then work on at their own level of engagement

20 seconds to 2 minutes for students to make sense of questions

Mathematics Methods for Early Childhood Copyright © 2021 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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## enVision Mathematics - Middle School Math Curriculum

enVision® Mathematics offers a unique combination of problem-based learning and visual learning to develop conceptual understanding. This instructional model has been consistently successful and effective for middle school students across the nation.

- Motivate students through ownership of their learning with Let’s Investigate!
- Harness storytelling for instruction with engaging 3-Act Math problems
- Leverage student interests with Pick a Project and enVision STEM Project’s varied contexts, modalities, and final products
- Easy accessibility to meaningful digital content on the award-winning Savvas Realize® LMS

## Grades 6-8 Math Program Built for Success

Set students up for success in your class and beyond with a math curriculum that meets today’s challenges.

## Student-Centered Mathematics

enVision ’s 3-Act Math, Let’s Investigate!, and Pick a Project components connect mathematical thinking to familiar real world scenarios so students stay engaged.

## Personalized and Adaptive Learning

Formative and summative assessments plus tools like MathXL® for School practice and enrichment and Savvy™ Adaptive Practice tailor assignments and content to each student’s interests and learning level.

## Monitor and Support Student Understanding

Assess students’ progress, customize content, and reach or exceed state standard proficiency through the Savvas Realize® platform.

## Comprehensive and Flexible Planning Materials

Editable lesson presentation slides allow teachers to present content and engage each student with customized content relevant to the students’ world around them.

## Middle School Math Built for Students, Teacher, and Families

Unique and innovative lessons, motivating student projects.

- Enlightening Interactivities powered by Desmos™

## Supportive Professional Development

Family engagement resources.

- Problem-Based Learning Real-world math problems foster collaboration skills. By evaluating options and presenting their own solutions, students stay engaged throughout the lesson.
- Productive Struggle Let’s Investigate! and 3 Act Math Modeling lessons allow students to experience productive struggle through inviting problem solving.
- Real-World Application Flexible student activities such as enVision STEM and Pick a Project provide opportunities for students to explore math concepts with real world application.

## Enlightening Interactivities powered by Desmos™

- Concept Visualization Embedded interactivities throughout lessons help students visualize concepts
- Instructional Support and Insight Professional development videos, such as Using Manipultives videos, provide valuable instructional support and insight into student learning.
- Family-Friendly Support Every topic and lesson comes with family-friendly support that offers compatibility with Google Translate so that families can access resources in 300+ languages.

## Award-Winning Digital Lesson Support

Savvas Realize® provides access to all the enVision Mathematics Grades 6-8 program’s digital resources and downloadable, editable print materials to meet every educational standard.

## Offline Accessibility

Learning does not stop when students have no internet access. enVision ensures access to resources offline, automatically updating their work when reconnected!

## Further Enhancements to the enVision 6-8 Math Program

- Meet Your Students Where They Are
- Personalized Programs
- Bilingual Support with enVision Mathématicas

## Savvas Math Screener & Diagnostic Assessments

An easy and reliable way to identify student needs, assign the right content, and measure growth, delivered on Savvas Realize®.

## SuccessMaker® Math helps learners at every level

This adaptive intervention program continuously personalizes math instruction for student growth or differentiation.

## Embedded Spanish-Language Materials

Spanish texts, audio, and video come fully integrated within the Grade 6-8 courseware.

## School Stories

In these inspirational stories, you'll learn about what schools and districts from across the country are doing to help students succeed and shape the future of education.

## Frequently Asked Questions

enVision® Mathematics © 2024 for grades K-5 is the only middle grades math program that combines problem-based learning and visual learning to deepen students’ conceptual understanding. enVision is used by classrooms across the country and around the world. The latest enVision is even better with new digital Let’s Investigate! lessons which provide students with opportunities to take ownership of deeper exploration into problem-based learning. Ensure successful implementation with the comprehensive teacher support based on the 5 Practices.

enVision packs a unique one-two punch. Lessons start with Problem-Based Learning (PBL), where students must think critically about a real-world math problem, evaluate options, collaborate, and present solutions. This is followed by Visual Learning to solidify the underlying math concepts. It’s the best way to help kids better understand math ideas.

The program is made up of the following program components:

- Teacher’s Edition - Available in digital or print, the Teacher’s Edition includes wrap-around pages that provide direct instruction and teaching suggestions to engage students. The Interactive Teacher’s Edition online features annotation models and downloadable lesson resources.
- Student Edition - Interactive Student Edition—available in digital or print write-in format.
- enVision Digital - enVision digital courseware on Savvas Realize® includes robust digital tools that give teachers flexibility to use a digital, print, or blended format in their classrooms. Teachers can customize the program to rearrange content, upload their own content, add links to online media, and edit resources and assessments. All program resources, including personalized practice, remediation, and assessments are available in one location for easy lesson planning and presentation Students will use technology to interact with text and activities, and they can write directly in their digital Student Edition to make interaction with text more meaningful. Students will engage in activities that will inspire conceptual understanding, classroom discourse, and build their mathematical thinking skills, while learning to formulate and defend their own opinions.

The learning model in the enVision program—problem-based learning, visual learning, and data-driven differentiated instruction—has been researched and verified as effective. Core instruction used for every lesson has been shown to be effective for developing conceptual understanding.

enVision Mathematics features comprehensive differentiated instruction and intervention support to allow access for all students. The program’s balanced instructional model provides appropriate scaffolding, differentiation, intervention, and support for a broad range of learners, and is designed to facilitate conceptual understanding of mathematics for students at a range of learning levels.

Comprehensive, built-in differentiation resources support all levels of learners, including those with learning disabilities and ELLs, through personalized, adaptive learning. The program meets a variety of student needs and provides Response to Intervention (RtI) during each lesson, at the end of each lesson, at the end of each Topic, and any time as indicated in the Teacher’s Edition. A description of RtI tiered instructional resources for the program is included in the Teacher’s Program Overview for each grade. The following are examples of tiered instructional support found online for each lesson.

Tier 1 ongoing Intervention includes the following resources that can be used during the lesson:

- Prevent Misconceptions. During the Visual Learning Example, a remediation strategy is included to address a common misconception about the lesson concept.
- Error Intervention (If... Then...). During Practice & Problem Solving, error intervention identifies a common error and provides remediation strategy
- Reteaching Set. This set is provided before independent practice to develop understanding prior to practice.
- MathXL for School: Practice & Problem Solving, during the lesson, includes personalized practice for the Practice & Problem Solving portion of the lesson, along with Additional Practice or Enrichment; auto‐scored with on‐screen help, including Help Me Solve This and View an Example tools, tutorial videos, Math Tools, and one‐click animated glossary access.

Tier 2 strategic intervention includes the following resources that can be used at the end of the Lesson:

- Intervention Activity. This supports teachers working with small groups of struggling students.
- Reteach to Build Understanding. This provides guided reteaching as a follow‐up to the intervention activity.

Tier 3 intensive intervention instruction is delivered daily outside of the core math instruction, often in a one‐to‐one situation. The Math Diagnosis and Intervention System can be used for this purpose, for example.

- Variety of Instructional Strategies
- Multisensory instruction is provided in online Solve & Discuss It!/Explore It!/Explain It! activities that include audio, Visual Learning
- Animation Plus, Virtual Nerd videos, interactive MathXL for School: Practice & Problem Solving, Additional Practice, and Enrichment, online digital math tools, and online math games.

The authorship team is made up of respected educational experts and researchers whose experiences working with students and study of instructional best practices have positively influenced education. Contributing to enVision with a mind to the evolving role of the teacher and with insights on how students learn in a digital age, these authors bring new ideas, innovations, and strategies that transform teaching and learning in today’s competitive and interconnected world.

- Dr. Robert Q. Berry, III is an Associate Professor at the University of Virginia in the Curry School of Education with an appointment in Curriculum Instruction and Special Education. A former mathematics teacher, he teaches elementary and special education mathematics methods courses in the teacher education program at the University of Virginia. Additionally, he teaches a graduate level mathematics education course and courses for in-service teachers seeking a mathematics specialist endorsement.
- Zachary Champagne taught elementary school students in Jacksonville, Florida for 13 years. Currently he is working as an Assistant in Research at the Florida Center for Research in Science, Technology, Engineering, and Mathematics (FCRSTEM) at Florida State University.
- Dr. Randall Charles is Professor Emeritus in the Department of Mathematics at San Jose State University, San Jose, California. His research interests have focused on problem solving with several NCTM publications including Teaching and Assessing Problem Solving, How to Evaluate Progress in Problem Solving, and Teaching Mathematics Through Problem Solving. In recent years Dr. Charles has written and talked extensively on Big Ideas and Essential Understandings related to curriculum, teaching, and assessment.
- Francis (Skip) Fennell, PhD, is emeritus as the L. Stanley Bowlsbey professor of education and Graduate and Professional Studies at McDaniel College in Maryland, where he continues to direct the Brookhill Institute of Mathematics supported Elementary Mathematics Specialists and Teacher Leaders Project. A mathematics educator who has experience as a classroom teacher, principal, and supervisor of instruction, he is a past president of the Association of Mathematics Teacher Educators (AMTE), the Research Council for Mathematics Learning (RCML), and the National Council of Teachers of Mathematics (NCTM).
- Eric Milou is a Professor in the Department of Mathematics at Rowan University in Glassboro, NJ. He is an author of Teaching Mathematics to Middle School Students. Recently, his focus has been on approaches to mathematical content and the use of technology in middle grades classrooms.
- Dr. Jane Schielack is Professor Emerita in the Department of Mathematics and a former Associate Dean of Assessment and PreK-12 Education in the College of Science at Texas A&M University. A former elementary teacher, Dr. Schielack has pursued her interests in working with teachers and students to enhance mathematics learning in the elementary and middle grades. She has focused her activities for improving mathematics education in two main areas: teacher education and professional development and curriculum development.
- Jonathan Wray has involvement and leadership in a number of organizations and projects. His interests include the leadership roles of mathematics coaches/specialists, access and equity in mathematics classrooms, the use of engaging and effective instructional models to deepen student understanding, and the strategic use of technology in mathematics to improve teaching and learning.
- How do I sign up for an enVision digital demo? enVision digital courseware on Savvas Realize® includes robust digital tools that give teachers flexibility to use a digital, print, or blended format in their classrooms. Teachers can customize the program to rearrange content, upload their own content, add links to online media, and edit resources and assessments. Program resources, personalized practice, remediation, and assessments are available in one location for easy lesson planning and presentation.

enVision Mathematics is designed to achieve a coherent progression of mathematical content within each course and across the program, building lesson to lesson. Every lesson includes online practice instructional examples as the progression of topics builds, allowing students additional practice with these skills and to develop a deeper conceptual understanding.

At the beginning of every topic, teachers are provided with support for the focus of the topic, how the topic fits into an overall coherence of the grade and across grades, the balance of rigor in the topic, and how the practices enrich the mathematics in the topic. Carefully designed learning progressions achieve coherence across grades:

Coherence is supported by common elements across grades, such as Thinking Habits questions for math practices and diagrams for representing quantities in a problem. Coherence across topics, clusters, and domains within a grade is the result of developing mathematics as a body of interconnected concepts and skills. Across lessons and standards, coherence is achieved when new content is taught as an extension of prior learning—developmentally and mathematically. (For example, Solve & Share at the start of lessons engages students in a problem-based learning experience that connects prior knowledge to new ideas.)

Look Back! and Look Ahead! connections are highlighted in the Coherence part of Topic Overview pages in the Teacher’s Edition.

The Topic Background: Rigor page shows teachers how the areas of rigor will be addressed in the topic, and details how conceptual understanding, procedural skill and fluency, and application builds within each topic to provide the rigor required.

On the first page of every lesson, the Lesson Overview includes sections titled Focus, Coherence, and Rigor. The Rigor section highlights the element or elements of rigor emphasized in the lesson, which may be one, two, or all three. Features in every lesson support each element, but the emphasis will vary depending on the standard being developed in the lesson. The core instructional model features support for conceptual understanding, procedural fluency, and application during both instruction and practice, as described below.

- Problem-Based Learning Step 1 Problem-Based Learning supports coherence by helping students connect what they already know to a problem in which new math ideas are embedded. When students make these connections, conceptual understanding emerges. Students are given time to struggle to make connections to the mathematical ideas and conceptual understandings. They can choose to represent their thinking and learning in a variety of ways. Physical and online manipulatives are available.
- Visual Learning Step 2 Visual Learning further develops understanding of the lesson ideas through classroom conversations. The Visual Learning Example features visual models to help give meaning to math language. Instruction is stepped out to help students visually organize important ideas. Students perform better on procedural skills when the procedures make sense to them. Procedural skills are developed through careful learning progressions in the Visual Learning Example.
- Assess and Differentiate Step 3 Assess and Differentiate features a Lesson Quiz and a comprehensive array of intervention, on-level, and advanced resources for all learners, with the goal that all students have the opportunity for extensive work in the state standards. Leveled practice with scaffolding is included at times. Varied problems are provided and math practices are identified as appropriate. Higher Order Thinking problems offer more challenge. Students have ample opportunity to focus on conceptual understanding and procedural skills and to apply the mathematics they just learned to solve a range of problems.
- How does the program identify performance gaps? At the start of the school year, schools have the opportunity to implement norm-referenced and validated assessments to identify students’ strengths and areas for growth. The new award-winning Math Screener and Diagnostic Assessments and Growth work directly with the enVision Mathematics course on Savvas Realize to inform instruction and provide robust student data. As a result of the Diagnostic assessment, teachers are armed with flexible instructional recommendations personalized to every student.

enVision Mathematics portrays diverse individuals and groups in a variety of settings and backgrounds. The program has been reviewed and approved for unbiased and fair representation. The selections in enVision Mathematics include a wide variety of contemporary, classic, and multicultural authors.

Our educational materials feature a fair and balanced representation of members of various cultural groups, including racial, ethnic, and religious groups; males and females; older people; and people with disabilities. The program integrates social diversity throughout all of its lessons, and includes a balanced representation of cultures and groups in multiple settings, occupations, careers, and lifestyles. We strive to accurately portray diverse groups within our society as well as diversity within groups. Our programs use language that is appropriate to and respectful of our cultural diversity. We involve members of diverse ethnic and cultural groups in the concept development of our products as well as in the writing, editing, illustration, and design.

- What is Pick a Project? Pick a Project is one of the motivating activities in enVision Mathematics , giving students a choice by letting them pick from a selection of math projects. Pick a Project launches each enVision topic and engages students in a real-world math project that accommodates different learning styles and interests. Students work independently, with a partner, or in small groups. The math problem activates prior knowledge and is a great way to deepen understanding during the entire topic.
- How does the relationship between enVision Mathematics and Desmos benefit students? Exclusive integration of Desmos into Savvas Realize® offers a groundbreaking interactive experience designed to foster conceptual understanding through highly visual interactives that bring mathematical concepts to life. Embedded interactives powered by Desmos and animated examples engage students and deepen conceptual understanding. Allowing students to manipulate data and see an immediate effect on graphs, number lines, etc. clarifies concepts as students are learning new content. Unique to enVision , the Desmos best-in-class graphing calculator and brand new geometry tools are available to middle and high school enVision students anytime, anywhere, both online and offline through Savvas Realize.

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## What can QuickMath do?

QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

- The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
- The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
- The inequalities section lets you solve an inequality or a system of inequalities for a single variable. You can also plot inequalities in two variables.
- The calculus section will carry out differentiation as well as definite and indefinite integration.
- The matrices section contains commands for the arithmetic manipulation of matrices.
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## Fluency, Reasoning and Problem Solving: What This Looks Like In Every Math Lesson

Neil almond.

Fluency, reasoning and problem solving are central strands of mathematical competency, as recognized by the National Council of Teachers of Mathematics (NCTM) and the National Research Council’s report ‘Adding It Up’.

They are key components to the Standards of Mathematical Practice, standards that are interwoven into every mathematics lesson. Here we look at how these three approaches or elements of math can be interwoven in a child’s math education through elementary and middle school.

We look at what fluency, reasoning and problem solving are, how to teach them, and how to know how a child is progressing in each – as well as what to do when they’re not, and what to avoid.

The hope is that this blog will help elementary and middle school teachers think carefully about their practice and the pedagogical choices they make around the teaching of what the common core refers to as ‘mathematical practices’, and reasoning and problem solving in particular.

Before we can think about what this would look like in practice however, we need to understand the background to these terms.

## What is fluency in math?

What is reasoning in math, what is problem solving in math, mathematical problem solving is a learned skill, performance vs learning: what to avoid when teaching fluency, reasoning, and problem solving, what is ‘performance vs learning’, teaching to “cover the curriculum” hinders development of strong problem solving skills., fluency and reasoning – best practice in a lesson, a unit, and a semester, best practice for problem solving in a lesson, a unit, and a semester , fluency, reasoning and problem solving should not be taught by rote , the ultimate guide to problem solving techniques.

Develop problem solving skills in the classroom with this free, downloadable worksheet

Fluency in math is a fairly broad concept. The basics of mathematical fluency – as defined by the Common Core State Standards for math – involve knowing key mathematical skills and being able to carry them out flexibly, accurately and efficiently.

But true fluency in math (at least up to middle school) means being able to apply the same skill to multiple contexts, and being able to choose the most appropriate method for a particular task.

Fluency in math lessons means we teach the content using a range of representations, to ensure that all students understand and have sufficient time to practice what is taught.

Read more: How the best schools develop math fluency

Reasoning in math is the process of applying logical thinking to a situation to derive the correct problem solving strategy for a given question, and using this method to develop and describe a solution.

Put more simply, mathematical reasoning is the bridge between fluency and problem solving. It allows students to use the former to accurately carry out the latter.

Read more: Developing math reasoning: the mathematical skills required and how to teach them .

It’s sometimes easier to start off with what problem solving is not. Problem solving is not necessarily just about answering word problems in math. If a child already has a readily available method to solve this sort of problem, problem solving has not occurred. Problem solving in math is finding a way to apply knowledge and skills you have to answer unfamiliar types of problems.

Read more: Math problem solving: strategies and resources for primary school teachers .

## We are all problem solvers

First off, problem solving should not be seen as something that some students can do and some cannot. Every single person is born with an innate level of problem-solving ability.

Early on as a species on this planet, we solved problems like recognizing faces we know, protecting ourselves against other species, and as babies the problem of getting food (by crying relentlessly until we were fed).

All these scenarios are a form of what the evolutionary psychologist David Geary (1995) calls biologically primary knowledge. We have been solving these problems for millennia and they are so ingrained in our DNA that we learn them without any specific instruction.

Why then, if we have this innate ability, does actually teaching problem solving seem so hard?

As you might have guessed, the domain of mathematics is far from innate. Math doesn’t just happen to us; we need to learn it. It needs to be passed down from experts that have the knowledge to novices who do not.

This is what Geary calls biologically secondary knowledge. Solving problems (within the domain of math) is a mixture of both primary and secondary knowledge.

The issue is that problem solving in domains that are classified as biologically secondary knowledge (like math) can only be improved by practicing elements of that domain.

So there is no generic problem-solving skill that can be taught in isolation and transferred to other areas.

This will have important ramifications for pedagogical choices, which I will go into more detail about later on in this blog.

The educationalist Dylan Wiliam had this to say on the matter: ‘for…problem solving, the idea that students can learn these skills in one context and apply them in another is essentially wrong.’ (Wiliam, 2018) So what is the best method of teaching problem solving to elementary and middle school math students?

The answer is that we teach them plenty of domain specific biological secondary knowledge – in this case, math. Our ability to successfully problem solve requires us to have a deep understanding of content and fluency of facts and mathematical procedures.

Here is what cognitive psychologist Daniel Willingham (2010) has to say:

‘Data from the last thirty years leads to a conclusion that is not scientifically challengeable: thinking well requires knowing facts, and that’s true not simply because you need something to think about.

The very processes that teachers care about most—critical thinking processes such as reasoning and problem solving—are intimately intertwined with factual knowledge that is stored in long-term memory (not just found in the environment).’

Colin Foster (2019), a reader in Mathematics Education in the Mathematics Education Center at Loughborough University, UK, says, ‘I think of fluency and mathematical reasoning, not as ends in themselves, but as means to support students in the most important goal of all: solving problems.’

In that paper he produces this pyramid:

This is important for two reasons:

1) It splits up reasoning skills and problem solving into two different entities

2) It demonstrates that fluency is not something to be rushed through to get to the ‘problem solving’ stage but is rather the foundation of problem solving.

In my own work I adapt this model and turn it into a cone shape, as education seems to have a problem with pyramids and gross misinterpretation of them (think Bloom’s taxonomy).

Notice how we need plenty of fluency of facts, concepts, procedures and mathematical language.

Having this fluency will help with improving logical reasoning skills, which will then lend themselves to solving mathematical problems – but only if it is truly learnt and there is systematic retrieval of this information carefully planned across the curriculum.

I mean to make no sweeping generalization here; this was my experience both at university when training and from working in schools.

At some point, schools become obsessed with the ridiculous notion of moving students through content at an accelerated rate. I have heard it used in all manner of educational contexts while training and being a teacher. ‘You will need to show ‘accelerated progress in math’ in this lesson,’ ‘School officials will be looking for ‘accelerated progress’ etc.

I have no doubt that all of this came from a good place and from those wanting the best possible outcomes – but it is misguided.

I remember being told that we needed to get students onto the problem solving questions as soon as possible to demonstrate this mystical ‘accelerated progress’.

This makes sense; you have a group of students and you have taken them from not knowing something to working out pretty sophisticated 2-step or multi-step word problems within an hour. How is that not ‘accelerated progress?’

This was a frequent feature of my lessons up until last academic year: teach a mathematical procedure; get the students to do about 10 of them in their books; mark these and if the majority were correct, model some reasoning/problem solving questions from the same content as the fluency content; give the students some reasoning and word problem questions and that was it.

I wondered if I was the only one who had been taught this while at university so I did a quick poll on Twitter and found that was not the case.

I know these numbers won’t be big enough for a representative sample but it still shows that others are familiar with this approach.

The issue with the lesson framework I mentioned above is that it does not take into account ‘performance vs learning.’

The premise is that performance in a lesson is not a good proxy for learning.

Yes, those students were performing well after I had modeled a mathematical procedure for them, and managed to get questions correct.

But if problem solving depends on a deep knowledge of mathematics, this approach to lesson structure is going to be very ineffective.

As mentioned earlier, the reasoning and problem solving questions were based on the same math content as the fluency exercises, making it more likely that students would solve problems correctly whether they fully understood them or not.

Chances are that all they’d need to do is find the numbers in the questions and use the same method they used in the fluency section to get their answers (a process referred to as “number plucking”) – not exactly high level problem solving skills.

This is one of my worries with ‘math mastery schemes’ that block content so that, in some circumstances, it is not looked at again until the following year (and with new objectives).

The pressure for teachers to ‘get through the curriculum’ results in many opportunities to revisit content being missed in the classroom.

Students are unintentionally forced to skip ahead in the fluency, reasoning, problem solving chain without proper consolidation of the earlier processes.

As David Didau (2019) puts it, ‘When novices face a problem for which they do not have a conveniently stored solution, they have to rely on the costlier means-end analysis.

This is likely to lead to cognitive overload because it involves trying to work through and hold in mind multiple possible solutions.

It’s a bit like trying to juggle five objects at once without previous practice. Solving problems is an inefficient way to get better at problem solving.’

By now I hope you have realized that when it comes to problem solving, fluency is king. As such we should look to mastery math based teaching to ensure that the fluency that students need is there.

The answer to what fluency looks like will obviously depend on many factors, including the content being taught and the grade you find yourself teaching.

But we should not consider rushing them on to problem solving or logical reasoning in the early stages of this new content as it has not been learnt, only performed.

I would say that in the early stages of learning, content that requires the end goal of being fluent should take up the majority of lesson time – approximately 60%. The rest of the time should be spent rehearsing and retrieving other knowledge that is at risk of being forgotten about.

This blog on mental math strategies students should learn at each grade level is a good place to start when thinking about the core aspects of fluency that students should achieve.

Little and often is a good mantra when we think about fluency, particularly when revisiting the key mathematical skills of number bond fluency or multiplication fluency. So when it comes to what fluency could look like throughout the day, consider all the opportunities to get students practicing.

They could chant multiplication facts when transitioning. If a lesson in another subject has finished earlier than expected, use that time to quiz students on number bonds. Have fluency exercises as part of the morning work.

Read more: How to teach multiplication for instant recall

## What about best practice over a longer period?

Thinking about what fluency could look like across a unit of work would again depend on the unit itself.

Look at this unit below from a popular scheme of work.

They recommend 20 days to cover 9 objectives. One of these specifically mentions problem solving so I will forget about that one at the moment – so that gives 8 objectives.

I would recommend that the fluency of this unit look something like this:

This type of structure is heavily borrowed from Mark McCourt’s phased learning idea from his book ‘Teaching for Mastery.’

This should not be seen as something set in stone; it would greatly depend on the needs of the class in front of you. But it gives an idea of what fluency could look like across a unit of lessons – though not necessarily all math lessons.

When we think about a semester, we can draw on similar ideas to the one above except that your lessons could also pull on content from previous units from that semester.

So lesson one may focus 60% on the new unit and 40% on what was learnt in the previous unit.

The structure could then follow a similar pattern to the one above.

When an adult first learns something new, we cannot solve a problem with it straight away. We need to become familiar with the idea and practice before we can make connections, reason and problem solve with it.

The same is true for students. Indeed, it could take up to two years ‘between the mathematics a student can use in imitative exercises and that they have sufficiently absorbed and connected to use autonomously in non-routine problem solving.’ (Burkhardt, 2017).

## Practice with facts that are secure

So when we plan for reasoning and problem solving, we need to be looking at content from 2 years ago to base these questions on.

You could get students in 3rd grade to solve complicated place value problems with the numbers they should know from 1st or 2nd grade. This would lessen the cognitive load , freeing up valuable working memory so they can actually focus on solving the problems using content they are familiar with.

## Increase complexity gradually

Once they practice solving these types of problems, they can draw on this knowledge later when solving problems with more difficult numbers.

This is what Mark McCourt calls the ‘Behave’ phase. In his book he writes:

‘Many teachers find it an uncomfortable – perhaps even illogical – process to plan the ‘Behave’ phase as one that relates to much earlier learning rather than the new idea, but it is crucial to do so if we want to bring about optimal gains in learning, understanding and long term recall.’ (Mark McCourt, 2019)

This just shows the fallacy of ‘accelerated progress’; in the space of 20 minutes some teachers are taught to move students from fluency through to non-routine problem solving, or we are somehow not catering to the needs of the child.

When considering what problem solving lessons could look like, here’s an example structure based on the objectives above.

It is important to reiterate that this is not something that should be set in stone. Key to getting the most out of this teaching for mastery approach is ensuring your students (across abilities) are interested and engaged in their work.

Depending on the previous attainment and abilities of the children in your class, you may find that a few have come across some of the mathematical ideas you have been teaching, and so they are able to problem solve effectively with these ideas.

Equally likely is encountering students on the opposite side of the spectrum, who may not have fully grasped the concept of place value and will need to go further back than 2 years and solve even simpler problems.

In order to have the greatest impact on class performance, you will have to account for these varying experiences in your lessons.

## Read more:

- Math Mastery Toolkit : A Practical Guide To Mastery Teaching And Learning
- Problem Solving and Reasoning Questions and Answers
- Get to Grips with Math Problem Solving For Elementary Students
- Mixed Ability Teaching for Mastery: Classroom How To
- 21 Math Challenges To Really Stretch Your More Able Students
- Why You Should Be Incorporating Stem Sentences Into Your Elementary Math Teaching

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade – 3rd grade – 4th grade – 5th grade – 6th grade Why not learn more about how it works ?

The content in this article was originally written by primary school lead teacher Neil Almond and has since been revised and adapted for US schools by elementary math teacher Jaclyn Wassell.

## [FREE] Word Problems Grade 5 Place Value and Rounding

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## Problem Solving Maths Lessons

September 1, 2022, problem solving is at the heart of engaging and inspiring mathematics lessons. .

In the past couple of years, my faculty have gone through deep-dive OFSTED inspectionsâ€”a trial inspection with three HMIs and an actual inspection . The trial inspection showed all students have a love of maths. Later, in the words of my headteacher, the real inspection simply blew the lead inspector away. In each review, we showed how we teach mathematics through problem-solving. mathematics through problem-solving.

Shortly after blogging about the inspections, I received hundreds of messages from teachers and Heads of Maths asking for advice about embedding such an approach. My answer is simple, make problem-solving questions a feature of every lesson for students of all abilities.

To help with this, I would like to share four problem-solving lessonsâ€”these range from applying the rules of arithmetic to solving real-life problems with compound interest. In addition, every resource comes with a worksheet and tutorial video that models the solutions.

## Sample Problem Solving Mathematics Lessons

## Perimeter and Area

Students are challenged to solve problems involving perimeter and area of rectangles and triangles.

Go teach it

Students are challenged to apply their understanding of the mean, mode, median and range to solve problems involving datasets.

## Written Methods

Students are challenged to apply the rules of arithmetic to a series of real-life, functional problems.

## Compound Percentages

Students are challenged to apply compound percentage changes to calculate an original amount after a percentage change, a percentage rate and compare real-life investments.

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## Math Problems For kids

Math lessons and practice are mostly never on top of a kid’s to-do list for the day. They would rather do anything other than solve math problems for kids or sit through a math class. But math is an important part of a child’s education, it’s crucial children learn math and do well in the subject. In this regard, math problems for kids must be given extra attention in order to enhance their mathematical skills. To start with, explore easy math problems for kids for better understanding. Besides this, explore math games for kids for long-term enjoyment and engagement.

Check out more interesting math problems for kids online so that they get a variety of equations to solve. At the age of 5-6 kids, you must focus on addition math problems for kids and later with other complex math problems. To enhance their performance, you can conduct math activities where math problems for kids can be solved easily. Must say! There are quite simple math problems for kids available here. Read on to find out.

## How To Make Simple Math Problems For Kids Easy?

Math problems for kids are applications of concepts of mathematics. Learning math and solving math problems for kids can never be ignored. So, however reluctant your child may be, you still need to get them to learn and practice math. The best way to get an uninterested child to learn math is to make it more interesting.

- Kids learn better when they’re engaged with hands-on and fun learning activities and worksheets. Entice your child to sit through a math lesson or a session of problem-solving by including fun activities in the lesson.
- Parents and educators often assume that a child who doesn’t do well in math has no aptitude for the subject. However, this is far from the truth. The problem doesn’t lie in the child’s aptitude, it lies in the way of teaching, lack of understanding and lack of practice.
- Hands-on activities and educational play will help children understand lessons better. But only when they practice what they’ve learned, they get better at the concept. According to several studies, kids who practice solving math problems tend to score well in the subject.
- Incorporate real life examples to teach math problems for kids. Look around the scenarios around the house or in the classroom to prepare simple math problems for kids.
- Include interesting materials or props in the classroom or at home for teaching math problems for kids. For example, building blocks, popsicle sticks, balls or any other materials that help you in teaching math problems.

## Tips To Teach Math Problems For Kids

All through their school years, children learn several of these math concepts. The lessons are reinforced by solving problems. But sometimes, these problems can get too complicated and stump the kids. Here are some tips to help kids tackle math problems easily.

- Plan Strategies: Each problem needs to be looked at from a different viewpoint and choose the appropriate strategies for that problem. For that, you need to first understand the problem, work out strategies to solve the problem and effective strategies to check the answers. Once kids understand these strategies, they’ll be able to solve most mathematical problems on their own.
- Understanding the problem: When faced with a math problem, several students have trouble figuring out what it’s asking them to do. They have trouble assessing what they need to do in a problem. Often, this becomes the greatest hurdle for most kids. In this regard, you can assist them by giving clues so that kids understand the problems more easily.
- Read the question multiple times: Often, the answer to a math problem lies in the question itself. Kids often miss out on vital information when they skim through instead of completely reading the problem. So, encourage the kids to read the question several times until they figure out what they need to do.
- Identifying vital information: When kids look at a mathematical problem, they sometimes don’t focus on the vital information in the question. Without the relevant information, they struggle to solve the problem. Teach students to identify relevant and vital information in the question and highlight it. Then ask them to use this information to solve the problem. A great way to do this is to swap out unnecessary information about the problem, like names or situations. Keep the numerals intact and solve the problem. Help them understand that altering the scenarios or names or objects does not change the end result. This helps them understand what should be the point of focus while solving problems.
- Solving the problem: Often, the simplest of things help us solve the most complicated of problems. It’s all about choosing the right strategy to solve the problem on hand. When solving math problems for kids, employing very simple and basic strategies will help kids come up with the solution.
- Visualizing: Sometimes, when you’re faced with an abstract problem, visualizing it helps to solve the problem. A simple thing like drawing pictures or tally marks could help children figure out how to solve a problem.
- Finding patterns: Most math problems for kids have a pattern. Once kids learn to find and exploit the pattern, they’ll be able to find the necessary information to solve the problem. To find the pattern, kids must list all the relevant information in a problem. Then compare this information to locate the missing fact and solve the problem.
- Work backwards to solve the problem: Sometimes, you’re faced with a problem or mathematical sentence that wants you to find a missing number. In such cases, work backwards from the answer to find the missing number and solve the sum. For example: 12 – x = 8. In this problem, students need to find the value of x.
- Start with the answer to the problem. So, here we start with 8.
- Shift the x to the right and 8 to the left in the equation. So, it can be rewritten as: 12 – 8 = x.
- Subtract 8 from 12. 12 – 8 = 4.
- Therefore, x = 4.
- Check the answers:

Checking the solution is one of the most important parts of solving math problems for kids. Often, kids rush through the process of solving a problem to get an answer but forget to check if the solution is correct. And most often, this misstep causes errors. Checking the steps followed and the solution is an important step of problem-solving. It helps pinpoint areas of difficulty and to fix any errors they’ve made in the process.

- Checking with peers: Comparing answers with peers is a great way to check if the solution you arrived at is right. Sometimes, it also helps students learn different methods of solving the same problem. A peer can help point out errors in working out the problem and troubleshoot ways to fix the errors.
- Backtracking to check and fix mistakes: Teach students to check the solution to the problem step-by-step to find errors in the process. Then, they can fix the errors to find the correct solution to the problem. These simple strategies can make solving even the most complicated math problems simple. Kids need to get comfortable with solving math problems. Additionally, solving more sums will help them gain more confidence. This, in turn, will improve their problem-solving skills.

## Examples Of Math problems For Kids

Check out a few examples of math problems for kids addition and subtraction given below:

- The target score set for the bowling game is 100, Sam scores 65 points and Dan scores 50 points in the first round. How many points do Sam and Dan need to reach the target score?

Answer: Sam needs to score 35 points and Dan needs 50 points to reach the target score.

- There were two brothers living across the street in New York City. The elder brother was carrying 5 apples and 2 pears in his hand from the grocery store, whereas the younger brother carried 3 apples and 3 pears. How many apples and pears did both brothers carry?

Answer: The total number of apples and pears they carried are 8 and 5 respectively.

- Fill in the missing numbers for the questions mentioned below:

100- 20= ———— – 40 = —————– – 15 = —————-

Answer: 100- 20= 80- 40= 40-15= 25

- Solve the following addition problem given below:

10= 3 + ———–

Answer: 10= 3 + 7

- Solve the following subtraction problem given below:

9 – 3 = ————

Answer: 9 – 3 = 6

- There are three girls playing on the swing and two girls playing on the slide. What is the total number of girls playing in the park

Answer: There are five girls playing in the park.

- There are five monkeys sitting on the tree. If one monkey goes down to fetch some food. How many monkeys are sitting on the tree?

Answer: There are four monkeys sitting on the tree.

- There are around 15 chairs kept for the musical chair competition. When one of the participants is out of the game. How many chairs are kept for the game to continue? 4

Answer: 14 chairs.

- There are 200 vehicles parked near the mall. 80 out of them are cars and the rest are bikes. How many bikes are there in the parking lot?

Answer: There are 120 bikes parked in the parking lot of a mall.

- In an orchard, there are 500 trees. 300 are apple trees and the rest are orange trees. What is the total number of orange trees in the orchard?

Answer: 200 orange trees.

- Complete the counting series given below:

14, __, __, __, __, 24, __, 28, 30

Answer: 14, 16, 18, 20, 22, 24, 26, 28, 30.

- There are 20 balloons blown for the birthday party. Arthur bursted 12 balloons while playing. How many balloons are left for the party?

Answer: 8 Balloons

- Sam is keeping 25 pencils in a box. How many pencils do Sam require for 10 boxes?

Answer: 250 pencils required for 10 boxes.

- Complete the even number series given below:

2, __, __, 8, 10, __, __…..

Answer: 2, 4, 6, 8, 10, 12, 14….

- Rumi has 19 popsicle sticks with him. If he gives 15 popsicle sticks to his friend. How many popsicle sticks are left with him?

Answer: 19-15= 4

- Your friend has arranged a birthday party for you. He has got a yummy cake with 5 red candles, 10 pink candles and 8 blue candles? What is the total number of candles your friend gets?

Answer: 5+10+ 8= 23 candles.

- There are 6 slices of pizza. Your friend has eaten 4 slices of pizza. How many total slices of pizza are left?

Answer: 2 slices of pizza

- 10 Boys are there in the 4th of July parade. Later, 15 more boys join the parade. How many boys have joined the parade?

Answer: 25 Boys

- Jack has got eight puppies for his pet shop. Two puppies were taken by the customer. How many puppies are left with Jack?

Answer: 6 Puppies.

- Solve the following problem given below:

40 + ————-= 50 x ———= 500 – 100

Answer: 40 + 10 = 50 x 10 = 500 – 100

- William has 20 goldfish in a water tank. He observes that 10 goldfish have died after 10 days. Currently, how many total goldfish are there in the tank?

Answer: 10 Goldfish

- Tim has gone to buy ingredients for making a sandwich. He purchases bell pepper for 2$, tomato sauce for 3$ and cheese for 4 $. What is the total cost of all the ingredients?

Answer: 9 $

- If you take 30 mins to walk one mile, How long will it take to walk five miles?

Answer: 1 hour 50 minutes

- If a square has four sides, how many sides does a triangle have?

Answer: Three

- Fill the missing numbers given below:

25 + —–= 50

Answer: 25

- There are four vendors selling dry fruits across the street. One of the vendors packed his stuff and left the place. How many vendors did you see still selling the dry fruits?

Answer: 3 vendors

- How many sides does an octagon have?

Answer: 8 Sides

- There are 30 cotton balls stuffed in a soft toy. If you remove 20 cotton balls, how many will be remaining inside the toy?

Answer: 10 cotton balls

- Multiply the following number given below:

30 x 12 = 360

Answer: 360

- There are 250 bags of rice, 200 bags of corn and 300 bags of millets kept in the store. What is the total number of bags in the store?

Answer: 750 Bags

## Benefits of Math Problems For Kids

Some of the benefits of free math problems for kids are mentioned below:

- Solving math problems will help your child become more confident in the subject and help them develop several skills. They are great tools through which kids learn to apply their math skills and solve a range of mathematical problems. Solving these math problems will also help them develop their critical thinking and problem-solving skills.
- They start developing interest in acquiring math skills with utmost confidence and dedication.
- Solving problems helps kids to think out of the box scenarios to come up with logical solutions to problems.
- Math problems for kids improve their academic performance.
- Kids become highly skilled in solving problems associated with any mathematical concepts accurately.
- Math problems for kids enables them to manage time, speed and accuracy while solving any equations.

Check Osmo for more activities, games to aid in your kids learning – math riddles for kids , coding games for kids and writing games for kids .

## Frequently Asked Questions on Math Problems for kids

What are the math problems for kids.

The Math Problems for kids are fill in the missing numbers, 10, 20, _, 40, _, _, 70, _, 90, _. What is the product of 5 x 6 x 7?, etc.

## How to teach Math Problems for kids?

You can teach Math Problems for kids in the interesting and fun ways such as, helping kids identify the patterns, study the questions repeatedly and understand the problems, then help them to visualize on how to solve the problems.

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- » 5 Best Free Math Problem Solvers

## 5 Best Free Math Problem Solvers

By Casey Allen, 06 Jun 2023

Math problems allow students to learn new concepts and strengthen problem-solving skills. But many learners feel confused or frustrated if they can’t find the correct solution. A math problem solver is a handy tool that helps students doublecheck their work and identify errors.

However, not all math problem solvers are created equal. Here are the top five math solvers for K-12 and college students.

IntMath’s free online math solver offers comprehensive help for math and science problems. This innovative technology blends cutting-edge artificial intelligence language models with a mathematical computation engine. As a result, the math problem solver provides the quickest and most accurate answers.

This math solver answers questions from every branch of math, from introductory algebra to calculus. It can also interpret and solve complex word problems. Plus, students can use the solver for other challenging subjects like chemistry and physics.

IntMath’s solver also offers the most flexibility. The tool analyzes natural human language to answer questions a calculator wouldn’t understand. It’s also more accurate than standard AI interfaces, so users always get the correct answers. And the math equation solver can answer questions more quickly than a tutor at any time of the day.

## 2. Microsoft Math Solver

The Microsoft Math Solver provides step-by-step answers for pre-algebra, algebra, trigonometry, and calculus problems. It also links video tutorials, worksheets, and similar math problems posted online.

Users can manually type problems into the Microsoft Math Solver or take pictures of them with their smartphone. This tool also has a sketch calculator, which lets students handwrite problems using their fingers or a stylus. You can download the Microsoft Math Solver as a mobile application or use the built-in web application in Microsoft Edge.

Mathway is a math equation solver developed by the online homework service Chegg. This mobile and web application answers many types of math problems, including calculus, finite math, graphing, and physics. Students can take a picture of the problem or manually input it with the built-in keyboard.

The free version of the Mathway app only provides solutions. If you want to see the step-by-step process to answer problems, you’ll need to purchase an annual subscription for $9.99 monthly or $39.99 annually.

## 4. Photomath

The Photomath mobile application allows students to take pictures of math problems with their smartphone camera and offers multiple solution methods. This tool is designed for K-12 students and covers subjects like geometry, statistics, and word problems. Math teachers vet all problem-solving methods.

Photomath has a free version that provides answers but doesn’t explain how to solve the problems. Photomath Plus is a subscription plan that costs $9.99 monthly or $69.99 annually and gives animated tutorials for each solution.

## 5. Socratic By Google

An honorable mention goes to Socratic by Google . This tool uses artificial intelligence to locate relevant online explanations and resources for math problems. However, Socratic doesn’t solve problems itself. As a result, Socratic works best for students who want to learn how to solve problems but get the answers on their own.

## Get More Help With IntMath

Math word problem solvers are valuable homework aids, but live math tutoring can help students master challenging math subjects. IntMath’s expert tutors provide personalized online tutoring to help all learners excel and gain lifelong math skills. Contact us today to get started.

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OpenAI Made an AI Breakthrough Before Altman Firing, Stoking Excitement and Concern Save 25% and Read now

## OpenAI Made an AI Breakthrough Before Altman Firing, Stoking Excitement and Concern

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One day before he was fired by OpenAI’s board last week, Sam Altman alluded to a recent technical advance the company had made that allowed it to “push the veil of ignorance back and the frontier of discovery forward.” The cryptic remarks at the APEC CEO Summit went largely unnoticed as the company descended into turmoil.

But some OpenAI employees believe Altman’s comments referred to an innovation by the company’s researchers earlier this year that would allow them to develop far more powerful artificial intelligence models, a person familiar with the matter said. The technical breakthrough, spearheaded by OpenAI chief scientist Ilya Sutskever, raised concerns among some staff that the company didn’t have proper safeguards in place to commercialize such advanced AI models, this person said.

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## How to Solve Math Problems

Last Updated: May 16, 2023 Fact Checked

This article was co-authored by Daron Cam . Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary's College. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 574,843 times.

Although math problems may be solved in different ways, there is a general method of visualizing, approaching and solving math problems that may help you to solve even the most difficult problem. Using these strategies can also help you to improve your math skills overall. Keep reading to learn about some of these math problem solving strategies.

## Understanding the Problem

- Draw a Venn diagram. A Venn diagram shows the relationships among the numbers in your problem. Venn diagrams can be especially helpful with word problems.
- Draw a graph or chart.
- Arrange the components of the problem on a line.
- Draw simple shapes to represent more complex features of the problem.

## Developing a Plan

## Solving the Problem

## Expert Q&A

- Seek help from your teacher or a math tutor if you get stuck or if you have tried multiple strategies without success. Your teacher or a math tutor may be able to easily identify what is wrong and help you to understand how to correct it. Thanks Helpful 1 Not Helpful 1
- Keep practicing sums and diagrams. Go through the concept your class notes regularly. Write down your understanding of the methods and utilize it. Thanks Helpful 1 Not Helpful 0

## You Might Also Like

- ↑ Daron Cam. Math Tutor. Expert Interview. 29 May 2020.
- ↑ http://www.interventioncentral.org/academic-interventions/math/math-problem-solving-combining-cognitive-metacognitive-strategies
- ↑ http://tutorial.math.lamar.edu/Extras/StudyMath/ProblemSolving.aspx
- ↑ https://math.berkeley.edu/~gmelvin/polya.pdf

## About This Article

To solve a math problem, try rewriting the problem in your own words so it's easier to solve. You can also make a drawing of the problem to help you figure out what it's asking you to do. If you're still completely stuck, try solving a different problem that's similar but easier and then use the same steps to solve the harder problem. Even if you can't figure out how to solve it, try to make an educated guess instead of leaving the question blank. To learn how to come up with a solid plan to use to help you solve a math problem, scroll down! Did this summary help you? Yes No

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Step 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

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1. Link problem-solving to reading. When we can remind students that they already have many comprehension skills and strategies they can easily use in math problem-solving, it can ease the anxiety surrounding the math problem. For example, providing them with strategies to practice, such as visualizing, acting out the problem with math tools ...

(10 minutes) Bring students together in a circle, either seated or standing. Bring blocks with you to the circle. Show the student the blocks and ask them to watch you build a tall castle. After you build it, bring out two figurines that you would like to play with in the castle. Say out loud, "Hmm....there seems to be a problem.

in mathematics? A problem is "any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific 'correct' solution method" (Hiebert, et. al., 1997).

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One way to use problem-solving activities in your math lessons is to help introduce a new concept. For example, when we were learning about even and odd numbers, we started our math lesson by playing the Odds vs. Evens game from Beast Academy Playground. This simple math problem solving activity is a variation on the game Rock Paper Scissors.

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Subtract 8 from 12. 12 - 8 = 4. Therefore, x = 4. Checking the solution is one of the most important parts of solving math problems for kids. Often, kids rush through the process of solving a problem to get an answer but forget to check if the solution is correct. And most often, this misstep causes errors.

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Featuring original free math problem solving worksheets for teachers and parents to copy for their kids. Use these free math worksheets for teaching, reinforcement, and review. These math word problems are most appropriate for grades four and five, but many are designed to be challenging and informative to older and more advanced students as well.

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Draw a graph or chart. Arrange the components of the problem on a line. Draw simple shapes to represent more complex features of the problem. 5. Look for patterns. Sometimes you can identify a pattern or patterns in a math problem simply by reading the problem carefully.