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## Center for Teaching

Teaching problem solving.

Print Version

## Tips and Techniques

Expert vs. novice problem solvers, communicate.

- Have students identify specific problems, difficulties, or confusions . Don’t waste time working through problems that students already understand.
- If students are unable to articulate their concerns, determine where they are having trouble by asking them to identify the specific concepts or principles associated with the problem.
- In a one-on-one tutoring session, ask the student to work his/her problem out loud . This slows down the thinking process, making it more accurate and allowing you to access understanding.
- When working with larger groups you can ask students to provide a written “two-column solution.” Have students write up their solution to a problem by putting all their calculations in one column and all of their reasoning (in complete sentences) in the other column. This helps them to think critically about their own problem solving and helps you to more easily identify where they may be having problems. Two-Column Solution (Math) Two-Column Solution (Physics)

## Encourage Independence

- Model the problem solving process rather than just giving students the answer. As you work through the problem, consider how a novice might struggle with the concepts and make your thinking clear
- Have students work through problems on their own. Ask directing questions or give helpful suggestions, but provide only minimal assistance and only when needed to overcome obstacles.
- Don’t fear group work ! Students can frequently help each other, and talking about a problem helps them think more critically about the steps needed to solve the problem. Additionally, group work helps students realize that problems often have multiple solution strategies, some that might be more effective than others

## Be sensitive

- Frequently, when working problems, students are unsure of themselves. This lack of confidence may hamper their learning. It is important to recognize this when students come to us for help, and to give each student some feeling of mastery. Do this by providing positive reinforcement to let students know when they have mastered a new concept or skill.

## Encourage Thoroughness and Patience

- Try to communicate that the process is more important than the answer so that the student learns that it is OK to not have an instant solution. This is learned through your acceptance of his/her pace of doing things, through your refusal to let anxiety pressure you into giving the right answer, and through your example of problem solving through a step-by step process.

Experts (teachers) in a particular field are often so fluent in solving problems from that field that they can find it difficult to articulate the problem solving principles and strategies they use to novices (students) in their field because these principles and strategies are second nature to the expert. To teach students problem solving skills, a teacher should be aware of principles and strategies of good problem solving in his or her discipline .

The mathematician George Polya captured the problem solving principles and strategies he used in his discipline in the book How to Solve It: A New Aspect of Mathematical Method (Princeton University Press, 1957). The book includes a summary of Polya’s problem solving heuristic as well as advice on the teaching of problem solving.

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- Lesson plans

Problem solving

## Problem solving lesson plan

Time to complete

Download the full lesson plan pack including all related resources

Choose to download one or more individual resources

Problem Solving: Lesson plan

Problem solving: Presentation slides

Demonstrating your skills quick fire activity

Problem solving in practice: Interactive worksheet

Our problem solving content focuses on one of these skills and develops understanding of the six stages of problem solving, as well as identifying different types of situations in which young people might already be using these skills. Furthermore, it encourages them to use an adaptive approach, explaining that different types of problems can be approached in different ways.

The activities on this page support your teaching of these skills through an independent activity, quick activities or a full length, curriculum-linked lesson plan.

## Teaching resources:

- Problem solving: Lesson plan and presentation slides – full lesson plan including icebreaker for use with a group of students in the classroom
- Demonstrating your skills: Quick-fire activity – 10 minute activity for a group of students in the classroom, can be used as an icebreaker for the lesson plan
- Problem solving in practice: Interactive worksheet – activity for independent learning whether remote or in class

## Lesson plan

(60 -75 minutes)

This lesson is designed to equip young people with an adaptable approach to solving problems, large or small. It includes a short film and scenarios that encourage development of practical problem solving skills which can be useful for learning, day to day life, and when in employment.

By the end of the lesson, students will be able to:

- Identify problems of different scales and what is needed to solve them
- Illustrate the use of an adaptable approach to solving problems
- Understand that problem solving is a core transferable skill and identify its usefulness in a work setting
- Work on a problem solving activity in a team

The lesson aims to reinforce students’ understanding of the potential future applications of this skill as they move into the world of work, particularly in an activity differentiated for an older or more able group on creating new opportunities.

## Quick-fire activity

(5 - 10 minutes)

The demonstrating your skills quick-fire activity focuses on helping young people understand the key skills that are needed in the workplace, including the importance of problem solving.

Students will be asked to name the skills being demonstrated in a variety of scenarios, and identify ways they’re already using those skills in this short activity.

You might find it useful as a starter or icebreaker activity to begin a lesson, or at the end to allow students to put what they have just learnt in the Problem solving lesson into practice.

## Interactive worksheet

(20 - 25 minutes)

Please note that students below the age of 14 cannot sign up for their own LifeSkills account. Any independent tasks must be printed or downloaded and provided digitally for them to complete as they are currently hosted on educator pages.

The Problem solving in practice interactive worksheet introduces some of the themes from the full lesson plan and gives students some practical strategies for problem solving, including introducing the six stages of problem solving. The worksheet can be printed or completed digitally, so can be used flexibly to give students practise putting their problem solving skills into action. You might choose to assign it:

- As homework following the Problem solving lesson
- For independent study
- For remote learning

## Looking for more ways to boost self confidence with LifeSkills?

Other lessons that may prove useful for students to build on these activities include the Adaptability and Innovation and idea generation lessons. Alternatively, consider encouraging them to apply their skills through Steps to starting a business or the Social action toolkit .

Why not build problem solving in as a focus in your students’ wider curriculum? Refer to our Content guide to find out how this resources can be used as part of your teaching.

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## Why not try one of these next?

## Staying positive (resilience)

Staying positive and learning through experience are key to succeeding in challenging situations. Try this lesson and help your students succeed at work.

Good leadership styles and effective teamwork can help students excel in their future workplace. Read more about team leader skills in this lesson.

## Social action toolkit

Build a comprehensive social action programme and support young people to access enriching experiences that build transferable skills for work.

## Lesson Plan

Problem-solving steps, view aligned standards, learning objectives.

Students will be able to identify a problem and engage in problem-solving steps to come up with a solution to strengthen their responsible decision-making competency.

## Introduction

- 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. This castle is too small for my toys. How can I make this bigger?"
- Take a few big breaths into your belly and ask the class to breathe with you.
- Explain that deep breathing can help you stay calm and focused when solving a problem.
- Ask the class, "How can I make this castle bigger?"
- Ask, "Would someone like to come up and think of a new way to solve this problem?"
- Invite one student to come up to solve the problem (i.e., make a larger castle).
- Ask your student volunteer, "What would you do here? How would you test a solution?"
- Give the student time to come up with a solution to make a larger castle.
- Identify the problem.
- Look at what solutions have already been tried.
- Think of new ways to solve the problem.
- Try it out!
- Thank your student volunteer for showing you how to solve the problem and send them back to their seats.
- Say, "All of us sometimes come across problems we have to solve, and sometimes friends can help us, too!"

- AP Calculus
- AP Statistics
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Online Math Class

Mr. Math Blog

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Problem Solving - Organize Data - Lesson 2.1

Use Picture Graphs - Lesson 2.2

Make Picture Graphs - Lesson 2.3

Use Bar Graphs - Lesson 2.4

Make a Bar Graph - Lesson 2.5

Solve Problems Using Data - Lesson 2.6

Use and Make Line Plots - Lesson 2.7

Number Patterns - Lesson 1.1

Round to Nearest Ten or Hundred - Lesson 1.2

Estimate Sums - Lesson 1.3

Mental Math Strategies for Addition - Lesson 1.4

Use Properties to Add - Lesson 1.5

Use the Break Apart Strategy to Add - Lesson 1.6

Use Place Value to Add - Lesson 1.7

Estimate Differences - Lesson 1.8

Mental Math Strategies for Subtraction - Lesson 1.9

Use Place Value to Subtract - Lesson 1.10

Combine Place Values to Subtract - Lesson 1.11

Describe Plane Shapes - Lesson 12.1

Describe Angles in Plane Shapes - Lesson 12.2

Identify Polygons - Lesson 12.3

Describe Sides of Polygons - Lesson 12.4

Classify Quadrilaterals - Lesson 12.5

Draw Quadrilaterals - Lesson 12.6

Describe Triangles - Lesson 12.7

Chapter 12 Performance Task Review For Test

Problem Solving - Compare Fractions - Lesson 9.1

Compare Fractions with the Same Denominator - Lesson 9.2

Compare Fractions with the Same Numerator - Lesson 9.3

Compare Fractions - Lesson 9.4

Compare and Order Fractions - Lesson 9.5

Model Equivalent Fractions - Lesson 9.6

Equivalent Fractions - Lesson 9.7

Divide by 2 - Lesson 7.1

Divide by 10 - Lesson 7.2

Divide by 5 - Lesson 7.3

Divide by 3 - Lesson 7.4

Divide by 4 - Lesson 7.5

Divide by 5 - Lesson 7.6

Mid-Chapter 7 Checkpoint on Division Facts and Strategies

Divide by 7 - Lesson 7.7

Divide by 8 - Lesson 7.8

Divide by 9 - Lesson 7.9

Problem Solving - Two-Step Problems - Lesson 7.10

Order of Operations - Lesson 7.11

## Problem Solving - Model Division - Lesson 6.1

Size of Equal Groups - Lesson 6.2

Number of Equal Groups - Lesson 6.3

Model (Division) with Bar Model - Lesson 6.4

Relate Subtraction and Division - Lesson 6.5

Mid-Chapter 6 Checkpoint

Model (division) with Arrays - Lesson 6.6

Relate Multiplication and Division - Lesson 6.7

Write Related Facts - Lesson 6.8

Division Rules for 1 and 0 - Lesson 6.9

Chapter 6 Review for Test - Understanding Division

Multiply with 2 and 4 - Lesson 4.1

Multiply with 5 and 10 - Lesson 4.2

Multiply with 3 and 6 - Lesson 4.3

Distributive Property - Lesson 4.4

Multiply with 7 - Lesson 4.5

Associative Property of Multiplication - Lesson 4.6

Patterns on the Multiplication Table - Lesson 4.7

Multiply with 8 - Lesson 4.8

Multiply with 9 - Lesson 4.9

Review For Test on Chapter 4

## Describe Patterns - Lesson 5.1

Find Unknown Factors - Lesson 5.2

Problem Solving: Using the Distributive Property - Lesson 5.3

Multiplication Strategies with Multiples of 10 - Lesson 5.4

Multiply Multiples of 10 by 1-Digit Numbers - Lesson 5.5

Chapter 5 Review on Multiplication Facts

## Third Grade

Math

- Second Grade Math
- Third Grade Math
- Fourth Grade Math
- Fifth Grade Math
- Sixth Grade Math
- Sixth Grade Math (CA)
- Seventh Grade Math (CA)
- Eighth Grade Math (CA)
- Integrated Math 1
- Integrated Math 2
- Integrated Math 3
- PreCalculus
- AP Statistics Exam Prep
- Elementary Statistics
- ELM Practice
- Percents and Decimals
- Sixth Grade Math (Big Ideas)

Model Perimeter - Lesson 11.1

Find Perimeter - Lesson 11.2

Find Unknown Side Lengths - Lesson 11.3

Understanding Area - Lesson 11.4

Measure Area - Lesson 11.5

Use Area Models - Lesson 11.6

Problem Solving - Area of Rectangles - Lesson 11.7

Area of Combined Rectangles - Lesson 11.8

Same Perimeter - Different Area - Lesson 11.9

Same Area - Different Perimeter - Lesson 11.10

Chapter 11 Review for Test on Perimeter and Area

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Count Equal Groups - Lesson 3.1

Relate Addition and Multiplication - Lesson 3.2

Skip Count on a Number Line - Lesson 3.3

Problem Solving - Model Multiplication - Lesson 3.4

Model with Arrays - Lesson 3.5

Commutative Property of Multiplication - Lesson 3.6

Multiply with 1 and 0 - Lesson 3.7

Time to the Minute - Lesson 10.1

A.M. and P.M. - Lesson 10.2

Measure Time Intervals - Lesson 10.3

Use Time Intervals - Lesson 10.4

Problem Solving - Time Intervals - Lesson 10.5

Measure Length - Lesson 10.6

Estimate and Measure Liquid Volume - Lesson 10.7

Estimate and Measure Mass - Lesson 10.8

Equal Parts of a Whole - Lesson 8.1

Equal Shares - Lesson 8.2

Unit Fractions of a Whole - Lesson 8.3

Fractions of a Whole - Lesson 8.4

Fractions on a Number Line - Lesson 8.5

Relate Fractions and Whole Numbers - Lesson 8.6

Fractions of a Group - Lesson 8.7

Find Part of Group Using Unit Fractions - Lesson 8.8

Problem Solving: Find the Whole Using Unit Fractions - Lesson 8.9

Copyright 2013. All rights reserved.

## Eedi Newsletter

## A little bit of problem-solving in every lesson: 10 tips to make it happen

Ideas to embedd proble-solving consistently in your curriculum.

This newsletter is made possible because of Eedi. Check out our brand-new set of diagnostic quizzes, videos, and practice questions for every single maths topic, ready to use in the classroom, and all for free, here .

Welcome to a series of posts where I will share 10 ideas to supercharge some of my favourite student practice activities. We will cover:

Venn Diagrams

SSDD Problems

Intelligent Practice

A little bit of problem-solving in each lesson (this post)

If you find these posts useful, the best way to support this newsletter is to share it with your colleagues. Thanks so much!

## A little bit of problem-solving in every lesson

I wrote recently about the work I have been doing with a maths department to ensure that problem-solving is embedded in their curriculum so that all students get regular opportunities to try unfamiliar problems. I recommend you read that post first as it describes:

What I mean by problem-solving

Why embedding problem-solving is a challenge for many departments

Approaches that don’t work

The approach we developed

What happened when teachers tried it

Then come back to this post where I will share 10 practical tips to help you do the same.

## Problem-solving: 10 top tips

Find good sources of problems. As maths teachers, we are so lucky with the quantity of high-quality, free resources available. But, of course, there is a downside: the time it takes to sift through them all to find what we are after. So, for the type of problem-solving I am talking about here - short, sharp problems that can be done in around 10 minutes - I limit myself to three main sources:

Eedi: We have four problems for every maths topic on one giant spreadsheet.

Open Middle: Fill in the gap challenges amidst various constraints

NRICH short problems: questions adapted from the UK maths challenge

Do the problems yourself as a mathematician. Don’t be tempted to just give the problem a cursory glance. It is only when we sit down and do a problem that we notice fun twists and turns, can predict where our students might struggle, or get inspired to think of variations and extensions. If doing this in a departmental meeting, I would recommend 5 minutes working on a problem on your own, and then 5 minutes sharing where you have got to with a colleague.

Then put your teacher hat on. It is important not to get too engrossed in the problem. I have arrived at many lessons excited to share a problem with my students, only to find that my students struggled, finished early, or simply did not enjoy it. The mistake I made was not pausing to think about how to help my students get the most out of the problem. So, after you have wrestled with a problem, I find these are two good questions to consider:

How would you support a student who is struggling?

How would you challenge a student who has finished?

Set an alarm when there are 10 minutes of the lesson left. Time files in lessons, and if we are not careful, we will not leave ourselves sufficient time to get the most out of the problem. Setting an alarm on my phone helps remind me to get cracking with the problem.

Plan how you will pitch the idea to students. Students find unfamiliar problems challenging, and thus may be reluctant to engage with them for fear of failure. So, how we introduce the problem is important. The pitch will be different from class to class, but here are some things to consider:

Keep the stakes low

Emphasise the importance of effort

Share your belief that they can do it

No talking or writing for 30 seconds. I am sold on this approach after trying it a few times. No speaking stops the inevitable I don’t get it . No writing stops students from diving straight in, forcing them to pause and think. Students are likely to be reluctant at first, so sharing the rationale is important. But eventually, you will find they appreciate the headspace.

Give students a mini-whiteboard. Mini-whiteboards are the ideal vehicle to tackle unfamiliar problems. Students have less fear about writing their ideas down as they can be rubbed away if needed. And as I wrote about here , the simple act of putting two mini-whiteboards together provides the catalyst for focussed, positive student collaboration.

Use the 4-2 approach. You want students to work independently on a problem, but also to reap the benefits of collaboration. I find the 4-2 approach works well here, with students spending 4 minutes working on their own to see how far they can get, and then having 2 minutes to collaborate with their partner to compare progress and approaches.

Ask questions to find the best pairs to call upon. After the 2-minute paired discussion is a good time to start collecting ideas about how to solve the problem. Here are three good prompts to help choose which pairs to hear from:

Put your hand up if you disagree with the answer of your partner

Put your hand up if you changed your mind during your discussion

Put your hand up if your partner said something that you found interesting

Showcase students’ work. It is always a good idea to share students’ work with the rest of the class, especially when tackling unfamiliar problems. Students can benefit from thinking hard about the approach of others. There are loads of different ways to share students’ work, including under a visualiser, recreating it on the board, or simply holding it up. My favourite is to use the infinite canvas on Jake Gordon’s Maths Universe website . It is completely free, and it is amazing!

## Three of my favourite problems:

Challenge your students with some of these… or enjoy trying them yourself. And remember, you can find all the Eedi problem-solving resources here :

Angles in parallel lines

Equations of straight lines

Mean from a list of numbers

What is your experience with problem-solving?

Do you like any of these ideas?

Do you have any extra tips to share?

Let me know in the comments below!

## 🏃🏻♂️ Before you go, have you…🏃🏻♂️

… checked out our incredible, brand-new, free resources from Eedi?

… read my latest Tips for Teachers newsletter about hiding your tell when calling upon students to explain their thinking?

… listened to my most recent podcast about feedback cycles, lesson observations and Exit Tickets?

… considered booking some CPD, coaching, or maths departmental support?

… read my Tips for Teachers book ?

Thanks so much for reading and have a great week!

Ready for more?

## Problem solving with 5 Whys

- Business Skills

Asking questions

## LESSON OVERVIEW

In this lesson, we want to focus on a very popular problem solving technique called 5 Whys (5W) . If your students know something about Six Sigma or Lean, they should be familiar with this technique. Otherwise, they will learn a useful method for problem solving and practise asking questions .

## DISCUSSION & VIDEO

The lesson starts with a quick warm-up speaking task about problems and how students approach solving them. Then, they watch a short video introducing 5 Whys , a problem-solving method developed by Sakichi Toyoda, a Japanese inventor and industrialist, and answer the questions.

Afterwards, your students will practise using the technique based on an example. First, they need to study the example and then fill in the other graph by asking 5 why questions to get to the root cause. Answers in this task may vary. Let your students be creative there. The aim of the task is to get them familiar with using 5 Whys. Finally, in the last task, students will use the technique to find root causes for problems they’ve had at work.

## RELATED LESSON PLANS

This worksheet goes well with the following lesson plans:

- How to use questioning techniques to get better answers
- Questions no one knows the answers to

Subscribe to unlock these and many other Standalone lesson with the Premium plan

## Leave a Reply Cancel reply

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This lesson uses a number line to solve problems involving time intervals.

Lesson 10.5 Practice and Homework COMMON CORE STANDARD—3.MD.A.1 Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects. 4. WRITE Math Write a multistep word problem that has at least two amounts of elapsed time. The problem may require finding a starting time or ending time. Include a ...

TED-Ed lessons on the subject Problem Solving. TED-Ed celebrates the ideas of teachers and students around the world. Discover hundreds of animated lessons, create customized lessons, and share your big ideas. ... Lesson duration 10:42 56,796 Views. 05:18. Mathematics Can you solve the time traveling car riddle? Lesson duration 05:18 ...

Make students articulate their problem solving process . In a one-on-one tutoring session, ask the student to work his/her problem out loud. This slows down the thinking process, making it more accurate and allowing you to access understanding. When working with larger groups you can ask students to provide a written "two-column solution.".

Lesson plan. (60 -75 minutes) This lesson is designed to equip young people with an adaptable approach to solving problems, large or small. It includes a short film and scenarios that encourage development of practical problem solving skills which can be useful for learning, day to day life, and when in employment.

Lesson 10.5 COMMON CORE STANDARD—1.MD.C.4 Represent and interpret data. Complete each sentence about the tally chart. Write greater than, less than, or equal to. 3. The number of children who chose is _____ the number who chose . 4. The number of children who chose is _____ the number who chose . Chapter 10 six hundred three 603

LESSON 10-5 Practice and Problem Solving: A/B 1. kite; P ≈ 15.2 units; A = 12 units2 ... P ≈ 23.6 units; A = 31.5 units2 6. P ≈ 25.1 units; A = 34.5 units2 Practice and Problem Solving: C 1. P ≈ 13.1 units; A = 4 units2 2. P ≈ 22.3 units; A = 12 units2. Created Date: 6/16/2014 12:59:43 PM ...

compare parts of the data to the whole. show frequency of data divided into equal intervals. show change over a period of time. compares two sets of categorical data. Line plot. show frequency of data with a number line. Study with Quizlet and memorize flashcards containing terms like Bar Graph, Box Plot, Circle Graph and more.

606 Chapter 10 Circles 10.5 Lesson WWhat You Will Learnhat You Will Learn Find angle and arc measures. Use circumscribed angles. Finding Angle and Arc Measures Finding Angle and Arc Measures Line m is tangent to the circle. Find the measure of the red angle or arc. a. A B m 1 130° b. K m L J 125° SOLUTION a. m∠1 = 1— 2 (130 °) = 65° b ...

Introduction. (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.

PROBLEM SOLVING Lesson 5.1 Chapter 5 eighty-one P81 Add or Subtract Make a model to solve. 1. Stan has 12 pennies. Some pennies are new. 4 pennies are old. How many pennies are new? 8

Count Equal Groups - Lesson 3.1. Relate Addition and Multiplication - Lesson 3.2. Skip Count on a Number Line - Lesson 3.3. Problem Solving - Model Multiplication - Lesson 3.4. Model with Arrays - Lesson 3.5. Commutative Property of Multiplication - Lesson 3.6. Multiply with 1 and 0 - Lesson 3.7. Time to the Minute - Lesson 10.1.

Using the Language of Mathematics to Communicate Learn with flashcards, games, and more — for free.

This Go Math video covers the topic of finding varied combinations of rectangular prisms, but learning how they will all have the same volume. Students will ...

Lesson 10: Angle Problems and Solving Equations. Students use vertical and adjacent angles and angles on a line and angles at a point in a multi-step problem to write and solve simple equations for an unknown angle in a figure. In Lessons 10 and 11, students apply their understanding of equations to unknown angle problems.

Practice and Problem Solving: A/B . Draw and classify each polygon with the given vertices. Find the perimeter and area of the ... T(1, −2), U(4, 1), V(2, 3), W(−1, 0) LESSON 10-5 . Name _____ Date _____ Class_____ In Exercises 5-9, use the diagram. 5. Find the perimeter of square ADEF . 6..Find the perimeter of BCD. 7. Find the area of ...

What I mean by problem-solving. Why embedding problem-solving is a challenge for many departments. Approaches that don't work. The approach we developed. What happened when teachers tried it. Then come back to this post where I will share 10 practical tips to help you do the same. Problem-solving: 10 top tips

Print. Name Problem Solving Customary and Metric Conversions Solve each problem by making a table. PROBLEM SOLVING Lesson 10.6 COMMON CORE STANDARD CC.5.MD.1 Convert like measurement units within a given measurement system. Number of Quarts Number 12 of Cups 32 P211 2. 3.

4 (Week 4 & 5) Solves Problems Involving Measures of P. sitionAbout the ModuleThis module was designed and written with you in mind. It is here to. help you master the skills in solving problems involving measures of position. The sc. pe of this module permits it to be used in many different learning situatio.

This video lesson is created for my students to help them understand their modules better. Its content is based from the module provided by RO/DO.At the end ...

In this lesson, we want to focus on a very popular problem solving technique called 5 Whys (5W). If your students know something about Six Sigma or Lean, they should be familiar with this technique. Otherwise, they will learn a useful method for problem solving and practise asking questions. B2 / Upper Intermediate 30 min Standard Lesson ...

G10-MELC5-WEEK 5-Lesson-Exemplar - Free download as Word Doc (.doc / .docx), PDF File (.pdf), Text File (.txt) or read online for free. This document provides information about an online/modular English lesson for Grade 10 students. The lesson focuses on analytical listening and problem solving. It discusses the objectives, content and performance standards, and most essential learning ...

Art of Problem Solving AoPS Online. Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12 ...