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Tasks from JoBoaler

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The Inside Problem Solving problems are non-routine math problems designed to promote problem-solving in your classroom. Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity. The problems were developed by the Silicon Valley Mathematics Initiative and are aligned to the Common Core standards.

To request the Inside Problem Solving Solutions Guide, please get in touch with us via the feedback form.

Why Problem Solving?

Problem solving is the cornerstone of doing mathematics. George Polya, a famous mathematician from Stanford, once said, "A problem is not a problem if you can solve it in 24 hours." His point was that a problem that you can solve in less than a day is usually a problem that is similar to one that you have solved before, or at least is one where you recognize that a certain approach would lead to the solution. Bu t in real life, a problem is a situation that confronts you and you don't have an idea of where to even start. Mathematics is the toolbox that solves so many problems. Whether it is calculating an estimate measure, modeling a complex situation, determining the probability of a chance event, transforming a graphical image or proving a case using deductive reasoning, mathematics is used. If we want our student s to be problem solvers and mathematically powerful, we must model perseverance and challenge students with non-routine problems.

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Inside Problem Solving | Inside Mathematics

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The Inside Problem Solving problems are non-routine math problems designed to promote problem-solving in your classroom. Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity.

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That leads me to when should you use which one? So to me, this is based on how I've been using the two. You may disagree with me, so let me know in the comments, but to me, I like to use Number Talks when I'm working with a group of students who already have strategies. If I put a problem and you get the "deer-in-the-headlights" look, then the Number Talk may not be the best thing because they don't have different strategies and they are all solving it the same way. That makes it hard to do a talk about the different ways kids are solving it.

So in a Number Talk, you have to have a group of kids who have some strategies and then you can talk about the different strategies and how they are alike and how they are different. That's one of the powerful parts of a Number Talk, is that you're not just showing all of the different ways that kids solved it because that can be overwhelming for a lot of kids. To do a number talk correctly, you also need to make connections. If Johnny did it one way and Jose did it another way, we've got to talk about how those ways are still alike and how they are different. What did Johnny do that's different than what Jose did and but yet still connected.

For our struggling kids, sometimes a Number Talk can be very overwhelming because they see six different ways that people solved the problem and they don't get any of them. So you've got to spend some time in Number Talks to be able to make connections between the different ways that kids are solving them.

In a Number String, I like to use them to help develop strategies. If I'm working with kids who don't have strategies already, a string of problems is super powerful to help build those strategies. So let's take a look at this example.

This can definitely be different depending on the grade level that you work. However, in this example, I could start with the string off to the left if my kids don't have a strategy or strategies for that bottom problem. So I might start out just showing 9 + 7 and talking about how they solved that problem. Then I would show 29 + 17 and see if they're using the same type of strategy. They may have a different strategy for that problem but I want to make the connection of how they solved 9 + 7 connects to how they can solve 29 + 17. Then how that connects to how you can solve 498 + 267. Finally, I get to the problem which is what I actually want my kids to develop at this grade level. When I'm adding decimals, can I use those same strategies? You bet!

If they haven't made those connections, I'm going to use a string to help bridge the gap between the things that they learned prior to what I'm wanting them to learn now. If I just throw up 3 and 99 hundredths plus 17 hundredths, and they don't have strategies already and they haven't made connections, then I'm probably going to get every kid solving it the same way or not knowing how to solve it.
So again, a Number String to me is a way to help develop strategies whereas a Number Talk is to talk about the different strategies kids have for that one problem, how they connect together, how they're alike, and how they are different.
I hope that that's helped you build your math mind, so you can go build the math minds of your students. If you would like to learn more about using Number Strings, I talk about them as Number Talks because most of you are familiar with Number Talks.

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Number Talks Versus Number Strings - The Recovering Traditionalist

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That leads me to when should you use which one? So to me, this is based on how I've been using the two. You may disagree with me, so let me know in the comments, but to me, I like to use Number Talks when I'm working with a group of students who already have strategies.

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3 Act Math Tasks By Kyle Pearce, Dan Meyer and Others

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3 Act Math Tasks by Kyle Pearce, Dan Meyer and many others to spark curiosity and fuel sense making.

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search

Enriching mathematics for all learners

A free-to-use collection of mathematics activities, lessons and problems designed to nurture curious, resourceful and confident learners of mathematics. Produced by The University of Cambridge.

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The Nrich Maths Project Cambridge,England. Mathematics resources for children,parents and teachers to enrich learning. Problems,children's solutions,interactivities,games,articles.

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This is Which One Doesn't Belong?, a website dedicated to providing thought-provoking puzzles for math teachers and students alike. There are no answers provided as there are many different, correct ways of choosing which one doesn't belong. Enjoy!

Which One Doesn't Belong?

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This is Which One Doesn't Belong?, a website dedicated to providing thought-provoking puzzles for math teachers and students alike. There are no answers provided as there are many different, correct ways of choosing which one doesn't belong. Enjoy!

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The Progression of Addition and Subtraction HD

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(Challenge #) Challenge Title

Algebra

Equations, linear

(8)Stamps, (30)Smiles,(58)Gone Fishin',(74)The Race

Modeling problems

(1)Line Up!, (29)Salaries,(42)Rose Bowl,(44)Table for 19,(77)Movie Money, (80)Gifts Galore

Patterns, representing

(6)Which Way?, (8)Stamps,(11)Play Ball!,(41)Radar Bills, (80)Gifts Galore

Problem Solving using Symbols

(21)Body Mass,(30)Smiles, (58)Gone Fishin',(60)Number Tricks

Representations, comparing

(7)Double or Not,(11)Play Ball!,(21)Body Mass,(29)Salaries, (60)Number Tricks

Geometry

2-D and 3-D figures, properties

(3)Popcorn, (4)Don't Fall In,(15)Big Trees,(35)Fire Hydrant,(54)Cut the Cake,(55)Decorating Boxes,(76)Visualizing

2-D representations of 3-D figures

(9)Mirror Mirror,(40)Putt-Putt, (54)Cut the Cake,(55)Decorating Boxes,(57)Envelopes

Connections to other disciplines

(3)Popcorn, (4)Don't Fall In,(5)Upside Down?,(37)Map Colors,(43)Seeing Stars,(51)Bar Codes,(56)Drip Drops, (62)Keeping Cool

Geometric models of algebraic concepts

(45)Three Squares,(46)Time Zones,(54)Cut the Cake

Networks

(6)Which Way?

Scale Drawing
(61)Statue of Liberty

Similarity/Congruence

(59)Japanese Floors, (61)Statue of Liberty

Transformations

(5)Upside Down?,(9)Mirror Mirror,(40)Putt-Putt, (41)Radar Bills, (49)Angolan Designs,(73)A Shard or Two,(79)Morphing

Measurement

Angles and Length

(9)Mirror Mirror,(10)What's My Angle,(39)Patios, (40)Putt-Putt,(44)Table for 19,(66)Airport Runways

Area and Circumference

(15)Big Trees,(23)Fractals,(32)Windshield Wipers,(39)Patios, (54)Cut the Cake

Estimation

(1)Line Up, (2)Beating Heart,(3)Popcorn, (10)What's My Angle,(15)Big Trees,(47)VCR

Metric and customary units

(34)Faster, (67)When in Rome

Rates and velocity

(1)Line Up, (2)Beating Heart,(16)Beanie Babies,(18)Tern Turn

Units, appropriate

(3)Popcorn, (15)Big Trees,(34)Faster, (74)The Race

Number

Compare and order decimals, fractions, percents

(14)Battling Averages,(17)Perplexing Percentages,(25)Grape Juice Jungle,(36)Majority Vote

Large numbers and notation

(2)Beating Heart,(7)Double or Not,(11)Play Ball!,(28)Pigeonholes, (56)Drip Drops

Ratio and proportions

(14)Battling Averages,(24)Gasoline Tanks,(25)Grape Juice Jungle,(47)VCR

Solve problems with decimals, fractions, percents

(3)Popcorn, (8)Stamps, (14)Battling Averages,(15)Big Trees,(47)VCR, (71)Archeology

Statistics & Probability

Collecting Data

(26)I Win!, (29)Salaries,(64)Wheel of Fortune

Graphical represenations of data

(20)Basketball Picks,(48)Working Hours,(53)Census,(64)Wheel of Fortune,(70)Soda

Make conjectures about differences

(27)Life Expectancy,(29)Salaries, (71)Archeology,(72)TV Ratings

Make conjectures about relationships

(27)Life Expectancy,(29)Salaries, (68)Bones,(72)TV Ratings

Probability, Apply

(19)Two Points, (26)I Win!, (52)Capture-Recapture,(63)Matching Birthdays, (69)Misaddressed

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Figure This! Math Challenges for Families - Challenge Index

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(Challenge #) Challenge Title Algebra Equations, linear (8)Stamps, (30)Smiles,(58)Gone Fishin',(74) The Race Modeling problems (1)Line Up!, (29)Salaries,(42)Rose Bowl,(44) Table for 19,(77) Movie Money, (80) Gifts Galore Patterns, representing (6)Which Way?, (8)Stamps,(11)Play Ball!,(41) Radar Bills, (80) Gifts Galore Problem Solving using Symbols (21)Body Mass,(30)Smiles, (58)Gone Fishin',(60) Number Tricks Representations, comparing (7)Double

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A number string is a set of related math problems, crafted to support students to construct big ideas about mathematics and build their own strategies (Fosnot & Dolk, 2002). Typically, a teacher gathers her students near the board and presents the problems - one at a time - providing plenty of time for students to think. Students first solve the problems mentally, then share their strategies with the class while making sure they understand others' thinking. Meanwhile, the teacher is representing students' strategies on a model and facilitating conversation among students about the sense they are making within and between the problems. Strings are not a rigid recipe but a flexible classroom routine - used by teachers daily or weekly with both small groups and entire classes.

Here is a narrative of a first grade number string or a video of a fourth grade number string to get you started.

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A number string is a set of related math problems, crafted to support students to construct big ideas about mathematics and build their own strategies (Fosnot & Dolk, 2002). Typically, a teacher gathers her students near the board and presents the problems - one at a time - providing plenty of time for students to...

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51 Esti-Mysteries - Steve Wyborney's Blog: I'm on a Learning Mission.

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My growing list of free real world problem-based lessons for grades 2 through high school. Includes links to download multimedia and teacher guide.

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Number talks were developed for classroom teachers to engage students in "mental math" through grappling with interesting mathematics problems. Educators can use number talks regularly as introductions to the day's mathematical practice, as "warm ups" for other lessons, or as stand-alone extended engagements with mathematical concepts. The teachers in these videos are from schools participating in the Silicon Valley Mathematics Initiative.

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Number talks were developed for classroom teachers to engage students in "mental math" through grappling with interesting mathematics problems. Educators can use number talks regularly as introductions to the day's mathematical practice, as "warm ups" for other lessons, or as stand-alone extended engagements with mathematical concepts.

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ASKING STUDENTS TO CHOOSE A PATH AND JUSTIFY IT WITH MATH

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Home

CHALLENGING MATH PROBLEMS WORTH SOLVING

DOWNLOAD OUR FAVORITE PROBLEMS FROM EVERY GRADE LEVEL

WANT GOOGLE SLIDE VERSIONS OF ALL PROBLEMS?

HERE'S OUR GROWING COLLECTION

WANT TO SHARE OPEN MIDDLE WITH OTHERS?

CHECK OUT THESE FREE WEBINARS TO HELP TEACHERS RETHINK CLASSWORK

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CHALLENGING MATH PROBLEMS WORTH SOLVING DOWNLOAD OUR FAVORITE PROBLEMS FROM EVERY GRADE LEVEL Get Our Favorite Problems Take The Online Workshop WANT GOOGLE SLIDE VERSIONS OF ALL PROBLEMS? HERE’S OUR GROWING COLLECTION Get Google Slide Versions WANT TO SHARE OPEN MIDDLE WITH OTHERS? CHECK OUT THESE FREE WEBINARS TO HELP TEACHERS RETHINK CLASSWORK Elementary Version

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Number talk resources

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Due to school firewalls and security restrictions, students may not be able to access resources directly from this site. To give a student access, please first save a copy of the resource to a teacher-owned Google account, then connect the student to the teacher-owned copy. See the video tutorial on "How to share Google files with students" for more information.

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Due to school firewalls and security restrictions, students may not be able to access resources directly from this site. To give a student access, please first save a copy of the resource to a teacher-owned Google account, then connect the student to the teacher-owned copy.

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Steve Wyborney's Blog: I'm on a Learning Mission.

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Welcome to Splat! You are only moments away from a VERY POWERFUL, highly interactive number sense strategy that can be used at any grade level!

This post includes 50 (fifty!) free, downloadable PowerPoint math lessons!

Watch the video, download some (or all) of the lessons, and experience what happens when you unleash this resource in your classroom!

I've been excited to click the Publish button on this post for several weeks! It's time!
UPDATE: The Fraction Splat! Series (with 20 more downloadable lessons) is now available!
After reading this blog post, take a look at THE 12 MOST POPULAR MATH STRATEGIES AND DOWNLOADS ON THIS BLOG.

Splat Through 10
How many dots are under the Splat?

Splat Through 20
How many dots are under the Splat?

Multiple Splats
(Note: Splats that are the same color must cover the same number.)
Splat 3.1 - multiple splats Splat 3.2 - multiple splats Splat 3.3 - multiple splats Splat 3.4 - multiple splats Splat 3.5 - multiple splats
Instant Multiple Splat
(Note: An additional feature of this level is the opportunity to ask, "What could the total be?")

Splat 4.1 - Instant Multiple Splats Splat 4.2 - Instant Multiple Splats Splat 4.3 - Instant Multiple Splats Splat 4.4 - Instant Multiple Splats Splat 4.5 - Instant Multiple Splats
2-Color Splat
(Splats of different colors must have different values.)

Splat 5.1 2-color Splat Splat 5.2 2-color Splat Splat 5.3 2-color Splat Splat 5.4 2-color Splat Splat 5.5 2-color Splat Splat 6.2 Instant 2-color Splat Splat 6.3 Instant 2-color Splat Splat 6.4 Instant 2-color Splat Splat 6.5 Instant 2-color Splat
Instant 2-Variable Splat
(Splats of different colors must have different values. Splats of the same color must have the same value.)
Splat 7.1 Instant 2-variable Splat Splat 7.2 Instant 2-variable Splat Splat 7.3 Instant 2-variable Splat Splat 7.4 Instant 2-variable Splat Splat 7.5 Instant 2-variable Splat
Number Splat
(Note: Only the numbers circled on the screen may be used.)
Splat 8.1 - Number Splat Splat 8.2 - Number Splat Splat 8.3 - Number Splat Splat 8.4 - Number Splat Splat 8.5 - Number Splat Splat 9.1 - Instant Number Splat Splat 9.2 - Instant Number Splat Splat 9.3 - Instant Number Splat Splat 9.4 - Instant Number Splat Splat 9.5 - Instant Number Splat
Instant Number Splat with 2 Variables
(Note: Only the numbers circled on the screen may be used, and Splats of different colors must have different values.)
Splat 10.1 - Instant 2-Variable Number Splat Splat 10.2 - Instant 2-Variable Number Splat Splat 10.3 - Instant 2-Variable Number Splat Splat 10.4 - Instant 2-Variable Number Splat Splat 10.5 - Instant 2-Variable Number Splat
NOTE: I wrote each of these lessons using PowerPoint, so I recommend playing them in PowerPoint. If you do not have PowerPoint, you may want to download the PowerPoint viewer so that you can play the lessons.

UPDATE: The Fraction Splat! Series (with 20 more downloadable lessons) is now available!
Other Posts and Resources Which May Be of Interest Tiled Area Questions The Maze Hundreds Chart
Introducing Cube Connectors
Provide Massive Space to Notice
Primary Tile Questions
3 Powerful Tile Strategies (and 40 new downloadable pages)
Please take a moment to subscribe to the blog so you'll be notified each time new resources are posted!

I would treasure hearing about your experiences using Splat! to promote number sense in your classroom!

All my best,
Steve

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Splat! - Steve Wyborney's Blog: I'm on a Learning Mission.

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Welcome to Splat! You are only moments away from a VERY POWERFUL, highly interactive number sense strategy that can be used at any grade level!

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Maine Learning Results - Mathematics

The Maine Learning Results, which incorporates the Common Core State Standards, is a single set of clear educational standards for mathematics. These standards are designed to ensure that students graduating from high school are prepared to go to college or enter the workforce, and parents, teachers and students have a clear understanding of what is expected of them.

K-12 by Strand

By Grade Span

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The Maine Learning Results, which incorporates the Common Core State Standards, is a single set of clear educational standards for mathematics. These standards are designed to ensure that students graduating from high school are prepared to go to college or enter the workforce, and parents, teachers and students have a clear understanding of what is expected of them.

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Supporting documents for the Mathematical Teaching Practices

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Principles to Actions Professional Learning Toolkit - National Council of Teachers of Mathematics

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These grade band specific professional learning modules are focused on the Effective Teaching Practices and Guiding Principles from Principles to Actions Ensuring Mathematical Success for All. Presentation, presenter notes, and required materials are provided in each module to support professional learning with

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Welcome to KENKEN!

No matter what you're looking for - learning or fun - KenKen.com has it! Free online puzzles. Cool math games. Number games. (More addictive than Sudoku or Kakuro? You decide!) Educational games for kids. Visit our For Teachers section for math teacher resources, our free KenKen Classroom program, and a message from Tetsuya Miyamoto, the Japanese educator behind KenKen. Miyamoto, chess master David Levy, and the rest of Team KenKen have crafted these math puzzles into great games for learning and brain training. Need even more KenKen? Check out our Will Shortz Presents books. Try our mobile app on iPhone and iPad or our Kindle version. Play on the New York Times puzzle page and NCTM website. Regardless of why, how, or where you play, KenKen are the math puzzles that make you smarter!

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KenKen Puzzle Official Site - Free Math Puzzles That Make You Smarter!

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No matter what you're looking for - learning or fun - KenKen.com has it! Free online puzzles. Cool math games. Number games. (More addictive than Sudoku or Kakuro? You decide!) Educational games for kids. Visit our For Teachers section for math teacher resources, our free KenKen Classroom program, and a message from Tetsuya Miyamoto, the Japanese educator behind KenKen.

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Kakooma and puzzles

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Educational math games and math centers for elementary and middle school students

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Download PDF Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts By Jo Boaler, Professor of Mathematics Education, co-founder youcubed With the help of Cathy Williams, […]

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Google has announced that Jamboards will become "read-only" on October 1, 2024 and will disappear completely on December 31, 2024 ( https://support.google.com/jamboard/answer/14084927?hl=en). While we are disappointed in this news, KCM is dedicated to providing resources to support teachers. All KCM Jamboards will remain available for the 2023-2024 academic year. We have begun the process of researching options for converting or updating our current Jamboards to a new format and plan to have the revised resources available in the fall of 2024.

Chapter 1. The Five Fundamentals

Chapter 2. Foundational Addition and Subtraction Facts

Chapter 3. Derived Fact Strategies for Addition and Subtraction

Chapter 4. Foundational Multiplication and Division Facts

Chapter 5. Derived Fact Strategies for Multiplication and Division

Chapter 6. Assessing Foundational Facts

Chapter 7. Assessing Derived Fact Strategies and All Facts

Chapter 8. Families and Facts

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Google has announced that Jamboards will become "read-only" on October 1, 2024 and will disappear completely on December 31, 2024 ( https://support.google.com/jamboard/answer/14084927?hl=en). While we are disappointed in this news, KCM is dedicated to providing resources to support teachers. All KCM Jamboards will remain available for the 2023-2024 academic year.

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The Ministry is migrating nzmaths content to Tāhurangi. Relevant and up-to-date teaching resources are being moved to Tāhūrangi (tahurangi.education.govt.nz). When all identified resources have been successfully moved, this website will close. We expect this to be in June 2024. e-ako maths, e-ako Pāngarau, and e-ako PLD 360 will continue to be available.

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Free math fluency centers for K-5 to assess fluency, monitor progress, and plan instruction and intervention.

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Addition/Subtraction - SAME BUT DIFFERENT MATH

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… efforts to intensify attention to the traditional mathematics curriculum do not necessarily lead to increased competency with quantitative data and numbers. While perhaps surprising to many in the public, this conclusion follows from a simple recognition that is, unlike mathematics, numeracy does not so much lead upwards in an ascending pursuit of abstraction as it moves outward toward an ever richer engagement with life’s diverse contexts and situations.

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Error analysis

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These games use a deck of colorful cards with written numerals and dots, 0-10. The dots are arranged in different configurations: straight lines, rectangular arrays, triangular arrays, dice patterns, ten frames, circles, and scattered configurations. The variety of dot arrangements helps children develop different mental images of specific quantities. Dot Cards help children practice important early math skills, such as counting and number operations. In developing these materials, we found it was most helpful when teachers introduced a game or a minibook in the classroom and then sent the materials, directions, and the minibooks home for children to play and read with their families. Children are proud to share-and teach-what they learned in school. Then, children can come back to the classroom and share what they learned at home.

Materials Needed

Printable Dot Cards, Directions

What's the math in these games?

Dot cards help children practice important early math skills: including subitizing, counting with one-to-one correspondence, and recognizing written numerals. We often ask children to count a line of objects, but circular and scattered arrangements are harder to count one by one!

Download Directions for all Dot Card Games

Dot Card Games

Printable Materials Classroom Resources

To print at home: There are 7 types of cards in the dot card deck. Each type of dot card can be printed separately on colored paper. We have suggested these colors in the directions, but you can use any paper available to you. For example, you can use blue paper to print the blue circle dot cards.

Game Kits Coming Fall 2021!

Check back for a new way to play our games and access game materials at your fingertips.

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These games use a deck of colorful cards with written numerals and dots, 0-10. The dots are arranged in different configurations: straight lines, rectangular arrays, triangular arrays, dice patterns, ten frames, circles, and scattered configurations. The variety of dot arrangements helps children develop different mental images of specific quantities.

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[KH] IM K-5 Math Centers Navigation Tool

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Figure 4.5. Mathematical Teaching Practices Continuum.pdf

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Magic Vs printable sheet The problem is explained below, but you may wish to scroll to the foot of the page to watch a video of the NRICH team presenting this challenge.

Place each of the numbers 1 to 5 in the V shape below so that the two arms of the V have the same total.

How many different possibilities are there?
What do you notice about all the solutions you find?

Can you explain what you see?

Can you convince someone that you have all the solutions?

What happens if we use the numbers from 2 to 6? From 12 to 16? From 37 to 41? From 103 to 107?

What can you discover about a V that has arms of length four using the numbers 1 to 7?

Here is a video of the NRICH team presenting the challenge. You could just watch the start to check that you understand the problem, or you may like to pause the video and work on the task at various points.

Click here for a poster of this problem.

If you have enjoyed this problem, you might like to explore Magic Letters.

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Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?