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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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Peter Liljedahl's Research on Teaching Through Problem Solving

"What was missing for these students, and their teachers, was a central focus in mathematics on thinking. The realization that this was absent in so many classrooms that I visited motivated me to find a way to build, within these same classrooms, a culture of thinking, both for the student and the teachers. I wanted to build, what I now call, a thinking classroom - a classroom that is not only conducive to thinking but also occasions thinking, a space that is inhabited by thinking individuals as well as individuals thinking collectively, learning together, and constructing knowledge and understanding through activity and discussion."

Peter Liljedahl, Simon Fraser University, Canada

Additive Reading Extras

Multiplicative Reasoning Extras

Professional Development. Workshops. Presentations.

​​Celebrating 100 Years with Big Beautiful Problems (Grades 6-8) - Alicia Burdess and Jessie Shirley
100 Days of Professional Learning, Online with the National Council of Teachers of Mathematics

Big Beautiful Problems can be life-changing for teachers and students alike as they show how math becomes alive and is connected to our world. See how much fun math can be as we get caught up in the challenge, excitement, and flow of learning. Deepen your skills, confidence, and joy as we experience and explore some of our favorite problems.

Teaching Math Through Problem Solving in a
Thinking Classroom Online
2020

Mighty Peace 2020

Mathematical Playground Problems

Skyscrapers

Where Are All of the
Beautiful Problems?
MCATA 2019

Big Beautiful Problems
Geeks Unite
Teaching Middle School Math 2018

La Numeratie.
St. Gerard - Francais

Numeracy in GPCSD 2007-2018

Teaching Through Problems Worth Solving Mighty Peace 2018

Early Numeracy 2017

Afternoon at Holy Cross
A Conversation
About Math 2018

Friday afternoon MATH fun with Kateri 2016

Problem Solving in STM 2018

Problem Solving at Kateri 2016

Grade 123 New Teachers Day 2018

Mathematics Teaching and Learning at St. Mary Beaverlodge 2016

Diehard Jug Problem

Additive and Multiplicative Reasoning

Professional Development

Teaching Through Problems Worth Solving In A Thinking Classroom

Inquiry-based, Curriculum-linked, Differentiated Math Problems for Grade 2

Inquiry-based, Curriculum-linked, Differentiated Math Problems for Grade 3

Inquiry-based, Curriculum-linked, Differentiated Math Problems for Grade 8 Version 3.0

Numeracy Leadership Group - Building Capacity in our Schools

More Learning and Problem Solving with the Numeracy Leadership Group

Research and Articles

Visibly Random Groups

Building Thinking Classroom

Four Purposes of Assessment

Peter Liljedahl Research - Now You Try One

Peter Liljedahl Research -Homework

Learning, Teaching, and Loving Math - BOOKS TO READ

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Additional Teacher Resources | Magical Math Journey

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Magical Math Journey is a resource website for additional excellent teacher resources and downloads

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Virtual Math Manipulatives CUBE Collection

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Virtual Math Manipulatives CUBE Collection

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Virtual Math Interactives/Manipulatives - Cube For Teachers

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Building Thinking Classrooms Rocket PD Podcast

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Building Thinking Classrooms Rocket PD Podcast

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Episode 2: How to Build a Thinking Classroom (w/Peter Liljedahl)

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How do I get started? 1 USE THE FREE MATH LESSONS Estimation 180 Tasks 3-Act Tasks 2 SUBSCRIBE Email Newsletter Podcast

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Research and data to make progress against the world's largest problems

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7 Research-Backed Ways to Boost Working Memory in Math

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7 Research-Backed Ways to Boost Working Memory in Math

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7 Research-Backed Ways to Boost Working Memory in Math

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Short-term memory is finite and fills up quickly. Here are 7 ways we can free up space for clearer-headed mathematical thinking.

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Interactive Hundreds Chart | Toy Theater

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Use this online hundreds chart to practice counting to one hundred, for modeling math problems, for skip counting, and more! Try it, it’s free!

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Open-Educational Resources (OER) by Discipline

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Open-Educational Resources (OER) by Discipline

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ASCCC Open Educational Resources Initiative

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Open Educational Resources by Discipline - ASCCC Open Educational Resources Initiative

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Project Zero (PZ) Visible Thinking Resources

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Project Zero (PZ) Visible Thinking Resources

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Project Zero (PZ) Thinking Routines Toolbox

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Project Zero (PZ) Thinking Routines Toolbox

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Jack Diekman - Multilingual learner success in math webinar

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Jack Diekman - Multilingual learner success in math webinar

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Defining Success for Multilingual Learners in Math Webinar - CEMD

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Listen to why high-quality math teaching resources are essential for the success of multilingual learners.

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Take out the numbers to make word problems make sense Learn More Try one tomorrow Primary (K-2) Older Students Explore ready-made problems Problem Banks More Problems This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

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People shared examples of sticky notes used in a variety of ways in response to the November challenge. They are small, which forces concision, lend themselves to being easily rearranged and remove the constraints of our tools. Pick up a pack of post its the next time you find yourself needing to co

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Free Education Resources for Parents in California

No one knows your child or their needs more than you. With Starting Smarter, you now have access to information about your child's education that goes beyond just grades. You can use our test score reports to see the areas your child is excelling and where they can improve. All of this can begin a conversation with your child's teacher about offering better learning support at home and all year long.

Check out the Test Questions

Enter your child's grade and select a subject and area to see descriptions of the skills and sample test questions in each performance area.

It's natural for every student to have academic strengths as well as areas where they may need more support and improvement. The tests your child takes are meant to measure the most important skills in each subject. For all tests, we break down the score reports into different performance areas-like reading and problem solving-to identify where your child is excelling or needs more support.

Families will also receive more information on their child's reading and math performance with the new Lexile and Quantile report. A Lexile measure shows the level of text your child can read and understand. A Quantile measure shows what math skills your child knows and is ready to learn next. Learn how to use the scores to support reading and math.

Use this information to have a meaningful conversation with your child's teacher about the skills and content in the grade level and to support your child's learning at home.

Be a Learning Hero!

Visit the Learning Heroes website to find resources and tools to help your child thrive in school.

Plan for your Parent-Teacher Conversations

Use this tool to have a meaningful conversation about your child's test results and to co-create a learning plan with the teacher.

Check out the test in action

Visit the CAASPP practice test to try out live, interactive questions and the testing features students can use.

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Free Education Resources | Starting Smarter California - CAASPP Starting Smarter

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No one knows your child or their needs more than you. With Starting Smarter, you now have access to information about your child's education that goes beyond just grades. You can use our test score reports to see the areas your child is excelling and where they can improve.

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Want to learn more? Take the free workshop. All 3rd 4th 5th 6th 7th 8th Alg 1 Geom Alg 2 Stat Calc Water Leak Will service arrive in time?Car Ramp How far did the ca…

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The progression videos are a series of videos that show how math concepts build and grow through the grade levels. Here is a post that gives a "behind the scenes" of what it takes to make the videos.

Click on the image below to watch the progression video and download the resources.

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Cognitively Guided Instruction "CGI" Support

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Cognitively Guided Instruction "CGI" Support

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Cognitively Guided instruction, or CGI, is based upon 30 years of research. It's a framework that helps teachers to understand how children's mathematical ideas develop, and provides an opportunity to build on the child's own thinking and understanding. CGI is not a curriculum. CGI is not a lesson. CGI is a framework, a way of thinking about how children learn math that is research-based. The central theme of CGI is that children come to school with mathematical knowledge. Children develop their math understanding by solving a variety of problems in a way that makes sense to them. The children build their own understanding and inform other children, and the teacher, as they share their thinking. As a result, children build connections and relationships among the mathematical ideas so that the math makes sense to them, and the math is meaningful in their lives.
Here are some videos of CGI in action with teachers from LAUSD: Kinder Video 3rd Grade Video 4th/5th Grade Video Videos from the Online course Navigating the CGI Math Classroom: The What and Why of CGI Multiplication and Division and CGI CGI and Place Value Supporting Teachers in CGI, Part 1 Supporting Teachers in CGI, Part 2
Resources and Links:

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Cognitively Guided Instruction "CGI" Support

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Cognitively Guided instruction, or CGI, is based upon 30 years of research. It's a framework that helps teachers to understand how children's mathematical ideas develop, and provides an opportunity to build on the child's own thinking and understanding. CGI is not a curriculum. CGI is not a lesson.

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Talking Math: How to Engage Students in Mathematical Discourse

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Talking Math: How to Engage Students in Mathematical Discourse

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The importance of engaging students in meaningful mathematical discussion has long been identified as an essential component of students' mathematics learning.

When students share and exchange their ideas, both they and their teachers benefit.
Dr. Gladis Kersaint
Understanding the Importance of Mathematical Discourse

Talking about mathematical concepts allows students to reflect on their own understanding while making sense of and critiquing the ideas of others. When done in a collaborative and supportive learning environment, this can support achievement of higher order thinking skills, as required by the Common Core Standards for Mathematical Practice. Students can make conjectures, link prior knowledge to current understanding, reason about mathematics, refine and amend their approaches, and take ownership of their mathematical knowledge.

Students benefit greatly from learning to use the tools of mathematical discourse-including words, symbols, diagrams, physical models, and technology-to present and defend their ideas. Imparting these new skills on students allows teachers to access, monitor, and evaluate students' mathematical understanding and development.

Mathematical Discourse in Action

One approach to giving students the information needed for a mathematics lesson involves the teacher simply telling the students what they need to know. Far more productive, however, is more a thoughtful approach to student engagement that encourage a productive dialogue, as seen in the example below. As educators, we need to structure lessons to encourage student interaction, address gaps in student understanding, and help students express mathematical concepts more precisely. Providing opportunity for conversation does just that.

Does this look like a classroom dialogue you've seen before? To further your understanding of the importance of facilitating mathematical discourse, ask yourself:

How did these students benefit by sharing what they knew as part of the mathematics discussion?

Why did the teacher allow students to evaluate the correctness of Julie's answer?

Why did the teacher ask another student to restate what James said?

Establishing a Discourse-Rich Mathematics Learning Community

To engage students in productive mathematics discussions, it is important to establish a learning environment that welcomes student involvement. The first step is setting the expectation that every student will contribute to the discourse community.

Students must be encouraged to use their problem-solving, reasoning,and communication skills to make conjectures, explore their own ideas and approaches, and find solutions to routine and non-routine mathematics problems.

In particular, the five process standards-problem solving, reasoning and proof, communication, connections, and representation-can be seen in action in a discourse-rich mathematics community as students interact, question one another and convey their understanding. During this dialogue, teachers can recognize errors in context and reinforce that they are natural occurrences that enhance learning.

What does a classroom rich with mathematical discourse look like? Walk into one and you'll likely see:

Students and teachers acknowledging and discussing errors and the reasons behind them to help students build greater understanding.

Students questioning each other using mathematics arguments to establish the correctness of solutions.

Students reaching and justifying conclusions based on their own mathematics knowledge without relying on the authority of teachers.

Students engaging in "productive struggle" with appropriate scaffolds for support.

Students encouraged to use a variety of approaches to convey their knowledge and solution strategies, including oral presentations; written explanations; and physical, graphical, pictorial, or symbolic representations.

Understanding Student and Teacher Roles in Mathematical Discourse

Facilitating student engagement in mathematical discourse begins with the decisions teachers make when they plan classroom instruction. The tasks they use, the ways in which they organize the classroom, and the behaviors they model communicate expectations for classroom norms, including the ways students are expected to engage in classroom discussions.

Depending on prior experiences, students might find these new expectations for engagement uncomfortable and may not be ready to plunge into mathematical dialogue feet first. Teachers must ease the transition to a dialogue-rich mathematics classroom and prepare students to engage in such discussions. It is also important to carefully consider the best ways to coordinate student interaction in pairs, small groups, or whole-class interactions to ease transitions and maximize learning.

To support students, teachers must help students create avision for expected behaviors and actions, prepare them for their roles by modeling or role-playing, and reinforce these behaviors consistently. Teachers need to monitor progress as students engage in mathematics discussions, supporting them as their mathematical knowledge grows and they become more skilled at expressing ideas coherently and using vocabulary, syntax, and semantics precisely.

Teachers should observe, listen to and monitor students to support instructional decision-making. Determining what questions to ask, which students to call on, when to intervene and when to extend student thinking provides opportunities to understand student thinking, monitor growth and assess knowledge. A discourse-rich classroom enables teachers to gain insights not only about what students know, but also about the approaches they use, how-and how well-they understand the ideas, and the ways they present their knowledge.

In addition to content knowledge, mathematical discourse allows teachers to monitor students' dispositions and gauge their developing confidence, interest, and perseverance. Teachers can use this information to determine areas of confusion or frustration in order to decide when an intervention might be needed. They also examine understandings and misconceptions revealed during classroom discussions and adjust lesson plans accordingly.

Asking the right questions can help support instructional decision making and direct student focus. Some examples are below:

Engaging Every Student in Mathematical Discourse

All students are mathematics language learners, regardless of their level of English language proficiency, and discourse allows ALL students to develop mathematical language.

While it is important to build scaffolds to support students who are learning English and mathematics at the same time, it is essential to recognize that mathematics is a technical language that all students must learn. Students who are English speakers also require support as they learn the language of mathematics.

As outlined in the Standards for Mathematical Practice, students should become fluent in mathematical language, including vocabulary, symbolic representations, syntax, semantics, and linguistic features. In addition, they must have ample opportunities to use the language of mathematics as they engage in various forms of communication. Discourse allows students to practice precision in multiple areas, including:

1.Vocabulary

English words with a different meaning in mathematics, such as "negative," "table," or "rational."

Specialized terms, such as "hypotenuse" or "trapezoid."

Terms with multiple meanings in mathematics, such as "median" or "base."

2. Symbols

Ways to read and interpret symbolic representations. For example, "a × b" can be expressed as "a times b," "the product of a and b," or "multiply a and b."

3. Syntax

Understanding the rules that govern the structure of sentences. For example, "A number y is 4 more than a number x" is translated symbolically to "y=x+4."

4. Semantics

The process of making meaning from language.

The same mathematics word may be interpreted differently depending on the context. For example, the median of a set of numbers versus the median of a triangle.

Teachers support mathematical language development by asking key questions and encouraging students to ask for clarity. By doing this, teachers motivate and encourage students and facilitate productive discussion, fostering a supportive classroom culture that maximizes learning.
This blog is part of a three post series on the importance of mathematical discourse from Curriculum Associates Dr. Gladis Kersaint is the Dean of the Neag School of Education at the University of Connecticut and an author of the Ready® Mathematics and Ready® Classroom Mathematics programs from Curriculum Associates. Stay in-the-know with all things EdTech and innovations in learning by signing up to receive the weekly Smart Update. , a Getting Smart Advocacy Partner, and Dr. Gladis Kersaint, the author of the recently published whitepaper Orchestrating Mathematical Discourse to Enhance Student Learning. Download your free copy here.
For more on Curriculum Associates, check out:

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Talking Math: How to Engage Students in Mathematical Discourse

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By Gladis Kersaint - Engaging students in meaningful mathematical discourse is an essential component of students' mathematics learning.

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Take out the numbers to make word problems make sense Learn More Try one tomorrow Primary (K-2) Older Students Explore ready-made problems Problem Banks More Problems This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

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The UDL Guidelines are a tool used in the implementation of Universal Design for Learning, a framework to improve and optimize teaching and learning for all people based on scientific insights into how humans learn. Learn more about the Universal Design for Learning framework from CAST. The UDL Guidelines can be used by educators, curriculum developers, researchers, parents, and anyone else who wants to implement the UDL framework in a learning environment. These guidelines offer a set of concrete suggestions that can be applied to any discipline or domain to ensure that all learners can access and participate in meaningful, challenging learning opportunities.

Reenvisioning the UDL Guidelines

Because the UDL Guidelines are meant to be informed by feedback from the field as well as new research, they have been updated several times in the past. We are in the process of updating the Guidelines once again in our UDL Rising to Equity initiative.

This update will focus specifically on addressing systemic barriers that result in inequitable learning opportunities and outcomes. CAST aims to develop a transparent, inclusive, and community-driven process. If you are interested in collaborating and staying updated on our progress, we invite you to complete a brief survey.

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The UDL Guidelines are a tool used in the implementation of Universal Design for Learning, a framework to improve and optimize teaching and learning for all people based on scientific insights into how humans learn. Learn more about the Universal Design for Learning framework from CAST. The UDL Guidelines can be used by educators, curriculum developers, researchers, parents, and anyone else who wants to implement the UDL framework in a learning environment. These guidelines offer a set of concrete suggestions that can be applied to any discipline or domain to ensure that all learners can access and participate in meaningful, challenging learning opportunities.

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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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California Coalition for Inclusive Learning

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Welcome to the California Coalition for Inclusive Learning (CCIL) universal resources. It is our intent that educators in a variety of roles, find resources to support their practice in universally designing learning experiences for all learners. In anticipation to the barrier of time, we’ve

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Literacy Across Disciplines Got a Minute?

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Literacy Across Disciplines Got a Minute?

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GaM? Slidedeck - Literacy Across Disciplines

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Data talks are short 5-10 minute classroom discussions to help students develop data literacy. This pedagogical strategy is similar in structure to a number talk, but instead of numbers students are shown a data visual and asked what interests them. In a data talk the teacher does not have to be an…

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Goals and Moves | Professional Development | The Inquiry Project

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The Inquiry Project takes a unique approach to a study of matter for grades 3-5, bringing together core ideas, scientific practices, and crosscutting concepts...

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Differentiated Instruction

Journal prompts with multiple entry points allow teachers to differentiate instruction for young learners.

Math journals will look different depending on whose classroom you are walking into. Some colleagues use math journals as a place for students to record problem-solving strategies before sharing out with peers. Others use math journals as a place for students to reflect on their learning or to synthesize the lesson objectives as a form of note-taking.

Journals can also support equity-based mathematics practices-especially when journal prompts have multiple-solution strategies and entry points. In our kindergarten classroom, where students may not have developed the same language or writing skills as some of our older students, we use prompts that invite students to see math in their everyday lives and to illustrate situations that lead to deeper conversations and connections.

What Do Math Journals Look Like in Kindergarten?

Over the past two years, our kindergarten math journals have been a way to capture students' understanding and growth over the course of the school year, allowing teachers, families, and students to see their progress. Kindergarten students arrive at school with a wide range of knowledge, interests, and readiness. To meet the diverse needs of a variety of learners, it's important to provide activities and tasks that allow students to play, discover, and explore while maintaining a high level of cognitive demand.

In our kindergarten classroom, students flow through math centers that include exploratory activities, manipulatives, and/or games. Approximately once a week, one of those centers involves a math journal prompt that connects to the current learning target. Journal prompts are open-ended and offer multiple entry points, allowing all students to start from where they are and feel successful in doing so.

Students are excited to use their journals as a place to draw, write, and express their ideas, and they feel a sense of ownership and pride over these journals. Before the journal is presented, teachers prepare follow-up questions that help push student thinking as needed. Students talk with each other as they work, discussing their ideas and pictures, which helps create a community of learners who view math as more than worksheets and answer getting. Teachers are available to take notes or voice-record as they listen in on conversations to document discoveries and noticings.

What Makes a Good Journal Prompt?

Kindergarten math prompts should be short, be easy to explain, and allow for students to connect math to their lives outside of school. Prompts that could be answered with multiple-solution strategies or allow students to approach the situations in different ways provide opportunities for rich discussions.

Even at this young age, we can see students developing the habits and behaviors of good mathematicians. In these journals, children are using pictures, words, and numbers to make sense of the problem, use what they know, and connect math to their everyday lives (National Council of Teachers of Mathematics Standards for Mathematical Practice MP2 and MP4).

Students might draw pictures or use objects to explain their strategies, especially when the language to do so might not yet be developed ( MP3). This type of problem-solving, sense-making, and communication are vital for supporting students' development as mathematicians.

Journal Examples

The Counting Sequence: One of our math journal prompts at the start of the year is " Write your numbers in order as high as you can go," which can provide a small window into student thinking and understanding. Teachers are able to look at how students write and organize numbers and track any reversals.

As students write their numbers, they are encouraged to use tools around the classroom, such as number lines, labels, and clocks, to help refer to numbers they may know how to say but not yet write. These moments, in which students make their own discoveries, allow them to feel empowered and have lasting benefits.

Composing and Decomposing Five: As students are learning how to decompose numbers to five, the following math journal prompt allows students to practice this skill: Sara made a train using five snap cubes. She used cubes of two colors. What might the train look like? Draw the train.

Students are presented with snap cubes to explore this task and make discoveries on their own before recording their findings in their journals. As students draw their combinations, follow-up questions include "Are there any other combinations you can make?" and "How do you know you have found them all?"

More and Less: As students are learning how to compare numbers to 10 and use vocabulary such as more and less, we use this journal prompt: Draw eight balloons. Draw less presents. This prompt has led to a lively discussion among the students about how many presents they want to draw as we ask them to prove how they know they have less. As students work on their responses, they discuss how their picture illustrates the prompt and how they know their response is true.

Opportunities for Meaningful Feedback

In addition to documenting progress, journals allow teachers to provide students with meaningful feedback, either written or orally, about their strengths in mathematics. Journals also create a portfolio of work samples, which help families understand what their child is working on in their classrooms and how they are growing and learning over the course of the year. Teachers, parents, and students alike have found math journals to be a wonderful addition to our kindergarten program.

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Using Math Journals in Kindergarten to Deepen Students' Understanding

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Using math journals allows kindergarten teachers to create prompts with multiple entry points, making it easier to differentiate instruction.

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Slow Reveal Graph is an instructional routine that promotes sensemaking about data. This highly engaging routine uses scaffolded visuals and discourse to help students (in K-12 and beyond) make sense of data. As more and more of the graph is revealed, students refine their interpretation and construct meaning, often in surprising ways.

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

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

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51 Brand New Esti-Mysteries! 4 Challenge Levels! All New Images! If you've used the Esti-Mysteries from the original Esti-Mysteries blog post, or the 15 additional Esti-Mysteries that I wrote for 20 Days of Number Sense and Rich Math Talk, you'll really enjoy this new set! My goal was to write and post 51 brand new...

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Learn about CAST’s Universal Design for Learning framework, fostering inclusive educational experiences and improving learning outcomes for all learners.

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Lesson Design Institute
Learn more about the Lesson Design Institute, where you will learn how you can support California educators in high quality lesson design to be shared and showcased with educators across the state.

Learning Resources
Classroom-ready materials from high-quality providers free for educators to blend with their own instructional strategies and share with students and peers.

Collaboration
Group tools help local teams collaboratively develop and share learning materials, instructional strategies, and leadership approaches.

Leadership Support
Resources, programs, and strategies to guide leadership teams in promoting educator excellence, student success, and community support.

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Lesson Design Institute Learn more about the Lesson Design Institute, where you will learn how you can support California educators in high quality lesson design to be shared and showcased with educators across the state. Learning Resources Classroom-ready materials from high-quality providers free for educators to blend with their own instructional strategies and share with students and peers.

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Best AI tools to increase productivity. An organized database that categorizes the top AI tools for you to achieve superhuman abilities. All the latest AI news, trends, and tools.

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Comprehensive Problem Solving 3-read method

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Comprehensive Problem Solving 3-read method

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Building Math Minds - The secrets of fluency

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Building Math Minds - The secrets of fluency

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With the implementation of new standards, assessments, and (for some) new curriculum, teachers are feeling bogged down and yet left out on their own to make sense of it all.

Gain knowledge of how kids learn math so that you can confidently close your textbook and do fun lessons that meet your students where they are.

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Build Math Minds | Learn to LOVE Teaching Math

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With the implementation of new standards, assessments, and (for some) new curriculum, teachers are feeling bogged down and yet left out on their own to make sense of it all. Gain knowledge of how kids learn math so that you can confidently close your textbook and do fun lessons that meet your students where they are.

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Kids compete - Open-answer responses

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The easiest most fun way to practice math facts in a classroom! Instantly engaging and perfect for all-class activities. No students' accounts required.

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"math practice" "math videos" "math lessons" "common core" "3rd grade" "4th grade" "5th grade" "6th grade" "7th grade" "8th grade" "9th grade" algebra geomet...

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Mashup Math is YOUR online guide for teaching and learning mathematics in elementary, middle, and high school. Mathematics can be challenging and you could use a guide. On our channel, you can access an extensive library of cool math videos for visual learners and professional development and math activities for teachers and parents.

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Post By: Anthony Persico

Are you looking for the best free 8th grade math resources, lesson plans, activity ideas, and worksheets all in one place? The following list shares some of the best, most popular, and free math resources (like free 8th grade math worksheets and answer keys) for 8th grade teachers and parents for in the classroom, remote learning, and homeschool as well. Enjoy!

Are you looking for fun and engaging activities for helping your kids understand 》See all 5 Scientific Notation activities. scientific notation? You can use the following activity ideas to differentiate your instruction and help your kids to gain a deeper conceptual understanding of this challenging topic.

Looking for More Awesome 8th Grade Math Resources?

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The Best Free 8th Grade Math Resources: Complete List! - Mashup Math

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Post By: Anthony Persico Are you looking for the best free 8th grade math resources, lesson plans, activity ideas, and worksheets all in one place? The following list shares some of the best, most popular, and free math resources (like free 8th grade math worksheets and answer keys) for 8th grade teachers and parents for in the classroom, remote learning, and homeschool as well.

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Post By: Anthony Persico

Are you looking for the best free 7th grade math resources, lesson plans, activity ideas, and worksheets all in one place? The following list shares some of the best, most popular, and free math resources (like free 7th grade math worksheets and answer keys) for 7th grade teachers and parents for in the classroom, remote learning, and homeschool as well. Enjoy!

When it comes time to help your kids gain a deep conceptual understanding of measures of central tendency (mean, median, mode, and range), engaging hands-on activities work best! This fun hands-on activity requires a deck of playing cards!
》Get your free Mean, Median, and Mode Activity Lesson Guide.

Looking for More Awesome 7th Grade Math Resources?

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The Best Free 7th Grade Math Resources: Complete List! - Mashup Math

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Post By: Anthony Persico Are you looking for the best free 7th grade math resources, lesson plans, activity ideas, and worksheets all in one place? The following list shares some of the best, most popular, and free math resources (like free 7th grade math worksheets and answer keys) for 7th grade teachers and parents for in the classroom, remote learning, and homeschool as well.

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Post By: Anthony Persico

Are you looking for the best free 6th grade math resources, lesson plans, activity ideas, and worksheets all in one place? The following list shares some of the best, most popular, and free math resources (like free sixth grade math worksheets and answer keys) for 6th grade teachers and parents for in the classroom, remote learning, and homeschool as well. Enjoy!

The following collection of the 5 best 6th grade algebra resources is a great starting point for helping grade 6 students to begin learning algebra concepts and topics such as solving equations, using algebraic properties, and graphing lines and functions. 》Get Your Free 6th Grade Algebra Resources.

When it comes time to help your kids gain a deep conceptual understanding of measures of central tendency (mean, median, mode, and range), engaging hands-on activities work best! This fun hands-on activity requires a deck of playing cards!
》Get your free Mean, Median, and Mode Activity Lesson Guide.

Use these supplemental resources to support your lesson on place value and writing numbers in standard and expanded form. This collection includes a free printable place-value chart and video lesson.
》 Get your free place value pdf chart.

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The Best Free 6th Grade Math Resources: Complete List! - Mashup Math

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Post By: Anthony Persico Are you looking for the best free 6th grade math resources, lesson plans, activity ideas, and worksheets all in one place? The following list shares some of the best, most popular, and free math resources (like free sixth grade math worksheets and answer keys) for 6th grade teachers and parents for in the classroom, remote learning, and homeschool as well.

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Post By: Anthony Persico

Are you looking for the best free 5th grade math resources, lesson plans, activity ideas, and worksheets all in one place? The following list shares some of the best, most popular, and free math resources (like free fifth grade math worksheets for all topics) for 5th grade teachers and parents for in the classroom, remote learning, and homeschool as well. Enjoy!

Do your 5th grade students need extra practice with solving math problems involving topics like multi-digit operations, measurement, rounding, and fractions? These free 5th grade math word problems are available as easy-to-print pdf files and include complete answer keys. 》Access your free 5th grade word problems.

Use these supplemental resources to support your lesson on place value and writing numbers in standard and expanded form. This collection includes a free printable place-value chart and video lesson.
》 Get your free place value pdf chart.

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The Best Free 5th Grade Math Resources: Complete List! - Mashup Math

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Post By: Anthony Persico Are you looking for the best free 5th grade math resources, lesson plans, activity ideas, and worksheets all in one place? The following list shares some of the best, most popular, and free math resources (like free fifth grade math worksheets for all topics) for 5th grade teachers and parents for in the classroom, remote learning, and homeschool as well.

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Post By: Anthony Persico

Are you looking for the best free 4th grade math resources, lesson plans, activity ideas, and worksheets all in one place? The following list shares some of the best, most popular, and free math resources (like free fourth grade math word problems worksheets) for 4th grade teachers and parents for in the classroom, remote learning, and homeschool as well. Enjoy!

Do your 4th grade students need extra practice with solving math problems involving topics like multi-digit operations, measurement, rounding, and fractions? These free 4th grade math word problems are available as easy-to-print pdf files and include complete answer keys. 》Access your free 4th grade word problems.

Looking for More Awesome 4th Grade Math Resources?

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The Best Free 4th Grade Math Resources: Complete List! - Mashup Math

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Post By: Anthony Persico Are you looking for the best free 4th grade math resources, lesson plans, activity ideas, and worksheets all in one place? The following list shares some of the best, most popular, and free math resources (like free fourth grade math word problems worksheets) for 4th grade teachers and parents for in the classroom, remote learning, and homeschool as well.

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Post By: Anthony Persico

Are you looking for the best free 3rd grade math resources, lesson plans, activity ideas, and worksheets all in one place? The following list shares some of the best, most popular, and free math resources (like free third grade math word problems worksheets) for 3rd grade teachers and parents for in the classroom, remote learning, and homeschool as well. Enjoy!

Do your 3rd grade students need extra practice with solving math problems involving topics like addition, subtraction, multiplication, division, and measurement? These free 3rd grade math word problems are available as easy-to-print pdf files and include complete answer keys. 》Access your free 3rd grade word problems.

Looking for More Awesome 3rd Grade Math Resources?

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The Best Free 3rd Grade Math Resources: Complete List! - Mashup Math

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Post By: Anthony Persico Are you looking for the best free 3rd grade math resources, lesson plans, activity ideas, and worksheets all in one place? The following list shares some of the best, most popular, and free math resources (like free third grade math word problems worksheets) for 3rd grade teachers and parents for in the classroom, remote learning, and homeschool as well.

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Post By: Anthony Persico

Are you looking for the best free 2nd grade math resources, lesson plans, activity ideas, and worksheets all in one place? The following list shares some of the best, most popular, and free math resources (like free second grade math word problems worksheets) for 2nd grade teachers and parents for in the classroom, remote learning, and homeschool as well. Enjoy!

Do your 2nd grade students need extra practice with solving math problems involving topics like addition, subtraction, measurement, and time? These free 2nd grade math word problems are available as easy-to-print pdf files and include complete answer keys. 》Access your free 2nd grade word problems.

Looking for More Awesome 2nd Grade Math Resources?

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The Best Free 2nd Grade Math Resources: Complete List! - Mashup Math

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Post By: Anthony Persico Are you looking for the best free 2nd grade math resources, lesson plans, activity ideas, and worksheets all in one place? The following list shares some of the best, most popular, and free math resources (like free second grade math word problems worksheets) for 2nd grade teachers and parents for in the classroom, remote learning, and homeschool as well.

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Post By: Anthony Persico

Are you looking for the best free first grade math resources, lesson plans, activity ideas, and worksheets all in one place? The following list shares some of the best (and 100% free) math resources (like free worksheets and math riddles) for 1st grade teachers and parents for in the classroom and homeschool as well. Enjoy!

Looking for More Awesome 1st Grade Math Resources?

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The Best Free First Grade Math Resources: Complete List! - Mashup Math

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Post By: Anthony Persico Are you looking for the best free first grade math resources, lesson plans, activity ideas, and worksheets all in one place? The following list shares some of the best (and 100% free) math resources (like free worksheets and math riddles) for 1st grade teachers and parents for in the classroom and homeschool as well.

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Pre-K games and activities

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Translate the Activities We have translations to Catalan, Chinese, Danish, Dutch, and Spanish in the works. If you are interested in volunteering to translate to other languages, or help speed us along in those five languages, please contact us.

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Jon Orr and Kyle Pearce

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The Make Math Moments Academy

YOUR ROADMAP TO AN INSPIRING MATH CLASS
Create memorable math moments daily with access to curiosity rich tasks, in-depth PD video trainings, and a system for making these moments a reality. Join the Making Math Moments Academy now. Your first 30 days are for free.

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Make Math Moments | Empowering Teachers with Math Activities

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The Make Math Moments Academy YOUR ROADMAP TO AN INSPIRING MATH CLASS Create memorable math moments daily with access to curiosity rich tasks, in-depth PD video trainings, and a system for making these moments a reality. Join the Making Math Moments Academy now. Your first 30 days are for free.

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Liesl McConchie

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math with the brain in mind Transform your teaching to close the achievement gap for secondary math students. FREE Special Report: 7 Powerful Tools to Motivate Secondary Math Students Get it NOW Want More? Get FREE articles with tools you can use today! Learn the science of learning from my book.

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Math Talks Hub - PYLUSD Rio Vista Elementary

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Math Talks Hub - PYLUSD Rio Vista Elementary

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If you love Tetris, you will LOVE this game that helps solidify math facts in a spatial way.

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Tutorial

Factris is a classic risk reward high score game. Just like in Tetris, you have to arrange falling blocks using the arrow keys, and when you fill an entire row, it disappears to make space for new blocks. However, there are two key differences:

The blocks are always rectangles of varying sizes. Rather than rotating them, you can "refactor" them. For example, a block might start out as 12×1 tiles, but by pressing the up key, you can successively change its size to 6×2, 4×3, 3×4, 2×6 and 1×12.

You can exponentially more points if you complete multiple rows at once. Every tile in the first row is worth 10 points, every tile in the second row is worth 20 points, then 40 points, 80 points, 160 points, and so on.

There is no undo, so you need to pay close attention to the size of the rectangles coming next. Improve your numeracy, packing and knowledge of factors with Factris!
Factris was invented by Richard Lissaman at MEI. This version was designed and developed by Philipp Legner.

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Factris is a fun game that teaches about simple arithmetic, divisibility and factorisation.

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From Prime Factorization through Fractions

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Learn about alternate ways to introduce the Math Building Blocks to students. Download a version of the building blocks grid that uses symbols instead of colors. Watch slide shows that use the blocks to illustrate a modified version of the Sieve of Eratosthenes. Play a prime factorization game on the computer.

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12 Reasons to Notice and Wonder in your Classroom

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12 Reasons to Notice and Wonder in your Classroom

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12 reasons for Notice & Wonder routines in your class | Maths Pathway

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We know that Notice & Wonder is a great value add to the maths teaching toolkit. Let's recap before we delve into more ideas around the practice.

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Christopher Danielson

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Parents know that we need to read 20 minutes a day with our kids.

But teaching math to your kids doesn't have to be intimidating. It can be fun. Just as children say goodnight to the moon, they can say hello to numbers, shapes and measurement.

Talking Math with Your Kids provides ideas and products that get children and caregivers talking about math with creativity and joy.

Whether you're reading books on the couch, cooking dinner, running errands, or scheduling your day, opportunities for math talk abound.

This website collects research about, and examples of, math talk with children. Each post offers suggestions for adapting the examples for your own children, interests, and lives.

Enjoy and let me know how those conversations are going in your home. Use the contact form to ask questions, send feedback or to send along a conversation for highlighting on this site.
Thanks for reading. Read more about Christopher

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Parents know that we need to read 20 minutes a day with our kids. But teaching math to your kids doesn't have to be intimidating. It can be fun. Just as children say goodnight to the moon, they can say hello to numbers, shapes and measurement.

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Make Math Playful is an unofficial slogan here at Talking Math with Your Kids. An important part of play is that there is not one right answer. Through Which One Doesn't Belong, I showed a way to make geometry playful. Now with How Many? I'm working on a way of making counting playful.

The idea has grown out of the TED-Ed video I did a while back, and the more I play with it, the more I see it in the world around me. My goal is to help parents, teachers, and especially children see it too.

Most counting tasks tell you what to count. Whether it's Sandra Boynton's adorable board book Doggies, or Greg Tang's more sophisticated The Grapes of Math, the authors tell you what to count-or even count it for you.

How Many? is a counting book that leaves possibilities open and that seeks to create conversations. Creativity is encouraged. Surprises abound.

The premise is simple. Every page asks How Many? but doesn't specify what to count. Each image has many possibilities.

An example. How many?

Maybe you say two. Two shoes. Or one because there is one pair of shoes, or one shoebox. Maybe you count shoelaces or aglets or eyelets (2, 4, and 20, respectively). The longer you linger, the more possibilities you'll see.

It's important to say what you're counting, and noticing new things to count will lead to new quantities.

Another example. How many?

A few possibilities: 1, 2, 3, 4, 6, 12, 24, 36. What unit is each counting? Maybe you see fractions, too. 2/3, 4/6, 3/4, 1/12....others? What is the whole for each fraction? The number 3 shows up more than once-there are three unsliced pizzas, and there are also three types of pizza. Are there other numbers that count multiple units?

All of this leads to two specific invitations.

Let me come talk with your students.

(It turns out my schedule filled very quickly, and I'm no longer seeking new classrooms to visit right now-thanks to everyone for your support!)
If you are within an hour of the city of Saint Paul and work with children somewhere in the first through fourth grades, then invite me to come test drive some fun and challenging counting tasks with your students. I have set aside November 17 and 18 and hope to get into a variety of classrooms on those two days. Get in touch through the About/Contact page on this blog.
Join the fun on Twitter.

I've been using, and will continue to use and monitor, the hashtag #unitchat, for prompts and discussion of fun and ambiguous counting challenges. Post your thoughts, your own images, the observations of your own children or students, and I'll do likewise.

How Many? A counting book will be published by Stenhouse late next year.

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How Many? An invitation to #unitchat | Talking Math With Your Kids

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Make Math Playful is an unofficial slogan here at Talking Math with Your Kids. An important part of play is that there is not one right answer. Through Which One Doesn't Belong, I showed a way to make geometry playful. Now with How Many? I'm working on a way of making counting playful.

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Mash Up Math

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Writing about math helps kids to organize their thinking, use key vocabulary, and communicate mathematically-which leads to deep and meaningful understanding.

Over the past few years, math teachers are incorporating more writing activities into their lesson plans-a trend that is being driven by the use of highly engaging think-notice-wonder writing prompts that spark deep mathematical discussion and are highly effective as warm-up or cool-down activities.

This strategy has recently been endorsed by the National Council of Teachers of Mathematics:

By asking What do you notice? What do you wonder? we give students opportunities to see problems in big-picture ways, and discover multiple strategies for tackling a problem. Self-confidence, reflective skills, and engagement soar, and students discover that the goal is not to be "over and done," but to realize the many different ways to approach problems.

How does it work?
Math teachers often struggle to find topics for their kids to write about. Sometimes the best way to encourage creativity and exploration is simply posting an image and asking students to describe what they think, notice, and wonder about what they are seeing.

The best way to use think-notice-wonder activities is to choose an image every day and project it as large as you can at the front of your classroom.

Then, have students write 3 sentences about the image starting with:
* I think... * I notice... * I wonder...
✔ You may want to have students share their entries in a daily math journal. This practice will get them used to writing about mathematics regularly.

✔ Try not to give too many prompts. You'll be surprised by how creative and detailed student responses will become over time!

✔ Try to choose images that work with the day's topic/theme
What would think-notice-wonder look like in YOUR classroom?

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Are You Using This Genius Strategy for Math Writing? - Mashup Math

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Writing about math helps kids to organize their thinking, use key vocabulary, and communicate mathematically-which leads to deep and meaningful understanding. Over the past few years, math teachers are incorporating more writing activities into their lesson plans-a trend that is being driven by the

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Tapping into Teen Minds

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Exploring Fraction Constructs and Proportional Reasoning

Fractions are a beast of a concept that causes struggles for many adults and students alike. While we all come to school with some intuition to help us with thinking fractionally and proportionally, the complexity quickly begins to increase as we move from concrete, to visual, to symbolic and from identifying, to comparing, to manipulating. Fractions are formally introduced in the Ontario Math Curriculum when students begin dividing whole objects into pieces and identify these pieces using fractional names (e.g.: halves; fourths or quarters) and continues to promote the development fractional fluency concretely through each grade. Interestingly enough, it is often said that students struggle much more with numbers represented in fraction notation than those represented in decimal notation, yet the word "fraction" appears in the curriculum document 99 times beginning in grade 1, while the word "decimal" appears only 69 times beginning three years later in the 4th grade. While it might be true that fractions tend to intimidate, I wonder if our dependence on the calculator has tricked us into believing we are more fluent with quantities represented in decimal form than is reality.

Paying Attention to Fractions

As I did with the Progression of Proportional Reasoning, I'd like to reference the Paying Attention To Mathematics series released by the Ontario Ministry of Education Literacy and Numeracy Secretariat (LNS) called Paying Attention to Fractions. These guides are a great start to help you wrap your head around big ideas in mathematics and thus, this post will attempt to expand on the ideas shared in this particular document.

Why Fractions?

As if the struggles our students experience when working with fractions aren't enough justification, I like this quote shared in the guide:

"No area of elementary school mathematics is as mathematically rich, cognitively complicated, and difficult to teach as fractions, ratios, and proportionality. These ideas all express mathematical relationships: fractions and ratios are 'relational' numbers. They are the first place in which students encounter numerals like ' 3/4 ' that represent relationships between two discrete or continuous quantities, rather than a single discrete ('three apples') or continuous quantity ('4 inches of rope')."

(Litwiller & Bright, 2002, p. 3)

What Is A Fraction?

A fraction is a number.

While many think of fractions and fractional notation as a means to represent quantities that are not whole, all quantities can be represented as a fraction.

While these descriptions are simple on the surface, they do not appropriately communicate the complexity throughout the progression of fractions and the many big ideas connected to this progression.

Connecting Proportional Reasoning and Fractions

If you recall from my last post, proportional reasoning takes place when one compares two quantities in relative terms rather than absolute terms. Thus, it can be said that a fraction provides a means for representing that relationship between two numbers. However, it is important not to confuse that while the numerator and denominator can be thought of as two quantities that are related, a fraction is a single quantity that can be represented on a number line.

So where does the progression of fractions begin?

As it turns out, pretty early on!

Young learners begin their journey through the progression of fractions in tandem with the progression of proportional reasoning as they make comparisons between two (or more) quantities such as:

this bag is heavier than that one;

I'm "this much" shorter than you are; and,

there are 5 more red candies than green.

While these early comparisons are likely absolute (i.e.: additive) in nature, we are one step closer to comparing these quantities relatively (i.e.: multiplicatively). This multiplicative comparison between two or more quantities is called a ratio.

In these early stages of the progression of fractions, young learners often begin thinking fractionally as they explore geometric quantities like volume and area; comparing a part relative to its whole.

Counting Cartons: Part-Whole Relationships

Since the progression of fractions begins for most by thinking of a part relative to a whole as a means to measure as well as the lack of conceptual clarity related to fractions and proportional reasoning, it should be no surprise that many think of fractions narrowly as a part-whole relationship alone.

Let's start our exploration of fraction constructs beginning with the most common part-whole fraction.

Have a look at the image below:

I might ask students:

How many eggs are there?
How many different ways can you represent that quantity?

While the question is clearly asking students to determine the quantity of eggs, asking students to share how many different ways they can represent the quantity encourages students to potentially think beyond counting relative to 1 egg. While we can expect that many students will be able to communicate that there are 18 eggs, others may get creative and unitize (i.e.: 3 groups of 6; 9 groups of 2; etc.) or even reference the number of eggs relative to egg cartons. By asking students to represent the quantity in different ways, we are promoting proportional reasoning by changing the whole that the quantity of eggs is being related to.

After students share out with a partner, I might ask students more specifically:

How many cartons of eggs are there?

This prompt maintains the same "part" (18 eggs), but changes the whole (1 carton rather than 1 egg).

After students have some time to think independently, I would have them share out with their table groups. Some interesting responses are sure to come up such as:

You may notice that by asking students to state how many cartons of eggs there are, there is room for interpretation. Some students might simply state that there are two egg cartons without considering the number of eggs present, some might give a quantity relative to two egg cartons, while others might state the number of eggs relative to a single carton.

After students have been given an opportunity to defend their thinking, I might ask students:

How many cartons are full of eggs?

This fraction construct would be an example of a part-whole relationship. Students might consider how many whole and partial cartons there are in the form of a mixed fraction (1 and 1/2), improper fraction (3/2) or in decimal form (1.5). Some common representations for these different fractional forms might be using a set model, area model or number line. The following animated gif shows each of these representations, sometimes more than one simultaneously:

You might have noticed the double-number line included in the animated gif. This idea can be very useful when making the jump from thinking fractionally to proportional reasoning.

Doritos Roulette: Part-Part Relationships
Note: This section was updated on February 27th, 2021 to better communicate my current and more clarified understanding of fractions and proportional relationships that I continue to develop.
In the Doritos Roulette 3 Act Math Task, students are asked questions that stem from the relationship of "hot" chips to "not hot" chips based on the image on the front of the bag:

I might ask students to consider the image above and ask them this question:

What ratio of hot chips to not hot chips are there in a bag of Doritos Roulette?

and,

Can I represent this ratio as a fraction?

This string of questioning is helpful for emerging some of the muddiness that exists when it comes to understanding fractions and proportional relationships.

Despite the explicit request for a part-part relationship between "hot" chips (1) and "not hot" chips (6), we often see and/or hear many share the relationship between "hot" chips (1) and the whole (7).

When asking a group of students or educators to discuss with their neighbours about whether a part-part ratio relationship can be represented a fraction or not. Great (and sometimes heated) discussions can arise from this question, which makes this exciting to facilitate.

Based on the explanation given in the Paying Attention to Fractions document referenced earlier, I used to believe that a ratio is a fraction. However, after much learning and finally coming across the work of John Van De Walle et al and their book Elementary and Middle School Mathematics: Teaching Developmentally, I now realize that while a ratio can be represented using a fraction bar, it is not a fraction.

It is worth noting that while a part-part ratio can be represented using the fraction bar, note that this construct is representing two separate quantities where the fraction bar is used as an operator to divide.

In other words, when we see fraction notation used to represent this part-part ratio as 1/6, what this is representing is:

1 hot chip ÷ 6 not hot chips

Partitively dividing these two quantities in ratio will reveal a rate:

1 hot chip per not hot chip

The rate above represents a single quantity with a compound unit of hot chips per not hot chips.

Fun stuff, right?

I know!

All of this thinking is what fuelled me to create a full self-paced online proportional relationships course to help unpack the roadmap of proportional reasoning from spatial comparison all the way to proportional relationships.

After exploring part-whole fraction construct and representing a part-part ratio using the fraction bar for the purpose of dividing, you may begin to notice how important clarity around what a given fraction represents prior to attempting any comparison or manipulation of that fraction.

While the use of a set model and area model remain fairly similar regardless of whether you are dealing with a part-whole or part-part relationship, the use of a number line changes significantly.

Recall how the number line used in the part-whole counting cartons example made identifying the part and the whole fairly intuitive:

When we attempt to use a number line with a part-part ratio relationship, you'll notice that we cannot locate a single value on the number line because a part-part relationship is two separate quantities representing two different places along the number line:

Note that in the representation above, two parts are clearly shown and the sum of those two parts represents the whole.

Although the number line representation might not be as intuitive in the part-part case, you can visualize both parts and their sum represents the whole:

Extending from a single number line to a double-number line can help the learner see the part-part ratio relationship and easily scale both quantities in tandem. At any point along the double number line (or later, the ratio table) you can use partitive division to divide the parts to reveal a rate:

1/6 hot chips per not hot chip or 6 not hot chips per hot chip

In this case, there are one-sixth as many "hot" chips as there are "not hot" chips. Alternatively, there are 6 times as many "not hot" chips as there are "hot" chips.

Baking Brownies: Fraction as Quotient

As mentioned above, a part-part ratio relationship can be represented with a fraction bar when the intention is for two separate quantities to be divided. In other words, when we are using fraction as a quotient. The Paying Attention to Fractions document uses an example of kids fair sharing some brownies, so we'll stick to that and add some visuals to help.

Suppose there are 6 brownies that are to be shared amongst 4 friends fairly. How much should each person receive?

Like many situations in life, there are multiple ways to share these brownies fairly. Let's represent a few of these methods.

Using Partitive Division to Partition the Brownies into Fourths

Since there are 4 people to share the 6 brownies, we can divide the 6 brownies into fourths and then pass them out, one-by-one:

After splitting the brownies into quarters, you'll notice that each of the four people should receive 6 quarters. Hence why we can represent this fraction as quotient as the improper fraction, 6/4.

Sharing Full Brownies Equally, Then Partitioning Remainder

Some people might find it useful to share whole brownies equally and then when there are not enough whole brownies to share fairly, consider partitioning up the remaining brownies appropriately:

In the case with 6 brownies, we can share one brownie each and then partition the remaining brownies into four halves to share. This method would be equivalent to the mixed fraction of 1 and 1/2 brownies per person.

Partition Brownies Into Four Equal Portions

This method involves considering all 6 brownies and where cuts must be made to create four equal groupings:

This method would be another representation of the mixed fraction, 1 and 1/2 brownies for each person.

For each of the methods above, we could also consider the use of a number line to support the partitioning of the brownies to distribute a fair amount to each person. Here are a few examples:

Fraction as Operator

Another common construct is using a fraction as operator where a fraction is used to act as a function to:

scale a quantity larger;

scale a quantity smaller; or,

find equivalent fractional quantities.

Using a fraction as operator to scale a quantity larger or smaller feels pretty intuitive.

Consider a $100 investment.

Over time, this investment could increase in value or decrease in value.

If the investment decreases to $50, we could say that the investment "halved" from its initial value of $100.

In other words:

1/2 of $100 = $50

It is also possible that the $100 investment increases in value to $150.

In that case, we could say:

3/2 of $100 = $150

In both of the above cases, the fraction is still a part-whole relationship, however the intent is to use the fraction for scaling.

While it seems natural in both cases to think of the operation as simply multiplication, we can approach both cases through multiplication and/or division.

For example, in the second case, we could arrive at the equivalent result of $150 through each of these approaches:

3/2 of $100

$100 multiply by 3, divide by 2

$100 divide by 2, multiply by 3

Thus, the intent of the fraction 3/2 was to use as a function for operating.

Exploring Fractions Developmentally

Now that we have an understanding of the different fraction constructs and when we might bump into those situations, many are likely wondering how students will develop their understanding of fractions.

As mentioned earlier, students will likely bump into fractions through experiences in the world around them. For example, seeing that half of a brownie tray is missing, one fourth of a glass of milk remains, or each of three children will receive one third of the share of chocolates.

Exploring part-whole fractions beginning with unit fractions through counting can be helpful for building an understanding of the relationship of the part to the whole. This can be done through mathematical modelling using area models such as a bar model to encourage representing, comparing and ordering of fractions.

How Many Rolos: Adding and Subtracting Fractions

Let's have a look at another problem. We mind as well keep the context as food, because that is just yummy.

There are 7 pieces in every full roll of Rolo chocolate.

Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces.

How many full rolls of Rolo are there?

If we use concrete manipulatives or visuals, like we will in this instance, it is much easier to see how to add and subtract fractions. In this case, we'll show a few different representations including how using a number line can be helpful:

So in this particular example, you can see that 5 pieces plus 4 pieces will yield 9 pieces, which is more than a full 7 piece roll; 9/7 or 1 and 2/7.

Gimme a Break: From Adding and Subtracting to Multiplying Fractions

If you're a 3 act math fan, you might be familiar with my Gimme a Break 3 act math task involving the use of fractions as operators. I should note that while there are some useful pieces in that task, it has really evolved over time and I just haven't had the time to update the post yet. Here's some of the pieces I've added along the way.

The task begins with an opportunity for students to notice/wonder after watching a video of me opening a KitKat bar.

Later in the task, the following video is shown asking students to consider what operation could be taking place:

Since the video is open ended, I'm always hoping that all students will be able to share something mathematical about the situation. For example, we might see some responses like these:

However, the new learning I'm hoping to spend some time on is the multiplication of a fraction by a whole number. In this case, a student might think of this situation as 4 groups of one quarter of a whole bar:

Alternatively, we could think of this situation as one quarter groups of four whole KitKat bars:

In either case, the use of arrays and/or area models can be very useful when trying to help build an understanding of multiplication with fractions. In the following example, we extend this idea to the multiplication of two fractions:

While I find that multiplication of fractions seems to be the one operator I hear the least noise about, my gut tells me that it is more about being able to remember the algorithm and not due to any conceptual understanding. In my experience, many students can manage to remember to "multiply the tops, multiply the bottoms" without necessarily having any real conceptual understanding to fall back on in times of doubt.

Dividing Whole Numbers by Fractions and Fractions by Fractions

On the other hand, division of fractions seems to be a sore spots for many students (and adults) because the algorithm is a bit more complicated and less intuitive. Pair that with a very low level of conceptual understanding as to why the algorithm actually works and you'll end up with confusion and frustration for many. The gap between the pure memorization of steps and the conceptual understanding of what dividing fractions really means is often so large that many educators simply follow suit and focus purely on the algorithm. While I believe it is so important to build the conceptual understanding behind a complex idea like dividing fractions, I know that the task can be quite difficult and many teachers aren't so sure how to approach it. When we rush to an algorithm before constructing conceptual understanding, students (and adults) are often very difficult to engage in the heavy thinking required to connect the dots. This leaves many teachers believing that successful use of the algorithm is enough (or all they have time for) and they just move on to the next topic.

Let's look at the Gimme a Break context in terms of division of fractions. With manipulatives on the table such as pattern blocks or relational rods, I might ask students:

What would 1 ÷ 1/4 "look like" in terms of the KitKat task?

Can you show a neighbour what that looks like to you?

Hopefully students come up with some representation to model what a whole KitKat bar would look like an a quarter of a KitKat bar:

Then, I would ask them to determine how many quarter pieces there are in a whole KitKat bar. With this in mind, students should be able to "see" and manipulate their representation on the table to come up with an answer of 4.

Then, I might consider asking students to represent a couple more similar situations using whole numbers divided by a certain number of quarter pieces to build some confidence.

For example, 2 divided by 1/4:

When ready, students might be ready to start stepping outside of whole numbers to situations like 3/4 divided by 2/4:

Then, once students are really comfortable dividing fractions with common denominators, you might start moving to situations where the fractions have uncommon denominators.

Note that context is always very important for students to build a conceptual understanding. If we are trying to explain abstraction through abstraction (i.e.: explaining a rule algebraically based on another algebraic rule), then it is unlikely to stick. However, if we, the educators, spend the time necessary to understand mathematics conceptually, it will be a huge pay off for the understanding of our students. Only once students are able to visualize what mathematics "looks like" does it make sense for us to move towards more efficient methods like algorithms or more abstract problems where context is stripped away.

Throughout this post we have tackled the idea of fraction constructs by using a similar approach to how I introduce most big ideas in math class: using math tasks that are contextual, visual and concrete in order to build a conceptual understanding.

This post is by no means the "fractions rulebook", but rather a journal outlining my own journey to understanding fractions conceptually. As a secondary mathematics teacher, much of what our curriculum is built on is the assumed understanding of topics like fractions. However, the more I reflect on my own mathematical understanding, the more I realize that I never truly had a conceptual understanding of most concepts that I was teaching.

Let's avoid the rush to the algorithm and slow down to let kids truly experience and understand mathematics. I'd appreciate any pieces I should add to this post in the comments section.

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The Progression of Fractions | Understanding Fractions Conceptually

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Fractions are a beast of a concept that causes struggles for many adults and students alike. While we all come to school with some intuition to help us with thinking fractionally and proportionally, the complexity quickly begins to increase as we move from concrete, to visual, to symbolic and from identifying, to comparing, to manipulating.

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There is a lot of talk in math education circles about the power of noticing and wondering. And (pardon the pun) it's no wonder! When we ask math students to notice and wonder, we shift the focus from teachers' explanations to students' ideas. Observing (noticing) and questioning (wondering) are simple, powerful habits that enliven and enrich every aspect of instruction. For the teacher, they support formative assessment, lesson design, classroom discourse, differentiation, and the potential for greater rigor. For the student, they develop self-reliance, confidence, curiosity, perseverance, problem-solving, conceptual understanding, and reasoning.

Does it seem strange that such a simple idea has such profound potential? Try it! As you and your students spend more time observing, questioning, and creating, you develop new habits of mind that transform the learning environment, leading to more productive beliefsabout math and how we learn it.

So where to start? A fun and practical approach is to begin with Creative Math Prompts. These are just images-sometimes very simple ones-that you show to students in order to elicit their observations and questions. Images like these open a space for mathematical creativity. The immediate goal is simple-to get students comfortable expressing their thoughts and realizing that their ideas matter. Ultimately, you would like the habits of noticing, wondering, and creating to extend beyond the prompts and become an everyday part of your teaching and your students' thinking.

Please bear in mind that, in keeping with the theme of this website, the Creative Math Prompts below are designed with advanced learners in mind. However, prompts like these are appropriate for all learners, and I encourage you to try them with others! You will discover students whose mathematical insights and creativity blossom even though their strengths may not show up well in traditional tasks and assessments. If you find that some prompts are too complex or advanced, you can usually modify them to suit other needs. All learners benefit from noticing, wondering, and creating!

Read about tips and suggestions for Using Creative Prompts.
See the Click on an image to get more information about a prompt. CMP Content Guide to help you align Creative Math Prompts with the content that you teach.

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There is a lot of talk in math education circles about the power of noticing and wondering. And (pardon the pun) it's no wonder! When we ask math students to notice and wonder, we shift the focus from teachers' explanations to students' ideas.

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NCTM Resources

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When students become active doers of mathematics, the greatest gains of their mathematical thinking can be realized. The process of sense making begins when we create classrooms full of curious students' thoughts and ideas.

By asking "What do you notice? What do you wonder?" we help our students see problems in big-picture ways.

Self-confidence, reflective skills, and engagement soar, and students discover that the goal is not to be "over and done," but to discover multiple strategies for tackling a problem.

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Notice and Wonder - National Council of Teachers of Mathematics

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What is Notice and Wonder? When students become active doers of mathematics, the greatest gains of their mathematical thinking can be realized. The process of sense making begins when we create classrooms full of curious students' thoughts and ideas. By asking " What do you notice? What do you wonder?"

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The Brain Roots of Unproductive Struggle in Math (and 3 Tools to Ensure it Stays Productive) - Liesl McConchie

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You've likely seen it a hundred times. A student is working diligently on a math problem. They're leaning in, scribbling possibilities frantically. They pause to look up toward the ceiling for a moment. Then back to their paper with renewed gusto. And then it happens. The pencil is forcefully dropped on the desk.

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Ditch that Textbook

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Virtual field trips allow us to take our students on adventures we never thought possible. We can take our students under the ocean, across the globe and

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Interesting Subtraction Problems - Harmonic Triangle Challenge

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Interesting Subtraction Problems - Harmonic Triangle Challenge

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Grades 5+

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I'm exploring making a different kind of instructional video right now, to help motivate some rich and interesting approaches to skills practice for all the kids now at home.

Here's the idea: rather than watch a video on how to subtract fractions, followed by a page (or website) of exercises, what if there were an interesting challenge that would motivate you to want to learn about how to subtract fractions? Then the video could connect the skill and the challenge, and everyone would be more motivated to learn it.

That's the theory, anyway. Doing it this way over video is a bit of an experiment. There's a fair amount of finesse that I can employ in person in a classroom. But I'm hoping this will help!

Please comment to let me know if this format works.

I'd recommend starting with just the PDF of the question. If that's too hard, that's okay - I've got a video explanation below that. Actually learning to subtract fractions in a meaningful way and getting a bunch of practice is the main goal. But there's a fascinating question to motivate it, and making predictions and conjectures as you and your kids notice patterns make the experience richer.

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Harmonic Triangle Challenge & fraction subtraction

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I'm exploring making a different kind of instructional video right now, to help motivate some rich and interesting approaches to skills practice for all the kids now at home. Here's the idea: rather than watch a video on how to subtract fractions, followed by a page (or website) of exercises, what if there were an interesting challenge that would motivate you to want to learn about how to subtract fractions?

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Interesting Subtraction Problems - Diffy Squares

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Interesting Subtraction Problems - Diffy Squares

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Grades 2+

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When you want a problem that won't quit, it's hard to beat Diffy Squares. This crazy little enigma has had more folks doodling squares and subtracting through boring meetings, searching for a diffy square that will last longer than any other.

There's a lot of deep mathematics here, and it's possible to find strategies and solutions to "solve" diffy squares. But the best part about diffy squares is trying to beat your old record, and hitting a wall at level 7 or 8. This will give your kids an epic reason to keep doing subtraction practice. So I'm not going to give an answer, but please share your best Diffy records in the comments. To get started, check out the video and read the PDF below.

Great for 2nd grade and up.

(It'll be a serious challenge for motivated students who know some algebra to find real numbers to put in a diffy square so that it NEVER terminates.)

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When you want a problem that won't quit, it's hard to beat Diffy Squares. This crazy little enigma has had more folks doodling squares and subtracting through boring meetings, searching for a diffy square that will last longer than any other.

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Interesting Subtraction Problems - Subtracting Reverses

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Interesting Subtraction Problems - Subtracting Reverses

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Grades 2+

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Another video challenge to share, called Subtracting Reverses. Once again, a strange little mystery can motivate a ton of subtraction practice. The subtraction here is focused on 2-digit minus 2-digit examples. However, by coloring a hundred chart as you go, you can find really interesting patterns, many of which you may be able to explain.

If you want to take this problem to the next level, try reversing 3-digit numbers and see what happens when you take the difference!

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Another video challenge to share, called Subtracting Reverses. Once again, a strange little mystery can motivate a ton of subtraction practice. The subtraction here is focused on 2-digit minus 2-digit examples. However, by coloring a hundred chart as you go, you can find really interesting patterns, many of which you may be able to explain.

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Interesting Subtraction Problems - Flower Petal Problems

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Interesting Subtraction Problems - Flower Petal Problems

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Grades 1+

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Draw the "flower" drawing and explain that it is possible to put the numbers 1 through 7 into each "petal," using each number exactly once, so that each straight line adds up to 10. Take suggestions from students about which number could do where, making sure that a "1" does NOT go in the center. With student input, get as far as you can until you are stuck. Then distribute the template and let students try on their own.
Example Conversation
Teacher: I've heard that it is possible to put the numbers 1 to 7 into the petals of this flower to make every line of three petals add up to 10. How can we do it, I wonder? What number could we put in the middle?
Student: Three?
Teacher: Let's try it. I'll cross the "3" off my list of
numbers, since I want to use each number only once. Now what could I put in this next petal?
Student: I don't know.
Teacher: I don't either. But let's try something and see if it will work. We can always erase later.
Student: What about six?
Teacher (writing in 6, and crossing it out from the list of
numbers): Ok, now we have a 6 and a 3 in one row. What should we put in the last petal to make 10? In other words, if we add up 6 and 3, what more do we need to add to make 10? Think about it, then pair and share.

[Kids think and discuss]
Student: It has to be 1.
Teacher: How do you know?
Student: Because 6 + 3 is 9, and then we need one more to make 10.
Teacher: Aha! I'll put that in too, then. So what can go here?

[Teacher continues until it becomes clear that the flower petal puzzle CANNOT be solved.]
Teacher: Hmmmm... it looks like this isn't going to work. Maybe we should have put a different number in the middle.

What number would you try? [Students give suggestions.] You know what, let's go ahead and try it out on our own. I've got empty flower petal pictures right here (passes out the empty template). Try out different numbers in the petals and see if you can get each row to add up to 10. When you can solve that, I've got some followup challenges for you too.

Challenge 1: All lines add to 10.

Challenge 2: All lines add to 12.

Challenge 3: All lines add to 14.

Challenge 4: Students make their own Flower Petal Puzzles.

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Draw the "flower" drawing and explain that it is possible to put the numbers 1 through 7 into each "petal," using each number exactly once, so that each straight line adds up to 10. Take suggestions from students about which number could do where, making sure that a "1" does NOT go in the center.

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SFUSD Math Elementary School Closure Lessons

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Learn to Think Games

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Brilliant - Build quantitative skills in math, science, and computer science with hands-on, interactive lessons.

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Kids are distracted when they are in discomfort [bored/anxious].

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It's a deafening roar: 35,000 screaming spectators have waited all year for this event. No, it is not baseball or football. It is an epic, old-school tug-of-war across the 2nd longest river in the U.S.A. Every August, a half-mile of rope (9 football fields) is stretched across the Mississippi River between Illinois and Iowa. Even if you're never actually "played" or watched a tug-of-war you likely can guess how it works. Illinois residents tug on one side of the rope as Iowa residents do the same from the other side of the river. For one weekend, teams work to pull the other team off balance and knock them off their feet. The reward? Aside from bragging rights, there's a beautiful Alabaster Eagle trophy ... and it's a great fundraiser.

This year, the epic match of tug-of-war is in your brain between focus and distraction. Perhaps it has never been tougher, both for you and your students. You may have felt "pulled off-balance" more than usual.

This month's newsletter focuses on the science of distraction. Why are we narrowing in on distractions, and not just how to better focus? Because understanding your "opponent" is a vital step to winning any battle. We'll divulge the actual source of distraction, along with the science of how and why the brain gets distracted. Ultimately, we'll share a powerful, yet simple 3-part tool to make the "focus magic" happen.

What is a distraction?

A distraction is anything that pulls you AWAY from what you are doing.

Imagine a neurological game of tug-of-war involving multiple systems of the brain - the attentional system, emotional systems, working memory, executive functions, and more. As you've noticed in the past, they don't always play nice together (Shafer & Dolcos, 2012).

Whether these systems are working for you (giving you a healthy, "sticky" traction) or against you (as an annoying distraction) depends on the skills you possess. You and your students can retrain your brain to get what you want.

WHAT Pulls Students Away from the Task

Contrary to popular belief, students are not distracted by some " THING" that is capturing their attention. What usually pulls students away from the task at hand is a feeling of discomfort. Our brain is typically moving away FROM perceived boredom/pain TOWARD pleasure/novelty (Kafkas & Montaldi, 2018).

Here is a ground-breaking shift in thinking for you: Next time you feel distracted, instead of naming the thing that is distracting you, identify the distracting feeling.

A student who veers away from an assignment might feel bored enough to search for the latest dance craze on TikTok or see what is trending (novelty) on Netflix. A student who feels lonely or disconnected from friends will often reach for their social media app of choice for a sense of connection. The device itself is not the distraction. The distraction that students choose can temporarily alleviate their boredom or feeling lonely or... you name it.

If students perceive their learning experience as boring or painful, they will naturally seek out something more enjoyable or novel.

WHY Students Get Pulled Toward Distractions

Oftentimes students are distracted by something that is fulfilling a greater need. The well-supported Self-Determination Theory outlines three core psychological needs that drive student engagement and motivation (Ryan & Deci, 2013).

Competence - a sense of being good at something

Autonomy - having a perceived sense of choice

Relatedness - a feeling of connection to others

For many students right now, competence is wavering as they navigate new ways of learning. More restrictions are being placed on them, many of which impact their ability to connect with peers.

When students don't feel good at something (competence), feel they have no choice in the matter (autonomy), and have little/no opportunities to connect with their peers (relatedness), they will be drawn toward distractions that meet those needs. It could be a video game they feel good at (competence), violating a rule so they feel in control (autonomy), or listening to their favorite YouTuber for hours (relatedness).

As you might guess, the goal is to incorporate feelings of competence, autonomy, and relatedness within the focus task.

HOW the Brain Gets Distracted

It would be great if the brain's attentional systems naturally fell in line with your prioritized to-do list. Unfortunately, the priorities of your post-it note often don't align with the priorities of your brain. The brain's attentional systems are biased in layers. The first prioritized layer is survival, everything else is secondary.

A middle school student will be distracted by the movement of nearby students walking outside their class. Their brain says, "Moving objects might be unsafe" and "Are those students safely in my herd or not?". A K-5 student might be distracted by contrast (lights go on or off, a horn, voice, or clapping sound says, "Wake up"). A high school student may be distracted by emotions - laughter, screams, or tears. The untrained brain will always prioritize attentional (safety or danger) triggers.
Once core needs such as safety are felt, and the students feel like they belong, a whole new layer of triggers become relevant. The brain's attentional systems are naturally drawn toward stimuli/tasks that are , , and (Fortenbaugh, DeGutis, & Esterman, 2017).

When the brain perceives a task is highly engaging, it can focus on that goal for extensive amounts of time. You can see this in students spending hours deeply engaged in their music, dance, sports, or theater training.

The brain thrives on relevancy. When pursuing something meaningful, the brain's attentional systems will naturally suppress distractions and give extra focus to a relevant task (Stemmann & Freiwald, 2019). The significance of perceived relevance can be seen in the student who ignores the 50-problem math worksheet to go practice 50 layups in the driveway. The student in the driveway likely believes practicing their layups is more relevant to their current and future goals.

What we pay attention to is driven by rewards. The brain craves rewards (regulated by the dopamine reward system). When the reward is appealing enough, the brain will exert extra resources in pursuit of the reward (Anderson, 2019). Many martial arts programs are built on a system of rewards as you work to earn your next belt. Music classes have the rewards of getting to play in a band or orchestra. In athletics, practice can help you get better. When the coach sees that, you may see the reward of more playing time in the game.

To help your students win the TUG-of-war between focus and distractions, teach them these three simple tools to stay focused. When they sense they are getting distracted, they can reach for these tools and TUG their focus back to the task at hand. Your acronym is T-U-G which stands for "Tiny, Up the Why, and Get up."

Boost the feeling of competence by asking yourself, "What's the smallest chunk I can do right now?" Break a daunting task into tiny chunks. Celebrate the completion of each mini-task. These frequent bursts of dopamine will foster greater motivation to stay on task (Lloyd & Dayan, 2015).

Support students in breaking assignments into small chunks. Instead of writing the assignment as "Complete problems 1-39 odd", consider assigning "1-9 odd, then celebrate; 11-19 odd, then dance for 15 seconds; ..."

Model this approach to chunking in your instruction as well. Deliver small pieces of instruction (7-10 minutes) and then provide students an opportunity to apply, rehearse, or analyze what they just learned. Then jump back into your next chunk of content delivery.

Up the relevance of each task by asking yourself, "Why is this important to me? Why should I do this?" By upping the value of your why you are playing into your brain's natural attentional bias.

Guide your students to reflect on WHY an assignment, subject, or school, in general, is important to them. Encourage them to create a visual cue to remind them of WHY it matters. They could add a doodle to the top of their assignment to remind them why it matters.

Here are visual examples of how to use "Why" reminders: 1) change the background of their phone/computer screen to a picture of the college they aspire to attend, 2) use a photo editing software to put their face into a picture of someone doing the profession they hope to be and 3) create a mini vision board of the change they hope to make to their community or world.

The main idea here is to help students see a greater meaning beyond the individual task. This could also be done by helping students complete statements such as:
This is helping me to ... By doing this I am one step closer to ...
Push back on the discomfort that drives distraction by asking yourself, "How can I re-energize my mind and body right now?" Give your body a short surge of adrenaline and cortisol with a quick movement activity. These work together to boost attention and focus (Donnelly et al., 2016).

Whether you are in-person or virtual, infuse short energizer breaks in your instruction every 15-20 minutes. Here are a couple of new ones you can try:

Have all students stand up and run in place.

Instead of creating more polls on Zoom, have students select their multiple-choice responses by morphing their body into that letter (A, B, C, or D). Think of it as the YMCA of multiple-choice options.

The tug-of-war between focus and distraction can be won. Carve out 5-10 minutes this week to teach your students how to TUG their way to a win for them, for you, and their learning.

Citations:

Anderson, B. A. (2019). Neurobiology of value-driven attention. Current opinion in psychology, 29, 27-33.

Donnelly, J. E., Hillman, C. H., Castelli, D., Etnier, J. L., Lee, S., Tomporowski, P., ... & Szabo-Reed, A. N. (2016). Physical activity, fitness, cognitive function, and academic achievement in children: a systematic review. Medicine and science in sports and exercise, 48(6), 1197.

Fortenbaugh, F. C., DeGutis, J., & Esterman, M. (2017). Recent theoretical, neural, and clinical advances in sustained attention research. Annals of the New York Academy of Sciences, 1396(1), 70.

Kafkas, A., & Montaldi, D. (2018). How do memory systems detect and respond to novelty?. Neuroscience letters, 680, 60-68.

Lloyd, K., & Dayan, P. (2015). Tamping ramping: algorithmic, implementational, and computational explanations of phasic dopamine signals in the accumbens. PLoS computational biology, 11(12), e1004622.

Ryan, R. M., & Deci, E. L. (2013). Toward a social psychology of assimilation: Self-determination theory in cognitive development and education. In B. W. Sokol, F. M. E. Grouzet, U. Muller (Eds.), Self-regulation and autonomy: Social and developmental dimensions of human conduct (pp. 191-207). Cambridge, England: Cambridge University Press.

Shafer, A. T., & Dolcos, F. (2012). Neural correlates of opposing effects of emotional distraction on perception and episodic memory: an event-related FMRI investigation. Frontiers in Integrative Neuroscience, 6, 70.

Stemmann, H., & Freiwald, W. A. (2019). Evidence for an attentional priority map in inferotemporal cortex. Proceedings of the National Academy of Sciences, 116(47), 23797-23805.

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How to Win the Attentional Game of Tug-of-War - Liesl McConchie

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It's a deafening roar: 35,000 screaming spectators have waited all year for this event. No, it is not baseball or football. It is an epic, old-school tug-of-war across the 2nd longest river in the U.S.A. Every August, a half-mile of rope (9 football fields) is stretched across the Mississippi River between Illinois and Iowa.

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RtI Materials for ELA and Math

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Response to Intervention | RTI | RTI Resources | Intervention Central

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Intervention Central is the leading resource for Response to Intervention (RTI) tools and resources, including academic and behavior interventions for classroom management.

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Algebraic thinking and number sense

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Beautiful Math Games and Activities including Multiplication by Heart!

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These websites, apps and magazines showcase the incredible breadth and beauty of mathematics.

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When I look back to my elementary years, one of the most difficult things I was asked to do was memorize my multiplication tables. There were so many of them, and they seemed like strings of random numbers that I had to keep in order and in context with the numbers they were being multiplied by. I managed with the help of my retired teacher mother and some quality time with flash cards, but it wasn't until I was much older that I realized how critical the mastery of multiplication tables is.

Several years ago, I was trying to assist a couple of my neighbor girls who were in middle grade resource classes for failing math. They would come over with their math books and sit at my kitchen table. After a few minutes of watching them stare at the problems with a hopeless look in their eyes, I was shocked to realize the issue went well beyond the current material. They didn't know how to multiply without counting it out. Somehow the multiplication tables they should have learned 4-5 years previously had evaded them, and they were essentially crippled in math.

The trouble doesn't seem to be with the 2's, 5's, and 10's. These tables have an obvious pattern and are much easier to learn, but there is a serious slow-down as kids hit the 3's, 4's, and 6's. By the time they get to the 7's, 8's, and 9's, I've watched children decide that multiplication is way too hard, and math just isn't their thing.

Fortunately for me, a wonderful 3rd grade teacher gave us the solution. It helped that he was male, and my 8-year-old daughter had a major crush on him. She ate up every word that man said, so when he taught her to skip count to some familiar tunes she came home singing her math just for fun.
What is Skip Counting?
Skip counting is counting at intervals of whichever number you choose. For example, if I was to skip count by 7 I would count, 7, 14, 21... and so on. Using this method you can do single digit multiplication with ease. Say the teacher asks you to multiply 4 x 7 = __. You simply skip count 7's, 4 times, 7, 14, 21, 28. The answer to 4 x 7 = 28.

That still leaves the problem of memorizing a string of numbers, and here's where the magic is. Teach children to skip count to the tune of a simple song. Have you ever wondered why children can learn the lyrics to songs so easily? It is because music can be used as a mnemonic device, that is a strategy to assist with memorization. If you can associate the numbers with the sounds in a song, the children will not only learn them faster, but retain them much longer than they would memorizing them exclusively through repetition.

While there are several options out there for skip counting songs, my favorite are the skip counting songs to familiar children's tunes. That way your child doesn't have to worry about learning a tune, and can focus immediately on the numbers. My second daughter, now 8-years-old, was asked to learn her 8's this week. After singing the song just four times, she was already starting to skip count by herself. I've now started teaching them to my youngest children as early as kindergarten age because they love to sing, and it's fun.

These songs have been a game changer for us, and we wanted to share them with parents and teachers around the world. My oldest daughter and I recorded the songs for you and created simple animated videos to help the children sing along. Since posting the videos, they have been used in tens of thousands of classrooms and homes across the globe, helping children master their multiplication tables. At the special request of teachers, I have also created simple sheet music that can be printed and sent home for extra practice. You can find the printable sheet music here.
*Photo Credit for the feature image goes to Poppies & Posies Photography! Check them out. They're amazing!!
***

Amy Nielson is the education-obsessed mother of five who blogs at PlanningPlaytime.com. The daughter of a retired teacher mother, she learned at an early age an absolute love of learning, and she tries to bring a bit of joy and fun to everything she teaches. Having previously taught reading, preschool, and music, she is now at home full time teaching her preschooler to read and creating engaging, effective resources that her kids come home and fight over. Check out her work in her TpT store. You can also find her sharing great ideas on Facebook, Pinterest, and Instagram.

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The Easy Way to Memorize Multiplication Tables (An Amazing Hack!) | The TpT Blog

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When I look back to my elementary years, one of the most difficult things I was asked to do was memorize my multiplication tables. There were so many of them, and they seemed like strings of random numbers that I had to keep in order and in context with the numbers they were being multiplied by.

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Open Source Content-based FREE curriculum

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Implemented in Three Daily Blocks

Bookworms consists of three 45-minute segments that can be scheduled in any order:

1. English Language Arts

- Teachers read above-grade level text, modeling inferential thinking and comprehension strategies.

- Focused grammar instruction and related written responses require students to write about what they're reading.

- Students read grade-level texts out loud with support of the teacher to foster fluency, vocabulary, and comprehension.

3. Differentiated reading instruction

- Teachers use assessments to group students and provide targeted instruction to help them excel.

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ELA Curriculum | K-5 Reading & Writing English Language Curriculum

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Implemented in Three Daily Blocks Bookworms consists of three 45-minute segments that can be scheduled in any order: 1. English Language Arts - Teachers read above-grade level text, modeling inferential thinking and comprehension strategies. - Focused grammar instruction and related written responses require students to write about what they're reading.

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Open Source Content-based ELA and Math Curriculum

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Open Source Content-based FREE curriculum

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The Core Knowledge Curriculum Series™ provides comprehensive, content-rich learning materials based on the Core Knowledge Sequence. Student readers, teacher guides, activity books, and other materials are available for Language Arts, History and Geography, and Science. Materials for Math are in development.

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By making many of our Core Knowledge curriculum materials freely available, we work to put into practice the principle that every child in a democracy should have access to shared, enabling knowledge.

You will need Adobe Acrobat Reader 5.0 or newer to view our materials, which is available as a free download on the Adobe website.
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Download Core Knowledge curriculum, including teacher guides and activity books, for Language Arts, History and Geography, and Science.

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Open Source Content-based SCIENCE-focused FREE curriculum

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"It's deeper, and a lot more rigorous, than what they've been used to. But my kids have really been doing well, and I feel like a better teacher when I'm doing it."

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"It's deeper, and a lot more rigorous, than what they've been used to. But my kids have really been doing well, and I feel like a better teacher when I'm doing it." Josel Williams, 3rd grade teacher School 8, Rochester, NY "We're leaving the workshop seeing the possibility of brilliance in every student in every school, every day."

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Dan Myers "Math Class Needs a Makeover!"

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Dan Myers "Math Class Needs a Makeover!"

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Math class needs a makeover -  Dan Meyer

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Dy/Dan
Dan Myers Blog

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2016 Aug 6. Here is video of this task structure implemented with elementary students.

2013 May 14. Here's a brief series on how to teach with three-act math tasks. It includes video.

2013 Apr 12. I've been working this blog post into curriculum ideas for a couple years now. They're all available here.

Storytelling gives us a framework for certain mathematical tasks that is both prescriptive enough to be useful and flexible enough to be usable. Many stories divide into three acts, each of which maps neatly onto these mathematical tasks.
Introduce the central conflict of your story/task clearly, visually, viscerally, using as few words as possible.
With Jaws your first act looks something like this:

The visual is clear. The camera is in focus. It isn't bobbing around so much that you can't get your bearings on the scene. There aren't any words. And it's visceral. It strikes you right in the terror bone.

With math, your first act looks something like this:

The visual is clear. The camera is locked to a tripod and focused. No words are necessary. I'm not saying anyone is going to shell out ten dollars on date night to do this math problem but you have a visceral reaction to the image. It strikes you right in the curiosity bone.

Leave no one out of your first act. Your first act should impose as few demands on the students as possible - either of language or of math. It should ask for little and offer a lot. This, incidentally, is as far as the #anyqs challenge takes us.
The protagonist/student overcomes obstacles, looks for resources, and develops new tools.
Before he resolves his largest conflict, Luke Skywalker resolves a lot of smaller ones - find a pilot, find a ship, find the princess, get the Death Star plans back to the Rebellion, etc. He builds a team. He develops new skills.

So it is with your second act. What resources will your students need before they can resolve their conflict? The height of the basketball hoop? The distance to the three-point line? The diameter of a basketball?

What tools do they have already? What tools can you help them develop? They'll need quadratics, for instance. Help them with that.
Resolve the conflict and set up a sequel/extension.
The third act pays off on the hard work of act two and the motivation of act one. Here's act three of Star Wars.

That's a resolution right there. Imagine, though, that Luke fired his last shot and instead of watching the Death Star explode, we cut to a scene inside the Rebellion control room. No explosion. Just one of the commanders explaining that "the mission was a success."

That what it's like for students to encounter the resolution of their conflict in the back of the teacher's edition of the textbook.

If we've successfully motivated our students in the first act, the payoff in the third act needs to meet their expectations. Something like this:

Now, remember Vader spinning off into the distance, hurtling off to set the stage for The Empire Strikes Back. You need to be Vader. Make sure you have extension problems (sequels, right?) ready for students as they finish.
Conclusion
Many math teachers take act two as their job description. Hit the board, offer students three worked examples and twenty practice problems. As the ALEKS algorithm gets better and Bill Gates throws more gold bricks at Sal Khan and more people flip their classrooms, though, it's clear to me that the second act isn't our job anymore. Not the biggest part of it, anyway. You are only one of many people your students can access as they look for resources and tools. Going forward, the value you bring to your math classroom increasingly will be tied up in the first and third acts of mathematical storytelling, your ability to motivate the second act and then pay off on that hard work.

1. I gave this post a try a year ago.

2. Also, Breedeen Murray has a lot of useful things to say about storytelling, though I can't endorse her enthusiasm for "confusion."

2011 Dec 26: The Three Acts of a (Lousy) Mathematical Story is also on the syllabus.

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2016 Aug 6. Here is video of this task structure implemented with elementary students. 2013 May 14. Here's a brief series on how to teach with three-act math tasks. It includes video. 2013 Apr 12. I've been working this blog post into curriculum ideas for a couple years now.

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This is a great math/literacy activity for group discussion.

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Whenever and however we come back together as math classes this Fall, we're going to need to spend considerable time building up students' mathematical identities. Chances are students are going to be entering your classroom with a wider array of math learning experiences over the prior six months than ever before. Therefore we need a cache of absolutely accessible math tasks with opportunity for robust student discourse (low floor, high ceiling tasks). I think this task - or some version of it - can contribute.

This infographic shows the distribution of English letters and where they are located in words.

(You can find the methodology here. There are some interesting tidbits to note, including the fact that the words are accumulated from a text corpus, rather than a dictionary. That means that the analysis takes into account how often the words are used. The example the author gives is that "the" influences the distribution much more than "theocratic.")

(Another note: I don't recommend leading off with the phrase " text corpus" if you're trying to showcase a low floor task.)

There are lots of fun things to notice. It's interesting (to me) that 'x' occurs near the beginning of every word it's in, but rarely is it ever the first letter. A lot of these distributions are even intuitive. For example, it makes sense that there would be a lot of 'e's at the end of words (especially considering how common "the" is). It also makes sense that we see a lot of 's's at the end of words.

This infographic combines mane of the things I look for in a quality task.

The math is robust, but doesn't get too hung up on the numbers

Much of it meshes with intuition

It combines math and literacy

It's accessible by native English speakers and English Language Learners (more on that in a moment)

It's highly visual

It makes sense to turn this into a card sort.

I wouldn't recommend just tossing all letters together for a mega-card sort of 26 matching pairs. I'd recommend perhaps 5-6 matches at a time. You can just take them alphabetically, or pick your favorite 5-6.

I also wouldn't recommend just kicking off with a card sort. There needs to be some processessning and sensemaking to ensure that everyone knows what they're looking at. I'll offer the following suggested lesson plan, but feel free to comment how you'd tackle this open-access card sort.
Lesson: Letter Distribution
Launch. "Think about the letter 'y'. Think about all the words that contain the letter 'y'. If you were to take every word with the letter 'y' where would 'y' most often be? Toward the front? Toward the middle? Toward the end?"

Turn and Talk

Show distribution for letter 'y'.

Now let's think about 'q.'

With your group try to come up with as many words with the letter 'q' in it as possible.

Where in each word does the letter 'q' sit?

Make a quick distribution scatterplot/histogram with the words you chose. Example:

Share your list and your histogram with another group.

Explore, Hand out a subset of cards for the card sort. If possible, have groups examine each other's answers.

If students/groups are getting stuck, you may wish to reveal the little nugget that the colors are representative of their overall appearance in words. More red = more common.

Discuss. Debrief:

Reflect on your group process. What was your group's process?

Celebrate a teammate who demonstrated a moment of brilliance.

How did you work together? Did everyone have a say in contributing?

Which norms did we excel at? Which did we not do as well at?

Optional: Self-assess on a group collaboration rubric.

Show and discuss other examples of statistical distributions.
Epilogue
As I suggested earlier, I think this is a good activity for students learning the English language given that it allows students to draw upon the English they already know. However, I acknowledge it is English centric. If someone can point me to a similar analysis of a language other than English, that'd be fantastic. It also might be an opportunity for students with a native language other than English to share with the class how the distribution might differ in, say, Spanish. For example, I'd guess that 'e' would be more prevalent at the beginning of words such as estè.

For more on supporting English Language Learners, check out this post.
The Goods
Images and files (including examples and frequency hint) ( Letter Distribution - All Files)

Also, here's the activity in Desmos form, if that's your jam: Letter Distribution Card Sort (Desmos)

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Where does a letter occur in a word? A matching activity

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Whenever and however we come back together as math classes this Fall, we're going to need to spend considerable time building up students' mathematical identities. Chances are students are going to be entering your classroom with a wider array of math learning experiences over the prior six months than ever before.

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Browse through over 60 complete K-12 math units of study chalked full of problem based investigations, 3 act math tasks, visual number talks, consolidation prompts and purposeful practice worksheets to Spark Curiosity & Fuel Sensemaking in every student.

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Real World Math Lessons | 3 Act Math Tasks | Math Worksheets & Practice

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Contextual problem based real world math lessons with rich math tasks, multi-day 3 act math tasks, math practice materials and math worksheets

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Digital Manipulatives

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The most popular manipulatives at your fingertips. Powerful yet easy-to-use.

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Number Sense Maker

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Hosted by Chris Shore of The Math Projects Journal

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Make any video your lesson

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Easily create beautiful interactive video lessons for your students you can integrate right into your LMS. Track students' progress with hassle-free analytics as you flip your classroom!