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Hands-On Math Goes Digital In and Beyond the Classroom!

Explore Our Collection of FREE Virtual Manipulatives

From Number Lines to Base Ten, Pattern Blocks to Place Value, Disks our collection of free Virtual Manipulatives gives educators and families access to visual and hands-on digital resources that lead to understanding. Students can use these resources in class and at home to practice math concepts, reinforce their growing math understanding, technology they love in the process.

Free Resources & Instructions for Teachers and Families

We've developed these resources for easy access and easy use:

How-To Videos

There's a short video for each of the Virtual Manipulatives in our collection so teachers can use these digital materials easily and confidently with students.
Watch Now

Activities

Activities from the Didax books that teachers use most for easy download. Engaging challenges address core math operations, concepts and skills, from early numeracy all the way to introductory geometry and algebra.
Access Now

FAQ

Download this document of FAQ's to answer your Virtual Manipulative questions.
Download Now

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Demonstrate key math concepts in your classroom using our new virtual manipulatives!

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Good blogs understanding progression of visual maths

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Interactive tools and activites to aid the teaching of mathematics. Hundreds of randomly generated questions and answers.

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This section of the nzmaths website has problem-solving lessons that you can use in your maths programme. The lessons provide coverage of Levels 1 to 6 of The New Zealand Curriculum. The lessons are organised by level and curriculum strand. Accompanying each lesson is a copymaster of the problem in English and in Māori.

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The Problem of the Week is designed to provide students with an ongoing opportunity to solve mathematical problems. Each week, problems from various areas of mathematics will be posted here and e-mailed to teachers for use with their students from grades 3 and up.

Current Problems

Subscribe Now!

Information for Teachers

Problems are organized into themes. A problem may have more than one theme.

These problems may be used in a variety of different ways including:

posting the problems in a classroom,

discussing the problems with students, and

integrating the problems into the curriculum.

We know that teachers will use these problems in many creative ways!

Solutions to the problems will be sent out to teachers one week after the problems are e-mailed.

Problems & Solutions from 2023-2024

Problem and Solution Archive

The following table has links to booklets containing all the problems and solutions from particular years. The problems are organized into themes, grouping problems into various areas of the curriculum. A problem often appears in multiple themes.

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Design - Graphics, University of Waterloo

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The CEMC has become Canada's largest and most recognized outreach organization for promoting and creating activities and materials in mathematics and computer science.

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Viewing Difficulties in Mathematics Through a Cognitive Lens

Wednesday, March 27 at 3:00 pm Pacific Time

Students with learning difficulties, including dyscalculia, struggle to learn and show their potential in the usual ways expected in schools or other aspects of their lives. This webinar places a focus on how weak cognitive function can be the root cause of difficulties with mathematics and learning in general. It will discuss the approach of the Arrowsmith Program, a cognitive program which identifies, targets and strengthens the weak cognitive functions that affect learning. The premise of the Arrowsmith Program is that weaker cognitive functions that contribute to learning difficulties can be strengthened with cognitive exercises. After viewing this webinar, participants will know more about:

Specific cognitive dysfunctions that cause learning difficulties including difficulties related to performing mathematics

Specific cognitive functions that are involved with mathematics and learning

The Arrowsmith Program approach to learning difficulties, driving cognitive change using the principles of neuroplasticity

Observed outcomes from students participating in the Arrowsmith Program

Register now, space is limited! (After the live webinar takes place, a recording will be available on this page.)

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Viewing Difficulties in Mathematics Through a Cognitive Lens Wednesday, March 27 at 3:00 pm Pacific Time Students with learning difficulties, including dyscalculia, struggle to learn and show their potential in the usual ways expected in schools or other aspects of their lives.

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

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Steve Wyborney's Blog: I'm on a Learning Mission. - Sharing Thoughts About Education

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ASKING STUDENTS TO CHOOSE A PATH AND JUSTIFY IT WITH MATH

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Number talks ( Kindergarten, 1st Grade & 2nd Grade) offer a bridge between conceptual understanding and mental math. During mini-lessons and Guided Math, I am ALL about manipulatives. Base 10 pieces, unifix cubes, cuisenaire rods - give me all the math things. Especially in 1st grade, manipulatives offer a concrete way to represent a number and their thinking. At some point, this concrete thinking has to begin to transition into abstract and mental math processes. Number Talks offer a daily, short, structured way for students to talk about math with their peers.

What is a Number Talk?

Number Talks are short (10ish minutes), daily exercises aimed at building number sense. Number sense is the ability to play with numbers meaning students can visualize problem solving, perform calculations quickly, and are flexible in their mathematical strategy. Students who have strong number sense solve problems in more than one way and check that their answers make sense. As a part of this routine, students are thinking, asking their peers questions, and explaining their own thinking all while the teacher records the thinking.

Sherry Parish has written the book Number Talks (Amazon affiliate link) and it offers awesome ideas for Kindergarten to 6th grade. It is definitely an expensive (up to $60 if bought new) but offers great ideas and real-life examples of Number Talks.

What Do Number Talks Look Like?

When possible, I love slipping a number talk ( Kindergarten, 1st Grade & 2nd Grade) in at the beginning of our Guided Math block. If we're running late, I'll slip a Number Talk in during an awkward transition or at the very end of the day. A traditional talk follows this routine -
Present the class with a problem. Students are seated in a common area with no math tools. Number Talks are mental exercises. I store my number talks ( Kindergarten, 1st Grade & 2nd Grade) on binder rings next to my Document Camera for easy access.

Give think time. (When a student raises their hand, it tells other students they were not fast or good enough in their thinking. To account for this, students give me a sly thumbs-up on their chest to signal that they have thinking they want to share. Then, they put up multiple fingers (on the same hand) to indicate they have multiple ways to answer/solve the problem.

3. Call on a student - "_______, what are you thinking?" Have the student explain their thinking and then give their answer. My focus is always on the process not the answer.

4. Offer the strategy to the class. Student will elaborate (if correct) or ask questions (if misguided). If students aren't sure they understand a friend, they will say "I'm a little confused. Could you tell me in a different way?" If students agree with the student sharing they will do sign-language for 'me too' (thumb near chest and pinky pointing at friend). This builds the student's confidence and offers the student sharing positive affirmations.

5. Invite other students to share their strategies.

Scaffolding Students

If students have never been expected to explain their mathematical thinking, students may be very resistant to share their own strategies. So, how do I encourage students to share their own thinking/strategies?

Model, Model, Model - The first 2 days of Number Talks, I do a lot of modeling. After this, I completely release the Number Talks to the students.

Sentence Stems - "In my head I saw..." "My first step was..." "At first, I decided to try..." "I needed to..." "Since I wasn't sure what to do, but I noticed..."

Force It - Awkward silence is your friend. Sometimes I will wait (for extended periods of time) for students to share their thinking. Even if a friend is STRUGGLING with a problem or sharing their thinking, I do not come to the rescue. Growth comes when friends are able to sort out their own thinking and my 1st grade friends are able to do this...even if it is a little painful for me.

Examples of 1st Grade Number Conversations

So, it all sounds great, right?? But what do number talks actually look like? My first talks ( Kindergarten, 1st Grade & 2nd Grade) each year are always subitizing patterns (dot patterns). They are the perfect introduction into composing and decomposing numbers. Slowly that transitions into the 100s chart and eventually into word problems and balancing scales.

In the 2nd grade number talks, the focus is on building fluency within 20 and 100, as well as, layering skills in different ways. From missing addends, to algebraic thinking, over 400 prompts offer students dozens of chances to spiral review math skills each month.

Subitizing Quick Images

Subitizing ( Kindergarten, 1st Grade & 2nd Grade) is the ability to quickly identify the number in a group of objects without individually counting the objects. It is a concrete way to explore composing and decomposing numbers. Students will see dot patterns in different ways, and that is a great thing. When showing students 6, made from a pyramid students might see 3 and 3, 5 and 1, or 3 and 2 and 1. These are all ways to build 6 and lead students to understand that numbers can be composed and decomposed in many different ways.

Composing Numbers

Flexibility in number thinking is a critical skills in beginning mathematicians. Students must understand that 7 can be made from 4 and 3, 2 and 5, 6 and 1, or 7 and 0. Below students are provided with a "Target Number" and ask students to use some of the provided numbers to hit the target. Students then have to decide how many numbers to use, which operation to use (addition or subtraction), and then, decide if number order matters. ( 1st Grade Prompts, 2nd Grade Prompts)

Flexibly Composing AND Decomposing Numbers

An important part of teaching primary students is to build number fluency and flexibility. Students should be able to encounter numbers in a variety of formats and engage with them in a variety of ways. During our Representing Numbers conversations ( 1st Grade & 2nd Grade), I encourage students to branch outside of the typical ten frame, plus one, etc. These are great strategies when we first start, but after the first few days students can build on these basic representations to develop different relationships and understanding.

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Number Talks: How and Why? - The Brown Bag Teacher

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Number talks ( Kindergarten, 1st Grade & 2nd Grade) offer a bridge between conceptual understanding and mental math. During mini-lessons and Guided Math, I am ALL about manipulatives. Base 10 pieces, unifix cubes, cuisenaire rods - give me all the math things. Especially in 1st grade, manipulatives offer a concrete way to represent a number and their thinking.

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with special thanks to Christopher Danielson and his Which One Doesn’t Belong - A Shapes Book.

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with special thanks to Christopher Danielson and his Which One Doesn’t Belong - A Shapes Book.

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Mathematical Number Talks, Rio Linda, California. 13,817 likes · 420 talking about this · 30 were here. This page is dedicated to helping educators increase their proficiency with Number Talks and...

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ESTIMATION 180 is a place for both students and teachers to enjoy making sense of math and develop their number sense together.

Pick from over 200 visual estimation tasks that invite all your students to be part of rich math conversations and learning.

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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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Mashup Math activities make your math lessons fun and engaging for students

K-8 math activities designed to be highly engaging for students.

We inspire kids to learn math in a way that isentertaining and fun.

Our unique approach boosts academic achievement and participation.

Over 100,000 math teachers rely on Mashup Math resources to supplement their lessons.

Standards-aligned math resources for grades K-8+

Gain On-Demand Access to All of our K-8 Math Puzzles, Worksheets and Activities

Inspire Your Students to Learn Math in a Fun and Engaging Way

Join our mailing list to get free resources in your inbox every week

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Math Education Teacher Resources Common Core Cool Video Lesson Free Worksheets Elementary Middle High School Homeschool Algebra Geometry Lessons Help Tutor Homework

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"Mommy? Daddy? Read me a word problem," is probably not a request that many parents hear. Yet if a school child's parents replace a bedtime story with a math discussion even one night a week, the child's math skills may improve markedly compared with peers who listen to nonmathematical stories, a new study shows. The effect is sizable: Over the course of one 9-month school year, students who do bedtime math gain on average the equivalent of a 3-month advantage over their peers, researchers report online today in Science. The approach even works if the parents have math anxiety and generally shy away from discussing math with their children.

Educational experts generally applaud the new work, though they note it will take more investigation to understand why the strategy works. "I think it's a fantastic study," says James Stigler, an educational psychologist at the University of California, Los Angeles, who was not involved in the work. "But it is just the beginning." Andee Rubin, a mathematician and computer scientist at TERC, a nonprofit educational researcher and development company in Cambridge, Massachusetts, agrees. "I'm interested in teasing it apart and seeing what makes this effective."

Most parents understand that to help their children develop academically they should read to them, says Sian Beilock, a cognitive psychologist at the University of Chicago in Illinois and an author of the new study. But parents often assume that the school will take care of math instruction, she notes. "Our hope is that this study helps change the notion that math is the purview of the school and shows that talking about math at home is helpful," Beilock says.

To evaluate the effects of a little bedtime math, Beilock, psychologist Susan Levine, and colleagues at the University of Chicago recruited 587 first-graders from 22 schools, public and private, richer and poorer, in the Chicago metropolitan area. The parents of each child were given a tablet computer with which to read to the child at bedtime. Four hundred and twenty families were told to use it to work through word problems related to counting, shapes, arithmetic, fractions, and probability using a freely available and independently created app called Bedtime Math. Another 167 families were instructed to use a reading app. With a standardized test, the researchers assessed all the subjects' mathematics performance at the beginning and end of the school year.

Not surprisingly, use of the reading app made little difference to the children's math performance. In contrast, doing math at bedtime had a significant effect: Children who used the app two or more times per week outpaced peers whose family rarely used it. "It's like they've had 3 months more of math instruction," Beilock says. "In the real world that's a pretty big effect."

Perhaps most important, use of the app brought students whose parents said they were anxious about math up to par with those whose parents were at ease with the subject. Among children whose family rarely used the math app, those with math-phobic parents made only half as much progress as the children of parents comfortable with math. But doing bedtime math even once a week eliminated the performance gap.

Stigler says the study is particularly impressive for its size and for being done outside a laboratory setting. "You don't have to guess whether [the technique] will work in everyday life because the study was done in everyday life," he says. However, Janet Bowers, a math education researcher at San Diego State University in California, notes that 70% of the subjects came from middle- and upper-middle class families and questions whether the result can be generalized. "You're not going to bring it to a low-income school," she predicts, "because there just aren't parents who sit there and read."

Why the technique was so effective remains to be determined, Rubin says. "What makes this different from helping your kid with their homework?" she asks. "Is it because it's on an iPad? Is it because it's in place of a story? Why did the people who used it use it?"

Beilock says she suspects that the delivery mechanics-computer or paper-don't matter, but that the key is talking to your child about math. That should become as routine as bedtime reading, she says. Stigler says it's an attainable goal. "I don't think you're going to see a huge cultural shift," he says. "But a study like this, if it's well publicized, can make a difference."

Bowers notes that the study was funded by Overdeck Family Foundation, whose chair, Laura Overdeck, established the nonprofit Bedtime Math foundation, which makes the app. Beilock acknowledges that connection, but adds that the authors have no financial interest in app and the foundation had no control over the data. "It's all in the paper," she says, "so people can draw whatever conclusion they want."
*Update, 9 October, 11:02 a.m.: This story has been changed to avoid the implication that primarily fathers discuss math with their children.

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Bedtime problems boost kids' math performance

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"Mommy? Daddy? Read me a word problem," is probably not a request that many parents hear. Yet if a school child's parents replace a bedtime story with a math discussion even one night a week, the child's math skills may improve markedly compared with peers who listen to nonmathematical stories, a new study shows.

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interactive times tables | multiplication square | whiteboard | fish | visnos

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Interactive multiplication tables, simply view multiplication facts with their visual representation. Toggle multiplication square. Help students to learn times tables.

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Model your word problems with Thinking Blocks! Only at MathPlayground.com.

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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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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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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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math video game 3d educational ed-tech edchat mathematics visual learning kids

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Create and share games with Super Math World

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Cool Math Games - Free Online Games for Learning and Fun

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Coolmath Games is a brain-training site for everyone, where logic & thinking & math meets fun & games. These games have no violence, no empty action, just a lot of challenges that will make you forget you're getting a mental workout!

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Note: To ensure you get the best experience and for analyzing overall site usage, our website uses cookies. By continuing to browse our site, you are agreeing to this process. Learn more about the cookies we use.

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How you can be good at math, and other surprising facts about learning | Jo Boaler | TEDxStanford

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ABCya! * ABCya! * Learning Games and Apps for Kids

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Educational games for grades PreK through 6 that will keep kids engaged and having fun. Topics include math, reading, typing, just-for-fun logic games... and more!

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

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20 Days of Number Sense & Rich Math Talk - for all grades from K-12. That is the goal of this series. Entirely new, exclusive content fills this post in a ready-to-use format that will allow you to deeply explore 4 rich math routines.

By downloading each day, you can have a clear pathway to presenting rich number sense and math discourse opportunities to your students.

Additionally, each day contains multiple levels, so that a 2nd grade teacher, a 5th grade teacher, and a high school teacher can all be using the same file - and can all have access to other grade levels. The opportunities to differentiate across levels abound in this series as well as the opportunities to see into other grade levels from K-12.

Days 1-5
I've opened the series with a brand new set of Estimation Clipboards. The Estimation Clipboard strategy is a very invitational way to draw students into rich number sense discussions as it layers in more and more context and continues to provide students with estimation opportunities. The Estimation Clipboard strategy is also where a very important math instructional reality becomes abundantly clear: Writing is a powerful springboard into math discourse. As students WRITE their estimates (particularly for images 3 & 4) the math talk in your classroom will become very rich. You'll find the power in the pattern of (1) students estimating (2) writing their estimate (3) explaining what their estimate is, and why they chose it. Then right before each reveal, your classroom will become very quite in anticipation of the reveal. Don't be surprised if there is a lot cheering as the answers are revealed.

Days 1-5 have set the stage with The Estimation Clipboard. Not only have the 20 Days of Number Sense and Rich Math Talk begun, but both students and teachers have detected the power of invitations, estimation, writing, sharing their ideas aloud, and experiencing the joy and wonder of math.

Day 1 - The Estimation Clipboard
Day 2 - The Estimation Clipboard
Day 3 - The Estimation Clipboard

Day 4 - The Estimation Clipboard
Day 5 - The Estimation Clipboard Days 6-10
After the highly invitational nature of the Estimation Clipboard, opportunities to richly explore numbers continue in the Splat! images which are all brand new - and are all designed to work in both PowerPoint and Google Slides (just like everything else in this series - so feel free to use either one).

Ranging from Splats through 10 through Fraction Splats! and now including 10 of the most challenging Splats! that I have ever publicly released, your students will have very rich opportunities to lean into number sense and richly discuss their mathematical thinking while also hearing from other students. They will be able to see and explore multiple strategies for decomposing numbers, adding, subtracting, multiplying, dividing, and connecting many concepts together.

Since every day includes materials across a wide range of grade levels, there is also a really good opportunity to take a look at content from other grade levels and offer even more challenges to your students.

Let's take a closer look at each of the Splat! levels include in 20 Days of Number Sense and Rich Math Talk.

Splat! Level A includes numbers through 10 and is ideal for grades K-1. In these sequences the dots appear so that students can count them. Then, when you click, the total number appears in the corner. On the next click, the SPLAT appears and covers some of the dots. The immediate question is, "How many dots are under the Splat?" You will be amazed at how many different ways students see this, solve this, and even how they mentally move the dots around to decompose and make sense of the numbers. If you look at the images below, you'll see that some of the Splats separate the visible numbers and create a Part-Part-Part-Whole scenario. Splat Level A appears in all 5 days.

Splat! Level B includes numbers through 10 and is ideal for grades 1-2. It also provides the first example of a grade level (kindergarten) that may recognize the opportunity to venture to higher levels. Just like in Level A, the first click introduces the dots, the next click brings in the Splat and the question, and then the Number Sense and Rich Math Talk begins. When I use numbers through 20 in the classroom, I sometimes quickly move through the counting step by saying, "Here are some dots. There are 19 of them. This is the Splat!" I find that counting so many dots can slow down the opportunities that the Splat introduces. Of course, counting is critical, and so is finding many different ways to count, so at other times I spend more time on a variety of grouping and counting methods. The reality is that there are so many opportunities in these slides that you have many opportunities to focus on a wide range of concepts.

The images in Splat! Level C are some of the most wide-ranging images I have ever worked with. It offers opportunities across so many grade levels. I've labeled it as primarily matching grades 2-5, but I've seen it used very effectively in many grade levels. Level C is where multiplication and division begin to play a much stronger role. While students in younger grades may access the image above (the one that shows 14) by adding, subtracting, grouping, counting on, repeatedly adding 3 to 5, or repeatedly subtracting 5 from 14, I've seen other strategies across grade levels. For example, fourth grade students sometimes view it as 14 ÷ 3 with a remainder of 5. Likewise, high school students have viewed is as 3x + 5 = 14. Imagine the power of the conversation among the students across those grade levels!

Splat! Level D

Splat! Level D

A very interesting option opens up in Splat! Level D. In the images above you are seeing the first of the reveal clicks. In the moment before this - when all of the Splats are opaque, and none of the dots beneath the can be seen - students approach a new question, which is "What could the total be?" In the first image above, before it is revealed that each Splat is covering 3 dots, students find themselves working with the concepts that the total must be 3 more than a multiple of 3, so the total must be a multiple of 3. However, in the second picture above (again before the first Splat is revealed like you see in the picture) students can see that the total is unknown, that there are 4 visible dots, and that there are 7 splats. By rule, each splat is covering the same number. So students can work with the concepts such as that total must be 4 more than a multiple of 7. As the identify possible totals, they can see that it could be 4, 11, 18, 25, ... Then when you choose to reveal the value of a Splat by clicking, you'll see a single Splat turn transparent (which is what you see in the images above) and the students can determine what the total number of dots is. You may expect the students to use multiplication to do so, and many will, but they will also surprise you with a wide number of strategies that you may never have expected. They are showcasing both Number Sense & Rich Math Talk!

Level E is a favorite of many teachers. In this level, all of the whole dots and all of the fractional parts of a dot can be seen. Then when you click, the Splat appears, and covers some of the whole dots and fractional parts. The immediate question becomes, "What is the value beneath the Splat?" I am amazed at how students approach these questions.

Level F is similar to the previous level with one exception. While, Level E allow the students to see all of the images before the Splat appeared, Level F simply presents the total, the Splat, and the visible parts of the total all at the same time. Students once again bring very impressive insights into the conversation. It's Number Sense and Rich Math Talk and the way that students contribute to one another's insights is eye-opening.

Level G allows the students to see the all of the whole numbers and all of the fractions on the screen and gives the students the opportunity to determine the total. When you click the total will appear. Then, with the next click, multiple Splats will appear on the screen. By rule, each of the Splats is covering the same number (the same value), although those numbers may be represented in different ways. The students task is to determine the value of the Splats (not the specific pieces) under each Splat.

Level H is similar Level G, accept in Level H the Splats all appear at the very beginning, so the first image the students see are just line the ones that are showing above. The question is, "What is the value of each Splat?" Within Level, H (on days 9 & 10), I have included 10 of the most challenging Splat questions I have ever publicly released.

Remember that I've written slides for ALL levels on EVERY day. Download the day you are on, find the level you want to work with, and let your students exercise their Number Sense and engage in Rich Math Talk.

Also, consider challenging your students by trying out 2-3 slides from another level.
Day 6 - Exclusive Splat Sequences
Day 7 - Exclusive Splat Sequences

Day 8 - Exclusive Splat Sequences

Day 9 - Exclusive Splat Sequences

Day 10 - Exclusive Splat Sequences
Days 11-15
Exclusive Esti-Mysteries

15 brand new Esti-Mysteries fill these 5 days, and each day includes 3 different levels, ranging from kindergarten to advanced math concepts. As a bonus, the first 3 Esti-Mysteries include appearing charts on the screen. Additionally, 2 of the Esti-Mysteries include multiple perspectives and hidden clues that can be found in the 2nd image.

Esti-Mysteries strongly call upon number sense and promote rich math talk while also giving students rich opportunities to COMBINE estimation with a range of math concepts in the context of rich math talk. Students will be discussing their estimates - much like in the first 5 days when the work with the Estimation Clipboard - but will also be looking carefully at math concepts and seeing how those concepts are interrelated.

Day 11 - Exclusive Esti-Mysteries

Day 12 - Exclusive Esti-Mysteries
Day 13 - Exclusive Esti-Mysteries Day 14 - Exclusive Esti-Mysteries Day 15 - Exclusive Esti-Mysteries Days 16-20
Exclusive Cube Conversations

The highly visual cube conversations strategy is used in the final 5 days to promote number sense and rich math talk. In addition to brand new Cube Conversations Images, the downloadable set of matching pages will give students clear opportunities to annotate and describe their insights to one another.

As with all other days in this series, there is a very good opportunity to see into other grade level bands, and this can be use to differentiate content, challenge students, and promote even deeper number sense and rich math talk.

At the end of each day there is also a challenging task, which is placed in the highest grade band. The printable PDF includes 2 different versions of each task. One version includes 4 copies so that students will have an opportunity to try out 4 different entry points into the task, while the other page simply offers a larger, easier-to-use version of the image.

The beginning level of Cube Conversations provides very good opportunities to visualize and decompose small numbers in a wide variety of ways. The middle image shown above is animated to showcase 3 different ways that students may see how to decompose the shape. Likely, they will find many more than the 3 examples, but the animated examples typically provide a nice momentum and a clear invitation to seek out many different ways to decompose the number. Then another image appears (like the third one shown above) with 4 additional copies below it so that ideas can be shared with the entire class. As with all levels, every image is also available on the downloadable PDF so that each student can decompose the images on their own and then share their ideas with one another.

Deepening Cube Conversations (grades 2-5) introduce some layers of complexity that allow students to work with additional concepts, such as visualizing sets in multiple ways, visualizing shapes that overlap (so the overlapping cubes may need to be subtracted), and shapes that appear to have holes in them.

The 2nd image above shows layers that will appear when you click (although on the slide you download the images will all be in a single location on the slide). If you look at the middle image closely, you will see how the image has been decomposed into 9 + 3 groups of 3, then 2 groups of 5 and a group of 3 and 5 groups of 1. Still focusing on the blue and yellow shape in the middle image above, you can see why students describe it many ways. For example, a student might say, "I see 3 groups of 5 in the blue and yellow image." If you look carefully, you'll also see 3 groups of 5.

Then moving the the last image in the center picture above (the image which is red, yellow, and blue in the bottom right corner) you can see how a student might say, "I see three groups of five plus three more - but I see it in a different way." Remember, that is just the example page and it is followed by the wide open opportunity presented on the next slide. When you look at the third picture above you can see how a student might describe two groups of 9, two groups of 8, or even 2 groups of 7 while other students describe groups of 2, 3, or 5. You may also hear that a student sees a 25 minus two groups of 4 and 2 groups of 1. The possibilities go on and on as your provide students with rich opportunities to explore Number Sense and to explore and describe their insights together.

The routine continues - but at a deeper level - in Textured Cube Conversations. Remember, every level is available on every day in this blog post, and all content in 20 Days of Number Sense and Rich Math Talk is exclusive to this post.

The first image will present a structure, several examples of how the structure can be visually decomposed, and will finally reveal the number of unit cubes. Then a new image will appear with 4 opportunities on the bottom of the screen for students to show their thinking to each other.

All of the Cube Conversations images are available in PDF format so that you can provide each student with their own copy to annotate - and then to share their ideas with one another.

Advanced Cube Conversations will feel very much like a rich task. Two of the Advanced Cube Conversations are illustrated in the pictures above.

The thinking that comes from these tasks is tremendous!

On the downloadable PDF you will find a page with a large image of the structure an an additional page that provides 4 smaller versions of that picture so that students will have multiple opportunities to try out a variety of entry points.

In each case the question is, "This structure is composed of how many unit cubes?"
Day 16 - Exclusive Cube Conversations
Day 17 - Exclusive Cube Conversations

Day 18 - Exclusive Cube Conversations

Day 19 - Exclusive Cube Conversations
Day 20 - Exclusive Cube Conversations
I hope you enjoy 20 Days of Number Sense & Rich Math Talk. If you do, please leave a comment. You can even share who is using it. Perhaps we'll see some classrooms from around the world leaving comments like, "We are using #20DaysNS in Hunters, Washington!"

As always, you can find me on Twitter @stevewyborney

I'll be keeping an eye on #20DaysNS on Twitter.

I wish you well as you work with your students and give them opportunities to develop their numbers sense and engage in rich math talk.

All my best,

Steve

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20 Days of Number Sense & Rich Math Talk (K-12)

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20 Days of Number Sense & Rich Math Talk - for all grades from K-12. That is the goal of this series. Entirely new, exclusive content fills this post in a ready-to-use format that will allow you to deeply explore 4 rich math routines.

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UPDATE 3/17/19: The updated cards can be found in this post (Multiplication Subitizing Cards: An Upgrade to Building Fluency)

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Learning basic facts through tricks or a mnemonic song lead students down a path of memorization, not understanding. I previously discussed the idea of conceptual development here. When students practice their multiplication facts, they're expected to move from concrete to abstract too quickly. This takes time!!!

Moving too quickly forces memorization and avoids any possibility of multiplicative thinking. First students are asked to model 3 rows of 6 using colored tiles...

Then after a day or two of exploring arrays, students are magically expected to remember that...

Intensions are good but strategy development is the key underpinning of automaticity. In the past I've asked myself the following questions:

When do students practice multiplication that's not written as AxB? (for the purpose of automaticity)

When/how is multiplicative reasoning fostered?

When do multiplication strategies become automatized?

When do number strategies become number knowledge?

My answer to all 4 questions....I use multiplication subitizing cards with my daughter whose in 2nd grade so I figured I'd videotape and share.

My takeaways:

It's easy to identify which facts she's comfortable with and has automatized.

At 1:15- she demonstrates how she used multiplicative thinking for 6×6.

At 2:00- she could use her understanding of the commutative property for multiplication to build fluency. She was quickly is able to recognize that 7×5=35 at 1 minute, but it took some time to figure out 5×7. She owns this strategy for addition so I'm waiting for it to click with multiplication....no rush:-)

At 2:20- I missed a fact when she was wrong (doh!)

At 2:48 and 3:55 she explains her multiplicative reasoning.

5 minutes spent purposefully building fluency and I gained so much information about my daughter. The great thing is that it works for every 8 year old...even the ones in your school.

Disclaimer: I didn't create the cards and don't know who did but they deserve a virtual math hug! If you know who did please let me know so I can give proper attribution!

Cheers!

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Subitizing to Foster Multiplicative Thinking

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UPDATE 3/17/19: The updated cards can be found in this post (Multiplication Subitizing Cards: An Upgrade to Building Fluency) ---------- Learning basic facts through tricks or a mnemonic song lead students down a path of memorization, not understanding. I previously discussed the idea of conceptual development here.