One million hits on my blog…? Amazing. 🙂
I am truly grateful for the support, encouragement and professional curiosity of those of who who have visited my blog since its inception so many years ago… I am indebted to those of you who have told friends and colleagues about the blog and grateful for my “regulars” — those of you who return to it from time to time, looking for resources. Your enthusiasm buoys me up!
It is indeed a great privilege to be a part of this international community of teachers of mathematics. Thank you.
Wow. I have had the most extraordinary summer. Truly extraordinary. And somehow between engaging in a series of remarkable, life-affirming adventures I have managed to write another teacher resource book… 😊
It’s all about Place Value (as I’m sure you’ve figured out!) and is intended for teachers of kindergarten through grade 2, with special accommodations for those who teach in combined grades settings. There are 230 pages of developmentally framed lessons designed to address the diversity in our primary classrooms. Each one supports students to represent and describe quantity, to compare and order sets, to use referents to estimate and to skip count. Lessons devoted to measurement — an ideal practical application of place value in the world — are also featured. Whole class lessons, centres tasks and games for practice allow students to connect these important concepts in a seamless way, and can be used both as a unit or spread throughout the year to build and consolidate understanding.
(PS… A companion volume for Grades 2 to 4 is in the works – expect it later this fall!)
I thought it was time to post another game for those of you who are looking to support your intermediate students. This is another classic game from BEAM. It’s called the Game of Remainders — but don’t be fooled! It’s about far more than simple division. There are connections to be made to skip counting and the multiples here that are worth talking about!
As a tool for thinking and for identifying the important patterns inherent in this game, consider giving students a hundred chart to begin. Have them shade or highlight all the multiples of 6 (6, 12, 18, 24, 30, 36, etc) before playing the game.
Then, as they land on a number in the wheel (like say 49), they can refer to the chart and see that the number 49 is not coloured, so it’s going to have a remainder. Looking further, the will notice that it is in fact one more than a multiple of 6, which means there will be 1 remainder.
Be sure students gave a chance to talk about what they’re noticing in the chart as they use it. The more we describe our thinking, the clearer it gets and the more connections we make!
I’ve made a few other versions of this game if you’re interested in downloading them. They follow the same format, but address divisibly of 3, 4 and 5.
All the best as we count down to summer!
Welcome back, everyone!
I’d like to share a classic game from BEAM – a website that has, unfortunately, ceased to exist in a format that is easy to navigate.
Like the best of the BEAM games, Cross Over is a strategic one. Players start on opposite sides of the hexagonal game board and roll a regular 6-sided die, then choose whether to add the number shown or to subtract it.
They move their counter onto a space with the corresponding sum or difference, but only if that space touches the one on which they find themselves.
The goal of the game is to land on the opposing player’s start space.
It’s actually easier done than said… Really!
Students working on mastering addition and subtraction will enjoy playing Cross Over, and the combination of strategy and luck will ensure that everyone finds the game accessible and fun.
Just when you thought the math had gone out of the season…
There are twelve days of Christmas. And according to the song, there were a lot of gifts given over those 12 days.
Which means there’s a math problem lurking in there!
Have students think about ways to calculate the total number of gifts given on the 12th day (12+11+10+9…) or even calculate the total number of gifts given throughout the entire gift-giving season (the gifts from the 12th day added to all the gifts given on the 11th day…). Some may even apply ideas borrowed from Gauss to effortlessly calculate the sum of the arithmetic sequence.
And then if you want to get especially sassy, challenge students to calculate the daily cost of giving each of these extravagant gifts! Tamara B. shared this awesome website with prices for the items given over the 12 days – including shipping and handling changes – so that you too can figure out what it would cost to give 5 gold rings, 4 calling birds, 3 french hens, 2 turtledoves and a partridge in a pear tree… not to mention financing the maids a-milkin’!
The site is only updated to 2003, but at least it will provide a starting point. But better yet… If inflation has increased on average 2% a year for the last 12 years… well, the mathematical possibilities are endless! Consider checking out the Statistic Canada webpage for the actual inflationary numbers. And hold onto your calculators!
Enjoy — and happy ho ho!
PS – For an incredible, fun, fast and mathematically extraordinary experience, watch this video by the inimitable Vi Hart. She will blow your socks off while she explores the 12 Days of Christmas, math style…
Tonight marks the first evening of Hanukkah – the Festival of Lights in the Jewish tradition. Between candle-lighting and spinning tops, there is a great deal of math in this 8-day festival! Consider the following seasonal activities for your students in the days ahead. I hope you enjoy them!
There are 8 days of Hanukkah. For each of the 8 days, a candle is lit and placed in the menorah – one on the first day, 2 on the second day etc. Sounds simple, right? Well, yes and no. By the end of each day, these candles burn out and have to be replaced, which means that there are many candles used throughout the 8 days of Hanukkah. To top it all off, each of these candles is lit by another candle, called the shamash, which is then placed in the centre of the menorah.
So. If you had to buy enough candles for this year’s Hanukkah celebration, how many would you buy in all?
Young children might enjoy using materials to model and solve the problem. Older students might use some Gauss-ian logic and/or ideas around triangular numbers to generate a solution…
Playing the dreidel game is another fun – and mathy – way to celebrate Hanukkah. If you’re looking for the complete rules and a bit of history of the game, click here for more information.
In essence, this is a gambling game that involves a set of prizes. In the classroom, consider using counters, cubes or beads as “ante”. Players spin the 4-sided dreidel top and follow the instructions listed on each side, taking all, none or half of the pot – or adding counters to keep the game going! Even young children will enjoy playing this game and practicing concepts of addition, subtraction and “half of”.
Happy holidays, everyone!
I wanted to share some fun puzzle templates with you — one set for primary classrooms and another for intermediate. Both require the same reasoning skills: students must use a complete set of 0 to 9 tiles and place them on each of the blank spaces provided. Some of the forms are more complex than others, and will require students to not only know their facts but also to reason through the logic of placing the tiles in the correct position.
While students work, wander the classroom and ask them how they are making decisions about the placement of the number tiles. Students’ rationale may surprise you!
Some of my colleagues have suggested different ways to make the tiles:
- by cutting up vinyl placements into 1′ by 1″ squares and using Sharpie to record the numerals
- by writing directly onto colour tiles with a Sharpie
- by using this handy line master copied onto construction or bond paper
I hope you enjoy them! I know your students will!
PS — If you purchased a first edition copy of the Multiplicative Thinking book, these puzzles replace those included in the resource. 🙂
I am truly excited to announce the release of my newest teacher resource book: Multiplicative Thinking: From Skip Counting to Algebra (Grades 3 to 8). This book is designed for teachers of the intermediate grades and is focused on the teaching and learning of multiplication. This resource addresses multiplication deeply — what it means to multiply, when to use multiplication in problem-solving situations, as well as how to manipulate whole number, fractional and decimal factors using strategies like the distributive property.
Lessons on skip counting, patterns in the multiples, factoring, and on prime and composite numbers are included in this 220 page teacher resource. Algebraic thinking is explored as well, from T-charts and input-output machines to solving equations, from graphing linear relations and extrapolation to finding the slope of a line. Students engage with visuals and real-world problems involving proportionality, rates, discounts and taxes to build their understanding of multiplicative thinking and see its very real application to their everyday lives.
Each of the 40 lessons features a connection to prior knowledge, whole class and small group explorations of the Big Math Ideas, guided conversations about the mathematics with key vocabulary, opportunities for meaningful practice, tasks for consolidation and customized assessment tools. Skill building lessons are interspersed throughout the book, ensuring students recall and continue to practice the essential skills needed to apply multiplicative ideas.
And of course literature links and games for practice are — as always — included!
Multiplicative Thinking: From Skip Counting to Algebra (Grades 3 to 8) is available for $40 + $10 expedited shipping. To order, click here or on the link at the right. From there you can also order other titles, including Mastering the Facts: Multiplication, a resource dedicated to the teaching and mastery of the critically important multiplication facts. It’s a perfect complement to this new volume and one that can be used in advance — or concurrently — to build a solid foundation.
Thank you for your support. All the best for a remarkable school year!
Why Multiplicative Thinking?
Multiplicative thinking plays an enormous role in elementary and middle school mathematics. So much bigger than simply knowing the facts — a critically important aspect — the ability to think multiplicatively is essential for success with almost every other mathematical concept, from ratio and proportionality to algebra. It is the operation most often used in “real life” to make sense of large quantities, of taxes and discounts, of income per hour and kilometres travelled. It’s the operation we use when we figure out how much paint or carpet to buy or what a tank of gas is going to cost; when we convert currency for a holiday away or sort out how much to tip on a meal. No matter where we look, multiplicative situations abound. We can’t spend too much time on the teaching and learning of these critical concepts!
In writing this resource, I have attempted to introduce multiplicative thinking — both the operation itself and the bigger concept of multiplicative reasoning — in a sense-making way. Through stories, models, pictures and words, students are introduced to the idea of multiplication as “groups of” and as “rows of”. Problems are posed to support learners in connecting what they know about patterns in the multiples to proportional situations. The associative and distributive properties are introduced and applied. Algebraic concepts — input and output machines, graphing and exploring the rate of change in linear relations — round out the topic and provide a preview for multiplicative reasoning at the middle and high-school levels.
Hello to my friends near and far… It has been a long time since I have posted anything to my blog, and for that I am truly sorry. To my faithful visitors, I offer you my thanks. I hope this latest instalment has been worth the wait!
My newest collaboration — as always, co-created with the inimitable Sandra Ball — is entitled Financial Literacy In Primary: Thinking About Money in the Canadian Classroom. This full-colour, 50 page resource is free to download – but you’ll need a password to access it. Please use the form below to contact me for the secret code. Read on to learn more. Enjoy! Carole
Financial Literacy — An Important Aspect of Numeracy
Beginning in the fall of 2016, concepts addressing financial literacy are being introduced to the Western Canadian mathematics curriculum. Although they have been included at the middle school level for a number of years, this is the first time that financial literacy as been highlighted in elementary — and most notably at the primary level. This raises some important questions. What is financial literacy at the primary level anyway? Certainly it is more than recognizing and naming Canadian coins! Structured instead around notions of earning, saving and spending and giving money, curricular outcomes dedicated to financial literacy are intended to look more deeply at what it means to be financially responsible.
Our youngest learners need support and explicit teaching to reach these goals. Engaging, authentic and meaningful tasks are critically important. Learning about money should be fun, of course, but should include notions of social responsibility — earning, saving and giving — as well as spending!
Playing with Money: Connecting to Number Sense
In this new resource, Sandra and I have crafted a series of lessons to support teachers in introducing and developing ideas around financial literacy in Kindergarten through Grade 3. Currently there are few — if any — resources devoted to these ideas for primary students. We are hopeful this resource will fill that need!
Rich tasks, literature connections and games for practice in this resource are laid out along a continuum and are intended to be used across grades. In alignment with our new curriculum, these lessons are connected to both the core competencies (Thinking, Communicating and Personal and Social Responsibility) as well as to the curricular competenciesfor mathematics. By integrating financial literacy into our math program in primary, we create meaningful contexts for the math we are learning.
Think of the mathematical expertise our students will build as we represent and describe money amounts, compare and order values, skip count with coins and bills, use place value understandings and add and subtract dollar amounts!
The Math Big Ideas for Financial Literacy
This teacher resource is structured around a set of Big Math Ideas for earning money, saving money and spending and giving money. These enduring understandings apply across the primary grades and beyond. By addressing these Big Math Ideas in developmentally appropriate and engaging ways we can ensure students have fun while they lay the foundations for ensuring financially responsible decision-making.
Sandra and I hope that this resource proves useful to you and fun for your students!
To download the materials, visit my online store (https://mindfull.ecwid.com) and click on the FREE DOWNLOADS icon.
In Kindergarten and Grade 1, students need practice subitizing. That is, being able to recognize at a glance and name familiar arrangements of objects without counting. It’s an important precursor to estimation, skip counting and multiplication, and depends on students’ understandings of conservation — that 5 is 5, no matter how it is arranged.
In this simple partner game, students roll a standard die and then find a cell with the same number of dots. They cover the dots with a counter in their colour and then give their partner a turn. Three in a row in a single colour wins the game.
Small groups or even the whole class can play the Bingo version of this game. Each student needs a bingo card and a small handful of counters in a single colour. Have the “caller” roll a die and call out the number to be covered. As in traditional Bingo, three in a line (across, down or diagonal) wins the round.
Click on the links below to download the partner game and/or the Bingo version of this game.
I’ve said it before… Cuisenaire rods rock. I was first introduced to Cuisenaire rods in 2006 by my dear friend and mentor John Van de Walle. In February of that year I was invited to join him in Spokane, Washington and to observe as he did demonstration lessons in primary and intermediate classrooms. His problem-based lessons featured Cuisenaire rods, and while I watched the students responding to the openness of the tasks and the richness of the manipulatives I knew I was witnessing something powerful. The proportional relationships between the pieces and the colour wheel connections make Cuisenaire rods both aesthetically pleasing and mathematically significant. I was hooked. It took some time and experimentation to figure out how to best introduce the materials and to sort out what tasks and questions would promote thinking across the grades. But in the end, I’ve managed to compile a series of open-ended tasks, games, lessons and practice opportunities that are developmentally sequenced and laid out from Kindergarten to grade 7. I’ve compiled these lessons in 2 full-colour volumes – a primary resource (grades K-3) and an intermediate resource (grades 4-7). All of the tasks have been tested and refined to ensure they are classroom-ready and engaging for all! So it is with great humility (and no small sense of accomplishment!) that I announce the release of my latest resources – Remarkable Cuisenaire Rods: Mathematical Tasks for Primary Classrooms and Cuisenaire Rods Rock: Exploring Multiplication and Proportionality in Grades 4-7. These resources would not exist if it weren’t for the mentorship of John Van de Walle. He shaped my mathematical practice more than he knew in his all-too-short lifetime. And so I dedicate my student-centered and pedagogically grounded efforts to his memory. I hope you will enjoy the tasks and games, the investigations and the open-ended problems posed in these resources. They are intended to promote big thinking in elementary – from addition and subtraction to skip counting and multiplication, from fractions to division and more… The full colour primary resource, Remarkable Cuisenaire Rods: Mathematical Tasks for Primary Classrooms is intended for Kindergarten through Grade 3 can be ordered by clicking here or on the image on the left. The resource is $40 plus shipping. Click on the link below to preview lesson titles and the intended grade levels for the tasks. Cuisenaire Rod Resource K-3 Table of Contents The full colour intermediate resource, Cuisenaire Rods Rock! Exploring Multiplication and Proportionality in Grades 4-7 is designed for intermediate students and can be ordered by clicking here or on the image at the right. This resource is $50 plus shipping. Click on the link below to preview lesson titles and the intended grade levels for the tasks. Cuisenaire Rod Resource 4-7 Table of Contents As always, thank you for your support! Carole
My colleague Sandra Ball and I have completed another resource for primary teachers!
Read A Story: Explore The Math promotes the teaching of important math concepts through the context of great children’s books… The lessons span K-3 and some are even appropriate for grades 3/4 classrooms. Lessons involving number sense and operations, data management, measurement and more are included in this 65 page resource. Scan through the list of titles and corresponding math concepts to sort out which children’s books would be a best fit for you and your students. The complete set of ISBN numbers for each of the stories is included so you can easily share ordering information with your teacher librarian…!
To download, please visit my online store (https://mindfull.ecwid.com) and click on the FREE DOWNLOADS icon.
For those of you who have been looking for some of the teacher resources and student materials that I have used in my demonstration lessons, I’ve opted to try and put a collection of them all in one post… This collection of materials are intended for teaching place value, for use in lessons involving partitioning (addition, subtraction and multiplication), for comparing and ordering whole numbers as well as decimals, and for the teaching of fractions. As you’ve seen modelled in the lessons I’ve taught, these materials work best in concert with visuals (ten frames, base ten blocks, etc) and with plenty of opportunities for students to write equations, describe their thinking orally, build with models and create real-world situations to match.
The money and Cuisenaire Rods are best printed in colour, of course. I’d recommend sending the pdf’s to Staples.ca for printing. You can specify the weight of the paper (I like 80lb gloss cover) – and they’ll have the materials ready quickly for a reasonable price.
Consider putting magnetic tape on the back of these materials to allow them to be displayed on the white board. Check out the dispenser of magnetic tape available from Poster Pals. It’s great stuff!
I hope these prove helpful.
Happy New Year, Everyone!
I am pleased to announce the release of my latest resource, Sums and Differences – Teaching Addition and Subtraction in Grades 2&3. This teacher resource is matched to the WNCP curriculum and addresses the operations of addition and subtraction to 100 for grade 2 and to 100o for grade 3. Designed to be used by teachers of combined grades – or by anyone who has a range of learners in their classrooms – these lesson sequences focus on the big math ideas of adding and subtracting! Each lesson asks students to engage with place value in concrete, pictorial and abstract ways, while practicing and developing fluency with the operations. Word problems, games and written practice are included to ensure students hone their skills and deepen their understanding of addition and subtraction with bigger numbers.
The resource includes all the line masters, game boards, written practice and teaching materials required to support your students in becoming proficient with addition and subtraction in ways that are consistent with the curriculum and which promote number sense.
This 220 page resource is just $40 plus shipping. Click here to order!
PS – The companion resource for grades 1&2 is also available for purchase. Read about it by clicking here.
PPS – My sincere apologies. I have discovered 2 errors in the book. One comes on page 167, in the game called “Three in a Line – Subtracting hundreds, tens and ones”. The wheel at the top of the page and the differences below don’t match. 😦 I’ve attached the replacement game here for you.
Likewise, I’ve had some feedback about the “I have…, Who has…?” game in the early pages of the resource. I’ve re-created it and uploaded the replacement here.
Thanks for your patience and understanding…
The NCTM (National Council of Teachers of Mathematics) has a set of online math resources worth looking into. Check out Calculation Nation (calculationnation.nctm.org/) for a series of challenging and engaging games for students in grades 3-9. Students will explore factors, prime and composite numbers, multiplication, area and perimeter, operations on fractions, solving algebraic equations and geometric concepts like tessellations and symmetry. Students can play these games against the computer or even against another player somewhere else in the world!
A simple log in is all that’s required in order to play.
I’m sure you’ll find some worthwhile tasks in this set of carefully crafted materials. Enjoy!
Sandra Ball and I have been at it again!
We have just put the finishing touches on another assessment tool for use in fall of the year. It’s a companion to the other “What Do They Know” assessments featured above, but are designed for students in grade 2 and 3.
Focussed on place value, partitioning and skip counting, this tool highlights number sense and the operations. The tool is administered by the classroom teacher and involves the whole class and small groups.
If you’re interested in piloting these materials, please go to my online store (https://mindfull.ecwid.com) and click on the FREE DOWNLOADS icon.
I hope it proves helpful to you in getting to know your learners and what they’re truly capable of!
Sandra Ball and I are pleased to announce the release of our newest collaboration, entitled Daily Math Investigations: Meaningful Math Routines. The resource is intended to present alternatives to a traditional calendar time – ways to keep our students actively engaged in the learning of mathematics in meaningful, hands-on and developmentally appropriate ways.
In the resource, you’ll find a set of thoughtful tasks that promote visual spatial capacity, number sense, operational sense, data and measurement concepts. Each is presented in the form of both entry tasks and rich routines, intended to keep your youngest students thinking and reasoning mathematically. The complete resource, including a selection of classroom-ready line masters, is available to download for free, by visiting my online store at https://mindfull.ecwid.com. Click on the FREE DOWNLOADS icon to access the PDFs.
We sincerely hope that this resource provides you with practical, fun and engaging ideas for use your primary classroom – ways to ensure all students are doing math every day!
Carole and Sandra
I have been busy writing this summer – putting together a volume devoted to teaching the operations in primary. And so, I am pleased to announce the publication of my latest teacher resource: Sums and Differences: Teaching Addition and Subtraction in Grades 1&2.
This resource includes a series of lesson sequences – open tasks, games, written practice, word problems and assessment support – for teaching the operations in a developmentally appropriate way. Beginning with place value explorations, these lessons increase in complexity while providing support for students across the grades. The lessons make explicit connections between concrete, pictorial and abstract representations of the math to ensure the operations are truly mastered.
The content in the book covers the grade 1 and 2 curricula and presents them in such a way that teachers of combined grades can use the lessons to work with their classes as a whole.
All the best!
Who knew it could happen.
My little blog has been clicked on, read, shared and accessed 500,000 times since its inception 6 years ago.
It freaks me out just a little, and makes me immensely proud at the same time.
A half a million anything is a lot – almost beyond comprehension. So the mathematician in me did a little searching. You should know that…
- 500,000 is a little less than the number of minutes in a year.
- If you started today and counted to 500,000, you’d finish about 2 weeks from now.
- 500,000 is the number of times you’ll blink in the next month.
- If you made a stack of $500,000 in $100 bills, the pile would be almost 60 cm tall.
To all of you from around the globe who have visited, my sincere thanks for stopping by.
…and come again. 🙂
I have attached a short list of some of my favourite math and literature connections for intermediate and secondary classes. It follows on the heels of a workshop I gave yesterday in Maple Ridge, in which we explored important mathematical concepts in a series of engaging reads. There is so much math potential in each of these stories that they can easily be shared with learners across the grades – either as a way to introduce a new topic or to present a context for a meaningful mathematical exploration.
I hope you find these titles – and links to the mathematical concepts they address – helpful.