In this revamped second edition of Mastering the Facts Multiplication, you’ll find a scope and sequence for teaching the facts, strategies for addressing each of the fact families, expanded whole group lessons, new games for practice and literature connections to ensure your students learn — and retain — the facts.
I would like to acknowledge and recognize those teachers from SD72 Campbell River who helped to develop these lessons for the original volume in 2012. Our collaboration continues to inspire!
Go to my online store and check it out.
For those who have been patiently waiting (and lovingly nagging!) I am happy to announce the publication of my newest math teaching resource: Proportional Reasoning in Intermediate for Grades 4-8, available now through my online store.
This resource for teachers of Grades 4 to 8 presents more than 250 pages of open-ended lessons, meaningful practice, games, literature connections and a wealth of problem-solving contexts for supporting students to make sense of fractions, decimals, percentages, ratios and proportions. Designed for today’s diverse classrooms, this resource offers a range of tasks to promote proportional thinking through intentional development of mathematical language and the use of key manipulatives. Colour tiles, Cuisenaire rods and tangrams are used to model and make connections within and between concepts.
In the first section of the resource, students will explore three models – set, area and linear – for representing and describing, comparing and ordering fractions. Students will learn to convert between fractions, decimals and percent and to apply these skills to problem situations, including measurement, tax, discounts and data management. Next, students learn to add and subtract fractions, to solve proportions using a range of strategies (involving both mental math and the appropriate use of technology) and then finally to multiply and divide fractions.
Assessment tools are threaded throughout the resource to allow teachers to keep track of student progress and to make instructional decisions.
Proportional reasoning IS the math we do every day. This resource provides an access point for all.
Finally … French translations of my latest books are now available!
Je viens de traduire mes deux ressources de bonnes questions — “A Year of Good Questions for Grades 2-4” et “A Year of Good Questions for grades 5-8” — afin que mes collègues d’immersion aient des ressources françaises avec lesquelles travailler. Merci pour vôtre patience!
In these new French translations of the English originals, you’ll find more than 200 open-ended and engaging problems for french immersion students from primary through middle school. All are posed in French and explore important mathematical concepts across the grades.
The first book: Bonnes Questions: Une année de bonnes tâches mathématiques pour les élèves de 2e à 4e is suitable for late primary students (grades 2-4) and features operational tasks, measurement tasks and pattern tasks of increasing complexity, posed in French.
The second volume: Bonnes Questions: Une année de bonnes tâches mathématiques pour les élèves de 5e à 8e is perfect for middle school immersion students (grades 5-8), with a focus on proportional reasoning, algebraic thinking as well as operations on integers, fractions and decimals to name a few.
Engaging problems and choice make these volumes the perfect conversation starter for our immersion classrooms, promoting oral language development and mathematical thinking …en même temps!
For those of you who have been waiting ever so patiently, I wanted to let you know that I have completed a primary companion to the Year of Good Questions for Grades 5-8 resource released this summer.
A Year of Good Questions for Grades 2-4 is the late primary version of this stand up calendar of problems — one for every day of the school year!
Like its intermediate counterpart, this compact but potent book comes with an easel so you can set it up on your desk and flip from one rich problem to the next, posing open-ended questions of your primary students.
Good Questions: A Year of Open-Ended Math Problems for Grades 2-4 is a problem-a-day resource that includes rich tasks ideal for grades 2, 3 and 4. Organized by topic and structured in problem sets of 5, this simple to use teacher resource includes 200 mathematically important questions to engage your students in deep thinking. For only $25, it’s a reasonably priced way to stimulate and promote mathematical conversation!
Operations, measurement, proportional thinking and patterns are featured in this calendar of problems. Each one engages students in thinking flexibly, critically and creatively to solve tasks of varying complexity.
Let the fun begin!
I hope that your school year is off to a fun — if not frantic — start!
This is a great time for teaching — when we get to know our learners and what they’re capable of (mathematically speaking) and when we establish a solid instructional plan for the first term. This is a time of pedagogical hope and possibility!
This is also a time for professional learning — the chance to develop our ability to structure engaging mathematical tasks and rich conversations for and with our students.
If you are interested in adding to your repertoire of instructional strategies in mathematics, I’d strongly encourage you to attend the BCAMT NorthWest Math Conference taking place in Whistler, BC.
It’s a remarkable gathering of mathematical minds — Northwest US and Western Canada’s finest teacher leaders in numeracy will be on hand, presenting their latest and greatest ideas to support rigorous mathematics instruction that addresses the diversity and richness of our classrooms.
Oh — and I’ll be there too… presenting twice on Friday October 19th. :o)
One session for primary: From Subitizing to Place Value: Developing Number Sense in our Youngest Learners From 10:30 to 12:10 pm in the MacDonald B Ballroom in the Fairmont Hotel
One session for intermediate: From Multiplicative Reasoning in Middle School: From Skip Counting to Algebra From 2:45 to 4:25 pm in the MacDonald C Ballroom in the Fairmont.
I will also have a booth at the exhibitor’s hall, where I will have resources for purchase, including my 2 latest resources:
A Year of Good Questions: Open-ended Problems for Grades 5-8
A Year of Good Questions: Open-ended Problems for Grades 2-4
…and featuring some of the finest “salesman” ever. :o)
The exhibitor’s hall will be open Thursday October 18th from 5-7 pm, all day Friday the 19th and until 2 pm on Saturday the 20th.
Please come by and say hello.
I’ll look forward to meeting you there!
I am pleased to say that — beyond spending every day on the water this summer — I DID manage to create a new teacher resource for my intermediate colleagues.
This time, it’s a stand up calendar of problems — one for every day of the school year!
This compact but potent book comes with an easel so you can set it up on your desk and flip from one rich problem to the next, posing open-ended questions of your intermediate students.
Good Questions: A Year of Open-Ended Tasks is a problem-a-day resource that includes
rich tasks ideal for grades 5, 6, 7 and 8. Organized by topic and structured in problem sets of 5 or more, this simple to use teacher resource includes 210 mathematically important questions to engage your students in deep thinking. For only $25, it’s a perfect back-to-school gift for yourself!
Proportional reasoning, measurement, operations and algebra are featured in this calendar of problems. Each one engages students in thinking flexibly, critically and creatively in the face of important and challenging mathematics.
Let the problem-solving begin!
My sincere appreciation to all of you who have waited for the publication of this book. As you know, I’ve had a pretty remarkable year. I hope you’ll forgive me, knowing that only good distractions delayed its completion!
But I am pleased as punch to announce the release of Place Value For Intermediate: Building Number Sense for Grades 3-5, available now from my online store for $50.
This resource for teachers of Grades 3 through 5 features lessons designed to support deep learning of number. A wide range of both open-ended and directed tasks focus on representing, describing, comparing and ordering numbers to 100 000, as well as explorations of decimal numbers to thousandths.
Measurement experiences make up a big part of this series of tasks. The metric system and all of its place value connections is featured in explorations of linear measurement (mm, cm, m, km), perimeter (cm, mm), area (square cm and square m), mass (g, kg) and capacity (mL, L).
Addition and subtraction of large numbers and decimals are also addressed in this volume. Lessons at the grades 4 and 5 level focus on multiplication of 1 by 2- and 3-digit factors as well as 2 by 2-digit factors using the distributive property (an area model).
Assessment tasks tap into students’ understandings of these numbers and their application in the real world. Being able to see and relate to big numbers and to very small ones, to understand their relative size and to capably use these numbers to estimate is the essence of number sense.
Set up in a developmental continuum intended to facilitate the teaching of combined grades, this 352 page volume is certain to contain material to meet the needs of all learners and to inspire fun and engagement with critically important place value concepts.
When you buy the book online, you also get access to almost 40 pages of digital files and resources, which will be emailed to you as a downloadable pdf!
Thanks for your continued support and inspiration…
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.