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.
I am pleased to announce the publication of my latest teacher resource book called Fair Shares – Teaching Division in Grades 4-7. The book features tasks, games and problems for intermediate aged students focussed on making sense of division.
Through stories, models, pictures and words, students are introduced to the idea of division as sharing and division as grouping. Lessons include opportunities for talk, for exploration and for practice in the form of games and engaging tasks across the grades. The lesson sequences are designed to address division of whole numbers and decimal numbers, to make meaningful connections to fractions and decimals in context and to support students in seeing patterns in quotients. Lessons map out how to use manipulatives to model division situations, and literature connections to introduce great division contexts. Match to the WNCP curriculum, Fair Shares – Teaching Division in Grades 4-7 outlines a range of assessment tools to allow teachers to gather evidence – quickly and without stress on the part of the students – to show what their learners know and can do.
Thank you, as always, for your support.
I discovered this engaging game a week or so ago and wanted to share. It’s called Deep Sea Duel. In the game, you match wits with the computer in the form of an octopus called Okta. (She’s the same character as in the iPad game called Pick-a-Path, below). When you square off against Okta, you can choose the level of difficulty and her level of “nastiness”. Good thing too. It took me a while to sort out a strategy for winning when things were too easy, and I had to crank up the nastiness in order to really understand the game and its complexity. The thing that I hadn’t considered was that Okta could win with ANY of her 3 (4) cards – not just the first 3 – if that makes any sense. 😛
The goal of the game is to reach a target number by adding a set of cards together. You and Okta take turns. As you play, keep track of both your score and Okta’s to be able to block her attempts to collect three (or 4) cards with the specified total before you can! There’s a ton of mental math going on in this game – and, if you’re up for it, you can move into decimal numbers too…!!
Enjoy…! More games and links like these can be found on the NCTM Illuminations webpage.
I thought I’d post an autumn-themed game on the blog this week for my colleagues in kindergarten. The game is called Falling Leaves, and it’s based on a game from the BEAM website. In my version of the game, students start with 15 unifix or stacking cubes in their own colour. To begin, Player 1 rolls a regular 6-sided die and puts a cube on the leaf with that numeral. Then Player 2 has a turn. If there is already a cube in that leaf, students stack their cube on top of the one that’s there, to make a tower.
At the end of the game (when all of the cubes are used up), players scan to see which of the towers has their colour on the top. Those towers are collected and snapped together. The player with the tallest tower wins!
In this game, pink is playing green. Green collects all the towers with green on top. Pink collects all of the towers with pink on top.
Stacked together, it’s clear to see that pink wins!
Enjoy… And happy fall!
I wanted to follow up with my colleagues who attended the k and k/1 sessions in Langley on Friday. I showed some materials that I then promised to upload to the blog – and then promptly forgot! Here are the files… 🙂
For those of you who were not in attendance, the idea is simple. Young children need the opportunity to represent number in many ways to truly make sense of it. Our youngest learners need more than most to make sense of the squiggles we call digits by building, comparing, partitioning and learning to subitize amounts to five – and then from 5 through ten. Consider these cards, images and frames for representing number as part of your opening activities, a centre or as meaningful practice following on from a lesson. Students love the chance to roll a die and say how many – and then to build and record what happened! The files are below – and are included in French as well.
PS – Use the “finger cards” to create sets that make five like in the Room on the Broom task, below. Copy the cards, cut them out and then distribute them in pairs so that you know that every child in the room will be able to find their missing part (ie, be sire to hand out a 2 and a 3, a 4 and a 1, and a 5 and a zero…). You might consider NOT using the 5 and zero pairing – seems sort of unkind to leave a child with nothing in front of them!!
I wanted to send along a list of spooky books for math investigations for spreading the Hallowe’en math love. I hope you can find some or all of these in your school libraries… There are so many fun contexts to explore around this season – from notions of pumpkin circumference to skip counting, from growing patterns to playing with the operations and the complements of 5 and 10.
One of my favourite contexts for thinking about parts of 5 stems from a story called Room on the Broom. Just this week I worked in a K/1 classroom and explored the missing part – or complement – of 5. Then we read the book by Julia Donaldson, in which a witch and her friends fly about on a broomstick – adding a friend until there are 5 on the broomstick in all. We “built” some of the book’s illustrations in egg carton 5-frames, and talked about how much room was still left on our broom, if our brooms, like hers, had 5 seats.
Next, I whispered a number from 1-4 in each child’s ear (I left off zero and five for this initial exploration…) and had them build that number in their 5-frame broomstick. Then I asked the children “How much room is on your broom?”. The K/1 kids then had to find the person whose broom “completed” their’s… Click to take a look in the pictures below:
a child with 2 on his broom finds a child with three on her broom:
3 and 2
and they put their brooms together (one on top of the other) to fill it up.
The egg cartons I like best are clear plastic ones – and you can see why… looking through one 5-frame to the other is a powerful way to see the parts of 5!
After a couple of turns with this game, I asked children to record what they did and how they filled up the room on their broom with their partner. You are welcome to use the There’s More Room on my Broom! line master to try this with your students as well. In this photo, you can see see one of the grade 1 students working to show her thinking on the form…
Have a fun and spooky mathy season!
Here’s a fun little game for primary classrooms… The Magic Squares game provides students with a total for each row and column, as well as a few key starting numbers. Use the magic wand to place the correct digits from the set of numbers at the bottom of the screen. Double click to grab and then place the correct numerals in the grid.
Each game sets a different total for the rows and columns, so students can choose a number that makes sense for them before beginning.
The challenge of finding a sum for 3 addends is a good one for late grade 1 (when the digits without images to accompany them make sense) through grades 3.
I came across an interesting game today in my perpetual on-line search for quality math games that promote thinking. It’s called Mission 211 – Mental Maths.
A video transmission from mission control’s Caleb explains the tasks at hand. You must answer mental math questions as quickly as you can in order to collect biofuel rods and foil the evil roboids..!! Best of all, Caleb provides strategies for solving the problems, if you need his help. The strategies include “counting on”, “breaking down numbers” and “rounding” – what we might call compensation or friendly numbers.
The music and the heartbeat in the background (yes, really!) create a sense of urgency, and encourage you to complete the questions as quickly as you can. If you need help, pressing the “HINT” button produces a mental math strategy to one side of the screen. It’s a super helpful scaffold, and one that helps to make the numbers meaningful. As you progress through the game, there are true or false multiplication and division questions as well – a nice blend of methods and ways of presenting content.
I like it!
Now – back to the game. I’ve got some evil roboids to destroy. 🙂
PS – Play the game in FULL SCREEN MODE to avoid silly advertising…