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.
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
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 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…
Hard to believe the summer has flown by so fast. In the spirit of the season (new classes and freshly sharpened pencils and all that) I wanted to share a game that I put together last spring. It’s appropriate for students in late grade 1 (skip counting from zero) through grade 5-6 (using multiples).
To play, students pair up and each one chooses a colour of counter to play with. Player A spins the spinner (use a paperclip and a downward pointed pencil as a spinner) to find out what number she must count by. Player A puts a counter in her colour on any number in the lily pad grid that is a multiple of that number. So if Player A spins a 2, she can cover a 2, 4, 6, 8, 10, etc – but NOT a 5 or a 15… Then Player B has a turn.
Three in a row in one colour wins the game.
Oh – and if you spin a lily pad, you can put your counter anywhere at all!
Consider using this game as a beginning of the year start up task. Observe your students as they play and listen to their strategies. Chances are you’ll learn something new about your kids….
I had fun playing with place value concepts in Vernon last week with a FABULOUS group of grade 5 students. We modelled large numbers in a place value cart (ones, tens, hundreds, and thousands using magnetized colour replicas of Canadian bills – or rather loonies, tens, hundreds and thousand dollar bills! I built dollar amounts by placing bills of different denominations in their respective columns. It was easy for students to say the total value of the bills when presented in this way – easier still to describe them in terms of expanded form and standard form. :o)
Later we explored the idea of multiplication by 10 – and discovered that every time we multiply a number by ten, we move one place to the left in the place value chart. Kids had fun trading twenty loonies for 2 ten dollar bills and writing the multiplication sentence to match!
I’ve attached pdfs of Canadian bills and coins (thousands, hundreds, tens, loonies, dimes and pennies) so that you can explore place value with tenths and hundredths of dollars as well.