Tag Archives: middle school math

Another New Resource! Multiplicative Thinking – From Skip Counting to Algebra (Grades 3 to 8)

cover Multiplicative bookFor those of you about to return to another school year, welcome back!

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. 

Math and Literature for Intermediate and Secondary

by Cindy Neuschwander

by Cindy Neuschwander

Hello, all!

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.



Math and Literature for Intermediate and Secondary

Deep Sea Duel – A mental math and logic game for all

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.