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…
The iPad sensation is truly wild. I have one (of course) and use it often to present mathematical ideas and problems, stories with a mathy context and visual manipulatives to my students while I teach in classrooms around the province and territory. What I struggle with is the never-ending search for quality math games for the iPad that amount to more than digital drill… :oP Surely the technology can offer up something more thought-full??
I found an app this week that is worth sharing, called Pick-A-Path. It was released by the NCTM (National Council of Teachers of Mathematics) and features a number-puzzle for students to solve. The goal of the game is to navigate a maze, moving an octopus (Okta) through a series of numbers and operations, trying to create a maximum or exact amount. In the different levels, students use whole numbers, powers of ten, integers, fractions, exponents and decimals to solve the puzzles, gaining “starfish” as prizes. It had me hooked! Because for the different levels, I can see it being used from grades 2/3 through grade 9 — if you want to stick to the curriculum precisely — and beyond that, if you’re looking for a challenging game. Oh – and it’s free!
I just had to share! Here’s a game drawn from my new Mastering the Facts: Subtraction resource, called Lucky Ducky!
Before they start playing, children decide who will be the odd numbers and who will be the evens…
Each player subtracts from 18 on their turn.
Player A rolls the die and reads the number. She subtracts this number from 18 and puts a counter on the difference.
Player B has their turn, and play continues until all the counters are used up.
The player who is “Odds” collects all the counters that have been placed on odd differences on the board (9, 11, 13, 15, etc). The player who is “Evens” collects all the counters placed on the even differences (10, 12, 14, 16, etc).
The player with the most counters at the end of the game is the winner!
PS – This game was drawn from my Mastering the Facts – Subtraction resource. To order online, click here.
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….
Hello to my colleagues and friends…
Last spring, Sandra Ball (Surrey School District) and I crafted an assessment and instructional resource for kindergarten and K/1 classrooms. Focussed on subitizing, partitioning and patterning, this tool is designed to be administered in the fall and again in the spring of the year. Teachers work with the whole class or with small groups when performing the assessment. The tasks are drawn from story contexts to make them connected and authentic. Kids have fun showing what they know and can do! Hence the name – The “What Do They Know” Assessment. :o)
THIS spring, Sandra and I put together the companion resource for grade 1 and 2 classrooms. Again, we focus on subitizing and partitioning, but now we extend notions of patterning to include skip counting.
Best of all, an instructional component accompanies both the K/1 and Grades 1/2 resources, to help guide your teaching between assessments!
Sandra and I would like to invite you to download and use these assessments with your students this fall. They can be found by going to my online store (https://mindfull.ecwid.com) and click on the FREE DOWNLOADS icon.
Have an amazing year! Enjoy every minute.
Carole (and Sandra!)
To learn more about the K/1 resource, its design and intent, please click here to read an article from the September 2011 edition of the BC Association of Mathematics Teachers’ journal, Vector.
You know, some people holiday over the summer. Me, I seem to write teacher resources. :o)
I am very pleased to announce the release of my newest resource called Mastering the Facts – Subtraction: Lessons for Making Sense of Subtraction for grades 1 to 3. This teacher book includes 17 complete lessons aimed at supporting primary aged students in mastering the subtraction facts to 20. Each strategy-based lesson features:
- a 3-part direct instruction lesson
- a task for guided practice
- games and worksheets for independent practice
- open-ended story problems
- targeted fluency building opportunities
- an assessment task customized to match the facts learned
All the lines masters for games, written practice, flash cards, teacher materials and other instructional support are included in this 185 page resource. Organized by strategy, these lessons are designed to promote mastery of the facts, not just memorization! Teacher tips for using and organizing manipulatives, for supporting students who struggle and for working in a combined grades setting are also included.
Matched to the WNCP and BC math curricula, this book is designed for classroom teachers of grades 1 to 3 and primary resource teachers. Select lessons are suitable for kindergarten students as well.
Cost for the resource is $40 plus shipping.
If you’re interested in getting your hands on a copy, click here to order online.
Thanks for your ongoing support. I hope the book proves helpful.
PS – Please click below to download select colour line masters drawn from the resource. All other line masters are included and are to be copied onto black and white, but these ones deserved a little colour…!
I had the pleasure of working with K and K/1 teachers in Mission on Monday – a great group of teachers who somehow managed to summon up the energy to attend an after-school workshop with me this week!! Together we looked at ways to support their young students in subitizing and partitioning. Sounds complex, doesn’t it? 🙂 Truth is, children in early primary need opportunities to see numbers at a glance without counting (subtizing) and to recognize that we can break up sets and put them back together again and the set size is the same (partitioning). These concepts and skills are critically important for young children to develop – they underpin the ability to add and subtract, to multiply and divide…
Engaging young children in conversations about how they “see” sets of number is a great way to start. Present an arrangement of 5-8 objects in your daily opening activities, and ask children what they see and how they see it. Talk about the parts and label these smaller sets with numerals to make sense of the digits. Celebrate the fact that, no matter how you slice it, 7 is still 7!
Over time, you might want to make connections to the operations by using the attached “Missing Part Cards”. They include a numeral to indicate the set size, and then dots in familiar arrangements in the form of an equation. The important part of course is to cover up just one of the sets of dots before showing the missing part cards to the children! 🙂 A 6.5 cm x 6.5 cm square of thick paper (bond paper or construction paper – or even sticky notes doubled up) taped across the top creates a flap that will hide one of the parts from view, as indicated below.
Show the card and read it aloud with the children:
“Seven is the same as 4 and…?”
It’s a good idea to say “is the same as” and “and” for “equals” and “plus” here. “Equals” and “plus” are the names for the symbols and are less meaningful to learners than “is the same as” and “and” – which are words that describe what the symbols mean…
Have students say what they think is missing, and why they think so. You’ll be surprised at the strategies students will use to find the missing part! Older learners will benefit from seeing the equation written with a box to indicate the missing part – that is,
This is a great way to introduce algebraic thinking in a visual way!!
Feel free to download the Missing Part Cards for 5, Missing Part Cards for 6 and Missing Part Cards for 7 here. They are best printed in colour of course, and will hold up best if printed on card stock or bond paper. Credit for the idea goes to John Van de Walle, who first showed them to me years ago. A smart man, our John – and one I miss terribly.
PS – If you’re looking for more ideas like this for K and grade 1, consider purchasing a copy of my book: Number Sense – A Combined Grades Resource for K, K/1 and Grade 1 Math Classrooms. It’s set up to support teachers in addressing the number PLOs in mindful ways while keeping their Kindergarten and Grade 1 students together. Games, tasks, problems and meaningful practice opportunities are included in English and in French. To order online, click here.
Our math classrooms are more and more diverse each year. Learners come to us with a range of different experiences and levels of understanding of the mathematics that’s important to know. Meeting the needs can prove challenging. Dr. Marian Small’s book called Good Questions: Great Ways to Differentiate Mathematics Instruction is an excellent resource for learning how to craft questions to make the math accessible to all – challenging for those who need it, and simplified for others. Check it out for more rich and open tasks to engage children in thinking mathematically across the grades!
This is an incredibly useful teacher resource book. In it, Dr. Small poses open-ended questions across the strands as well as what she terms “parallel tasks”, which present the same math concept at 2 different levels of complexity. It’s a very clever way to include everyone in the mathematical discussion, and can really help when we’re planning for instruction in a combined grade setting. Oh – and did I mention that Marian Small is Canadian?? 🙂 Her work is in line with the WNCP math curriculum and so makes a great match for anyone teaching math here in Canada.
This selection of questions from Dr. Marian Small’s book are ideal for combined grades settings, since they address big math ideas that are common to side-by-side curricula. Invite your students to represent their thinking with models, pictures and words, and to share what they know with a peer, a small group and/or the class. These questions lend themselves to rich classroom discussions, and can give you as a teacher important assessment data to inform your planning.
PS – Marian Small has also authored a companion book for secondary math that’s well worth checking out: MORE Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction.
Here are some of my favourite Cuisenaire rod tasks for elementary. There are so many cool things to be done with these materials, I can’t begin to delve into it all here, but start with some of these ideas and see what kinds of thinking your students come up with. Remember it’s critical to record the numbers to accompany with your students’ constructions – modelling for them how a mathematician would record their reasoning is so very important. It allows students to formalize their learning and make connections to the “naked math”… (A phrase a dear friend of mine used to use often. Attention-getting, no??)
As well, I’ve uploaded are some Cuisenaire provocations — images to inspire creativity that your younger students may enjoy. To keep the play moving mathematically, try placing one or more of these pictures at the table where students are exploring the materials. You can suggest they might like to try making something like the image, but it’s much more interesting to simply place the image on the table and walk away. Your students will no doubt do something with the picture – and it’s oh-so-fun to observe them in action!
Look around your school for Cuisenaire rods – it’s not unusual to find them stashed away in a cupboard somewhere, forgotten. They are a classic manipulative and one with great possibilities. If you find them and want to figure out ways to use them, don’t hesitate to contact me. I’d be happy to provide a workshop for your school staff, or to do a series of demonstration lessons with students across the grades with these versatile materials.
My favourite place to order Cuisenaire rods is through Spectrum Educational. Be sure to get the wooden materials only – they truly demonstrate the relationships in the most compelling way. Here’s a link to a class set of wooden materials from their on-line catalogue. For those of you in the lower mainland of BC, be sure to call Collins Educational — or drop by to pick some up. They’re always happy to help.
Enjoy a lovely weekend.
I just had to do a shout out to the amazing teachers in the Yukon that I had the pleasure of working with this week. Not only are they dedicated and hardworking, they are gracious, too. My thanks to all, but especially to my awesome tour guide and math confidante Paula… You live in a beautiful place.
Enjoy your holidays!
PS – This photo was taken from the window of the plane as the sun set… Amazing.
I almost forgot to post this super fun game called “I have, Who Has…” This particular version allows students to think about place value in tens and ones using money and ten frames. To play the game, hand one (or more) cards to your students for them to read over and practice. It’s important that all the cards be distributed or the game won’t work!! :o)
Then, choose one student to begin by reading aloud both the “I have” side of their card, (“I have 3 tens and 5 ones.” or “I have 35.“) and then have them ask their “Who has…?” question: (“Who has one ten and six ones?”). The person with 16 on the left hand side of their card responds: “I have 16” and asks their question in turn.
If students are careful, the entire class will have a chance to read from their card and ask a question – and the game ends with the person who first read from their card. This is a great game since it links language, pictures, money and place value to make sense of tens and ones. I hope you enjoy it!
One of my all time favourite stories for introducing division is “Bean Thirteen” by Matthew McElligott. He is a mathematically clever author whose stories play with important concepts in accessible ways. “Bean Thirteen” is a story about 2 crickets who gather beans to eat, but find that, no matter how they share them, or with how many friends, there is always one left over! The story makes an excellent read aloud, and the pictures are worth sharing with students of all ages. I ask that learners copy the pictures on key pages, then record what a mathematician would write for that particular image. It inspires interesting conversations about doubles, groups of and division with remainders…
The next post contains some of the key images from the book so you can project them for your students to view and discuss.
Don’t forget to play “The Game of Remainders” as follow up! Rules can be found by clicking here:
PS – For another, more advanced look at division and sharing, consider “The Lion’s Share”, also by Matthew McElligott. In it, creatures share a single cake into increasingly small fractional pieces… Smart – and visually appealing for kids looking to truly understand multiplication by fractions!