Open-ended Problems for K-4
Hello all!
Last week I gave a session for teachers in Coquitlam looking to teach – and assess – problem-solving. We talked about what made a good problem, both in terms of content and wording, then worked through some samples across the grades. I have posted a selection of problems for you here, which I invite you to download and use with your students.
Remember that it’s important to collect students’ thinking in a variety of forms – numbers, pictures and words, to have them engage with models or manipulatives, and wherever possible to have students communicate their understandings about a concept by translating them to a problem of their own. The latter is no small task! :o) John Van de Walle’s diagram outlines the importance of not only including these representations but also connecting and bridging between them. Students learn deeply when they transform their learning from numbers to models, from words to pictures, from problems to numbers and back again…
Carole
Great games for developing fluency
Hello all! I wanted to upload a couple of my new favourite games for developing fluency with the facts.
Once a strategy for recalling the facts has been learned, these games will help students to apply those strategies more fluently. It takes time and practice to master the facts – practice with the strategies and then practice using the strategies to recall the facts themselves… We all know the facts are critical to success with math. How we master them matters too.
So, first is the game from BEAM called Add Nines. It depends on knowing the strategy of “compensation”. Compensation is an algebraic idea, in which we “take from one number and give to the other”. This strategy works because in every case we make a ten (or another round number).
Think of it like this:
If we add 9 and 7, then we can take 1 from the 7 and give it to the nine, to make 10 and 6. And ten and 6 is easy… 16!
This game invites students to practice “taking one from one addend and giving it to the 9 to make ten and some more…” While this SOUNDS tricky, if you imagine the following images of 9 and 7, it’s pretty evident:
The next game is for mastering the 2 x facts and the 4 x facts. It’s my own (adapted from another BEAM game) but with numbers accessible to students learning these facts in 4th and 5th grade. It’s called Double or Double-Double. The goal of the game is to practice the strategy of doubling (multiplying by 2) or “double-doubling” (multiplying by four).
Think of it this way. Double 6 is 12. Double 12 is 24. So that means that double-double 6 is 24… Mathematically speaking, it’s the same as 2 x (2 x6) or 4 x 6. The idea of “double-doubling” works for all numbers, too. I like to call it the Tim Horton’s strategy. :o)
I hope these games prove fun for you and your children… More than that, I hope they will help your kids to truly master these important facts!
Enjoy!
Carole
The cost of the 12 days of Christmas…
Hello all!
As we head into the holiday season, I thought I would post a link to a really fun site – The Christmas Price Index! It outlines the cost associated with each of the items mentioned in the “Twelve Days Of Christmas” song in an interactive way. Students travel through different locations, finding 6 geese a-laying and 5 golden rings, collecting each of the gifts. Inflationary trends in the cost of these items is also mentioned by the narrator, and the total cost is tallied at the end – that is, the total cost for JUST the 12th day… Students will have to use the information given to figure out the total cost for the 12th and 11th and 10th and 9th days – and so on! :o) Older students might investigate the increased costs in percentage, while younger students can simply add up all the gifts given on any particular day – the mathematical fun is endless! Additional instructional ideas for teachers are outlined on the main page.
Have a merry holiday!
Carole
Penguins everywhere!
Today I had a lovely time at Hillcrest Elementary working with some awesome kids in grades 1, 2 and 3. Our mathematical tasks grew out of the book “365 Penguins” by Jean-Luc Fromental (which is, of course, available in French as well!). Together we added flocks of penguins and created arrangements of penguins, exploring mental math strategies as well as multiplication and division!
The grade 1&2 penguin groupings are here as arelarger penguins for modelling “groups of” thinking. Help yourself, and do explore the book as well. There’s great mathematical potential in it! :o)
Carole
Place Value – with money!
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.
Enjoy!
Carole
Mastering the Multiplication Facts – Strategic thinking…
Hello to my colleagues in Coquitlam!
I wanted to share a template for a set of multiplication flash cards that are set up to be used by strategy. You’ll notice that in the attached multiplication fact cards file, each of the fact families appears in a different colour, to bring to mind the particular strategy that matches the fact. The hope here is that, like with Cuisenaire rods. the related colours will help to call to mind the relationships between the facts themselves. I recommend that students cut apart the cards and practice the facts by strategy, one at a time, as they learn and then recall them.
Here’s the idea…
Consider the “multiply by 5” facts. For these, we think of the 10’s facts, and then find half of it…
7 x 5… think 7 x 10 ÷ 2 or 70 ÷ 2
Because the 5 facts are related to the 10’s fact, their colours are likewise related in the flash cards – yellow for the 5’s and orange for the 10’s.
The 9’s are another fact related to ten. For the “multiply by 9’s” we think multiply by ten and subtract one group.
7 x 9… think 7 x (10 – 1) or 7 x 10 – 7
The 9 fact cards are in a related colour as well – red for the 9’s and orange for the tens.
In the same vein, the 2’s are green, the fours are dark green (since they are double twos or the “double-doubles”) and the 3’s are blue. All are related colours, and each family uses the 2’s in some way. For three, we think double and add one set, like so:
7 x 3…. think 7 x (2 + 1) or 7 x 2 + 7
As for the remaining colours, the 1’s facts are all in white to highlight their simplicity (What you see is what you get!!), the square numbers are purple (for no other reason than purple is my favourite colour and the squares are my favourite facts!) and the last three remaining facts – the most complex ones to master strategically – are in pink.
The facts that are the reverse of all of these (the commutative or “flip-around” facts) have been left uncoloured in the bottom left hand corner of the virtual chart, since if we master the coloured ones – and practice them both forwards and backwards – we will likewise have mastered these ones. You might choose to print and cut them out, or not.
Hope this proves helpful…
Carole
Actual Size… a book about measuring and comparing
Hello there!
A while back I was introduced to the book Actual Size by Steve Jenkins. The illustrations in the book are – you guessed it – the actual size of the creature being described. That makes it a wonderfully engaging and interesting book for children of all ages – and for anyone just learning English, the images are captivating enough to convey plenty of meaning.
I was asked to do a lesson in a grade 2 classroom in Richmond. The school was working onstrategies for non-fiction reading, and I was to focus on the strategies of visualizing and asking questions. We read the book together, talking about the images the children made (the pictures in my head) and the questions we had (what I wonder about). After reading and exclaiming :O at the illustrations for a while, I gave each child a card with a picture of one of the creatures and small bit of text on it. I asked them to look at their card, to read the text and then to do 2 things: to draw the picture they had in their head and to write one question they had about the image. Not surprisingly, every single question the kids had about the creatures was mathematical in nature, and their questions inspired many many math lessons after that day… 🙂 I’ve used the book in intermediate classrooms as well, where the notions of proportional reasoning and scale come in nicely!
I’ve been asked to include the Actual Size fact cards in this post so that others can try this task. I’ve summarized, in kid-friendly language, one fact about the animal on each the card and included (in METRIC) some information about the creature’s length, weight or height. Feel free to download them, understanding that they are drawn from Steve Jenkins’ work and should be credited to him. The file is very very big…!
Let me know how it goes in using these cards and this wonderful book!
Enjoy…
Carole
Introducing multiplication …with understanding
Our new curriculum asks that we introduce the concept of multiplication to our students in grade 3. Exploring the ideas of”groups of”, and “rows of” is an important starting point; connecting these ideas to meaningful situations is likewise critical. Students learn best when knowledge is constructed and maintained within a web of related ideas. Representing and communicating those ideas can take many forms – and the simple act of representing them can help to solidify learning for students. In his illustration (at right) John Van de Walle describes five ways to represent any mathematical concept – and explains that conceptual understanding comes from exploring the relationships between the representations.
The Conceptual Understanding Pentagon (as I like to call it!) suggests that students should build and represent multiplication using models, connect those models to words and translate them to pictures, write a multiplication number sentence and describe a real-world situation to match. There is great power in being able to translate between these representations; mathematicians do this kind of thinking seamlessly. Although students will not be asked to represent every multiplication sentence in 5 different ways, it is important that they use 2-3 of the 5 most of the time, particularly as they are making sense of multiplication in the early years.
There are several games for practice that will support students in continuing to make meaning in multiplication.
The first is called “Circles and Stars“. It’s a classic game from Marilyn Burns that focuses on the idea of multiplication as “groups of” something. Students roll a dice twice – or a double die just once. The first number tells how many circles to draw. The second number rolled tells how many starts to drawn inside those circles. Students should write a number sentence to match their picture and then solve the equation.
When playing Multiplication BINGO, students all use the same card. Like traditional Bingo, the idea is to complete a line (diagonal, vertical or horizontal). Each child in turn rolls the double dice (or 2 dice) and find the product of the numbers rolled. Students then cover the matching product on their own game card and pass the dice to the next player, who rolls just for themselves. Students should check each others work…! 🙂 For example, if I roll a 2 and a 4, then I can cover the 8 in either the I column (corresponding to the 2 facts) or in the G column (which corresponds to the 4’s facts) on my own card. My partner rolls 2 numbers and finds the product on their own card and play continues in this way. To keep it within the bounds of the curriculum – and to ensure the game cards work! – cover the 6 with a sticker or tell students to roll again if a 6 comes up.
Backwards Multiplication BINGO focuses on the commutative property – that is, that 3 x 2 is the same as 2 x 3… The game comes with 4 different cards (cards A through D). Students each take a card and some counters. Like traditional Bingo, the idea is to complete a line (diagonal, vertical or horizontal). In this game, a double die is rolled (or a single die is rolled twice). To keep it within the bounds of the curriculum – and to ensure the game cards work! – cover the 6 with a sticker or tell students to roll again if a 6 comes up… The product is read aloud. Students then find the multiplication sentence that matches the product. For example, if “12” is called out as a product, students could cover the 3×4 or the 4×3 spaces on their cards… Students will learn to be strategic as they play this game. Likewise, they will begin to see relationships between products and factors – an important idea in early division!
The BEAM game called Mice invites students to roll 2 dice and then to choose a number to cover, using either addition, subtraction or multiplication to create the numbers on the grid. The winner creates a line of three in their colour. Clearer instructions are included on the form itself, taken from the BEAM – Maths Of the Month site… I love that e-resource!
The game called “Around the World”, or “I have… who has…?” game is a great one to practice the facts and to make connections to visual representations of number. I have included two versions of the game – a game that matches arrays with their corresponding multiplication sentences, and a more abstract version which uses only numbers. Distribute all the cards and have one child read theirs aloud. In response to the question “Who has…?”, students should look at their own cards to find the match, then read their statement and ask their question. Play is over when you’ve gone “Around the World” (or rather around the room!) and ended with the same person who started the game. For the simpler version with the arrays, students should name the picture and the multiplication sentence using the language of arrays: “I have 3 rows of 4. Who has 2 rows of 3?”
It is important that students have contextualized experiences with multiplication. It is likewise important that they practice their knowledge and deepen their understanding through games and meaningful tasks – but not through timed drill. Memorization of the multiplication facts to 5×5 is not intended in the grade 3 curriculum. Spend time instead working to support your learners with mastering the ideas behind multiplication and developing fluency in meaningful and engaging ways…
I hope these tasks and ideas prove helpful…
Carole
Fractions, decimals and percents… grades 7&8
Hello to my colleagues in Mission!
As promised, here is the set of tasks linked to today’s demonstration lesson for use as follow-up in your grades 7 and 8 classrooms. Each is an open-ended (and therefore differentiated) task aimed at addressing key concepts around fractions, decimals, and percents: comparing, ordering and converting from one to another.
Open-ended problems fractions, decimals
Hope they prove helpful!
Carole
PS – Check out the “If the World Were a Village” Website!
Click Clack Moo resources!
Hello! Everyone has a favourite book to read in their primary classroom. Click Clack Moo, Cows Who Type is one that shows up in many of them! And, if so many people have the book, what better context could there be for fun, open-ended and challenging math tasks!
The file attached includes a series of math tasks that I put together to match the story – and the images – in this book. Best of all, it is available in French as Click, Clack, Meuh!
The cards attached here are for the game called “Get to 50!” described in the Click Clack file…
get to 50 cards – easy complements
Enjoy!
Carole
Centipedes 100 Shoes – resources
Hello all! Here are the pages from the Centipede book as well as the tasks to match! Enjoy!
Carole
Bridging through 10 and 20 – An online game!
A while back I came across a fun game – Bridging Shuttle – that is aimed at students in grades 1-3. The game is intended to show, through the use of a space shuttle’s flight path, how mathematicians “bridge” through friendly numbers like 10, 20, 50 and 100 while adding. When we talk to students about “making ten and some more” from two addends, we are “bridging” through ten. It involves, of course, breaking numbers up into parts and putting them back together again!
The image below shows the screen for the shuttle’s “flight path” called 8 + 4. Students begin at 8, type in the number that gets them to the next friendly number (2) and then hit the red button to proceed to the satellite. Next they type the number to take them the rest of the way – (2 again) and press the next red button to land.
Consider modeling the action of the space shuttle with ten frames with your students to consolidate the process… This game uses only digits and so is more abstract.
Try the harder versions of this game that bridge through 20 and 50!
Carole
Dot patterns and ten frames – Around the World for K/1/2
I thought I would post a copy of a game that I created (which I KNOW someone else out there has likely done before me!) to support young children in recognizing sets of number at a a glance. We all know how important this skill is in promoting number sense in young learners! In this game called “I have… Who Has…?” for grades 1 and 2, students start with a card (or more than one, if there are enough cards to go around!). Choose any child you like to begin the cycle. She reads from her card, from left to right, saying the number represented in the picture (“I have 14.”) and then asking the question (“Who has seven?”). Students listen for their number, then ask their question until all students have had a chance to read. The game is over when the first person to read, reads again. Be sure and hand out ALL the cards, or the cycle won’t work… 🙂
I have created a Kindergarten version of the “I have… Who has…?” game as well. The numbers go only to 10, and there are picture of fingers to help them “read” the number word. A small group – or partners – works best for this one, since there are only ten cards!
Have fun with this!
(And for my friends in Coquitlam who witnessed the spectacular initial fail of this game, rest assured that I’ve fixed it!)
Carole
Hallowe’en Math Games
OK, OK, Kim… Here you go!
It’s about time I did a post for my primary colleagues! Here are some seasonal games for your primary classrooms.
The first is called Ghost Blasters II. You play against a partner, working on finding sums of a number you choose. I picked 20 as my target number, but your students can play with any sum up to 50. This makes it a good game for Grades 1 through 4! One player clicks the Z key when they see 2 ghosts that sum to the target, while the other play types an M to score.
Have young students try the game called Ghost Blasters – Operation Even. Students click on even numbered ghosts and score points as they do. But be careful – if you’re not thinking and click on an odd numbered ghost, you’ll lose 10 points!!
For this last game, called Ghost Blasters I, students pick a “counting by” number to focus on – anything from 2 to 10. This becomes the multiple of the “unfriendly ghosts”, which you click on to evaporate! This is a challenging game – play it first to be sure that your students will be able to click in time…
And for a great couple of Hallowe’en math books, consider Pumpkin Town by Katie McKay – there’s lots to estimate, count and figure out in this book about a true bumper crop of pumpkins! It’s published by Scholastic and is available in French under the name Citrouilleville…
Consider also How many Seeds in A Pumpkin by Margaret McNamara. The characters explore many ways to can count seeds – in 2’s and 5’s and 10’s – and there’s an interesting surprise at the end! maybe even an algebraic exploration should anyone want to pursue it!
Enjoy!
Carole
Grade 10 tasks – Polynomials
Hello Coquitlam-ites…
Thanks for spending the day! I have attached the set of tasks related to polynomials and linear functions for you here.
The websites to mine for more information are:
MATHALICIOUS – open-ended, contextual and engaging digital problems with pdf handouts and thoughtful connections.
MATH INTERACTIVES SITE – This site features short videos to contextualize concepts from grades 6 through 9/10 in the WNCP curriculum. Each video has an interactive task for online explorations and a pdf practice sheet for student use. The main page includes all the strands and topics. This page makes links to how a DJ might use linear relations in his work…
DAN MEYER’S BLOG – Looks specifically for anything with the letters WCYDWT, which stand for “What can you do with this?”. Each of the WCYDWT tasks are digital, language free or language reduced, meaningful and contextual…
THE OFFICE BOUNCING SQUARE VIDEO – This is on Dan Meyer’s blog, but directly linked here…
BRIGHTSTORM MATH – A wealth of video tutorials on every subject in the secondary math curriculum… Great for distance ed students, for support for those who need another shot of content, or for students and resource teachers.
FREE NOTEBOOK SOFTWARE – so you can use Smartboard files like the ones included in your Pearson Teacher’s Guide to demonstrate concepts and move step-by-step through every lesson in the student text…
Hope these are helpful!
Carole
Grade 10 Math Games – Polymonials
Hello to my hard-working Grade 10 colleagues!
Please click here for the set of easy to prepare games for practicing expansion and factoring of polynomial expressions. My favourite is the Connect 4 game… Instructions are included for each, with the exception of the I have… Who has…? game.
To play have… Who has…? cut the game pieces into strips, so that each card reads “I have…” and “Who has…?”. Distribute BOTH FULL SHEETS worth of cards to your students (handing out less than the fill set will cause problems!!). Most students will have more than one card to begin the game. Choose one student to start. The should read the “I have” side of their card (ex: “I have x2 + 3x +2”) and then ask the question to the right of their statement (ex: “Who has (x-2)(x-3)?). The other students listen, multiplying the binomial factors that were just read aloud and respond only if they have the matching polynomial expression (in this case, x2 -5x +6). The game continues until all the cards have been read aloud (that is, until play returns to the first person!).
Challenge your students to run through the cards as a class as quickly as they can…
Enjoy!
Carole
Using manipulatives in math…
Hello to the teachers in Surrey! Thank you for spending the day with me, playing and talking about important mathematical concepts. Time seemed to fly by, as it does when we are truly engaged in thinking and learning together…
I wanted to provide a link to the article entitled “Marilyn Burns’ 7 Musts for Math Manipulatives“. In it, she describes ways to introduce, store and distribute math materials. She also answers key questions about manipulatives, including when and how to use them. It’s a classic article that will resonate with anyone trying to conscientiously implement the new math curriculum – supporting students to demonstrate understanding concretely, pictorially and abstractly!
Below you will find some photos of different ways to organize manipulatives for the classroom. The first is a “Math Cupboard” at an elementary school, with bins and shelves labeled with the names of the materials. Teachers looking to borrow the tubs have their names written on a wooden clothes peg, which they pinch next to the name of the manipulative borrowed on the master sheet.
This is a photo of a rolling trolley of materials used by an intermediate teacher. Each tub has a set of materials (protractors, multi-links, base ten blocks, pattern blocks, algebra tiles, etc) for a table group of 6 students. When it’s math time, students collect their math materials and get started. Having materials pre-counted and organized saves valuable class time!
Hopefully thse resources and ideas will prove helpful.
Have a great start up!
Carole
Play and wonderment in Kindergarten Math
Hello to the fantastic folks from Coquitlam and Mission! As promised, I am posting select slides from yesterday’s ever-so-hot session at the Early Learner’s conference. I hope you all sat sipping tall cool drinks today! It was great fun to spend a brief hour with you to begin the conversation around building play-based mathematical situations in kindergarten. I hope the attached slides, outlining Salma Wasserman’s Play-Debrief-Replay sequence as it related to math is helpful. We cannot underestimate the power of the questions we ask…
excerpt of play in k math session
Carole
Mathematical Thinking 