I’ve put together two free packages for you to download.
For the general public, the first is a collection of five advanced dot to dot puzzles for you to enjoy while you are dealing with self-isolation and social distancing.
For educators, the second package includes free samples of hard and easy versions of some of my puzzles, plus a teacher guide explaining how and why to use these puzzles with your students. Use them as part of your class work or include them in your remote/homeschool program.
Please feel free to share these documents with everyone, young and old, who might like them.
Both of these documents are PDFs and they open in new tabs.
*Read the directions carefully as these are not your ordinary dot to dots, and if you don’t follow the instructions the pictures won’t turn out.
Take care, everyone.
Please enjoy this Free Valentine Dot to Dot to Dot.
Click the image to download the PDF. Read the directions. Print off the “easy” or “hard” puzzle. Fill in the To: and From: Give it to a special someone for them to solve.
This freebie is a sample of one of my Dot to Dot to Dot skip counting number puzzles. These puzzles have a twist. If you don’t follow the pattern and skip over the extra dots, the picture doesn’t work. Learn or practise skip counting by 2s, starting at 1. Have fun and stay sharp.
This single puzzle is a Free Dot to Dot to Dot Thanksgiving Activity. Count by 3s to find the hidden picture. Click the image to download a copy of the puzzle.
These are not your mother’s connect the dot puzzles. They skip count by 1s, 2s, 3s, 4s, 5s, 6s, 7s, 8s, 9s, and 10s, and they have extra dots. Follow the pattern, skip the extra dots, and reveal the picture.
Each Holiday collection has over 13 different puzzles, with a “hard” and “easy” version of most of them. There is Halloween, Christmas, Valentine’s Day, and Easter, as well as a Spring & Summer and a Snowflake collection. Incorporate these puzzles into your number sense, operations, and patterning instruction and assessment. Use them in whole-class instruction, as part of your math centres, or for a fun but educational holiday activity to do with your class.
Get the Free Teacher Package that shows you how to manage the puzzles in the classroom, describes ways to include them in your math lessons, and how to analyze your students’ errors to know where they are on the Number Sense continuum.
Please leave your feedback to help me make these collections the best they can be.
Have you been really focused on number sense in your classroom? Are you using number strings, doing number talks, counting around the circle? If so, then you are thinking about numbers and the relationships between numbers. You are thinking about the patterns made by numbers, and the effect of repeated operations on numbers.
Have you ever considered using dot to dots as another way to play with and explore number?
Maybe not. Typically, they are very simplistic, only a few numbers, rarely going beyond the twenties, and only counting by ones; they are a challenge for only the earliest of learners. However, in the last couple of years, I’ve seen a renaissance in dot to dots, going hand in hand with the colouring book trend. There are a couple of very talented designers/artists out there who have created some brilliant dot puzzles, but they still have their limits as to their use in the classroom.
This past year, I’ve been playing with the puzzle design myself, tweaking the structure and mechanics to make something that teachers can use. My puzzles have several essential differences.
First, They skip count by more than just 1s. My puzzles run the gamut from 1s to 10s. This means they help to learn and practise skip counting and growing patterns, and, by extension, addition and multiplication for almost all the basic facts.
Second, they have different starting points. I have several puzzles that skip by 2s, but some of these start at 1 (1, 3, 5…). Some puzzles skip by 3s, but a few of them start at 1 or 2 (1, 4, 7, 10…; 2, 5, 8, 11…). Solving each number in the sequence in these unfamiliar patterns practices mental addition and number fluency.
Third, there are hard and easy versions of each puzzle, allowing teachers to differentiate for different children’s knowledge and skill levels. The “easy” version is useful for students who are struggling with the concept of skip counting and need some supports to get to each new number. The “hard” version throws distractions and red herrings onto the page in the form of extra numbers. Students doing a “hard” puzzle must really know how to apply the pattern, or risk connecting the wrong dots and creating an incomprehensible scribble.
If you want a more thorough explanation, I detail how these puzzles work and ways to use them in my Teacher Package, available for free here by clicking the link or the images, and on TeachersPayTeachers.com where you can see all the puzzles available. The package explains how to distribute or present the puzzles, tips for differentiating, suggestions for how to tie the puzzles to the number and patterning curriculum, and examples of errors that students might make with the puzzles and what these mistakes might be saying about a child’s skill and knowledge.
Have a look at the Teacher Package or download the free previews for the different Holiday themed collections, also available on this blog. Let me know if you try the puzzles with your students and give me feedback about how it went or suggestions you might have.
Is there really no such number as Eleventeen? Well, there isn’t, but bear with me for a minute.
I have been posting about how we math educators can play with language and make our slightly awkward number naming system a little more logical and a lot more transparent. Check out Part 1 to read about playing with number names to clarify our base-ten system. Check out Part 2 to read about how we can make place value and zero clearer. In Part 3, I am exploring how to make borrowing clearer to the new learner.
The number eleventeen was derived by some astute and logical child who figured that since 13, 14, 15, 16, et al. all end in TEEN, then so should 11, and 12 for that matter. Of course, the poor dear didn’t count on the linguistic mashup that is English. There is also every chance that some other child called 11 ‘Oneteen’ and 12 ‘Twoteen,’ bless her heart.
Wouldn’t it be nice if our number names made more sense? Well, what’s stopping us?
Read this list of numbers to yourself. Then tell yourself what comes next.
10, 20, 30, 40, 50, 60, 70, 80, 90
Did you follow Ninety with One Hundred? Of course you did.
Now, how about this string? Read it to yourself and say what comes next.
100, 200, 300, 400, 500, 600, 700, 800, 900
Did you say One Thousand? Or did you say Ten Hundred? Which one is right? What’s the difference?
Most would say that One Thousand is technically the correct response, but in common vernacular, Ten Hundred is acceptable too. We usually said Nineteen Hundred Ninety-Something a couple of decades ago, and we all knew what we meant.
My idea is to take this already common and, quite frankly, logical exception to naming a number, and apply it to clarifying the act of borrowing.
Look at this equation.
If we subtract these numbers in columns, without any deriving, manipulating, or expanding of the numbers (let’s go ‘old school’ for a moment), we immediately end up in a situation where we have to borrow. And how do we do that?
We can’t do 4-7, so we have to borrow 1 TEN from the Tens column, effectively breaking that decade unit into 10 ONEs. We add those to the 4 ONEs already in the Ones column. Now we have 14-7. But look at how we’ve notated that borrowing on paper. A new learner would look at that top number and lose it. That number now makes no sense. It was 234, now it’s 2214, which looks to me like Two Thousand, Two Hundred Fourteen.
But of course it isn’t actually. It’s 2 HUNDREDS + 2 TENS + 14 ONES.
So here’s my thinking. Let’s actually call it that. The original number was Two Hundred Thirty-four. Now it’s Two Hundred Twenty-fourteen.
You know, as in Twenty-eight, Twenty-nine, Twenty-ten, Twenty-eleven, etc.
If we can say Ten Hundred, why can’t we say Twenty-fourteen?
Typically we abandon what the Minuend (the number we are subtracting from) is called (thereby losing the value) the moment we break it up to borrow. We hash it up, cross things out, write things in. It’s no longer an identifiable number. However, while we broke the rule about how many digits are allowed in a column, and put more than 9 ONEs into the Ones column, we can try to keep order and at least follow the rest of the rules by naming the new number logically.
Rather than jumping in to solve 14-7, I’ve read that it’s a good idea to do all the borrowing before any subtraction is actually done. So, let’s keep borrowing and move on to the Tens.
We can’t take 5 TENs from 2 TENs. We must borrow 1 HUNDRED, break it up into 10 TENs, add them to the 2 TENs already in the column, and get 12 TENs altogether.
Now, we’ve done all the borrowing we have to do, and the minuend is completely mangled. Any thoughts on what we could call this new number? What do we name 12 TENs sitting in the Tens column?
Seventy, Eighty, Ninety, Tenty, Eleventy, Twelvety!
Therefore, I propose the new number we are subtracting from is One Hundred Twelvety-Fourteen.
One Hundred minus One Hundred is Zero Hundreds (or No Hundred, if you read Part 2). Twelvety minus Fifty is Seventy. Fourteen minus Seven is Seven. So the ‘answer’ is Seventy Seven.
Without naming the new number and making the borrowing obvious, there is so much potential for confusion. Children don’t understand how the carried over numbers apply. They think that the number has suddenly increased in magnitude. They forget to cross things out…
But, do you see how, by naming borrowed numbers this way, we are following through with the properties of our number naming system, staying true to the Hundreds, Tens, and Ones structure of base-ten, using the name system to impose order and make what we did with the borrowing process obvious?
Would I exclusively name numbers this way. Absolutely not. That does no one any good. However, by renaming the minuend, we are playing with language. Kids love playing with language. By playing with the language, we are making our number system clearer, exposing rules that have been hidden due to linguistic laziness/pragmatism over time. A clearer sense of number in this gaming context makes the learning stick in a meaningful, more flexible and fluid way. And, when we learn any language, our goal is fluency. The language of math is no exception.
If you want to read Part 1 where I discuss the number words we have for the TEENS and TENS, click here.
For Part 2 of tackling number naming conventions, another issue that stands in the way of many students truly understanding the base 10 system is the role of the zero, and what we call it.
Zero is a quantity, a benchmark, and placeholder. All students learn what zero is as a quantity. They connect 0 to the idea of having nothing very easily. They also catch on to the 0+n=n equation pretty fast; not just the idea of adding to ZERO but the formula itself, even at 8 years old. ZERO as nothing is easy.
As a benchmark, it’s iffy. It applies mostly to negative numbers which baffle teenagers sometimes. However, talks about rulers, scales, and centigrade thermometers can help students to at least see that we generally start counting from zero.
What is really tough is when it comes to place value, especially when borrowing and carrying in columnar adding, and regrouping when applying the distributive property. It’s overwhelming sometimes. And yet, without a ZERO, our number system is impossible. As such, it is imperative that students gain a strong understanding of how the ZERO works and what it means. Here’s a distracting link to Schoolhouse Rock you can show your class-but preview it first because you never know how people muck around with stuff.
So, how do we overcome some of this confusion?
Be aware the ZERO is there, and make sure students understand that it means something. When we have a number like 20, or 304, or even 2, the ZEROs go unmentioned, like they aren’t there; but they are, they have to be.
It’s not actually 20. It’s not even Two TENs. It’s Two TENs and ZERO ONEs.
It’s not 304. It’s Three HUNDREDs and ZERO TENs and Four ONEs.
The simple act of explicitly bringing the ZERO to a student’s attention, and giving a name to the position of the ZERO makes place value much more obvious.
Think of that ZERO as an empty box. Whenever we see a ZERO, there is a place we can put another number. We don’t think about it because we don’t say it (it takes too long to do such a thing practically, but for educational purposes…). If you think about it, even the number 2 has ZEROs. There are infinite ZEROs both before and after that 2, but for brevity and clarity’s sake we don’t mention them.
What’s important to understand is that empty boxes can be filled. You have experienced children who write 198, 199, 200, followed by 2001, 2002. This is because they have misunderstood that the ZEROs are placeholders for numbers that can be added to the HUNDREDs. They think that the ’00’ works like a plural S or some kind of suffix. It must be made clear that the first 0 after the HUNDRED is for any TENs that might come along, and the next ZERO is for any ONEs. What’s nice is the 0 looks like a cozy little container you can put stuff in.Again, labelling the ZEROs with their place value will support this understanding.
Here we have 200…Two HUNDREDs and ZERO TENs and ZERO ONEs. As well, we have Two ONEs. I’m going to add those ONEs in the empty ONEs container of 200 to make Two HUNDREDs and ZERO TENs and Two ONES.
I have Given ZERO a name. I know, ZERO is its name. So is nil, and cipher, and nought. But that’s not what I mean. If you think about 3, it can be THREE, it can be THIRteen, it can be THIRty, or THREE HUNDRED. And each name for 3 implies it’s location in the place value system (again, follow the link to Part 1 to see how to make this explicit). Well, I think we should do the same for ZERO.
So, we have the number 0 alone, and that is called ZERO.
What about for 10? First, we break 10 down and say it is One TEN and Zero ONEs. Then, when that is understood, or maybe in order to make sure it’s understood, we take things a step further, and name it One TEN None.
Did you see what I did there? To clarify, 21 is Two TENs ONE. Therefore, 10 is One TEN NONE. The TENS are no longer seen solely as large collections of ONES. By naming the empty ONEs column, we make the students aware a number is there, it has a purpose and a value even if its value is nothing.
If you keep this going, we have Twenty-None, Twenty-One, Twenty-Two….
We have Fifty-None, Fifty-One, Fifty-Two….
Next, 300 becomes Three HUNDRED, Empty-None. Get it? Fifty=Five TENS, Thirty=Three TENS, Empty=No TENS! (I’ve also toyed with Nonety or Nunty, but that might get confused too easily with Ninety. I’m still experimenting.) The point is, by naming the ZEROs, we emphasize the value of each part of the number as well as the ZERO’s placeholder role. This should prevent numbers like 60024 when writing about how many kids go to our school
Let’s do 2045? Two THOUSAND, NoHUNDRED, Forty-Five. Or perhaps you prefer None HUNDRED?
This can go on, but once we’re in the thousands, the irregularities of naming just repeat themselves. We, as a culture, haven’t used large numbers enough for the language to evolve shortened, lazier forms of big number names. How would you say 3 000 000?
Three Million, NoHundred Empty-None Thousand, NoHundred Empty-None.
Finally, I’ll give you a challenge that one of my students gave me. How would you write the number Empty-Six in numerals?
I’m pretty impressed that he understands that whole infinite ZERO thing, and I see this insight of his as proof that this idea of naming the ZEROs has some real legs. Let me know how it works for you if you try it, and tell me about any confusions or issues that popped up that I haven’t experienced yet.
I have come across a number of factors that stand in the way of children understanding our number system. One issue is our language (English, if you haven’t noticed yet), and the way we name numbers. Number names aren’t always logical, and can confuse many of our students. Here are some ideas to mitigate language confusion for those struggling to gain number sense and understand our base 10 system, and to enrich understanding for those who “get it,” but can go deeper.
You will likely have come across the fact that in some language families, such as Chinese and Algonquin, number names actually describe the number structure. Take 24 for example. In English it’s “twenty-four,” whereas these other languages say the equivalent of two-tens-and-four. The unitizing of TENS and the addition of ONES are plainly obvious. The very act of learning number words in these languages teaches the number system.
So, in my class, I make the language we use equally explicit. Here is how.
What is 10? In our system, TEN can be a series of or a pile of single ONES. It can also be a single unit that contains 10 ONES. I make the comparison between 10 pennies and a dime, or eating 10 cookies one at a time versus buying 1 bag of 10 cookies. I talk about having 10 birthdays and being 10 years old. This comes up when we are practicing skip counting, growing patterns, and working with money. Students must have a strong sense of TENness before they can understand base 10 and place value.
After 10, we have 20. While not as explicit as in Chinese, the word TWENTY has clues to how many it stands for. TW- comes from TWO. -TY implies ten. So TWENTY is Two TEN. We build on that. As a class, we look at THIRTY to see if there are any clues to how many it is. Then we look at FORTY, FIFTY, etc. Don’t forget to look at the words in relation to the numerals. To make it even clearer, I will list all the 10s that make the number in a column, and add them together in a sum below, counting the number of TENS, One TEN, Two TENS, Three TENS, that makes THIREE-TENS, or 30.
Now do 10s with 1s. Twenty-One means Two TENS and One ONE. Fifty-Four means Five-TENS and Four ONES. This ties nicely into columnar adding without carrying. Count the TENS, count the ONES, relate how many of each you counted to the final sum and the structure of that number’s name and numerals.
Now we look at the teens. There are several issues with the teens. One issue is that they use a different suffix to imply a group of ten.The language of the teens has very little in the way of a pattern to latch on to. You might have noticed children who can count by ones in the 20s, 30s, and higher, but still struggle to say what comes after 11. So, it is important to make TEEN explicitly understood as One TEN, and when we count by tens we are really counting One TEN, Two TENs, Three TENs, and so on.
What’s worse, ELEVEN gives no clue to how many it is. Neither does TWELVE. So we have to look at the numerals 11 and 12, and question why it is that we don’t say TEEN for these numbers, even though they have the same numerical structure. The numbers really should be said as ONEteen and TWOteen.
Notice for 20 we say the TENS first, then we say the ONES. However, when we use the TEENS, we say the ONES first, then the TEN. Many children reverse their numbers because of this. If the language of our number system was a little more logical, we might say Teen-eight, Teen-nine, Twenty. Or better yet, we could say Onety-Four, Twenty-Four, Thirty-Four. So, in my class, I actually do use this language. Not always, and not exclusively, but I do use it.
What’s Ten plus Nine? It’s Ten and Nine more. It’s Teen-Nine. But because we have crazy English, it’s Nine-Teen.
This discussion of the language makes students meta-aware of the structure of our numbers and how they express those numbers.
I use conventional language and the invented naming interchangeably. You might be worried that this would cause students problems in the future, with accountants who can’t be understood because they learned how to count from crazy Mister Reed. But this isn’t the case. The predominant convention will dominate. Making the exceptions of our naming system explicit, we create a stronger understanding of the rules, and create connections between the names, numerals, and quantities which creates deeper understanding of number.
This next week, I will be giving two presentations at the Ontario Association of Mathematics Educators conference in Toronto. One will be on using fluid hundred charts, which is essentially using 100 numbered tiles to teach about number sense concepts.
I’ve made a video that describes how to make a set of 100 tiles.
My plan is to post about the different ways to use the tiles, and to link those posts with other videos.
As part of a Math AQ course, I put together this Dos and Don’ts list for things to consider when working with students with autism. Many of the suggestions can be applied to more than math instruction.
However, there is a saying amongst those who work with these children. “If you’ve worked with one child with autism, you’ve worked with one child with autism.” Generalizations about this disorder, thanks to the spectrum nature of it, are impossible to make. Therefore, when referring to this resource, keep in mind that, while these are pretty sound suggestions, the degree of application of any particular suggestion must be tailored to each child individually.
In this list, the first Do corresponds with the first Don’t, the second Do with the second Don’t, etc. The downloadable PDF pairs them in columns.
I hope you find this useful.
- refer to general resources that support your math planning for students with ASD.
- get to know your student’s interests, because those will be the entry points to the math.
- make the math stimulating, fun, and interesting.
- use the same kind of differentiated methods of instruction as with other students.
- use visual aids and concrete materials to make the abstract real and accessible.
- break learning up into manageable parts.
- help the student to generalize math concepts and procedures.
- incorporate rote learning into your program: Many children with ASD are strong in this kind of learning.
- assessments that allow the student to be most successful in sharing thinking, playing to strengths.
- use precise praise for effort and success.
- assume the resource is definitive, e.g. most recommend visual aids, but some students with ASD are stronger auditory and/or kinaesthetic learners.
- be afraid to broaden the student’s horizons by introducing new things.
- over-stimulate, distract, or agitate the student: know the triggers and warning signs that tell you if things are about to take a turn.
- forget the communication skills such as vocabulary, and being able to explain the math, be it with pictures, numbers, or words.
- forget the hierarchy of visual aids: from most effective to least are real objects or situations, facsimiles or models, colour photographs, colour pictures, black and white pictures, line drawings, graphic symbols and written language.
- do the lesson without relating it to the other small parts and the whole.
- let manipulatives take on one fixed meaning or purpose: They are variable models that can represent many ideas.
- limit your lessons to rote learning: It’s important that this knowledge be connected to, and is in service of understanding the big ideas and concepts of math.
- forget to take the student’s weaknesses into account: be sure to consider stamina, mood, and external factors to decide when, how, and how much you assess at any one time.
- be stingy with the non math-based positive reinforcement that will foster a positive attitude and ideal brain state.
I recently posted instructions for a hands-on activity wherein the students make their own meter measuring tapes. I promised I’d explain how I’ve used them so far. Below is a description of ways to use the tapes to teach patterning. This will be followed in another post with ideas for using them with number sense and then, of course, measurement.
Patterning is the act of analyzing the relationships between elements in a string of repeated occurrances of those elements. 😛 In this case, we are looking at the relationship between one number and the next after something has been done to that number. We can focus on the individual numbers, or on the change happening to those numbers.
Have you ever noticed that your students can skip count, no problem, rambling on and on until they reach the biggest numbers they know? Have you noticed that they don’t actually know what 10+2 is, despite being able to count by 2s, or what 15 + 5 is, despite knowing how to count by 5s; they don’t actually know how these patterns are constructed or how they work?
This is because they have memorized these patterns the way they memorize the lyrics to the latest Katy Perry song; they know the words, but have no idea what they are actually saying!
To avoid this meaningless regurgitation, it is important to teach the students how the pattern works at the same time as teaching them the pattern. Better still, is teaching them about how changes happen in patterns and letting them figure out the different patterns that can happen because of those changes. Specifically, we can continually add on, or take off a fixed quantity, and say the new number we get, and eventually, we notice the same numbers happening over and over until we see a pattern.
To accomplish this learning, the skip counting must be accompanied by a concrete understanding of the quantity we start with, the quantity being added, and the quantity that we end up with, and the students must understand that it is a recursive process wherein the quantity we end with becomes the quantity we start with.
These measuring tapes (and any other number line example, such as thermometers, hundred charts, etc.) provide both a concrete example of quantity, and a way to figure out, check, and keep track of the pattern until it is memorized with meaning.
With the students sitting in a circle, with their personal measuring tapes in hand, start at zero, and go around the circle, adding the number you’re skip counting by. As you go around, the children mark the place on the number line with their fingers, and they move to each new number in sequence as they hear it called out.
If you followed my advice in the first post, and numbered only the 5s and 10s, then you should start the students practicing this activity with 5s. This will help those who struggle with counting, as the numbers are clearly marked.
Then you can do 10s, noting that you skip the 5s.
Then you can do 1s, easy because they should know the sequence orally, difficult because the numbers aren’t marked on the strip.
The students will lose their places on the tapes. That’s fine. Once they hear the number that came before, they will be able to catch up. Either, they will be forced to go right back to the beginning and count out (because they haven’t yet realized that the number on the tape always represents the same fixed quantity), which is valuable practice, or they will, hopefully, eventually, realize it’s easier to go to the closest anchor of 5 or 10, and work out from there where the next number is.
For example, “The previous number was 34, I can find 34 because it is one down from 35, and now add 2 more, that makes 36.”
Over and above developing a true understanding of the skip counting process, this activity builds estimating skills, builds fluency with number and the distributive property, and sets the ground work for operations with ‘friendly numbers.’
Follow the same activity to count by 2s.
In Ontario, grades 1 and 2 have to count by 1s, 2s, 5s, and 10s, starting from a multiple of those numbers. Grades 3 and higher have to be able to count by those numbers from any starting point, and be able to count by other factors as well. It’s easy to adapt the circle counting activity by changing the factor being added, and/or changing the starting point.
Using the tapes forces the students to figure out what comes next, by adding/subtracting the right amount to/from the previous number. This is better than just hearing the pattern and parroting it back. Also, as patterning becomes algebraic study in the later grades, the students become familiar with the recursive process, or the algorithm of skip counting (n=n+1), so the formula will have meaning when they eventually learn it.
What concrete thing are we counting exactly? Make sure that the students understand we are counting centimeters, “the spaces between the lines.” This might require some groundwork on what centimeters are, in measurement terms.
If centimeters are too abstract, use counters of some kind, small enough to fit between the lines on the tape. Place the correct number of counters, directly on the tape, with each new iteration of the pattern.
In and Out
There is a game played in the Junior grades where the players give a number to put in to the ‘machine’ and the leader gives the number that comes out (input and output). For example, a student says 5, the leader writes 10. A player says 10, the leader writes 15. You can use these tapes to figure out the algorithm for what happens to the input to get the output. By marking on the number line where the input is, and the output is, commonalities will reveal themselves, and a rule will be developed.
The patterning in this case lays in the repeated operation. What is always happening to the number given to get the new number. Incidentally, this is an A pattern. +1, +1, +1, etc. is A, A, A, etc..
Essentially, any number pattern game you can think of can be done with these measuring tapes. If you have games you use to explore the numbers on 100 charts, you might be able to adapt them to these number lines, taking advantage of the linear nature as compared to the array layout of the 100 chart.
If you have any other ideas for ways to use the measuring tapes to build number pattern knowledge, please share them in the comments below.