Dot Cards Revisited November 19, 2014

Number sense is an important thing for students to develop. Some do it naturally, but most do so as the result of activities and lessons that are designed to have students think about numbers and their relationships with each other.

I recently had an email from a teacher asking how she might use dot cards to help the students in her grade 2/3 split class develop number sense, so this post is specifically to address that question of Amanda’s.

I use dot cards extensively in K and grade 1 classrooms, but they can be used for grade 2 and 3 students as well, especially if those students have not been exposed to many visual representations of numbers. If students are rather unfamiliar with the visuals, going back to the “basics” and working with them regularly for short periods of time is profitable. There are many ideas for such activities in the handout from my Dot Plate Workshop. (Here is a link for my other posts that refer to using dot cards.)

Here are a few ways to use the dot cards for grade 2/3 students:

Estimation: have the dot cards in a pile, a basket, or other way to easily access them. Have students draw three or more cards and then estimate the sum. It is important to have students discuss why they made the estimation that they did. It is also important to not force them to get really close to the actual sum. Rather, the skill is to practice the thinking about the numbers to get a reasonable sum. There are many ways students might do this: they might look for pairs of cards close to 10; they might decide that every card is either over or under 5 by a bit, and thus just add 5 for every card; they might realize that no card has more than 10 dots, so the answer cannot be more than the number of cards x 10; they might use a combination of methods to get as close to the answer as they can. Again, it is the process of thinking that is important here.

Mental Math: again, have the dot cards ready to be accessed, but also have some “decade cards” marked with 20, 30, 40, 50, 60, 70, 80. 90, 100 (and higher if you wish, depending on the ability of the children.) Have the students draw a dot card and a decade card, and subtract the dot card from the other number: e.g., if a student draws the 60 card and a 4 dot card, he would do the question 60 – 4 = 5. We have students practice subtracting in basic facts (including 10 – x) but often we do not have them practice the mental math of subtracting from other two-digit numbers. This is a great place to begin that practice.

Match and Add: place the dot cards face up. Have students scan the cards for two that have the same visual someplace on the card. For instance, a student may pick two different cards that each have “5-on-a dice” on them, and then add the two amounts (In the illustration, the student would be adding 9 + 7 but may see the strategy 5 + 5 + 4 + 2).

Once students are quite familiar with the dot card/dot plate visuals, it is fine to do the exercises with numerals cards.

Mathematically yours,
Carollee

Dot Plate Make-and-Take: A Great Success! November 23, 2013

The Dot Plate Make-and-Take workshop was a rousing success! In spite of the nearly -30 C temperatures and the slippery road conditions, the registrants all showed up, some driving an hour or more to come.

We talked about number concepts for students in kindergarten to grade 2, mainly focusing on the “big four” relationships that we want students to understand regarding numbers 1 to 10:

• One more/one less (and two more/two less above K)
• Anchors of 5 and 10
• Whole-part-part
• Visual/spatial relationships

With those firmly established, we are able, over time and grades, to take those same relationships and help students make those same connections with other larger and more complex numbers.

Participants made a set of Dot Plates using desert-sized paper plates (we used sturdy ones), bingo daubers and a pattern guide. They also received copies of small dot cards (4 sets printed onto coloured card stock), a set of large card stock dominoes. All of these are visual tools that can be used to build the above “big four” relationships. We also discussed the use of ten frames (large, teacher-sized ones, small student-sized printed ones, and blank five- and ten-frames) which are particularly good for anchoring numbers to 5 and 10.

 

Download here:

• Handout from the workshop
• Large dominoes pattern pages

 

 

Go to the bottom of the Math Camp 2013 blog post to download the following:

• Dot plate pattern sheet
• Small dot card pattern pages
• Large dot card pattern pages set 1, set 2, set 3
• Student ten frames
• Teacher ten frames

 

Mathematically yours,

Carollee

Dot Plate Workshop: Early Numeracy Concepts October 9, 2012

This week some of the teachers in the district attended a workshop held here at the SD#60 board office. Our focus for the session was early numeracy, in particular, number relationships that are important for young learners. We focused on these “big four” relationships:
• One more/one less (extending to two more/two less)
• Visual/spatial relationships
• The benchmarks or anchors of 5 and 10
• Whole-part-part

I particularly refer to the whole-part-part relationship in that manner (as opposed to part-part-whole often used by others). I like stating “whole” first because the emphasis of that relationship is that a number can be pulled apart into two smaller parts, not the joining together of two parts to make a larger whole. This distinction is not just a matter of semantics, but rather a spotlighting of the pulling apart. “To conceptualize a number as being made up of two or more parts is the most important relationship that can be developed about numbers” (Van de Walle, 2005).

Understand that there is a lot of counting that must take place as children work to build these relationships. They must repeatedly work with counters as well as dice, dominoes, ten frames, dot cards, dot plates, and other things that show patterned arrangements of numbers to build a deep understanding about the numbers, first 1-10, then extending to 20, to 100 and beyond.

The workshop participants went home with a set of dot plates they made from small paper plates and bingo daubers (see photo). They also took home 4 sets of small dot cards printed on colourful cardstock. Lastly they took away larger paper plates with dot patterns on them (the patterns from either mini dot cards or mini ten frames) that could become spinners for games or made into activities for math centres.

One other tool that we talked about was a grid of tools that both the teacher and students could use for representing numbers. When one looks at the grid and possible combinations of materials, it is easy to see that having a few good tools on hand allows for many different ways for young children to be involved in representing number.

Download the Representing Number Grid here.

Download the dot card spinner here.

Download the ten frame spinner here.

I hope you will think deeply about the ways you are having your young learners interact with numbers! You are laying the foundation for later mathematical learning.

Mathematically yours,
Carollee

PS My apologies to the participants — I had intended to post this blog by the end of last week and did not get to it, and over the weekend I did not have access to the visuals I wanted to post with it. So, hopefully, this is a case of better late than never!

Reference
Van de Walle, J. and Folk, S. (2005). Elementary and Middle School Mathematics: Teaching Developmentally (Canadian Edition). Pearson: Toronto.

What’s Important to Have in a Grade 1 Classroom? October 2, 2014

I was recently contacted by a former colleague, Dawn, regarding what manipulatives a grade one classroom might need to have on hand to support effective learning math. It seems a friend of Dawn’s is in a classroom which really has nothing for the children to use for hands-on math learning and they were wondering what was needed.

First off the classroom needs counters — counters in different shapes, sizes, etc. They can be purchased ones (such as mini plastic teddy bears) or ones gathered from home (such as bread tags, but†ons, etc.). But the need to be abundant and available.

Students need a way to count efficiently, especially in tens and ones. Egg cartons cut down to 10 holes, blank 10-frames printed on paper or card stock, or commercially produced 10-frames can all be used. I even like using cookie sheets (non-aluminum) and marking them with coloured tape as a giant 10-frame for use with magnets.

Base 10 blocks are also great for young students. These a generally in the form of small 1 cm cubes for “ones”, sticks for the “tens”, and flats for the “100’s”. I do want to make a critical point here: students may be engaged in a game of trading 10 cubes for a stick, or 10 sticks for a flat with every appearance of understanding the “ten-ness” of our base-10 number system. But be careful here. Student can be following your rule of trading 10 for 1 without that understanding. They might be just as happy to trade 8 for 1 or 12 for 1. The manipulatives give a opportunity for students to develop that important base-10 understanding, but moving blocks around correctly does not necessarily indicate that the understanding has been built in the student’s mind.

I think a grade one classroom needs “pop cubes” (multi-link cubes) — those blocks about 1inch in each dimension that can be attached together. I like to store them sticks of 5. If students need a particular amount for an activity, say 18, we discuss how many sticks each student will need, and then go get them. I also use these in many quick number-sense building activities. If I have students hold up a certain number of blocks, I want them to do so to model a ten frame. If I ask for 9 blocks and a student were to hold up a single stick of 9, I, as the teacher, cannot tell from a distance if the student is holding 8, 9, 10, or 11 blocks. But if he holds up a five stick beside a four stick, I can tell at a glance that he has the correct number. Pop cubes can be used in a multitude of math activities and should be on-hand for regular use.

Another must-have in my book are pattern blocks. They are particularly great for patterning activities for exploring symmetry, not to mention the creativity factor! I love them!

There are a number of things that I think should be in the classroom that are “make-able” such as dot cards, dot plates, printed ten-frames, even printed dominoes (click for more info on these)— all useful in exploring numbers, in building number sense, and  in helping students develop the skill of subtilizing. Students need to SEE the numbers in math, and these materials can help develop that “seeing” in the children.

Of course there are many other things that are fun to have in the math classroom, such as dice, dominoes, blocks, playing cards, geoboards, plastic coins, bingo chips, square tiles, Cuisenaire rods, and two-colour counters, to name a few. But lots of math learning can take place with some thoughtfully crafted lessons and activities and just the basics.

I hope you will focus on the math understanding with whatever materials you have at your disposal!

Mathematically yours,
Carollee

Building Numbers: A Kindergarten or Primary Activity January 23, 2014

I visited Mrs. Merrill’s kindergarten class today. The focus of the math lesson was on building numbers to 10, we did this with a large organizing sheet (11” x 17”), some small dot cards (each student had a set of cards 1 to 10 — download below), and small bingo chips. Students were asked to place a dot card over each square on the paper, and then use the bingo chips to make another set the same size in each box.

Some students chose to lay their cards out in rank order, while others were happy to just lay the cards in any order they pulled the card out the small bag. We also noticed some students organizing and building left to right in the boxes, while others built randomly on the page.

The organizing sheet is very “generic” on purpose allowing it to be used in a variety of ways. The number for building can be generated by a dot card, by rolling a die (or dice), by placing a number word card (e.g., “two”) over the square, by dropping a bean onto a 100 chart, etc. Students can build the number with counters, with little ten frames (I find children love to work with tiny things!), with base-10 blocks, etc.

I have not figured out how to put a large 11 x 17 paper into Dropbox, but if you will print out the template you want onto legal-sized paper, you can then enlarge at a copier 121% and it will fit the 11 x 17 page rather nicely.

I have created templates for building 8 different numbers as  well as for building 10 different numbers. The latter could be used for 100 Day activities by building 10 in each space for a total of 100.

Dot cards can be downloaded here as well.

I hope you are able to use the activity in one of its “versions” in your classroom!

Mathematically yours,

Carollee

Math Camp 2013 Reflections… August 28, 2013

Wow! Math Camp 2013 was a resounding success! The focus each day was on how we can structure routine activities for our students that will allow them to build number sense. We also talked about Carol Dweck’s research about mindsets and looked at how we could help our students build a ‘growth mindse’t in mathematics and not be stuck in a ‘fixed mindset’. (If you have not read Dweck’s book Mindset, I encourage you to get a copy asap!)

We looked at visual routines, counting routines, and routines involving number quantity, and discussed how each of these can be utilized for learning.

Our visual routines involved using 10 frames, dot cards, dot plates, 100 dot arrays, fraction pocket charts, percent circles, base-10 grid paper, and number lines (I always have students draw these rather than use ones that are pre-drawn and pre-marked). See end of post to download the various tools.

Our counting routines involved choral counting, counting around the circle, and stop and start counting, and counting up and back.

Our routines for number quantity involved mental math, number strings, “hanging balances”, and decomposing numbers.

It would take too long to write here in one post about how best to use/do each of these ideas, but over time I will get to them. Are you interested in something in particular? Email me and let me know and I’ll get to that one right away!

All of the “math campers” went away with lots of ideas that can be implemented in the classroom right away. I’ll be excited to hear from them how it goes it their classrooms.

I’ll leave you with my favourite definition of number sense: “Number sense can be described as a good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms.” (Hilde Howden, Arithmetic Teacher, Feb., 1989, p.11).

There is much food for thought in that quote alone!

Mathematically yours,

Carollee

Click to download: student 10 frames , teacher 10 frames; student dot cards,  large 100 dot array, 12 small 100 dot arrays, 6 small 100 dot arrays, 4 small 100 dot arrays, teacher dot cards set 1, set 2, set 3; template for making dot plates;  base-10 grid paper, percent circles; directions for making fraction pocket charts;

Twain Sullivan Elementary, Houston, BC April 27, 2013

What a grand day we had at Twain Sullivan Elementary yesterday, in spite of a number of “complications” (including losing power in half the school and, oddly enough, half the room we were in). I did not even remember to tell the participants that when I first pulled into the school parking lot in the morning, there was a beautiful rainbow settled just off to the side of the school. It seemed a good omen for the day – there is something lovely about a rainbow!

We were able to persevere through the strange complications and ended up having great session. The main topic of the day was number sense, and we particularly focused on helping students build particular relationships between numbers, thus creating connections that would lead to greater overall math understanding.

There were a number of handouts that we used, and I promised “clean copies” of those. Some of those handouts are also posted elsewhere on the blog, but for the sake of simplicity, I shall repost them here in one spot for easy access for the Twain Sullivan folks (and anyone else who is interested):

Powerpoint handout of the day
Student ten frames
Mini blank ten frames 27 per page,  40 per page
Teacher ten frames
Large dot cards – set 1
Large dot cards – set 2
Large dot cards – set 3
100 dot arrays (4 & 12)
100 dot array (large)
mini blocks of 100 black dots
How far to 20?
How far to 30?
How far to 100?

The really large 10-frame that we used does not download properly – at least I have not figured out how to make it work.

If I have left anything off the list, let me know and I will add it on.

I hope I get to work with you all again in the future. I was impressed with your willingness to adapt what you are doing in math to make it better for your students. Thanks again for a great day in Houston, BC!

Mathematically yours,
Carollee

Spatial Activity: Tangram Art Bulletin Board November 19, 2012

There is compelling evidence (Reynolds, 1992 and Sfard, 1994) that imagery plays a significant role in mathematical reasoning and that mathematicians use imagery in powerful ways (Hadamard, 1949; Nunokawa, 1994; Sfard, 1994). Mathematics is not just a subject of logic and reasoning, but it is one that is laden with imagery.

Doing activities such involving tangrams (and other similar manipulatives such as pentominoes, dot cards, etc.) gives students a chance to develop their spatial sense in mathematics.

This past Wednesday the students in all of my math classes at Charlie Lake School did tangram art. I provided students with a set of tangrams die-cut from construction paper along with patterns for creating a variety of shapes. Over the years I have collected a variety of tangram patterns in books, but these days many patterns are readily available on the Internet. Put together on the hall bulletin board, the students’ tangram pictures make a delightful display.

When I have enough time, I prefer to have students (especially older ones – maybe not my grade 2’s) cut their own sets of tangrams from 10 cm squares of construction paper, but as my classes run on a fairly tight schedule on Wednesdays, I went with the pre-cut sets. (Download instructions for cutting a set of tangrams from a square here.)

I have found it interesting to observe that some students who struggle with symbolic notation in math “shine” when it comes to visual/spatial activities. I have also observed the reverse to be true: students who easily manipulate numbers cannot always move things in space so easily.

Although students sometimes perceive such as activities as “art” or a day away from doing “real math”, these kinds of activities actually build their ability to use imagery, thus building their math sense.

I encourage you to use some visual spatial activities with your students – of any age and grade level!

Mathematically yours,
Carollee

Math Camp: K-1 (2011) August 27, 2011

Thank you to all the wonderful participants in yesterday’s Math Camp session! Judging by the sense of excitement that was in the room by the end of the workshop, I know you were taking away with you some great ideas for the new school year.

Remember that much of what you do in Kindergarten and Grade 1 needs to centre around number relationships (primarily these: whole-part-part*; anchoring numbers of 5 and 10; one and two more/one and two less; and visual-spatial relationships.) It is as children have numerous opportunities to explore these relationships that they begin to develop number sense.

* Do note that I prefer the term “whole-part-part” to the more common “part-part-whole”. The emphasis in this number relationship is the pulling apart of a number, not the pushing together of two parts to make a larger whole. Primary teachers may make a connection to a similar distinction in reading, namely the difference between decoding and encoding words. They denote two very different processes. Traditionally in math the emphasis has been placed on “encoding” numbers, or adding them together, with little or no emphasis given to to “decoding” numbers or pulling them apart. Children need repeated practice in pulling numbers apart in different ways. We want them to notice that in different circumstances, different parts are more beneficial.

As promised in the workshop, I am posting links for the blackline masters that we referred to during the workshop. I hope up you make good use of them! (Click on any item below to download the file.)

dominoes (large) template
small dot cards template
large dot cards – 1       NOTE: these large, demonstration-sized dot cards appear to go off the page.
large dot cards – 2       That is normal. There are only two large cards fully on each page.
large dot cards – 3       Ignore the stuff on the sides! The two that matter are there!
student ten frames
teacher ten frames
blank 5 frames
blank 10 frames
folding whole-part-part cards

As always, let me know if I can be of more specific help.

Mathematically yours,
Carollee

Number Relationships: a Foundation for Number Sense April 22, 2012

In yesterday’s parent workshop (primary) session we talked about the importance of helping children build relationships between numbers. We focused on four such relationships, namely these:

• visual/spatial relationship
• anchors of 5 and 10
• 1 more/1 less (and 2 more/2 less)
• whole-part-part

Building these relationships for the numbers 1-10 lays the foundations for understanding other numbers, both larger and smaller. At every level of the place value system we can apply these same relationships. Students have the opportunity of taking something they already know and expanding it to something greater — a wonderful way for meaningful learning to happen!

In exploring these ideas and relationships yesterday we used or discussed some tools, which I will post here:

• 1 to 100 chart
• 0 to 99 chart
• ten frames
• 100 dot arrays in two sizes
• large blank 10 frame (this can be enlarged further and used as a frame for putting ten cards in each space for building numbers to 100)

Thanks to all the participants in both workshops yesterday. Contact me if I can be of any further hep to you.
Mathematically yours,
Carollee