# Focus on Math

## Helping children become mathematicians!

### GAD Workshop, Surrey, BCOctober 23, 2015

Thanks to the teaching staff of GAD Elementary in Surrey, BC, for their warm welcome and heartfelt participation as we delved into problem solving, math tools and strategies, and math processes (especially communication). Changing our teaching practice is not an easy feat, but if we commit to some small changes, practice them regularly, add more changes, practice those regularly, and keep on going in that manner, we can end up making a significant and lasting change that will benefit students greatly.

Remember, “math talk” does not just happen. We have to plan ways to incorporate it into each math lesson. It is a good idea to create math partners so students are responsible to talk to someone about their math thinking. Modeling (letting students hear YOU talk through a demonstration problem) is always a good idea. Responding to students with proper math language/vocabulary (when they have not used such) is helpful. Posting “sentence stems” is a great way to give them an easier start in speaking math. Additionally, try creating a “math words” chart with the students that they can use as an on-going reference in both their speaking and writing (click here to see an example of a “math words” chart.)

As promised, I am adding links from this post to the handouts from today’s session (see bottom of the post) and some that we just talked about.

I would LOVE to hear from any of the GAD staff of how things go in your math lessons in the next weeks. You all listed something that you could begin to do right away in your classrooms, and I hope you will share what you are doing and the effect it is having on the students.

Remember, understanding “lives” in the processes! Reflect on your teaching regularly to see if you are embedding those processes into math classes. It will make a big difference in students’ understanding if they are immersed in the processes!

Mathematically yours,

Carollee

100 dot array (teacher size)

100 dot arrays 4 per page

100 dot arrays 6 per page

100 dot arrays 12 per page

break apart number sheet – 2’s

break apart number sheet – 3’s

problem solving assessment rubric

10 frames (teacher size)

10 frames (student size)

10 frames blank mini’s

### “Number of the Day” Sheets: Choosing the NumberJanuary 5, 2015

I recently received an email from Stephanie, a grade 2 teacher in Newfoundland, inquiring about choosing the number for the Number of the Day sheets:

“I really love the Number of the Day sheets you have produced and the opportunities for differentiating the instruction. Just wondering how you set this up? Do children do this everyday or on designated days? How do you decide on the number for the day?”

I thought others might be asking this same question, so have decided to post an edited version of my response to Stephanie’s question:

As for setting up the Number of the Day sheets, things are really flexible. There is no one right way — you want what works best for your students and your time constraints. That being said, I have found that if you are able to have the kids do them really regularly (daily if possible) over a good number of weeks, the students are able to really get into the meat of them. By sharing about them after they have worked them, students get to hear what others have tried and will often stretch themselves to try to match what others are doing. They have a chance to really play and explore the number relationships that are brought out in the sheets’ activities.

The number can be picked in a variety of ways — everything from you choosing, a student choosing, drawing a number from a jar or dropping a bean on a 100 chart! Sometime I have chosen specific “repeats” (e.g., every number that week has a 9 in one’s place) sot the kids to see and compare what happens in such cases. What is the same as before? What is different? Or I might pick several numbers within a “decade” (e.g., 33, 37, 31, 35, 38) and again have students compare/contrast over those days.

No matter what number is chosen, one question that is really great to ask is “What do/did you notice?” When that is asked often in the math classroom, students get in the habit of paying attention to details, looking for patterns, making comparisons, and such.

I am happy with random numbers, too, but sometimes choosing numbers with a particular relationship is good so you can really draw out the depth of the relationship.

Mathematically yours,

Carollee

Level 1 (English and French)

Level 2  (English and French)

Level 3 (English and French)  (pictured above)

### 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

### How Many More to Make 30?February 12, 2013

This is an activity I created to use with two grade 2 classes that I work with at Charlie Lake Elementary. In BC, grade two students work extensively with numbers to 100. The activity, like “How many more to make 20?” (see post from Feb. 5, 2013),  is based on one of the foundational number relationships which is, for numbers 1 to 10, anchoring each number to 10.

30 was chosen as the focal point for this activity since multiples of 10 are also important anchoring numbers.

Once again I was delighted to put some special dice to use, in this case 30-sided dice.** Each child rolled the dice and then, using a set of 10 frames, created the number rolled at the top of the sheet, right over the blank ten frames there. Thus, if 14 were rolled, the child placed a full ten frame and a one showing four on the paper, and then recorded the number 14 in the roll column of the T-chart. Then he looked to see how many would be needed to make 30, in this case 6 to fill the partial ten frame and one more full ten. 16 was  recorded beside 14 on the T-chart (see picture).

Similarly, if 7 were rolled, the child placed a ten frame showing seven on the paper, and then recorded the number 7 in the roll column of the T-chart. He could see that to make 30 he would need 3 more to fill the partial ten frame and two more full 10 frames, and thus 23 was recorded on the T-chart (see picture).

As in the “How many more to make 20?” activity, some of the children stopped making the numbers with their ten frames soon into the activity. Clearly they could imagine the anchoring relationship in their minds and did not need to manipulate the cards to “see” the numbers. Other children needed the support for every roll, but they were still able to be successful because of the scaffolding the ten frames provided.

I hope you will try the activity with your students!

Mathematically yours,

Carollee

**If you do not have 30-sided dice, having students draw numbers from a bag or spinning numbers on a spinner will do nicely. You could even give students the page with the first column already filled in with numbers of your choice.

### How many more to make 20?February 5, 2013

This is an activity I created to use in a grade one classroom here my school district. (In BC grade one students work extensively with numbers to 20.) It is based on one of the foundational number relationships for numbers 1 to 10: anchoring each number to 10. A set of ten frames is a fabulous tool to help build this relational understanding with young children. The ten frames provide a visual representation of each number and clearly show how far away each number is from 10.

Along with 10 being an anchoring number, multiples of 10 are also important
anchors. With this in mind, I felt it was important to give grade one children the opportunity to practice anchoring numbers to 20.

I had some 20-sided dice that were perfect for the activity**. Each child rolled a die and then, using a set of 10 frames, created the number rolled at the top of the sheet, right over the blank ten frames there. Thus, if 14 were rolled, the child placed a full ten frame and a one showing four on the paper, and then recorded the number 14 in the roll column of the T-chart. Then he looked to see how many would be needed to make 20, in this case 6, and recorded it beside 14 on the T-chart.

Similarly, if 7 were rolled, the child placed a partially filled ten frame showing seven on the paper, and then recorded the number 7 in the roll column of the T-chart. He could see that to make 20 he would need 3 more to fill the partial ten frame as well as a full ten more, and thus recorded 13 on the T-chart.

Some of the children stopped making the numbers with their ten frames soon into the activity. Clearly they could imagine the anchoring relationship in their minds and did not need to manipulate the cards to “see” the numbers. Other children needed the support for every roll, but they were still able to be successful because of the scaffolding the ten frames provided.

I hope you will try the activity with your students!

Mathematically yours,
Carollee

**If you do not have 20-sided dice, having students draw numbers from a bag or spinning numbers on a spinner will do nicely. You could even give students the page with the first column already filled in with numbers of your choice.

### Candy Bars: a grade 2 or 3 questionNovember 15, 2011

I recently gave both my grade two classes at Charlie Lake School a question involving groups of six. We had been working on a variety of problem-solving strategies including these:

• using counters/objects
• drawing pictures
• using ten-frames
• using a 100-dot array
• using a 100 chart
• using a blank number line
• breaking numbers apart
• using operations such as adding and subtracting
• looking for patterns
• using charts or tallies

I posed this question to the students:
Luke is buying candy bars to share with his classes. They come in packages of 6. How many packages will he need to buy if there are 25 students in his class?

The students glued in their question strips and we read the question together to make sure they understood what was being asked.

I will stop here and give my opinion on an issue. I know there are a lot of math text books out there with lots of writing in them, and teachers have told me that some students who are not good readers, but are better with numbers, have little success using books that require a great deal of reading. I like using a problem-based approach to mathematics, and find that, when I am focusing the lesson primarily on a single “rich” question, then I can read the question with the students, make sure students understand what is being asked, and set the students to work. Although I am a great proponent of literacy and want students to be accomplished in that area, I do not want reading to hold a student back in my math classroom. A rich question, again in my opinion, is one with multiple possible solutions OR a single solution with multiple strategies for finding the solution (or both!). I use many of the latter, and encourage students to find as many solutions as they can. In many cases, the more ways they can solve the problem, the greater their understanding of the concept.

So, back to this particular question. The students went to work solving the problem of candy bars bought in packages of 6. The photo shows some of the strategies shared in the one class. The discussion was quite interesting. The students realized that buying four packages of candy bars would get Luke 24 bars, but most did not want him to purchase another full package. They were suggesting it would be better if he then went to a convenience store and bought only one more bar, which, in real life, is a great idea!. I asked the students, what if Luke were in a hurry and had to buy only packages, and they all agreed that he would need to buy 5 packages to have enough. There was then a discussion around the extra bars: he could save them; he could sell them; he could give them to his family. Lots of good ideas!

It is important to have those kinds of discussion around division and any remainders that come up, because in real life things are much more likely not to divide evenly than to do so. The remainder must be considered carefully. If students were doing this in a “standard” way, they might be likely to say that 25 divided by 6 is “4 remainder 1” without ever considering what the remainder of 1 would stand for. In this case, it would be a student without a candy bar! The actual answer to this question is 5, not 4 remainder 1. This is a case of the answer being forced up to the next whole number.

Remember, have your student discuss the remainders!
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