# Focus on Math

## Helping children become mathematicians!

### Patterns in 100 Charts: A Treasure of LearningMarch 21, 2013

Are you looking for a great activity to do with your elementary-aged children? Consider playing with a 1-to-100 chart (often called a 100 chart), or its first cousin, the 0-to-99 chart. These visual organizations of numbers are a staple in primary classrooms, but both bear looking at deeply and often.

One great thing to do is to examine the patterns are created when you colour or highlight various skip counting patterns on the chart. Often we think of skip counting by 2, by 5 and by 10. But if you consider each of the skip counting patterns (by 2, 3, 4, 5, 6, etc) you are also playing with the multiples of those numbers. When students in grades 3-6 practice multiplication facts, they usually do not go beyond x10 or in some cases 12. By looking at the skip counting all the way to 100, you can explore the repeating patterns that show up. Some of the patterns are particularly interesting! (Do you know how a knight moves on the chessboard? Try looking for that pattern in the skip counts.)

As an intermediate classroom teacher, I always had my class do these colouring patterns and we stapled the sheets into a little booklet that they kept in their “math toolkits”. We revisited them throughout the year.

Do some general exploring of patterns, too. Just let kids “notice” some things. If you need a few prompts, consider these:
What patterns happen in vertical columns?
What patterns happen in horizontal rows?
Compare the various diagonal rows?
What do you notice when you colour every number with the digit 4 in it? With 5?
What is the same in each of the above? What is different?

I have provided mini-100 charts and mini 0-99 here for you to download for use in your colouring and comparing activities. I hope you will try some with a class or with your own child. Enjoy!

Mathematically yours,
Carollee

### Number Relationships: a Foundation for Number SenseApril 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:

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

### 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

### Win Sum-Thing from NCTM Illuminations!September 20, 2011

I subscribe to a list serve from the National Council of Teachers of Mathematics‘ (NCTM) website Illuminations. If you have not visited there, it is a great source of lesson ideas for mathematics classes of all levels.

I am copying part of one of their recent emails where they are announcing a contest based on trying a particular posted lesson which explores patterns on a 100 chart. Even if this particular lesson does not “tickle your fancy” or you feel it is not at an appropriate grade level for your class, it is well worth your time to take a look at the Illuminations site. There are lots and lots of good lesson ideas there. Here is the information that came to me:

Win Sum-Thing!

Win a classroom set of A+ Tiles simply by trying your hand at an Illuminations lesson. In the lesson Do You Notice Sum-Thing?, students are asked to consider patterns that occur when various tiles are placed on a hundreds board. For a chance to win a full classroom set of the tiles or an individual set of A+ Tiles, try the lesson with some students and then do one of the following:

* Submit a picture (or link to a gallery of photos) of students participating in the lesson.

* Share a link of student work from the lesson.

* Post a link to a video of students participating in the lesson.

* Submit a write-up of things your students discovered during the lesson.

* Create an extension for the lesson.

Share your picture, video link, or write-up on our Facebook page, “NCTM Illuminations.” The link or photo that receives the most “likes” by 5pm ET on Thursday, October 6, will receive a classroom set of hundreds boards provided by A+ Compass. Additionally, ten other randomly selected entries will win an individual set of A+ Tiles.

Entries should be uploaded to our Facebook page, NCTM Illuminations. Or, you can submit to Christa Koskosky, ckoskosky@nctm.org, who will post any emailed entries to the Facebook page.

Mathematically yours,
Carollee

### Mental Math: Adding and Subtracting on the 100 ChartJune 17, 2011

In keeping with the theme of mental math, I would like to propose that students in grades 2+ be challenged with learning to add and subtract any two digit numbers. This can easily be supported by using the 100 chart until the number relationships become second nature to the students. After that students can either just do the mental calculations, or they can close their eyes, “see” the 100 chart in their minds, and calculate from the visual image they produce.

It is easiest to begin by adding/subtracting 1 and 10 from a given number. If students have done the 100 chart tic-tac-toe, this will be an extension of that. From there it is easy to move on to adding/subtracting more than one (2 to 9) and multiples of 10.

The answer to 38 + 10 is 48, which is immediately below 48. The answer to 38 + 20 is 58, found two rows below 38, which is the same as adding 10 twice. The answer to 38 – 1 is 37, found to the immediate left of 38 on the 100 chart. The answer to 38 – 5, 33, is found five spaces to the left of 38.

When it comes to adding and subtracting other two-digit numbers, the above procedures can be combined. To add 54 + 23 a student can move down two rows (adding 20) and then move right three squares (adding 3) to end up at 77.

Strategies can be developed for adding or subtracting numbers that would “wrap around” ends of the 100 chart. When adding 54 + 29 rather than move down two rows and try to count 9 to the right (which moves down to the next row) students can devise strategies that make use of “nice” or “friendly” numbers. In the above example it is much easier to add 30 to 54 and then subtract one for a total addition of the required 29.

My personal belief is that every student above grade 2 and every adult should be able to add and subtract two-digit numbers mentally with ease. If the skill is not there, it is only because it has not been developed and practiced.

Spending a few minutes every math class on mental math helps develop life-long skills. Most of us, as adults, do much more mental math and estimation than we do with paper and pencil for exact amounts. We figure time, mileage, money, etc. daily in our heads.