# Focus on Math

## Helping children become mathematicians!

### Looking at Numbers on a 144 ChartOctober 31, 2013

I was recently doing a math lesson in a grade 3-4 classroom here in my district. The teacher told me the students had been working with place value and patterning of numbers on a 100 chart. For my lesson I decided to have students consider how number patterns would look if the patterns were marked on a 144 chart instead of a 100 chart. More importantly, I wanted students to be thinking about WHY the patterns would appear differently.

The students had already coloured skip-counting patterns on their 100 charts. They had looked at the “easy” patterns of counting by 2, by 5 and by 10, but they had also looked at the patterns of counting by 3, 4, 6, 9, and 11.

With that prior investigation in place, I gave students a sheet with 4 of the 144 number charts on it so they could colour those same patterns on the new configuration of numbers and compare the visual patterns to those on the 100 chart. The students were asked to notice things that were different between the two charts and to think about why the differences were there. In particular, we wanted the students to think about why the patterns looked like they did on the 144 chart.

The students were enthusiastic in their discussion about the number patterns and we were able to lead them into a discussion about base 10 and place value.

A great follow up activity is to colour the same number patterns on a calendar (regular or extended) and notice again how the patterns look different, asking, of course, WHY?

One per page
Two per page

One per page
Six per page

Happy patterning!
Mathematically yours,
Carollee

### Ten Uses for Sticky Notes in Math ClassSeptember 9, 2013

Sticky notes (that wonderful — though accidental — invention that 3M first marketed as Post-It Notes©) are a wonderful tool to use in mathematics. I have always found that students enjoy using those little sticky pieces of paper! So here are ten ideas for incorporating them into math lessons:

1) Ordering numbers: Write 4 or 5 different numbers on sticky notes and have students work in pairs to put them in order from least to greatest. This is great practice for multi-digit numbers, for decimals, for fractions, etc.
2) Creating numbers: Write 4 or 5 different digits on sticky notes and have students work to create specific numbers: greatest possible; number closest to a particular given number (whole number or fraction); a number between two given numbers, etc.
3) Using operations: write several digits on sticky notes. Then use them and operations that the students know how to use to make number equations with as many different answers as possible or to get as close as possible to a particular answer.
4) Writing word problems: Give students numbers, operation signs, and possibly other math symbols (such at a percent sign) on sticky notes and have them create a word problem that uses all of the notes.
5) Graphing: Create the axis of a graph on chart paper or the chalkboard. Add your categories (4-6 work well). Give each student a sticky note and have them create a bar graph with then .(I usually have kids make their choice at their seats and write it on their notes so they don’t just add to a single category to make it “win”.)
6) Scavenger hunt: each student has a sticky note with math geometry word on it. Students must find an example in the room to represent the term and place the sticky note there (e.g., perpendicular lines, acute angle, sphere, etc.).
7) Measuring area: Cover a book or other object with sticky notes and calculate the area using the notes as the unit of measure. A particular book or surface may be covered by notes of one size, then by notes of another size and the area calculations compared.
8) Estimating on a number line: draw a number line on the board with only the endpoints marked (endpoints may vary according to grade: 0-10, 1-100, 0-1, 20-80, 1-1000, etc.). Give each student a number that appears on between the endpoints and have them come and place their number where they think it would go and explain their reasoning for placing it there.
9) Commenting on each other’s work: teach students to peer-evaluate problem solving work. Students can exchange their papers after working on an open-ended problem. The evaluator can make comments and ask questions regarding the strategies, visual representations, etc. by writing their comments on sticky notes.
10) Transformational geometry: Use sticky notes to show transformations, often called “flips”, “slides”, and “turns”. Light coloured sticky notes tend to be translucent. Using a sticky note, student can trace a shape from its original location on a grid and then use the sticky note to show the desired transformation (e.g., down two, left three; 90 degree clockwise rotation; reflection over a particular line).

I hope you try one or more of these in the next weeks. Let me know how it goes!
Mathematically yours,
Carollee

### Place Value not Face ValueApril 20, 2012

One cannot understand the numbers in our number system without understanding place value. Simply put, where a digit is situated in a multi-digit number, or its place in the number, determines the value of the digit. Thus, although 32 and 23 both are made using the same two digits (a 2 and a 3) we know that 32 and 23 are not equal. The 3 in 32 is in “ten’s place” and actually carries a value of 30 (or 3 tens) whereas the 3 in 23 is in “one’s place” and has only a value of 3. Each of the 3’s have a “face value” of 3, but one has a “place value” of 30. The same can be said for the other digit 2 in each of these numbers: in 23 the 2 carries a value of 20 (or 2 tens) whereas the 2 in 32 has only a value of 2 ones or just 2. Since 3 tens has a greater value than 2 tens, 32 is greater than 23.

We have traditional algorithms for doing each of the four main operations (addition, subtraction, multiplication, and division). An algorithm, according to Wikipedia, is a “step-by-step procedure for calculating.” It goes on to say, “More precisely, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating.” Thus, for each operation there is a list of rules or instructions which, when followed, effectively produce the solution. That is a good thing, right?

Well, the answer to that depends on whether the goal is to produce answers quickly or to understand mathematics. Following rules requires little or no understanding of the mathematical concepts underlying the procedure. Worse, all of those who do research regarding the brain and learning report the same thing: the brain remembers things that have meaning; isolated bits of knowledge that are not meaningfully connected to other knowledge that we have are discarded. Sadly, the rule-based approach to mathematics is rather devoid of meaning. The user often has no idea why the rule works. It is no wonder, then, that teachers spend weeks in the beginning of a new school year “reteaching” things that have been forgotten over the summer. Without meaning behind the rules, the rules are easily forgotten.

Let’s look at the thinking processes involved in some computational strategies using the problem 26 + 39. Most of us were taught in such a manner as to think the problem through like this (see example a): “6 plus 9 is 15, so put the 5 under the 9 and carry the 1. Now 1 + 2 + 3 equals 6, so put the 6 under the 3. Reading the bottom numbers together, the answer is 65.” Certainly that set of rules works, but do you notice that the “1” being carried does not, in reality, have a value of 1. It has a value of 10 — we just neglect to refer to it that way. In like manner we ignore the fact that the 2 and the 3 are worth 20 and 30 respectively, but we again never refer to this fact. You might argue that this is not really a problem — we are adding two 2-digit numbers and it makes sense to get a 2-digit number as an answer. But I will say, from many years experience as a classroom teacher and as a mathematics specialist, that the more complex the adding (or whatever operation) gets, the further removed from logic students get. It is, in fact, quite easy to make a mistake leaving out a whole place value spot and get an unreasonable answer, but students do not notice such things. They are not looking at the place value of any of the digits in the problem, and thus have no reference for really thinking about the answer.

I like to have students work with non-traditional methods of computing that keep the place value of digits. I offer three other such methods for your consideration (all of which, by the way, have been used to good effect by young students whom I have taught, and as recently this year!). There are many other ways of non-traditional computing — this is not meant to be an exhaustive list, but rather to give some examples of other possibilities.

Looking at example b, the thought process might sound something like this: “20 plus 30 is 50, so I will record that. 6 plus 9 is 15, so I will record that. But I see that 15 includes a ten which I can combine with the other tens, giving me an answer of 65.” For example c, the thinking might go like this: I see that 39 is very close to the friendly number 40. If I add one more I can use that friendly number. Now I can mentally add 26 plus 40 (adding 4 tens and 2 tens) and get 66. Since I added one extra to make the friendly number, so I must remove it**. My answer is 65. Finally, for example d, the thinking might be this: 39 is very close to the friendly number 40 — just one away. I can take one away from 26, leaving 25, and move that one to add it to 39 to get 40**. Now I just have to add 25 plus 40 (adding 4 tens onto 2 tens) to get the answer of 65.

Examples b, c, and d all keep place value of each number, a critical component, especially when first learning about the operations or different levels of the operations. Later, when understanding is firmly in place, students can move to the “traditional algorithms” and not lose sight of what is happening conceptually. But conceptual understanding needs to be developed for the operations — which takes time. It is not something accomplished in flash.

I could give similar examples of meaning-based methods of calculating for each of the other three main operations, and maybe in a later post I will do so. The point, however, is that we want to students to be thinking about the meaning of the numbers that they work with, to be thinking about the place value of each digit in the number and not just the face value of each digit. It is an important component of understanding mathematics.

What will you do tomorrow to ensure that your students working meaningfully with numbers?

Mathematically yours,
Carollee

** Both of these examples involve using the “zero principle” — that for any number when you and and then subtract the same amount, you do not change the number. Thus 39 + 1 – 1 = 39, because + 1 – 1 combine to be zero. This zero principle is actually a very powerful principle in addition and subtraction and can be used effectively to simplify many problems.

### Math Toolkits for Students — More Stuff to Add (part 3)May 19, 2011

There are more items that can be added to the toolkits for students, but these I will separate by primary (gr 1-3) and intermediate (gr 4-7) levels. Again, it is hard to just mention the contents without going into activities that use the tools to help students build mathematical understanding. Hopefully the tool itself will prompt you to think about some ways to use it.

Primary Tools:

• 25 chart, laminated (usually created in 5 rows of 5)
• blank 5-frame (with spaces big enough to put counters on)
• blank 10-frame
• blank double-10-frame (two blank 10-frames on one card)
• set of filled in 10-frames (1-9, multiple 10’s)
• bead bracelet (10 beads in two colours, 5 of each) to be worn draped over the fingers so the beads can be manipulated. Two bracelet may be worn to use for numbers in the teens.
• large flattened paper plate or cut out paper circle for making dot plate configurations with bingo chips
• mini bags of small coloured wooden sticks or other small materials for patterning
• teeny-tiny Hundreds Tens and Ones (HTO’s) — miniature place value pieces cut out of large plastic canvas (found in crafting stores)
• place value cards — overlapping cards that show, for example, 425 can be pulled apart to reveal 400, 20 and 5 (click on image above to print)

Intermediate Tools:

• booklet of mini 100 charts to be coloured in to show multiples (x2, x3, x4, etc.)
• metre tape (purchased or created by taping photocopied paper lengths together)
• fraction-bar card (a card with a fraction bar in the middle — students use numeral cards to place as the numerator and denominator)
• fraction percent circles (two different coloured circles partitioned off in hundredths each cut along one radius and then placed together so they “spin” over each other to show different percent values)

As you can see, there are many things that can be used as “tools” in the teaching of mathematics. Creating a toolkit with students is a wonderful way to make lessons engaging.

Mathematically yours,
Carollee