# Focus on Math

## Helping children become mathematicians!

### 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

### Basic Facts: Mental Math as a Foundation for Multiplication Fact StrategiesMarch 16, 2012

Basic Facts are still very important. Although newer curricula put a greater emphasis on problem solving, communication, reasoning, and representation of numbers, basic facts are still an integral part number sense in students. If a student is a good “memorizer”, then learning the multiplication facts will not be difficult. However, for many students the random bits information we call “facts” don’t stick well in the brain (the brain tends to remember information that is personally meaningful!), and thus it is important that we support those students in their learning by teaching and rehearsing thinking strategies.

Before we look at those particular strategies that are useful for learning the multiplication facts, there are some “prerequisites” to consider. Many of the strategies I will be suggesting use some kind of mental math to help students go from a known fact to an unknown fact.

The mantra for students is this: “Use something you KNOW to get to something you DON’T KNOW!” This is a comforting thing for students, particularly those who have struggled with learning their facts. They tend to feel that there is no hope for them. In some cases they have worked for a very long time, even several years, to memorize these facts, and at this point they feel like it is a hopeless task. We need to offer hope in the notion that they can begin their learning with things they do know, and build from there.

Consider working with your students to build these kinds of skills, remembering to tie them to concrete and/or visuals (such as ten frames or 100 dot arrays):
• Subtracting a single digit number from a multiple of ten (e.g., students use the known fact of 10 – 6 to solve 60 – 6. Tie into ten frames).
• Subtracting a double-digit number from a multiple of ten (e.g., build on the previous skill and have students solve 60 – 16 by subtracting first 10 and then 6.)
• Doubling any 2 digit number, using whole-part-part strategies if necessary. It may be easy to double 12 (think 10 + 10 + 2 + 2), but it will be harder to double 16. Students might consider 16 as 15 + 1, then double each of the two parts, and add back together (think 15 + 15 + 1 + 1 — look for “friendly numbers ending in 5’s or 0’s).
• Adding any single digit number to a double-digit number, particularly when the sum of the one’s place digits is greater than 10. E.g., 35 + 7 can be considered as 35 + 5 + 2; 48 + 6 can be considered as 48 + 2 + 6. (Pull apart the number to be added in a way that makes a group of ten.)
• Subtracting any single digit number from a double-digit number, particularly when “regrouping” would be required. E.g., 54 – 8 can be 54 – 4 – 4. (Again, break apart the number being subtracted into parts that make the work easier.)

It is well worth the time that you invest with students doing mental math. In As well as being a great life-skill, mental math allows students to be flexible with numbers and use powerful thinking strategies.

Mental Math and Basic Facts — don’t skip these important things!
Mathematically yours,
Carollee

### Math Camp (gr 2-5) 2011September 2, 2011

Once again I am delighted to say that math camp was a success! We had a productive day exploring learning and the brain, mental math, strategies for all 4 operations (addition, subtraction, multiplication, an division), and equality.

We discussed that both cortisol and adrenalin, when present in the blood stream, tend to “shut down” thinking and memory in the brain. Adrenalin induces the “fight or flight” reaction, while cortisol induces a state of stress when other things are more important than learning. If we want students to learn well in our classrooms, we must take the time to build a safe learning community and do all we can to reduce students’ stress.

The mental math we talked about was similar to what was presented in the Grade 6-8 math camp, so I will direct the readers there for some notes about that topic.

As for the basic operations, it is critical that students do these in ways that are meaningful for them. We shared many strategies for addition and subtraction, some based on numbers only and some based on tools (e.g., 100 dot arrays, blank number lines, base to blocks, etc.). One of the most important things that students should know about the operations of multiplication and division is that they are always, always, always about groups. In representing multiplication, the area model is very effective and can help students understand multiplication beyond basic facts. (Base 10 grid paper is useful for this).

Equality is an important for students to develop in the elementary years. Studies show that when students see the equal sign in an equation, they do not think of equality. Rather, they think it means, ‘put the answer here’ or ‘now do what the sign says to do’. I have had students tell me, when I wrote an equation such as 8 = 2 + 6 on the board, that I wrote it “backwards”. Students expect the single number to be on the left because that is how they always see it! Not only should they see equations written “backwards”, but they should explore equalities with “multiple parts” on both sides, such as 5 + 3 = 2 + 6. Here is a “balance scale” which can be useful in exploring equalities.

I wish you all a wonderful school year!
Mathematically yours,
Carollee

PS: I just remembered that I told all of you at the workshop to write on your 100-dot arrays because there would be clean copies available! So here are the links to the 100-dot arrays:  large 4 small, 6 small, 12 small.

### Math Camp (gr 6-8) 2011August 31, 2011

What a great group of participants we had at yesterday’s gr 6-8 math camp! I was delighted with all of the group sharing that we were able to facilitate — there is always so much that teachers can learn from each other.

The workshop yesterday focused on several things: mental math, integers, fractions, and algebra. These areas, I believe, are important for students to master if they are to be successful in subsequent levels of mathematics.

Mental math is a skill which is only developed if practiced, and we discussed some particularly useful strategies that might be incorporated into regular practice sessions. First, mental math begins with basic facts! From there we can have students practice thins such as these:

• adding multiples of tens (MOT’s) (20 + 50)
• subtracting single digits from MOT’s (50 – 8 )
• adding single digits to non-MOT’s (39 + 6)
• adding to get 100 or – from 100
• adding any two-digit #’s (47 + 39)
• doubling numbers
• halving numbers
• multiplying by 10, 100
• dividing by 10, 100
• multiplying by 20 (by doubling and then multiplying by 10)
• multiplying by breaking up numbers (using “nice” numbers)

Response boards are a great way to do immediate full-class assessment during mental math practice.

Our focus on integers was in using “chips” to have students learn about positive and negative numbers in a very visual way. It is helpful for students if they understand the power of zero in adding and subtracting integers, and zeros can be visualized by an equal quantity of positive and negative chips. Using the chips to solve addition, subtraction, multiplication, and division problems involving integers allows students to build a conceptual understanding of integers which goes far beyond memorized rules.

For fractions, we discussed the need for students to develop number sense regarding them. Using benchmarks to estimate fractions is one way to help facilitate this. We talked about making pocket charts for both the teacher and the students to use in practicing this. (Pocket chart directions.)

As for algebra, once again, visualization was the key to helping students make sense of this generally abstract area. By having students display and manipulate equations in a concrete way on a “balance scale”, they have the opportunity to learn what are acceptable or “legal” moves in solving algebraic equations. (Balance scale.) A hands-on, visual approach to algebra allows every student to be successful in this area!

One of the teachers at the workshop is going to email me a rubric that she used in her math classes last year, and I will up-date this post with that rubric once I have received it. (later) Here is the link for the rubric. When we discussed this, the “traffic light” part was really important. Remember that students can self-assess their understanding and record it as red (“I am totally stuck.), yellow (“I am able to work some on the problem but not I am not really sure about it.), or green (“I understand this well enough that I could teach someone else.)

In the meantime, I hope you will think about how you might better teach these areas of mathematics that are critical for students in these grades.

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.

Mathematically yours,
Carollee

### Mental Math: Finding Compatible NumbersJune 14, 2011

Using pairs of compatible numbers is a great way to do mental math. Learning about compatible numbers can begin in the early grades. Students who use 10 frames (see an earlier post for copies of those) to learn about numbers to 10 can visualize the number compatible for 10 easily. For instance, looking at the 6 card, it is clearly apparent that there are 4 “empty” spots on the card, and thus 6 and 4 are compatible for 10.

As in the last post, it is easy to work on finding compatible numbers for 100 using the 100-dot array. First, using whole rows, students make the same “to ten” connection as for small numbers, but using full rows of tens. Thus 60 (or 6 full rows of 10) can be seen to be compatible with 40 (or 4 full rows of 10). From there the 100-dot array can be use for pairs of compatible numbers: 55 and 45 become compatible, etc.

Older students can work on compatible pairs for 1000. The same principle of ten works for hundreds (600 and 400 are compatible, 630 and 370 are compatible, as are 639 and 361). It is helpful for students if they examine and discuss the pattens that appear: the one’s place digits add to 10, all other place value digits add to 9).

A great way to practice finding compatible numbers is to display on the chalkboard or overhead groups of numbers, say 10 or 12, in which there are compatible pairs. You can write the numbers so that every number has a compatible partner, or have some “distractors” in the group that have no match. Students can find pairs of compatible numbers and display these on response boards.

Once students have practiced finding compatible numbers and become comfortable with that process, the skill can be used for other mental math. Consider adding 78 + 33. If a student recognizes that 78 + 22 = 100, and 33 is 11 more than 22, then 78 + 33 = 111. Using larger numbers, when doing 880 + 250, a student can split apart the second number into 2 smaller numbers, one of which is the compatible number. So 880 + 120 + 130 = 1130.

Using the same principle, students can practice finding compatible decimal numbers (e.g., 0.6 and 0.4 are compatible) use those numbers for mental computation as well.

Students should also play with fractions that are compatible (e.g., 3/8 and 5/8 are compatible, as are 13/16 and 3/16) and do some mental computation with those at the appropriate level.

There is some definite benefits in being able to find and use pairs of numbers compatible to multiple of ten (and for fractions, compatible to a whole number). I hope you will consider spending a few minutes throughout your math week working on some mental math skills with your students.

Mathematically yours,
Carollee

### Mental Math: “How Many to 100?” on the 100-Dot ArrayJune 12, 2011

I mentioned in an earlier post that one of my favourite tools to use with students is the 100-dot array. I want to share one way that I use it to help students develop some mental math skills.

I have used this with students as young as grade two, but if older students have developed few mental math skills, this is a great way to add to their strategies. The fact that this mental math is grounded in the use of a visual tool allows every student to have success with the method.

I have traditionally done this strategy using an overhead and a transparency of the 100-chart, but if you have access to a document camera, you can easily use that. The other thing I usually do while doing this activity with students is have them write their responses on some kind of response boards (e.g., small chalk boards, small white boards, etc). It is a great way to incorporate formative assessment in the math classroom. As students write their responses, you can see at a glance who is “getting it” quickly and who needs a bit more help. I should mention, too, that the first time I use the 100-dot array with any group of students I always spend 10-15 minutes having the kids notice things about the array and talk about it. We pay attention to the 10’s, the 5’s the 25’s and the 50’s that are displayed on the chart. It is an important step if you are going to use the tool for any activity.

So, now to the actual activity. I begin by displaying the 100-dot array on the overhead, the I cover some rows with a piece of paper. I ask two questions: How many do you see? and How many to 100? I look for the answers in two forms: the number of tens the students see, and the number of dots they see. For instance, if I have covered 3 rows, I want students to say they see 7 rows of 10 dots, or 70 dots. The “how many to 100?” is answered by 3 rows of 10 dots or 30 dots. Each answer uses the relationship of 10 as an anchor, which is one of the foundational number relationships that students need to develop. Depending on the age and quickness of the students, I spend the first number of days doing this part of the activity.

The second stage is to use two pieces of paper to cover dots. With the first, I cover full rows as before. With the second I cover part of the last exposed row. So, from the previous example, if I were covering 3 full rows, I would go on and cover part of the next row, say 8 more dots. Now when I ask, “How many do you see?” the answer is 62 (6 full rows and 2 more dots). When I ask, “How many to 100?” students must “complete” two types of tens: in the individual row and in the number of rows. So, completing the row in the example there are 8, and then there are 3 full rows covered, for a total of 38 needed to make 100.

The year I taught grade 2 I did this activity regularly throughout the year. By late spring I could just say to my students (without showing the 100-dot array), if you start at 57, how many do you need to make 100? Across the whole classroom, almost as one, the eyes of every student would close as they would visualize the array. Heads would swivel and bob as students were completing the 10’s, and they all could find the needed amount to make 100.

It’s an easy way to build some mental math skills with your students. I recommend doing it fairly regularly as a warm-up activity. Build these skills, and then build some more!

Below  are some links to download some 100 dot arrays in different sizes. (Students use the small ones all the time in my class. When they are problem solving, if their strategy makes use of the array, they grab a small one from the basket at the back of the classroom and glue it onto the sheet they so they can represent their thinking. Choose a size that works for your students!)

one large 100 dot array

4 small 100 dot arrays

6 small 100 dot arrays

12 small 100 dot arrays

Mathematically yours,
Carollee

PS The link for the 100 dot arrays was not working for a while (thanks, Pat, for pointing out the glitch!), but I think I have it fixed now! — at least it downloaded OK for me. Let me know if you have more trouble.