Math Musings

February 13, 2015: Bumping into math-teaching through problem solving

The other day I came across this blog:  It is a very good explanation of problem solving in mathematics.  The phrase I’d like to chat about and that I am going to steal and use for evermore, is “bumping into the math”.  While there are many teachers experimenting with using problems in the math class, not all problems will do.  We need to be very intentional in the problems we choose so that students will use the math they already have in order to “bump into” the math they need to learn next.  And when they do bump into that math, your job as the teacher is to help students to recognize and organize that which they have bumped in to.

Let’s suppose that students in grade two have experience with adding and subtracting to 20 but we want them to move into double digit problems.  We need to design a problem that will help them “bump into” the concept of grouping by tens when adding or subtracting.  So, first we think about the math they already know that might help: familiarity with a 100s board,  making numbers with base ten blocks and counting by tens.  We design a problem that will cause students to add, for example, two 2-digit numbers:  Mrs. Smith’s class sold 46 cupcakes at the bake sale yesterday and 39 cupcakes today.  Did they sell all 95 cupcakes that they had donated?  We DO NOT teach kids how to add and regroup first.  But, we do ensure that hundreds boards and base ten materials are available.  We might even do some fun games with those materials as a minds on activity reminding students that they do know how to use those materials.  And then we watch to see who bumps into the concept of counting by tens.  We watch to see which students solve this problem  counting by ones and who is able to find more efficient ways to solve the problem by making jumps of ten.  During our consolidation or connect and reflect times, we help students to understand how grouping by tens is going to be effective in this problem.  We purposely showcase those solutions.  We start to build our anchor chart of strategies.

This concept of “bumping into the math” is brilliant and will help you to change how you approach problem solving.  Problem solving is not just being able to do word problems.  It is not being able to recreate an answer the way the teacher taught it.  In a problem solving model we want to extend and develop students’ mathematical knowledge by creating situations in which they can’t help but come across the ideas they need to learn.  But we need to be intentional in that practice.  We need to know what those mathematical concepts are.  We need to then help students to understand what they bumped into and give them time to practice with that new skill.  And, we need to recognize the difference between bumping into the concept and being able to organize it into an algorithm.  Returning to our grade two example, after a number of problem solving situations where students are regrouping with tens, the teacher will need to help students to see how the paper and pencil algorithms might work (and there is more than the old “carry the tens” algorithm that you remember).

Students will not bump into the long division algorithm or the invert and multiply rule for dividing fractions.  We do, however, want them to bump into the mathematical concepts that make those algorithms make sense.

January 8, 2015:  Building a mathematical toolkit instead of using mathematical tools

I can learn to hammer nails in to a board.  The next day I could learn to saw wood into planks.  That does not mean that I can build a house.  I can teach a student to add.  I can teach a student to subtract.  That does not mean that the student will necessarily know when to add and when to subtract.  We may be teaching students how to use all the tools in their mathematical toolkit, but are we teaching them to have confidence to choose the tools in  that toolkit to solve any mathematical problem?

In my everyday mathematical world it is the same:  when approached with a situation I need to determine whether I need to measure, add, subtract, draw a diagram, figure out a missing part and so on.  I delve into my mathematical toolkit and choose the appropriate tools.   In all my years no one has ever approached me on the street and said what is 324 multiplied by 16.  But when we teach mathematics we almost never require that students use their entire mathematical toolkit.  Rather, we give students a problem with the parameters already defined (we are doing a unit on the addition of decimals).  We hope that students will learn all of the individual bits and then be able to apply them in the future when required.  But, what teachers often find is that once the unit is over, students seem to completely forget the mathematics and do not necessarily make relationships between concepts.  Students develop a mathematical toolkit but need the teacher to tell them which tool to use.  What we want is for students to feel so comfortable when faced with any mathematical problem that they can rummage around in  their toolkit and choose the appropriate tool because they understand the mathematical problem.

If we change the way we approach the mathematics curriculum, perhaps we can help students to see mathematics as a whole, not the sum of many unrelated parts.  Currently most mathematics textbooks and most mathematics programs are broken into 2-6 week units of study.  Each unit of study is a specific topic in a specific strand of mathematics (Two weeks on perimeter, three weeks on multiplication, two weeks on money, three weeks on area, four weeks on division etc).  What if we changed that and taught in a more recursive way so that students were exposed to a variety of problems and topics many times of the course of the year?  What if we taught addition and subtraction simultaneously?  Might students learn to differentiate between the two operations and also notice their similarities?  What if we taught addition, subtraction, multiplication and division simultaneously?  Might students not develop a confidence and flexibility of thinking?  Not only would students need to learn HOW to do the mathematics but they would also learn WHY and WHEN to use the mathematics.  Students would be building a toolkit they could use instead of just learning how to use the tools.


November 28, 2014:  You have more time to teach math than you think

Frequently teachers lament (in Ontario, at least) that the math curriculum is too big and the time is too short and they will never get through it all.  And so, the spring arrives and we rush through fractions and then we wonder why kids don’t understand fractions.  Here are some ideas that will give you more time for math:

  • Not all concepts are equal.   Curriculum documents tend to be lists of content to be taught.  But usually such documents leave it to teachers to determine how and when to teach.  For example, many grade one documents have an expectation of learning the names of 2 dimensional shapes.  That isn’t very hard.  It doesn’t need a two week unit or a lot of worksheets.  In fact, you can probably teach it just by mentioning them, or doing art with shapes, or using them in your patterning unit.  In fact, in the primary years there are quite a few expectations that you can just do “by the way”.  Sit down with your grade partners and make a list.  You will gain some precious math time.
  • Some concepts just need to be taught more frequently and over the course of the year.  All grade one teachers know that you can’t have a two week unit on counting to 100.  It just won’t happen.  In grade one you have to count, and count, and count.  In grade one you count everything.  But there are lots of other expectations throughout the grades that are just like that.  Here are a few but I urge you to work with your grade team to think of others:
    • Telling time.  Few grade 2 students will master telling time to the quarter hour in a 2-3 week block.  Instead, introduce it and practice telling time once a day, every day for the whole year.  The same goes for lapsed time in the later years.
    • Learning your number facts.  You do need a block of time  or a few shorter blocks of time to learn about the four operations.  But, once students have some strategies to figure these “facts” out, then you want to practice them, for a little bit, every week all year long.  Number strings is a great way to do this.
    • Ordering and representing numbers, in all grade levels.  Not only should students have lots of time to practice this but as they learn new representations of numbers they need to be able to integrate that knowledge.  When junior students learn about mixed numbers, they need to place them on number lines and compare them to whole numbers; when intermediate students learn about square roots they should understand how they can represent them on a number line.
    • Factoring and Multiples.  This concept usually comes up in the intermediate grades.  It is a new concept with new weird words.  Long before I gave students problems that required them to use factors and multiples, I would have practiced with factor tress or ladders so that the procedural part was not holding them up.
    • Rounding Numbers.  This concept often baffles students.  They learn it in a “unit” and memorize a number of rules.  What would happen if even once a week, for 5 minutes, your class had a discussion about which number 56 was closest to and why?  Or 56.09? or 794?
  • Sometimes, in a rush to “cover” the curriculum, we start teaching without ensuring that students have ample background knowledge to understand the grade level concept.  So, the unit takes longer; kids and teachers are frustrated; students often don’t do well.  Instead, teachers can anticipate which areas of the curriculum are going to be tricky (regrouping in addition and subtraction, fractions, unit rate etc).  Next, think about the background knowledge you want your students to have prior to tackling that unit.  Then, for just a few minutes every once in a while, have the students do some work with that concept.  When you get to your big idea, your students will be more likely to be ready.  For example, if I know that I am going to teach the addition and subtraction of fractions in April, all year long I am going to very little lessons (5-10 minute warmups and that’s it) having students name fractions, represent fractions, count fractions etc.  In the end, your students will be more ready to conquer the new material and you will save time.
  • Stop and come back.  We expect tiny kid brains to learn, synthesize and remember new information that is presented to them in 3-4 weeks of study.  And, because it is new, and hard, and kids are kids, and there are interruptions like assemblies and field trips, the unit takes 5-6 weeks.  You will find that you have more time if teach a little bit of your unit and then stop and do something different for a while and then come back to your unit.  So, in a month of number sense and numeration you might do a little bit of fractions, a little bit of multiplication and a little bit of proportional reasoning.  There is a lot of research that says that students actually do better when they learn (mathematics and other things) in a spaced sequence, that is time between lessons, than in a blocked sequence where lessons follow one after the other (The Benefit of Interleaved Mathematics).  Another advantage is that students are often more keen when they return to a concept.  Haven’t you ever been struggling with something but then after a few days you return to it and it doesn’t seem so hard?
  • Does it have to be during the math block?  If you teach your class all the subjects, start to think about where else you can integrate the math.  In the very early primary years you do not need to do the worksheet on whether 3-D shapes stack, roll or slide.  Simply watch your students build with blocks.  Student who consistently use the sphere as the base for their castle will need some extra instruction!  As students are entering the class, make graphs, and tallies and predictions.  You don’t need to do a graphing unit.

So, maybe you have more time than you think.  Certainly when we try to cover the curriculum expectation by expectation, it is too big and there isn’t enough time.  But if you rethink how you approach the curriculum it doesn’t seem as daunting as you thought.

November 21, 2014:  The power of the Hundreds Chart

All primary math teachers (I hope) have a hundreds chart.  Hopefully they have a big one and a lot of little ones.  Only after I started to work with junior and intermediate students did I recognize how important an intimate understanding of the 100s chart is to a student’s sense of number.  Students who struggle with number fluency in the upper grades have likely missed some key understandings about how numbers go together.  So, while today’s suggestions may seem primary, you may wish to see if your junior/intermediate students can do them.  Certainly students in the junior and intermediate grades who are struggling need to do them.  If you are a junior/intermediate teacher, keep reading because at the end I’ve provided some suggestions just for you.

At first we use the 100s board to count by ones, tens, fives and twos.  When we do so we want students to pay attention to where the numbers are on the board.  But we also want to move into counting by tens starting a places other than 10.  Students with a good understanding of how the board works will quickly be able to count 12, 22, 32, 42, etc.  As students move on to mental math strategies such as adding 42 + 25, it is helpful that they can add groups of 10 to 42.

Once students in the primary years have a lot of exposure to the 100s chart, we want them to play with it.  Math students should be able to picture the board in their heads.  When a student can do that, he or she then has a better understanding of our base ten system and how the numbers are related to each other.  The following activities help students to develop their understanding of our number system through the manipulation of the 100s chart:

Where does the number go?  Start with a blank pocket hundreds chart.  Hand out numbers to students.  You may want them to discuss with their elbow partner or table group where the number should go.  Ask students to come and place their number in the correct pocket.  This can be done as a center activity as well with students drawing different numbers to place.  Watch for students that use the concept of 10 to decide where their number should go.

Which number am I pointing to?  With a blank 100s chart, point to a spot.  Ask students to write the number that belongs in that spot on their whiteboard.

I’m thinking of a number?  You will describe a number and the students will tell you where it is on the board:  I’m thinking of a number between 4 and 6.  I’m thinking of an even number less than 8 and greater than 4.  I’m thinking of a number that has 4 tens and 3 ones.  I’m thinking of a number that has 2 tens and 13 ones.

Fill in the missing number?  Provide students with puzzles such as the one below to figure out the missing number.  The puzzle pieces are part of the 100s chart.


 33 34   ?
? ? 45
53 ? ?

You can, of course, make it harder or easier.  Once students have the idea, have them create their own puzzle pieces for each other on the little white boards.

Blank 100s charts.  Have students begin with a blank square page or whiteboard.  Ask them to pretend that the whiteboard is a 100s chart.  Where would 10 go?  Where would you then put 19?  20?  67? 77?  Plan your questions at first so that the previous answer supports the next one.

You can use the 100s board framework for junior and intermediate work.  If you are learning about decimals you can begin with 0.01 and finish the board with 1.0.  For junior and intermediate students this familiar framework could be helpful and all the same activities apply.  Ask students what would happen if we created a 100s board starting with 0.1?  What would the last number be?

Could you create a hundreds board with fractions?  Could some of the fractions be equivalent fractions or percents or decimals?  If you created the board out of fractions, what could the denominators be?

If you had a regular 1-100 hundreds board in the intermediate grades, could the students replace any of the numbers with square roots, with improper fractions?

How would you create a hundreds board of negative numbers?

None of these activities should take up your whole math class.  But regular exploration of how numbers are related will develop number fluency and flexibility.


November 14, 2014:   Math Musings:  Ordering Numbers

Every day you should do a short 10-15 segment of math that reinforces important conceptual skills.  We call this part of teaching mathematics recursively.  If you only learn about fractions in April, chances are that by the next time April rolls around you aren’t going to remember much.  During this 10-15 block you can do a variety of short games and activities that will help students to develop number fluency and flexibility.  When you have number fluency and flexibility, you have mathematical confidence.

Right from the very beginning we want students to understand how numbers are ordered, and to have a sense of their magnitude.  As we move from whole numbers in the earliest years and move into using fractions, money, and decimals it is important for students to understand these relationship to the whole numbers.  We want intermediate students to know that 0.456 is greater than 0.452 but less than 0. 459.  Or, that 1.25 is greater than 7/8 but less than 1 ½.

Here are some activities to try that will engage your students in thinking about the magnitude of numbers and how they compare to other numbers.

  1. Put yourselves in order.  As students enter the room, hand them a card with a number on it. In the very beginning, I would just include whole numbers but as soon as students started to learn about money, I would include it and the same for fractions.  For example I might use the following for a grade 4 class:
    1/2 7 3 1/2 4/2 75₵
    $3.60 35₵ 2 15 $10.01
    $10.10 5 32/100 6 6 1/3 2/3
    12 11 3/4 $11.50 8 8 2/3
    9 3 2 1/2 13 42

    In the intermediate years I would include whole numbers, fractions, decimals, percents, integers, exponents and square roots.  For example I might have the following numbers for a class of 25 in grade 8:


1.0 -5 -13 2.25 10%
-0.39 9/10 300% √4 3.05
2 8/9 3 3/4 3.058 -3 1/4
5 √9 -0.42 99%
–         3 3/4 3.8 0.6 3.76 √6

Students have to arrange themselves along the front of the class in order.  Think about the conversations they would need to have.  Think about where students would stand.  What type of spaces would they need between themselves?  If you have more time, give the students 15 seconds to change numbers with someone else and have them repeat the activity.


Extensions and variations:


  • Give table groups the same activity to do.
  • Have students take their number and write a number that is just a little bit smaller on the back and then do the activity with that number.
  • Have students find two other people and just order the three numbers.
  • Have half the class do it (half the numbers are red) and then invite the other students to find their place in line.


  1. Create your own number line.  Ask students to write any number on their little white board between 10and 100, or between 1 1/2 and 3 1/2   or between -4 and +4, or between 2.35 and 2.36 and then put themselves in order.  In the older grades you can do the same task but give each table group different parameters:  table one writes fractions, table 2 writes exponents, table three writes decimals etc.
  2. Numbers in between.  Give students numbers and have them find a partner.  It can be a random partner or a partner who has a number that has a difference of at least 1 with you, at least 5 with you etc.  Your job is to write on the whiteboard 1 number, 5 numbers, 10 numbers that are in between your two numbers.


  1. Find your partner. Create cards of numbers that are equal in value but expressed differently (0.25 and ¼; 0.25 and 0.250; ¼ and 25%).  Hand out one card to each student and have them find their partner.  In the early primary years I would do the same thing but talk about tens and ones (3 tens  3 ones and 2 tens 13 ones; 4 tens and 3 tens 10 ones).


These activities won’t take very long.  In Ontario they reinforce the first overall expectation in number sense and numeration in each grade level (and we can’t be that different from other places!).  They are worth taking time on.  The benefit to your students is that the relationship between numbers begins to make sense and things like money, fractions, exponents and squares stop being “things” and start having their place in the number system.


October 31, 2014:  Benchmark Numbers and Developing a Sense of “About”

If you think about the math you do on a regular basis, lots of it is estimation.  How much will these groceries?  About how long is it to Kingston?  Another big portion of everyday math is about reasonableness:  Is my grocery bill reasonable?  When I doubled my recipe does 1 ¾ cups of flour make sense?

If we are thinking about increasing students’ flexibility in mathematical thinking and their number fluency, we want them to want them to have experience with the concept of “about”.  We need them understand about benchmark numbers.

Hopefully  you have decided to have a 10-15 minute block of time, at least 3 times a week if not daily, to review and consolidate concepts.  This is a great time to have students practice with understanding benchmark numbers and the concept of “about”.

In the early years 5, 10 and 100 are the first benchmark numbers.  As students become more comfortable with whole numbers we want them to understand the value of multiples of 10, multiples of 100 and larger numbers such as how much a million might be.  As we introduce fractions and money students need to have concepts of ¼, ½, ¾ and 10₵, 25₵, 50₵ and 75₵.  In measurement they should be able to show you visually how much is a centimetre, a metre and eventually be able to estimate a kilometre.  We want them to do the same with other measurement terms (grams, degrees, square centimetres).

Here are some ideas to get you started:

  1. Draw a number line.

10                       ?                                                                    200 (or 20, or 15, or 11)


  ½                                             1                                       ?


  ?                  ?                                    10 (or 5/6 or  45% or -13)

(You have to imagine that the number lines also have vertical marks to show you where the numbers and the question marks are since I still can’t do this in html yet).

As you change the numbers, change the spacing and have the discussions, students will begin to develop a sense of how numbers are related by using benchmark numbers.  Try mixing up whole numbers, integers,  fractions, decimals, square roots and percents all on the same number line (as appropriate for your students).  The conversations students have in justifying their answers is what develops their understanding of number.

  1. How much is it, about? I have been thinking about not even teaching the formal rules for rounding numbers until students have had a lot of experience with the concept of about.  If you start with rounding rules students learn a procedure without understanding why or what it means.  Even before learning about rounding think about the conversations students could have with these questions:

About how much is 44?

Is 44 about 40?  About 45?  About 50?

About how much is 713?

Is 713 about 700?  About 710? About 720?  Why might all of those answers be “correct”?

About how much is $0.93?

Is $0.93 about $0.90?  About $1.00?

You can also use the about question to look at fractional amounts (or decimal amounts or square roots).

Is 3/5 or 7/8 about 1?  About ½?  About ¾?  How do you know?

Which fractions are close to 1?  Close to 0?  Close to 5/4?  Close to 3?

If we know √4 is 2 and √9 is 3 then about how much is √7?

You can use the about concept for operations (which is really just estimation but seems to work better for kids than saying estimation and I wouldn’t give them any rules about how to estimate):

About how much is 17 + 23?

About how much is 158 ÷ 3?

About how much is 9/10 + 15/16?

About how much is 0.99 – 0.53?

You can use about it measurement as well.

About how long is your pencil?  Your desk?

About how much is the area of your paper?  The floor?

Show me about how much a metre is?  3 metres?

About how cold is it today?  About how cold is it in March?

Once you get started, you will find multiple places to use the concept of about and benchmark numbers.


October 24, 2014 :  Developing Student Confidence in Mathematics 

The ultimate goal for our math students is that they develop confidence in mathematics.  I think this goes beyond just getting a good mark on a test, or being able to perform the algorithms and procedures correctly and quickly.  Confidence in mathematics is developed when students believe that they have the mathematical knowledge to solve any problem; when they know how numbers relate to each other; when they are flexible in their thinking so that they will try a new strategy if the first one isn’t working for them; when they can understand multiple responses to the same question.

That’s a lot for your average math teacher!

But, if you wanted to start in just one place, I’d suggest daily number talks or number strings.  These are quick questions that get students thinking about how numbers go together.  This week I was working with a student and used this technique to help him access the mathematics he knew in order to solve a multiplication problem that was too difficult for him.

The question was 30 x 15.  The student clearly didn’t remember how to do the multiplication algorithm and looked at me hoping I would offer the calculator as an option (which, of course, might have been fine).  But we started like this:

10 x 10

20 x 10

30 x 10

3 x 5

30 x 5

And he got to 450.  Of course there are other strategies.  We could have done 30 x 30 and then halved the result.  We could have recognized that once we had 30 x 10 we could have halved 300 to get 30 x 5.  If I had done enough of these with students and various students were giving their strategies, students would begin to see that there are many paths to finding the answer to 30 x 5.  That is what creates mathematical fluency and flexibility.

It is not important that all students can demonstrate proficiency with all strategies.   I would never “test” my students to ensure they could solve 30 x 15 three different ways.  What we want students to do is to mathematically relax—that is to accept that numbers work in ways that relate to each other and that no one way is the “right” way.  When we teach students only one way to solve a problem, and they forget it, they don’t have the confidence or ability to figure out the answer.  As students begin to see the relationships between numbers, they will relax and learn to manipulate numbers in many ways.

Resources that will help:

Any of the minilesson books by Cathy Fosnot.

Number Talks by Sherry Parrish

Building Powerful Numeracy for Middle and High School Students by Pamela Weber Harris

Video clip: (grade 3)

Short PDF by Sherry Parrish

San Diego Middle School Routine Bank

October 17, 2014:  Shouldn’t they know their multiplication facts?

We used to drill the multiplication facts.  Most of us have memories of Mad Minutes, flashcards and Around the World games from grade school.  Ask any parent what the most important part of mathematics is and chances are they will tell you that knowing your facts is.  And they are right.  Knowing your facts is important and it makes other parts of mathematics a lot easier to do.  If I had to count by 8s every time I needed to know 8 x 7 I would be frustrated.  Especially since counting by 8s is hard.  It is efficient to know your multiplication facts but, it is important to know how to get the answer if you forget.

But then came “problem solving in mathematics” and the message, albeit misunderstood, seemed to be that learning your facts was not a good thing.  And teachers all over the western world either stopped drilling the facts or felt guilty doing so.  However, problem solving in mathematics never equaled not learning your multiplication facts (or addition/subtraction facts either).  The real message is that knowing your multiplication facts without understanding what they mean is not nearly as valuable as understanding multiplication before you learn your facts.  Another message was that timed drills of meaningless facts can be stressful for many students since if you don’t have any strategies to find the answer beyond your memory which has failed you, you really are out of luck.

So we need to teach students how the facts are related.  If I know 2 x 8, but I can’t remember 4 x 8, I don’t need to worry because I can double 16.  If I know 4 x 4 but I don’t know 4 x8, I don’t need to worry because I can double 16.  If I know 4 x 5 and I know 4 x3 I can figure out 4 x8 by adding 20 + 12.  Plus, I understand how all these facts are related because I can break 4 x 8 into different smaller arrays.

Number strings  are a great way to teach kids how these facts are all related.  In the end you want students to be able to think about multiplication facts as a series of connected arrays.  If we return to my original troublesome fact 8 x7, I don’t have to count by eights if I know 8 x4 = 32 and 8 x 3 = 23 and can add 32 + 23 = 56.  (If you want to learn about number strings, google Cathy Fosnot or get her books:  Minilessons for Early Multiplication and Division and Minilessons for Extending Multiplication and Addition or check out this video clip:

Once students understand these relationships, it is very beneficial for them to learn their “facts”.  A teacher I know has the students fill out the multiplication grid every Monday (Multiplication Mondays-we love alliteration even as math teachers!).  As opposed to having random facts, the multiplication grid can help students to use what they know to figure out what they don’t know once they understand how the facts can be related with arrays.

1 2 3 4 5 6 7 8 9
 1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 10 12 14 16 18
3 3 6 9 12 15 18 21 24 27
4 4 8 12 16 28 32
5 5 10 15 20 25 30 35 40
6 6 12 18 36 ?
7 7 14 21 49
8 8 16 24 ? 64
9 9 18 27 ? 81

Let’s suppose a student knew a number of the 2s, 3s, 5s and their doubles.  Once they have had practice with the arrays, they can use the grid to figure out the trickier “higher” facts.  Look at the coloured ?s and the related facts to see how students could use the grid to help them figure out the facts they didn’t know.

BUT, the careful caveat here, is NOT to just show the students a “trick” but to make sure they understand how each fact can be an array broken into smaller arrays.  Once they know that, making the connection to the facts within the multiplication table makes sense.  Then having the students learn their facts through short practice times is beneficial and will make their mathematical lives much easier.

And….should you time them?  My answer is yes and no.  Competition against yourself can be fun.  For some students, beating their time for how many they can do provides motivation.  But, for other students it causes undue stress.  I would probably say, we are going to practice for one minute, or 5 minutes but no longer.  After all, you don’t want some poor soul labouring over this for half an hour.  I would also have different grids available—a 5 x5  up to a 12 x 12 (or show the kids how to block it off).  For a student just beginning to know their facts, 12 x 12 is going to be daunting no matter how understanding you are.  But, I would never make the fastest kid the best kid.  Each student should celebrate their own accomplishments, be able to see growth, and have a way to figure them out without waiting for divine inspiration.

(I’m not quite sure how to do graphics in a blog post but if you click here you can see the original in .pdf and it has much better graphics:  shouldn’t we know our multiplication facts)

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