# Category Archives: Operations

## Multiplying & Dividing – Part Two – Algorithms

After the students have solved many problems in a variety of their own ways we move onto the algorithms (procedure for calculating).

Break-apart strategies use the distributive property of multiplication.  Don’t worry, I hadn’t heard of that before I started teaching math either.  I wish I had because it’s really important to understand (even if you don’t know the term). It is critical for algebra and it is also handy for mental math.   I talked about it in my post about math talks here because we can use it for our times table facts too.  This video does a really good job explaining it with visuals. I find that some of the students can find the break-apart method of multiplying a 2 digit number by a 1 digit number on their own and when they share their strategy with the class!  Wow!  The other kids eyes open wide with a Eureka moment!  What is the break apart method?  You break one or more of the numbers into parts to make the multiplication easier.  It also demonstrates why you would “carry” a number over to the next column. When I go on to teach the students the traditional way of multiplying they will be able to see why they are “carrying” the 2 (it’s really 2 tens in 25 and needs to be in the tens column).  The grade 4 students are working on this right now.  The grade five students have already done this with 2 digit by 2 digit numbers.

Division

In grade five we have been working on division.  We started with problems and pairs solving the division any way they wanted to.  Then we moved on to the alternate long division algorithm.  I like this method because it reinforces that division is the inverse operation to multiplication (they undo each other) and that it is repeated subtraction.  Here is an example: I started with 10 groups first because 10 is a nice easy number to multiply by but many kids quickly see that it is easier and faster if you multiply by larger “friendly” numbers).

I like this video explaining the method except that it does not have sound and the reading could be a problem for some kids – but we can read it to them.

Multiplication and division take a long time to learn well.  Students need time to solve problems, learn different ways to solve problems, practice their times tables, practice algorithms.  I think all students would benefit from doing one subtraction problem a day (and check their answer with addition) and one division problem a day (check their answer with multiplication) until they are fluent.

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Posted by on March 19, 2014 in Math, Operations

## Multiplication & Division – Part One – Concept & Skill Development

The grade fours have been working on multiplication and the fives on division.  All the ideas below apply to both grades as the fives are already expected to understand the multiplication (they need to also understand 2 digit by 2 digit multiplication) and the fours will be moving on to division later.

With all concepts in mathematics there is a progression from a concrete understanding to an abstract understanding.  There are phases of development that all students go through, some move from concrete to abstract quickly and others need to remain concrete for longer before they are developmentally ready to move to abstract thinking about the concept.  Concrete means needing manipulative and/or pictures to solve problems.

This developmental process is why I do not teach them the algorithm right away and when I do teach it, I insist they understand what they are doing.  A child can memorize a series of steps to do for multiplying or dividing without understanding the what or the why.  This can lead to problems in math in later years, particularly in algebra.  (as an aside, preschoolers can do pre-algebra easily, it seems the steps and formulas are what mess up students!)

I use a chart that shows movement through the phases for understanding the operations (+ – x \ ).  It was developed for Nelson by Dr. Marion Small.  The phases of operations development are across the top of the chart and the concepts and skills are down the left side of the chart. The five phases are:

Phase 1 – Beginners, focus on counting to solve problems.

Phase 2 –  Concrete – formal operations with numbers to 20; Concrete operations with numbers to 100

Phase 3 – Whole Number Comfort – formal operations with whole numbers; concrete operations with decimals

Phase 4 – More Abstract – Fluency (can use multiple methods to solve) with whole number operations; formal operations with decimals

Phase 5 – Flexible – Fluency with whole number and decimal operations; Concrete operations with integers and fractions

There are also 3 concepts and 3 skills on the chart but for this post I am only looking at:

Concept 2 – Multiplication and division are extensions of addition and subtraction.  Multiplication and division are intricately related.

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Concept 3 – There are many algorithms for performing a given operation with multi-digit numbers.

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Skill 3 – Computes with multi-digit whole numbers and decimals using pencil and paper without the aid of a calculator.

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I approach multiplication and division from a problem solving perspective.  This means I pose problems that the children can relate to and then I have them solve the problems in pairs using whatever method they would like to use.  I ask them to check their work using a different strategy.  The groups solve their problems on chart paper and then they share their solutions with the class.  The children are exposed to many different ways to solve the same problem and they see that there are many ways of thinking about and solving problems.  Every year I learn something new from the students, there are ways to solve grade 4 & 5 problems that I have not thought of yet after 12 years of teaching math!  That is awesome!

Here are a few examples of the grade 4s solving a multiplication problem:

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After we share our examples we talk about how efficient the methods are.  In this case, 23 x 3, drawing a picture is doable but not very efficient.  Repeated adding is fairly efficient.  But what if it were 233 x 23?  Would drawing a picture be efficient?  Repeated adding?  From here I get into algorithms (procedures for calculating).  I try to direct the children to come up with their own algorithms because they really remember those the best!  Naturally, none of the children has come up with the traditional algorithm on their own so I do have to teach those, but I want them to fully understand what they are doing.  But before I do that, I let them discover the break apart method!

The purpose of this post is to explain the development of students’ developmental learning in math.  Some students go through the phases quickly – some can multiply and divide quite well using several methods.  Others take longer and they are still not quite there.  This is normal and is one of the challenges of organizing children by age into grades.  All kids develop at their own rate (of course there are learning skills such as attention, work ethic, etc., that also come into play).  It is important for ALL kids to understand how to multiply and divide concretely before they move into the more abstract algorithms such as the ones we learned.

The next post will describe the break apart method of multiplying and the alternative long division algorithm.

Posted by on March 19, 2014 in Math, Operations, Uncategorized

## Math is so cool! I love number talks!

It really is!  I think I have learned way more about math as a teacher than I ever did as a student, and I enjoy is so much more now then I did then.  Wow.  This has been a journey that began ten years ago when I took a Math Solutions (Marilyn Burn’s) 5-day course on teaching elementary school math and I just continue to learn and grow.

I’ve been doing number talks for a long time.  A number talk is when the class has a discussion about numbers. I pose problems on the SMART board and ask the children for solutions and we discuss patterns and strategies in math.  For example, we often play silent multiplication, the game is silent until the end until we discuss patterns and strategies.  Here is an example:

2 x 4 =  8

4 x 4 =  16

8 x 4 = 32

16 x 4 = 64

What do you think I want the students to get from this?  If you guessed using doubling you are correct.  The students will notice that if one factor doubles then the product also doubles.  Today I asked the grade fives to predict what would happen is we doubled both factors.  What do you think?

Yesterday we had some PD (professional development) around math and I found there is still much to learn.   I’ve been trying to help the students learn about the associative and distributive properties of multiplication for years with varying levels of success.  What are those you ask?  Good question: You can break factors up into smaller numbers and still get the same product.  Well, I’ve used that for a long time with double digit numbers.  For example,

15 x 5 =

10 x 5 = 50

5 x 5 = 23

50 + 25 = 75

But some of the students struggle with this and yesterday I had an epiphany – they don’t really understand about breaking the numbers apart and I need to do it with single digit numbers first so today that’s what the grade 4s did.  All the students had tiles on their desks and I asked them to solve 2 x 7 = using the tiles to make an array.  Then I put up 4 x 7 =  and Courtney remembered the doubling strategy from another day (yay!).  Next I put up 3 x 7 and asked them to use some of the facts they already know on the board to solve 7 x7 = There were a couple of students who saw that they could use 3 x 7  & 4 x 7 to help them solve 7 x 7.  All they had to do was add the products 28 + 21 = 49 to get the answer.

7 groups of 7 is the same as 3 groups of 7 + 4 groups of 7.  They broke the first 7 into a 3 & 4.

Super cool!  We need to do some more work on this until all the students see it but we will get there!!!!!

I found some of the Math Solutions number talks on YouTube if you are interested in watching.  I will try to get someone to film one of ours so you can see our class in action!  Stay tuned.

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Posted by on March 4, 2014 in Math, Mental Math, Operations

## Multiplication & Division and Book Trailers

Right now both classes are focusing on multiplication and division through the lens of division.    The learning goal is to deeply understand the concept that they are inverse operations, they “undo” each other.  This is a critical concept for math.  We will be exploring this concept with manipulative such as colour tiles and through rich problems.

We started with a simple problem of splitting up brownies between friends.  We have 20 brownies and we want to split them equally between 5 friends.  How many brownies do we have?  I can be difficult to a student to translate that question into a division expression so we used the tiles.  Each student had 20 tiles to represent the brownies.  Most sorted them into 5 groups and found that each person gets 4 brownies.  Once the tiles are split up we can see the multiplication expression 5 x 4 = 20   5 people each get 4 brownies or 5 groups of 4 = 20.  Then we talked about what 4 x 5 means – 4 groups of 5.  Although the product is still 20, it’s slightly different because there are 4 groups of 5 instead of 5 groups of 4.  One might wonder why this is an important distinction – because they need to understand division.  If 20 brownies are divided between 5 people they each get 4 but if their divided by 4 people, they each get 5.  There is a difference.  A common error for children is to put the numbers in the wrong order for divison, for example:  5 divided by 20 = 4.  It’s important for them to understand that the dividend (20) is the total amount, the divisor (5) is the number of groups and the quotient is the number in each group.  Today each class worked in pairs to solve a division and were asked to prove their answer was correct (use multiplication to prove their division was accurate).  Tomorrow we will share strategies and I will post different methods here.

Book Reports

Both classes have been asked to read a novel and have it finished by February 18th in order to create a book report.  The grade 5s will all be doing book trailers, like a movie trailer but about a book.  The grade 4s will have several creative choices to choose from, more information to come.  All the work (other than the reading) will be done in class.   Here is an example of a book trailer for the excellent book Wonder.

Posted by on February 12, 2014 in Math, Operations, Uncategorized

## Multiplication & Revising

The grade 5s have wrapped up their first multiplication unit.  The grade 4s are just beginning the journey.  Both began the same way so that is where I will start.

First I posed a multiplication problem to see how much the students understand about the concept of multiplication.  I asked them to model their answers as many ways as possible.

The grade 4s had a choice of two problems and all chose the gummy bear problem: Here are some of the solutions:

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I was looking for the following concepts, Multiplication is:

Grouping numbers

Area (an array)

Part of Fact Families (with division)

Skip Counting (multiples)

The students presented their work to the class and now we are making multiplication posters showing the five different models for multiplying.  Some completed posters:  Next we will move into multiplying with double digit numbers as well as memorizing our multiplication/division fact families.  I will write another post about memorization strategies.

The grade fives started this way and moved onto different ways to model double and triple digit multiplication problems.  Their project was a mini-book in the math notebooks.

The traditional algorithm is great and works well but it’s difficult for students to understand how it works.  That is why I like to do it in tandem with the area model and break apart models which show the place value while multiplying.  Here is a photo of the area model and a video of the break apart model: Here is an in depth look at the lattice method.

We still need lots of practice with mental math, not just the times tables, but mentally estimating and multiplying using strategies like rounding and breaking numbers apart.  More to come on this.

Revising

The grade 5s are revising their descriptive paragraph.  Revising is critical for good writing so we talked about using word choice and sentence fluency as we revise.  Can we replace boring words with vibrant words?  Can we combine two sentences into one?  Do most of our sentences start with different words?  Does our writing flow?

Here is the paragraph we revised together.  Today and tomorrow the students are revising their own paragraphs.  I’ve told them I want to see the messiness, all the marking up of their page.  That is how I will know they are learning to revise.  If they erase their work then I can’t tell.   Their final versions will be on their personal blogs so check for them in a few days. 1 Comment

Posted by on November 7, 2013 in Descriptive, Math, Operations, Uncategorized, Writing

## Subtraction with Regrouping

Subtraction can be very difficult for students when they have to regroup (borrow).  I learned the algorithm as a child with no understanding of what I was doing.  We spend weeks practising and memorizing it.   It actually wasn’t until I became a teacher that I really understood what was happening when I “borrowed”.

These days we want the children to really, truly understand the concept of regrouping, adding, subtracting, multiplying and dividing.  We do more than teach algorithms.  We want the students to try to come up with ways to solve problems on their own, using what they already know.  Sometimes they come up with methods I’ve never thought of!  Students also need plenty of mental math strategies for the four operations but that’s a topic for another post.

I’m going to start with the traditional algorithm that most of us learned in school.  Here is a video using a two digit number and a single digit number.   Unfortunately I can’t figure out how to rotate the video – sorry!

Here is a video extending this method to 3 digit numbers:

There are many other ways to subtract.  The following photos show some of the ways that are possible (taken from Dr. Marion Small Big Ideas from Dr. Small):    Word problems create some problems for students as they try to figure out which operation to use.  Even is a child know how to add and subtract they need to be able to suss out what they need to do first.  Here is a chart that demonstrates the different structures of word problems: The most difficult problems for students are the unknown parts problems in the bottom two rows.  I am going to try to use a method I’ve learned that is used in the Singapore Math program of using rods to represent problems.  I’ll keep you posted on how that goes.    This video shows one type of problem solved using rods.  I do not have this program but would draw the rods.

Posted by on October 22, 2013 in Math, Operations, Uncategorized

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## More Descriptive Writing, Curriculum Expectations & Subtraction

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Today the class looked at the grade 5 expectations for writing from the Ontario Curriculum.  They highlighted the expectations that they thought applied to descriptive writing.  Tomorrow we will co-create success criteria for our descriptive writing.  Then we will look at the paragraph we wrote last week and we will revise it to match our success criteria.  Here are the students reading (there is some sophisticated vocabulary as it is written for teachers) the actual curriculum document and highlighting the expectations they will use.  I was very proud of their thought process today.

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The grade 4s & 5s are working on subtraction right now.  I have given both classes problems to solve that involve subtraction.  I have told them that they can only use the subtraction algorithm (the procedure most of us learned in school) if they are able to explain it.  So far I have found that very few can explain the algorithm, they can explain the steps that they are taking, but they cannot explain the algorithm.  As a result, I have asked them to solve the problem in different ways.  Here is an example of students who solved it using expanded form.  This problem did not require regrouping (borrowing is the inaccurate term most of us learned).    Tomorrow I will post more student work and explanations of their strategies.  I will also post about how I am trying to get them to figure out what is going on when they have to subtract with regrouping.