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Category Archives: Math

Algebra – from Simplifying Expressions to Solving Equations

While we were working on measuring circles; circumference, area of circles and volume of cylinders,  I noticed that manipulating the formulas was a mystery for some students.  For example, the formula for the area of a circle is:

most students could find the area when given the radius but if I gave them the area and asked them to find the radius they were stumped.  I could just show them how but I realized that the issue lay in algebra so that is where we have returned.

Algebra is often a huge stumbling block for students.  I’ve been doing a lot of reading and research that I may post about later but for now I’ll focus on what we are doing in class.

First we revisited that the equal sign means that both sides of the equation are balanced or the same.  Surprisingly, some people don’t really realize that.  To reinforce this idea we used Cuisenaire Rods as models.

The students made a “train” with the yellow and black rod.

They had to find other trains that were equal in length to the yellow and black together and then write down equations to express equality.

This photo shows our progression through simplifying the equation.  The letters represent the colours of the rods.  The rods can also be given numerical values.  The distributive property came out during the discussions!  I was so excited I actually jumped up and down and squealed a little bit!

At this point, I knew I had to be explicit about the commutative property of addition and multiplication and the distributive property of multiplication over addition (which you can see in the photo above).

I struggled with how to present it and luckily Kyle Pearce wrote a post that helped me out (and I attended three online webinars presented by Kyle).  These animations by Kyle really helped me to present the distributive property first with numbers (modeled by Cuisenaire Rods) and then with variables.  We did this on the SMART board with Cuisenaire Rods but, alas, I did not take photos!  I know some of the students did though so I may update this post with their photos.

Next the students practised a few times on their own (we NEED a Cuisenaire Rod App for the iPads!) like this (sorry I can’t embed it):

TapintoTeenMinds

Then, still following Kyle’s lead we did this:

I did a few more examples using Alge Tiles and then the students proceeded to practice making models and writing down the simplification of the expressions.  Some still needed a fair bit of scaffolding, but they are getting there.  I also had some students who did not like models and just wanted to simplify using the numbers and letters only.  I think it’s important to use the models though so they aren’t simply memorizing procedures.

This went fairly well, but there are still a few students struggling.  We will move forward into solving for the variable by performing the same operation on each side of the equation.  Then we will circle back to formulas for circles and apply what we’ve learned about algebra to measurement concepts.  Most posts to come to show you our path.

Don’t forget, we’ve also learned the origin of the root word circ in English!  It all ties together in room 133!

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Posted by on April 10, 2015 in Math, Patterning & Algebra

 

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Surface Area of Cylinders

Another 3 Act Math Lesson!

Act 1

I’m starting to make my own videos to match the grade 8 curriculum.

This is Act 2.  Ideally, I would have actually wrapped the Pringles can with paper but….sometimes time is a factor so I have an ongoing goal to keep adding videos when I have time.  🙂

 

Working on it:

 

We discovered all of this!

cylinder images surface area formula

 

 

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3 Act Math Investigations!

We are loving the 3 Act (or 4 Act) lesson investigations in math.  They start with a “trailer video” that the students watch first.  They generate a number of possible math questions that could go with the video.  This really gets them thinking about math in the real world and situation where they might use math.  It also activates prior knowledge they have.
Next we watch Act 2 which provides some more information and a problem to solve.  The students solve the problem, present their solutions and then we watch Act 3.
Here is the first one we did.  It was created by Kyle Pearce @mathletepearce at Tap Into Teen Minds
Cookie Cutter
Act 1
The students came up with many excellent questions including the questions we were going for:  How many more cookies can we make?
Act 2
Here is some video of two of my students sharing their solution.  I started to use the time-lapse feature on my iPhone and we found it is too fast.  I’ve downloaded an App now that I hope will be better for next time.
Finally, we watch Act 3
We had some discussion about why Kyle Pearce was able to get more cookies than we calculated.  What do you think?
I started blogging for this class on Evernote and Pistach.io but I found it didn’t have the features I need so I’m continuing on here.  If you want to see the first few lessons leading up to this check them out here.  They give some background on Pi, deriving the formula for the area of a circle and Mindset.
 
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Posted by on February 23, 2015 in 3 Act Math, Math, Measurement

 

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Fractions!!!!! The Big Ideas!

Both classes have been working on fractions for the past few weeks (with some time off when some were swimming!).   The grade 5s have a test coming up on Monday, May 5th and the 4s will have on Wednesday, May 7th.   The grade 4s need to understand up to # 5 and the 5s need to understand all of the Big Ideas discussed below.

Fractions – Big Ideas

  1. A fraction has a numerator and a denominator.  The denominator tells how many equal parts the whole is divided into and the numerator tells how many parts there are.
  2. Fractions can mean different things and be modeled in different ways:  part of a set, part of a region, as a measure, division & as a ratio.

types of fractions 2types of fraction 1

3.  A fraction is not meaningful without knowing what the whole is (if you only see the numerals when comparing fractions you assume the whole is the same).

For example, in class one day I asked the students if they would rather have half of a chocolate bar or one fourth of a chocolate bar.  Most students said half but a few knew I was up to something.

The fourth came from this type of chocolate bar:

498_PC_Milk_Chocolate_Bar_-_(EN)_-_(500x500)

and the half from this:

kitkat

4.  If fractions have the same denominator, the one with the greater numerator is greater.  The denominator tells the total number of equal parts in the whole, and the numerator tells the number of parts accounted for.  Since both pies are cut into 8ths all the pieces are the same size.  Therefore 5 pieces or 5/8 is more than 3/8:

like denominators

4.  If fractions have the same numerator, the one with the greater denominator is less.  The denominator tells the total number of equal parts that the whole is divided into, and the numerator tells the number of parts accounted for.  The larger the denominator, the smaller the parts are:

unlike denominators

6.  There are proper fractions, improper fractions or mixed numbers.  The numerator is larger than the denominator in improper fractions.  The mixed number has a whole number and a fraction.  See below.

improper

..and the grade 5s discovered an algorithm to convert a mixed number to an improper fraction!  The whole number x the denominator + the numerator.   2 x 4 + 1 = 9

They also figured out how to go from improper fraction to mixed number.  Divide the numerator by the denominator (fractions are another way of expressing division after all!).  9 / 4 = 2 1/4

7.  Fraction can have different names, these are called equivalent fractions:

 

equivalent fractions

 
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Posted by on April 30, 2014 in Fractions, Math

 

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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.

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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.

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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:

photo (5)

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

 

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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.

photo

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.

 

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

 

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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:

download (4)

 

 

 

 

 

 

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 =

Untitled_1

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

 

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