Finding the GCF & LCM of 3 or More Numbers using the Cake Method

I wrote a blog post 2 years ago about using the cake method to find the GCF & LCM of 2 numbers.  (I absolutely LOVE this method and my students have had alot of success with it!)  I have been meaning to follow up on that post by sharing how to use this method for more than 2 numbers, and since I just got a question about this last week, I figured now is the time to write that post. 🙂

For this example, I will find the greatest common factor & least common multiple of 12, 15, & 18.

Start by writing the 3 numbers next to each other and drawing a “layer of cake” around them.  Take out any number that you can divide all three numbers by (in this case 3).  Then divide the numbers by 3 and write the quotients under the original numbers, drawing another layer of cake around them.  If there is something else you can divide all 3 numbers by, repeat this process.  In this case, though, there is nothing (besides 1) that I can divide 4, 5, & 6 by so at this point I like to CHANGE COLORS.  Now you need to find if there is any number you can divide some of the numbers by.  In this case, I can divide 4 and 6 by 2, so I write a 2 on the side (in a different color) and then divide the 4 and 6 by 2.  Since I am not dividing the 5 by anything, I just bring it down.  Repeat if you can divide some of the numbers by another factor.  In this case, I can’t, so I am done.


Find the GCF by multiplying all the numbers on the left that were in the ORIGINAL color ONLY!  (In this case, it is only “3”, so my GCF = 3).  Find the LCM by multiplying all of the numbers in the “big L” around the cake.  (That includes the numbers in both colors).  (For this example the LCM = 3 x 2 x 2 x 5 x 3 = 180).

In case my explanation makes no sense, I have included a second example showing this process.

As I said, I absolutely LOVE the cake method and have found that the students love it too and find it MUCH easier than other methods for finding the GCF & LCM.  I hope this post helped explain how to use it for more than 2 numbers!

Thanks for reading,

Christina

You also may be interested in my GCF & LCM Halloween Riddle Sheet – available in my TpT store for $1.

 

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Discovery Lesson: Factoring Trinomials

Whenever possible I like to have students discover or figure out the lesson on their own (with some guidance from me, of course), rather than simply teaching them an algorithm.  One such topic is factoring trinomials.

I start my lesson on factoring trinomials with a = 1 by giving students 4 binomial multiplication problems and having them solve them, showing all of their work.

discover factoring pic1

I then tell them that factoring is the opposite of multiplying, so basically they are given the “answer” to a multiplication problem and they need to figure out the “problem”.  I have them look back at the previous 4 trinomial “answers” they got and to try to come up with a rule for factoring by figuring out where the b and c came from.

With a little time the students are always able to come up with the idea that the b is the sum of the 2nd terms in the binomials and c is their product.  So, when I give them a trinomial to factor they know that they will have (x +/- #)(x +/-#) and need to find 2 numbers whose product is c and sum is b to fill in the #s.

I like teaching factoring this way because the students understand that I didn’t just make up some rule.  They came up with the rule themselves by analyzing problems that they already knew how to solve.

I also teach factoring trinomials where a ≠ 1 this way.

discover factoring pic2

With these problems I have students come up with a rule to go from the trinomial answer to the “work” column.  They are usually pretty quick to notice that the first and last terms stay the same and that the middle term gets split into 2 terms.  When I ask how to know how to split up those middle terms they are able to come up with the idea that it needs to be split into 2 numbers whose sum is the original number but I usually have to push them to get them to see that the those 2 numbers also must have a product of ac.  I teach factoring by grouping earlier in the unit, so once students get to this point in the problem, they are able to just apply the factoring by grouping method to finish the problems.  (I know that there are other methods to factor trinomials where a ≠ 1, like guess & check and the “airplane” or “slip and slide” method but I personally find that factoring by grouping makes the most sense for students since they are using 2 things they already know: multiplication of binomials and the distributive property, in reverse.  I think this lends itself perfectly to helping students understand that factoring and multiplication are simply inverses).

Factoring is such an important part of algebra and is used in many different ways (solving quadratic equations, simplifying rational expressions, solving radical equations, etc.) that it is really important that students master it.  At least in my opinion, having students take some ownership of the process helps build their understanding and mastery.

[For students who understand the process of factoring but struggle to come up with the 2 numbers with the given sum and product I show them how to use a factor tree to help them find the numbers.  You can read about that here.]

Thanks for reading,

Christina

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Breaking Down 2 Step Equations

Today I’m writing about a simple idea that makes 2-step equations easy for kids – a box “trick”.

2 step equations box trick

Students obviously have already learned how to solve one-step equations before being introduced to two-step equations, so I introduce 2-step equations by giving students a simple one-step equation.  The only difference is that I use an index card instead of a variable in my equation.

box trick pic1

Put an equation like the one pictured above on the board and tell students to solve it for the index card, which they should be able to do easily since it is a simple one-step subtraction equation.

Once they solve the equation for the index card, lift up the original card to reveal what is underneath it (in this case 8x).  It also works if you write 8x on the backs of the index cards and just flip them over.

box trick pic2

So, since the index card = 8x they now need to solve the equation 8x = 56, which is another simple one-step equation that they should already know how to solve.

box trick pic3

Do another example or two with the class and then discuss how to decide which part of the equation goes under the index card (whichever part comes first using the order of operations).  Have students replicate the process in their notebooks by giving them a 2-step equation.  They need to draw a box around the part that would be under the index card.  First solve the equation for the box and then solve the new equation for the variable.

box trick pic4

I have found that this method really helps students make sense of solving 2-step equations by turning them into two 1-step equations.  How do you introduce 2-step equations in your class?  Do you do something similar?  Please share in the comments!

If you are in need of resources to supplement your lessons on one and two step equations you may be interested in the following activities in my TpT strore:

one-step equations integers pic1  2 step equations bingo pic1  2 step equations partner matching pic1

Thanks for reading,

Christina

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Hands-On Integer Operations (Part 3: Multiplication & Division)

Multiplying and dividing integers are, in my opinion, the hardest operations to teach to students and it took me a few years before I found a good way to have these operations make sense to students.  After I am confident that my students are comfortable adding and subtracting integers, I move on to the final two operations.  (Click here for my post on teaching integer addition.  Click here for my post on integer subtraction.)

I begin with a class discussion of what multiplication means.  Once the students are able to tell me that multiplying means “grouping,” I pull out the two-color counters again…

I start with a positive x positive:  2 x 3.  We discuss how this means “make 2 groups of 3”.  The students show me this with their counters and get 6 as an answer.

pos x pos

I move onto a positive x negative problem: 2 x (-3).  This means “make 2 groups of -3”. The students are able to do this easily with the two-color counters, as well, by making 2 groups of 3 red counters to get an answer of -6.

pos x neg

Now we get to harder questions:  negative x positive and negative x negative.  I ask the class “if a positive times a number means to make groups, what do you think a negative times a number means?”  Then we come up with the idea that a negative times a number must mean to take away groups.

So we do a negative x positive problem: -2 x 3, which means “take away 2 groups of 3”.  Obviously the students can’t take away groups when there are no groups to begin with, so I ask the students what to do.  Because they are familiar with having to add zero pairs to take away numbers from our lesson on subtraction, the students are able to tell me that I will need to first make 6 zero pairs and then take away 2 groups of 3 yellow counters, leaving them with 6 red counters, or -6.

neg x pos

Finally, we move onto a negative x negative problem: -2 x (-3), meaning “take away 2 groups of -3”.  The students are able to figure this one out on their own based on the last example, so they again add 6 zero pairs, but this time take away 2 groups of 3 red counters, leaving them with 6 yellow counters (positive 6) as their answer.

neg x neg

I give the students problems to try on their own and then we come up with our formal rules for our notebooks:

Same signs:  multiply the absolute values of the numbers and make answer positive

Different signs:  multiply the absolute values of the numbers and make answer negative

Coming up with “real-world” examples for adding and subtracting integers is easy – you can use elevators moving up and down, temperatures rising and falling, and gains and losses in football.  It was much harder for me to come up with a “real-world” example for multiplication to give my students, but I finally have one that I am happy with that makes sense to the students and that I can use to show all 4 types of multiplication problems!

temp

  • Positive x Positive: It is 0 degrees outside.  The temperature is rising 2 degrees every hour.  What will the temperature be in 5 hours?  [The students know to do 2 x 5 = 10 degrees, so they are multiplying the rate the temperature is changing, which in this case is positive since the temperature is rising, by the time, which is also positive since I am talking about the future]
  • Negative x Positive: It is 0 degrees outside.  The temperature is dropping 2 degrees every hour.  What will the temperature be in 5 hours?  [-2 x 5 = -10 degrees – the rate is negative since the temperature is dropping, and the time is positive]
  •  Positive x Negative: It is 0 degrees outside.  The temperature rises 2 degrees every hour.  What was the temperature 5 hours ago?   [2 x (-5) = -10 degrees – the rate is positive, since the temperature is rising, but the time is negative since I am talking about going backwards in time]
  • Negative x Negative: It is 0 degrees outside.  The temperature drops 2 degrees every hour.  What was the temperature 5 hours ago?  [-2 x (-5) = 10 degrees – both the rate and the time are negative]

 

I teach dividing integers just by using the inverse of multiplication.  For example, I give the problem -24 ÷ 4.   Since they already know multiplication of integers rules they simply ask themselves 4 x ? = -24, so the answer must be -6 since a positive x negative = negative.

Students then come to the conclusion that multiplication and division use the same rules since they are inverses.

Once the students understand the rules, they have no problem solving multiplication and division problems, as they find multiplying & dividing integers to be easier than adding and subtracting.  I have them complete worksheets and/or play review games with them for practice and then I move on to practicing all 4 operations together the next day.

Click the image below to download a FREE page students can add to their notebooks  for a quick reference on integer operations.

int ops rules

If you are looking for more resources on integer operations, check out my Integer Bundle, which includes 21 (differentiated) worksheets, 4 games, and 3 sets of task cards.

integer bundle pic1

I would love to hear how other teachers teach integer operations, especially real-world examples for multiplication and/or division.  Please leave a comment if you would like to share!

 

Thanks for reading,

Christina

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Hands-On Integer Operations (Part 2: Subtracting Integers)

After I spend 2-3 days teaching integer addition (click here to read my post on adding integers) and am confident my students are comfortable with it, I move onto subtraction.

I start by giving students two color counters again and put an easy question on the board for them to solve independently:

-5 – (-2)

Since students are familiar with subtraction meaning “take away”, they are able to solve this problem without any assistance.  They start with 5 red 2-color counters and take 2 away, so they are left with 3 red counters (or -3).

sub int pic1

I always hear “this is easy!” at this point, but that sentiment goes away quickly when I give them the next problem:

-5 – 3

This one is not so easy for them.  They start with 5 red counters again but this time they need to take away 3 yellow ones, which is obviously impossible since they don’t have any yellow ones to begin with.  When I ask the class how I am going to take away 3 yellow if I don’t have any yellow to begin with I usually hear the following two responses:

  1. “Just flip 3 of the counters over to the yellow side and then take them away”.  I ask the class why I can’t just flip 3 of the counters over and someone is always able to tell me that if I flip 3 of them over, I have now changed the -5 to a +1, so it’s a completely different problem now.  (Sometimes I have to actually demonstrate this for them to see it).
  2. “Just add 3 yellow ones to the -5 and then take them away”. Again I ask the class if this makes sense and someone is able to tell me that  I have now changed the -5 to a -2 if I add 3 yellows to it, so once again it is a completely different problem.

After seeing those two suggestions fail one of my students is able to figure out what to do…instead of just adding 3 yellow ones to it, I should add 3 yellow and 3 red to it because that is essentially just adding zero to the -5 three times, which isn’t changing it at all.  (This is why it is really important to drive home the concept of zero pairs when teaching integer addition!)   Once I add those counters to the -5, I am able to take away 3 yellow ones to finish solving the problem, and am left with 8 red counters (or -8).

sub int pic2

I usually do one more problem with the class to make sure they get the concept of adding zero pairs so that they can take away what they need to take away.  Then I have them practice several problems on their own:

-8 – (-6)

-4 – 2

3 – (-7)

10 – 12

-5 – 4

-9 – (-7)

1 – 5

2 – (-3)

Once they are done using the counters to solve the problems, we check the answers as a class, and I ask the class to come up with a rule that they can use to solve the problems without 2-color counters.  It sometimes takes a little guidance from me, but eventually we come up with the rule that subtracting integers is really the same as adding the opposite.  I have them write our “official” rule in their notebooks along with some examples.

subtracting integers notes

We shorten the rule to “keep, change, change”, since they are keeping the first integer the same, changing the subtraction to addition, and changing the second integer to its opposite.  The students love this rule because after they do “keep, change, change” they are just solving an integer addition problem, which they already know how to do!

I assign subtraction problems for homework that use small numbers so that they can use counters if they want.

The next class period, I have the students practice integer subtraction with partner worksheets and/or self-checking worksheets and I assign problems with large numbers for homework the second night so that they are forced to use the “rule” instead of 2-color counters.

The following day I have the students practice both integer addition and subtraction.  I never used to take an extra day to review both operations but I found that some students are good when they are just adding or just subtracting, but get mixed up when they have a combination of the two operations.

To review both operations, I break the class up into 3 groups and do centers.  I have one group working with me on mini whiteboards solving integer addition and subtraction problems.  I have another group working on integer addition & subtraction self-checking task cards.  (I love, love, love self-checking activities because they allow the students to see for themselves if they know what they are doing!)  I have the third group playing a Bingo game on adding & subtracting integers.  I have found this stations day to be very valuable in giving students the practice they need with adding and subtracting integers and the kids always enjoy it!

I move onto integer multiplication & division the next day (which I will write about in my next post).

If you are interested in checking out the self-checking task cards or bingo game I use on adding & subtracting integers, click the images below.

add and sub integers self check task cards pic1     int add sub bingo pic1

Thanks for reading,

Christina

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Unit Analysis in the Middle School Math Classroom

After a long break (due to craziness in my personal life and some technical difficulties with the blog)…I’m happy to be back blogging!

I’m going to jump back into blogging with a post about Unit Analysis…

unit analysis in the  middle school math classroom

I hate teaching customary system conversions the traditional way!  No matter how many times you explain to students that you multiply when converting from a larger unit to a smaller unit and divide when converting from a smaller unit to a larger one, there are always students who mix the two up.  And then when teaching students to convert rates, it’s a whole new process…

So, I prefer to teach Customary conversions with unit (or dimensional) analysis.

Here are a few of the reasons why I love unit analysis:

  • It’s ALWAYS multiplication (no need to figure out which operation to use)
  • It reinforces fraction multiplication skills
  • The same process can be used for converting rates (no need to learn a new skill)
  • Students get it (and like it)!!

 

Here is an example of a simple unit analysis problem:

Convert 7 miles to yards.  (This example is done with the assumption that  students do not know that there are 1,760 yards in a mile;  If they do know this, it is only a one-step unit analysis problem).

basic unit analysis

 

When I teach unit analysis I always have a discussion with the students on why it works.  It sometimes takes some encouragement, but students are eventually able to come to the realization that each fraction (other than the starting one) equals one since the numerator and denominator are equivalent.  Therefore, unit analysis ‘ works’ because they are just multiplying by 1 over and over again, which doesn’t change anything (as we know from the multiplicative identity property).

I typically start the lesson with a simple fraction multiplication problem to review cross-simplifying.  That makes the transition to cancelling out the “yd” in the numerator with the “yd” in the denominator easy for the students.

I always tell them that they know they are done when the only word remaining is the one they are asked for.

When I give the students rate conversion problems the next day they are able to solve them using unit analysis, as well.  It just sometimes takes some gentle reminding that every fraction should equal one (the numerator should equal the denominator) and then they are good to go, even for “hard” problems like the one below!

converting rates unit analysis

Thanks for reading,

Christina

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Finding the GCF and LCM using the Cake Method

Today’s lesson in my 7th grade math class was on finding the greatest common factor and least common multiple of a pair of numbers.

I started by gathering background information about what the students already knew about the GCF and LCM.  With a little bit of help the students were able to come up with definitions for each of them and described the method they had used in the past to find the greatest common factor and least common multiple of two numbers.

Previously the students had learned to list all the factors of 2 numbers and find the largest one they had in common.  Similarly, to find the least common multiple, the class had learned to list multiples of the numbers until they found one in common.

We went through examples of finding the GCF and LCM the “old school” way.

GCF LCM list method

 

Then we discussed the limitations of this method for finding the GCF and LCM, mainly the fact that if I gave them much larger numbers it would be very time-consuming to list all the factors or list multiples until they found one in common.  So…I then introduced the class to the “Cake Method”.  (Alot of people refer to it as the ladder method or division ladder, but I first saw it called the cake method in the book Math Doesn’t Suck by Danika McKellar, and I thought it was cute, so that is what I go by, since it resembles an upside-down cake).

The cake method is great for a couple of different reasons…you can make one “cake” and use it to find both the GCF and the LCM and it saves you alot of time over the more traditional method.

For anyone who hasn’t seen this method before, you start by writing the numbers for which you are trying to find the GCF and LCM next to each other and draw a “layer of cake” around them.  You then find ANY common factor of the two numbers and write that factor to the left of them.  You then divide the numbers by their common factor and write the quotients underneath the numbers.  This process repeats then with the new 2 numbers you have inside the cake.  You do this as many times as necessary until the 2 numbers at the bottom are relatively prime.  To find the greatest common factor, you multiply all the numbers to the left of the cake.  To find the LCM, I tell the students to draw a big “L” around the cake, and multiply all the numbers in the “L” together.

Here is the example we did in class to find the GCF and LCM of 24 and 36.

cake method

 

We had a discussion about how different people could make different “cakes” by choosing different factors, but that the final answer would be the same.  We also discussed how the bigger the factor, the less layers of cake there would be.

Overall, the lesson went great!  The students loved this “new” method of finding the GCF and LCM!

I am looking forward to showing them how the cake method can be used to simplify fractions, too.  (Make a cake for the numerator and denominator and the 2 relatively prime numbers at the bottom will be the simplified fraction.  For example, the cake example above could have been used to simplify the fraction 24/36 to 2/3.)

 

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My Spin on Symmetry

Today’s 6th grade lesson was on rotational symmetry.  I have found that students tend to be less familiar and less comfortable with rotational symmetry than they are with line symmetry (and sometimes mix the two up), so I try to find ways to make it more clear and understandable to them.

I had several different shapes cut out ahead of time: an equilateral triangle, isosceles triangle, rectangle, square, rhombus, regular pentagon, irregular hexagon, etc.  I had the students each pick a couple of different shapes and trace them on a piece of paper.

DSCN0922

After tracing the shapes, the students were told to use their pen or pencil to hold down the center of the shape.  I had them rotate the shapes and count how many times the cut out shape lined up perfectly with the traced shape (until they got to a full turn around).

DSCN0926

I asked the students to try to figure out how many degrees they were able to turn the figure to have it line up with their tracing.  They were able to reason that since a full turn was 360 degrees, they had to divide the number of times they could turn the shape into 360.

Their finished notes for the day looked like this:

img002

As an extension, I gave each of the students an angle measure.  They have to draw, color, and cut out their own figure that has that rotational symmetry.  (For example, the student to which I assigned 120 degrees is not allowed to draw an equilateral triangle.  They have to create their own, original figure that also has 120 degree rotational symmetry).

I am excited to see what they come up with!

Thanks for reading,

Christina

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A Little Sum-thing about Triangles

I started my unit on Geometry with my 6th graders before Christmas break.  We got as far as the basic vocabulary and different types of angles.  I decided to begin with polygons after break, so my first post-Christmas lesson was on triangles.

We started off by discussing the different ways to classify triangles – by their sides and by their angles.  The students made a chart in their notebook listing the different types of triangles and we did some example problems.

[Before the lesson, I had made up little slips of paper with different triangle types on them.]  After going over the basics, I had each student pick one of the slips at random.

triangle types

 

They had to draw whatever kind of triangle they picked on construction paper and then cut it out.  This served two purposes – it showed me if they understood the first part of the lesson, and it provided me with a wide variety of triangles.  I had each student number the angles in their triangle (1, 2, and 3).  I then asked the students to rip off each of the angles of their triangle.  Finally, I asked the students to line up their three angles so that they were adjacent to each other, and asked them what they noticed when they arranged their angles in that way.  A few students noted that they formed a straight line.

triangles wholetriangles cut

 

 

 

 

 

angle sum

When I asked the students how many degrees the three angles must be in all if they are forming a straight line, the light bulbs went off!  Since everyone had started with a different triangle, the students were able to conclude that the angle sum of ANY triangle is 180 degrees.

All in all the lesson went very well.  We finished by going through an example of finding the missing angle in a triangle.  The students’ notes for the day ended up looking like this:

img001

At the beginning of the class when we were first going over types of triangles, one of the students had asked why an acute triangle has 3 acute angles, but a right triangle only has 1 right angle and an obtuse triangle only has 1 obtuse angle.  I posed the question back to the class and their original answer was that the triangle wouldn’t close if it had 2 right angles or 2 obtuse angles.  At the end of class, however, one of the boys in my class said “Oh, that’s why there can only be 1 right or obtuse angle…there’s only 180 degrees in all so if you already have 90 degrees, if you had another 90 degree angle you wouldn’t be able to fit another angle!”

Don’t you just love it when students can reason out the answers to their own questions?!

 

Thanks for reading,

Christina

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