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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Math Races – turning boring practice problems into fun activities

Yikes!  I’ve been back in school for 3 weeks now and this is the first time I am blogging!  I am going to do my best to be better about it going forward…but no promises 🙂

I have already completed my first units in all of my classes and am now working on my second units.  In my next few posts I am going to share some of the lessons I have done so far this year, but right now I am going to share what I did today in my 7th grade Pre-Algebra class because it went SOO well!!

race-math-activity

I am working on rational number operations with my pre-algebra class this unit (positive & negative fractions and mixed numbers).  Yesterday I did adding & subtracting negative fractions without whole numbers and today I did adding & subtracting negative mixed numbers.  I have noticed over the years that students tend to struggle with this lesson since there are so many things they have to remember: integer rules, finding common denominators, borrowing with mixed numbers, converting improper answers to mixed numbers, and simplifying fractions.  Because I know that this lesson gives students trouble I wanted to give my class lots of practice without boring them to death.

We started by going over the steps as a class and writing them down in their notebooks.  I then had students complete some problems on mini whiteboards, step by step.  Having them show me each step really helped me catch and address any issues early on in the problems.  I then had the class split up into groups of 2-3.  (While I often choose groups for my students, I allowed them to make their own groups for this particular activity).

I had a set of self-checking task cards on rational number addition & subtraction that I made a couple of years ago, where the answer to each card leads students to the next card they need to complete.  If they answer all 20 cards correctly, the last card they do will lead them back to the card they started with, making them completely self-checking.  In the past I have had students simply work through them in small groups, which works well, but this year I had the bright idea to turn it into a race…and it was AWESOME!

Here’s how I ran the activity:

I printed two copies of the cards (so there wouldn’t be an issue of students not being able to get the card they needed) and spread all the cards out on a table in the front of my classroom.  I gave each group one card to start with.  Students had to work in their groups to get the answer to the card.  Once they had an answer they all agreed on, one person in the group had to run their card back up to the table and find the next card.

adding-subtracting-rational-numbers-race-activity

I could not be happier with how this activity went!  The students were sooo into it.  They were all working, engaged, and talking with each other to figure out where they went wrong.  They all wanted to win the race (despite the fact that the only “prize” was a sticker!)  They got lots of practice since there were 20 different cards in all.  Best of all, I heard multiple students say that it was the best math class ever as they walked out of my room today, so that is definitely a win in my book! 🙂

If you want to make a self-checking activity that you could turn into a race like this, you just need to write questions on index cards.  Put the answer to each card on the top of the next card to create a “loop” of questions.  If you don’t want to make your own, I have several sets of self-checking task cards available in my TpT store that you can check out, including a free mini set on the order of operations.

order-of-ops-self-checking-task-cards-pic1

If you try a similar race activity with your class, I’d love to hear about it!

Thanks for reading,

Christina

 

 

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Improving Number Sense with the Divisibility Rules

I decided to make divisibility my first lesson of the year for 7th grade next year for a couple of different reasons: it is a skill used in many different concepts throughout the year and it really helps promote overall number sense in students.

The divisibility rules (hopefully) help students be less dependent on their calculators, which is an area I am hoping to improve on this year.  So even though it isn’t the most exciting lesson of the year, it is an important one, and I think it’s a good way to begin the year.

I go over the rules for 1 – 10 with my students.  (I used to skip over 7 but every year students ask me if there is a rule for 7….so now I give them the rule along with an example for which the rule could be useful (i.e. 231) but then explain that 9 times out of 10 it is easier to just do the division than it is to use the rule).

In addition to the rules I give students tips, like if a number isn’t divisible by 2 then it isn’t divisible by any other even number.  Tips like that help with divisibility by 8, since that isn’t the most useful rule.  I tell students to only check for divisibility by 8 if (1) the number is divisible by 2 and then (2) if it is divisible by 4.

I made half-sheets the students can stick in their notebook with the divisibility rules.  You can download it free by clicking the picture below.

divisibility rules half sheets

To make the lesson more exciting I enlisted the help of my programmer husband.  He was able to build an interactive divisibility rules game that my students can play on their chromebooks, which I am super excited about!  In the “Divisibility Challenge” game, you can choose which rules you want to practice and then either play for mastery, speed, or just for practice.  I think I am going to originally have students play for mastery, where they need to play until they get 10 questions correct.  I plan to begin class the next day with a speed competition to see who can get the most questions correct in 3 minutes.  Competition always seems to get middle school students involved and engaged!  Click below to try a round free.  (It should open right in your browser).

divisibility demo

If you are interested in getting the full game for your class it can be purchased in my tpt store for $6.

divisibility pic1

 

Thanks for reading,

Christina

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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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Having Fun with Distance and Midpoint

How do you take a topic that isn’t overly exciting and get the kids to have fun with it?  I have found that getting students out of their seats and working with friends usually does the trick! 🙂

If you follow my blog you know that I have been sharing activity ideas (and free downloads) for a set of (free) ordered pair task cards that I posted last week.  Click here for my post on a slope activity and click here for my post on a coordinate plane activity.

Today I am sharing an activity on the distance and midpoint formulas:

  • Give each student an ordered pair card and worksheet (download links are below)
  • Have them grab a partner, find the distance and midpoint between their point and their partner’s point and then “check” their work by graphing the ordered pairs and seeing if their answers make sense.
  • Have them repeat with 2 other partners.
  • Challenge early finishers (or everyone) to then grab another partner. Let their point be an endpoint and their partner’s point be the midpoint.  They need to find the other endpoint.

distance and midpoint example pic

It’s a quick easy-to-implement activity that gets kids moving and working together.  Enjoy!

Click the image below to download the ordered pair task cards:

ordered pairs cards pic

Click the image below to download the Distance and Midpoint Partner Activity worksheet:

distance and midpoint worksheet pic

Thanks for reading,

Christina

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Moving Around the Room with the Coordinate Plane – Activity Idea

Here is activity #2 for the ordered pair cards I posted in my last blog post.  (If you missed my post on a fun, free activity for teaching slope, you can read it here).

This is an activity on graphing in the coordinate plane that gets kids up and moving around the room:

  • Give each student an ordered pair card when they walk in the room (free download link is below)
  • Give each student a worksheet (free download link is below)
  • Have them walk around the room and find someone who has an ordered pair that meets the given description. Once they find someone with an ordered pair that “works”, they need to write down that person’s ordered pair and have them sign their paper.  (The signatures ensure that the students are actually walking around the room to find ordered pairs and not just copying from a friend).  They are only allowed to have a student sign their paper once, so they will need to find 9 different people to sign their paper in order to answer all of the questions.
  • After they have gathered all of their ordered pairs and signatures, they need to plot all of the points they found on the coordinate plane on the bottom of the page, labeling each of them with the given letter and their own ordered pair with a star.

coordinate plane find someone who example

I think that this lesson will be a nice way to break the monotony of simply having students graph points on a coordinate plane and write coordinates for given points.  It also makes students think more about their points than they would if they were just graphing them.  They need to think about their x-coordinate, y-coordinate, and what quadrant/axis it is located in for this activity.

Click the image below to download the ordered pairs cards:

ordered pairs cards pic

Click the image below to download the “Coordinate Plane Find Someone Who…” worksheet:

coordinate plane find someone who

Also, if you are looking for a way to challenge your higher level students with the coordinate plane, you may want to check out my Coordinate Plane Challenge Task Card Activity.  It consists of higher level thinking task cards and a riddle sheet and is a good way to challenge students who find coordinate plane graphing easy.  It’s $2.50 in my Teachers pay Teachers store.

coordinate challenge pic2

Thanks for reading,

Christina

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5 Ways to Use Factor Trees in Middle School Math

I think factor trees are one of the coolest “tools” for middle school students!  Students tend to like them and they have several different uses in middle school math.

5 ways to use factor trees

Here are the 5 ways I use factor trees with my students:

1.  Prime Factorization
This is the most obvious use of factor trees and it’s typically the first way students learn to use them.

prime factorization

When I teach prime factorization I introduce the idea of the Fundamental Theorem of Arithmetic (no matter how many different factor trees students make for a number, they will always get the same prime factorization).  I also emphasize the two things they can/should do to check their answer:
– make sure all numbers in their prime factorization are PRIME numbers
– make sure the product of the numbers in their prime factorization is the starting number

2 & 3.  GCF and LCM
Using factor trees is my second favorite way to teach the greatest common factor and least common multiple of numbers and/or monomials.  (The cake method is my favorite method).

GCF and LCM factor trees

4.  Simplifying Radicals
Factor trees make simplifying radicals so easy for my pre-algebra and algebra students!  Students understand that finding the square root of a number means figuring out what number times itself equals the starting number, so in simplifying radicals they are simply pulling out one of each prime factor that is listed twice in the factor tree.

simplifying radicals factor trees

5.  Factoring Trinomials
The idea behind factoring trinomials is pretty simple.  Students need to come up with two numbers that have a certain product and sum.  Students are typically pretty good at factoring simple trinomials like this, since coming up with the two numbers is a breeze:

factor trinomial easy
But, many students struggle with problems like the one below, not because they don’t understand the factoring process, but because they can’t come up with the two numbers with the given sum and product.   That’s where factor trees come in handy!

factor trinomial factor tree

For students who have a tough time coming up with the numbers, I have them make a factor tree for 126.  Once they have it broken into its prime factors, they just need to break up the prime factors into 2 numbers every possible way until they find the ones with a sum of 23.

While this method does take some trial and error, it gives “stuck” students a starting place and a set number of possibilities to try.

Do you use factor trees in any other ways in your math classes?  If so, please share!

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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