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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Hands-On Integer Operations (Part 1: Adding Integers)

Integer operations are truly a make or break middle school math concept.  Students who “get” integers will be much better off than students who don’t, in terms of their future math success.  That’s why I really take my time when teaching integers each year.  I’m breaking up my blog post on teaching integer operations into 3 separate posts so that it’s not too ridiculously long. 🙂

There are a variety of ways to teach integer operations, but my absolute favorite strategy is using two-color counters.  I have found that some students struggle with using number lines and rules for integer addition, but if I give them two-color counters, every student is able to successfully add two integers.

I always start my lesson on integer addition with a discussion of zero pairs.  I usually do this by asking students what happens if I take one step forward (+1) and then take one step backwards (-1).  They tell me that I haven’t moved at all (or that I have moved zero spaces), so that’s how we form a conceptual understanding that 1 + (-1) = 0.

Once they get this, I pull out the two-color counters and tell them red is negative and yellow is positive.  I hold up one red and one yellow and ask what that equals, and they (usually) answer zero, which tells me they actually do get zero pairs (hooray!)   I then show them a handful of counters (some red and some yellow) and ask the class how many I have.  We discuss how all the zero pairs cancel each other out and whatever isn’t part of a zero pair is the answer. I like using virtual 2-color counters on my Board instead of trying to hold up actual counters since it’s easier for the class to see.  This free website is perfect for that purpose.

counters

Once we’re done with the introduction, I have the students work on some addition problems individually.  I put a bunch of problems on the board for the students to solve independently using the two color counters:

  1. -4 + 3
  2. -2 + (-6)
  3. 8 + (-12)
  4. -7 + (-2)
  5. 4 + (-1)
  6. -6 + 9
  7. -8 + (-1)
  8. 5 + -4
  9. -7 + 3

After the students finish showing each problem with the counters and writing down their answers, we go over the answers as a class.  I then ask the class to come up with integer addition rules based on the answers we got.  I start by having them come up with 3 rules (one for negative + negative, one for positive + negative, and one for negative + positive).

The students are typically able to come up with the rules using the examples from the board (which are color-coded):

  • Negative + Negative:  add the numbers & the answer is negative
  • Positive + Negative:  subtract the numbers & the answer is the sign of whichever number is bigger
  • Negative + Positive:  subtract the numbers & the answer is the sign of whichever number is bigger

We then have a discussion about why the 2nd and 3rd rules are the same. It usually takes awhile to get the answer I’m looking for… students tell me that it’s because it doesn’t matter what order you add numbers in and I ask them how they know that.  Eventually someone is able to come up with the answer of the commutative property of addition. 🙂

Then the students write down the two “official” rules in their notebooks:
adding integers notes

That takes me to the end of the class.  I give the students homework on adding integers (all with small numbers so that they can use 2-color counters if they want/need to).

The next day I don’t teach a new lesson.  I just have the students practice adding integers using the rules.  We do a few problems together after the do now and checking homework.  I then break them into pairs based on how well they seem to be doing with the lesson.  I have them do partner matching worksheets where one person does the 20 problems in the left column and their partner does the 20 problems in the right column and they have to find the matching answers.  If some of their answers don’t match, I have them redo those problems and work together to identify their mistakes.  I made 3 different levels of the worksheets so that each pair is challenged appropriately.  I love partner matching activities because they keep the kids engaged and the students are able to check their own work and help each other out, freeing me up to focus my attention on struggling students.

(For students who really struggle without the two-color counters,  I tell them to draw little + and – signs to illustrate the problem the same way they would have with the counters.  This helps remind them whether they should add or subtract and what sign their answer will be.)

For homework on day 2 of integer addition, I tell the students to study for a quiz tomorrow and I assign self-checking worksheets for written homework.  Then the next day I can ask if anyone has any questions before the quiz because with the self-checking sheets, they already know if they were successful with the homework or if they need help, without me having to check each answer with them.

For the integer addition quiz I give the class 10 problems on Socrative so that I can see the results instantly.  If the majority of the class did well, I move on to subtraction (I will write about how I teach integer subtraction in my next post).  If they struggled with the quiz I spend the rest of the period reviewing addition again.

While I usually teach a lesson a day in my math classes, I really drag out integers because they are so crucial to future lessons.  There is absolutely no point in moving on to subtraction until students have mastered addition.  Most of my classes have been good with it after just the two days, but I have taken the third day (after the quiz) with some classes who needed more practice.  I then requiz them the following day.

(If you are interested in purchasing the integer addition partner matching or self-checking worksheets I use, click the images below.  Each set of 3 differentiated worksheets is $1.75.)

integer addition partner matching pic1Integer addition self checking sheets pic1

Thanks for reading,
Christina

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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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