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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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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