Interactive Math Notebook: Factors and Multiples

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We have been busy working on determining whether one number is a factor or multiple of another number.  We started by building arrays both with tiles and on centimeter grid paper to determine a number's factors.  Becoming proficient at building arrays has really helped some of my struggling students feel successful in finding factors even when they aren't proficient in their facts or have many strong strategies to use to determine a fact.  After practicing many many times,  I gave each student a pair of die (some were six sided, some were more for my kids who are proficient in their facts and wanted a challenge!) and grid paper and instructed them to roll the die and create an array with dimensions that match the numbers on the die.  We then added this page to our notebooks.  Below is an example.  

Next, we used our Math Handbooks and its Table of Contents (oh hello, Language Arts skills! See....I'm learning how to incorporate it all!) to find the definition and examples of factor, prime number, composite number, and square number.  We used this information to create a foldable to glue into the right hand side of the page.  An example is below!

We are also learning about multiples.  After many days of discovering, discussing, and applying this knew knowledge, we finally were able to put the information into our notebooks.  I always try to wait until I feel they understand it to put it into the notebooks.  We used a hundreds chart to choose a factor, then highlighted all of its multiples.  We also used this factor to create a basic real world problem to show how to apply it to multiples.  

This student chose the factor two, and wrote the problem: A store has CDs for $2 each.  She then drew a picture to show that one CD would cost $2, two CDs would cost $4, three would cost $6, and so on.  This shows that the price of buying CDs are multiples of 2

This student chose the factor four.  His problem is about video games costing $4 each, so two would cost $8, three would be $12, and so on.  He goes on to begin to write a question associated with it!

Lastly, I really wanted to make sure we understood the difference between factors and multiples, as it can get very confusing.  We used markers to circle all the factors in a list and all the multiples associated with it.  Students were allowed to pick their own factor, or for those who are still not feeling comfortable with multiplication, were permitted to use the factor four as I did in my example.  Then, they showed an example of an array that shows one of their listed factors and multiples, as well as a non array (just to make sure they understood that an array is a rectangle and cannot have any pieces sticking off the end!!).  These last two pages might be my favorite!  

This student wrote all the problems for the number four.  He circled all of the fours to show that is the factor.  He wrote: 4 is a factor of any whole number that it divides evenly.  He also circled all the multiples to show that you can multiply any number by four to get a multiple of four.

This student did an excellent job of showing the difference between an array that four is a factor of one of its multiples, as well as a non example saying "21 is not a multiple of 4" and using his array as proof!

Wrapping up Notebooks: A time for Reflection

Last week was our last full week of school, so when Friday rolled around, we used our last full class together to wrap up our Interactive Math Notebooks.  When I told my students it would be our last entry, some loudly groaned and others said audible "Noooooo!'s"  I'll admit, I was sad too.  We had all grown fond of the days we would add a new entry.  They were all very excited that our last entry fell on page 100, which had been our goal for a while.  "Ms. McHugh!  We wrote a book that is 100 pages long!"  And it's true, we did write a book.  Who says Math and Language Arts don't go together??

Before adding our last page, I had created a word scramble page from SuperTeachers (currently a free feature).  Each of the words were math terms that could be found in their table of contents.  This was just a fun way to sum up our year together.  Next, we all sat on the floor (Kindergarten style, as I call it) and slowly flipped through each page of our Notebook.  This was by far the best part of that entire week.  They couldn't believe all we had learned and how far we had come!  Some mentioned that when we entered certain pages, they still had not understood it completely, but now at the end of the year, they "got it."  This was a great lesson to learn.  It doesn't always "click" for all our students at the same time.  But with a little bit of hard work and a lot of  perseverance, most students understood everything by the end of the year.  Our Notebooks were a great reminder of that.

After we reflected on the year as a class, I let them make their very last foldable.  This was a reflection on what they thought they were good at in math, things that were difficult or easy for them, and times that they had the most fun.  I must admit, most said they had fun when their teacher got distracted or off track.  What?  I have no idea what they are talking about......

Here are some of my favorites from that last day:

This one makes me smile :)

Who...me? No!

Fun is a theme in our room.  When we have fun, we are learning and it sticks!

Interactive Math Journal: Fractions of Fractions

I've been promising my kids all year that I was going to teach them how to multiply fractions, and now with only 5 teaching days left, I can finally get to it.  Thursday, I introduced the lesson by giving each student multiple half sheets of paper.  Our first task was to find what 1/2 of 1/2 was.  We folded the paper in half vertically, colored half with one color crayon, then folded it in half horizontally and colored that half with a different color crayon.  The piece that had the two overlapping colors showed us what 1/2 of 1/2 was, or 1/4.  We then did the same activity again, but this time wondering what 1/2 of 1/3 was.  I kept recording our findings on the board.  At one point, I heard a gasp from Abbie.  She looked at me excitedly and kind of started bouncing in her seat.  I knew she had discovered something, but I asked her to hold on to her thoughts just a little longer.  As we kept going with the paper, I heard more and more gasps and "oh!!!!!" coming from the class.  I could tell some kids were getting frustrated that the others were discovering something they were not, so I finally let Abbie tell us what she first discovered.  Of course, she saw that we were making arrays with our papers, and noticed that all we had to do was multiply the denominators.  It was a different student that noticed that in each of our examples, the numerators were also multiplied.  It was an exciting moment for the class!! 

Then, someone raised their hand and asked, "Are we going to put this in our notebooks?"  To be honest, I hadn't thought of that, but they had such a great idea!  They actually wanted to add to their notebooks on their own!  YIPEE!!!  Since this activity took so long, we had to wait a day to enter it into our notebooks, but that gave me a chance to type up some blank rectangles to record our drawings in.  Here is the end result: 

This student chose pink and yellow to color with, so the overlapping piece is orange and shows the end result!

This student showed that 1/2 is shaded in yellow, while 3/4 is in pink.  The orange shows that 1/2 of 3/4 is 3/12!

This time, we included a "What I know" and "What I learned" section.  It should be on the left hand page, but oh well.  We are still learning!

Interactive Math Notebooks: Adding and Subtracting Negative Numbers

With testing over, and the end in sight, we are working working working hard to stay on a routine and make sure we are ready for 6th grade!  We just finished a quick unit on negative numbers.  Manipulatives are great to use here and serve as a constant reminder of what is going on.  

First, we spent a couple days adding positive and negative numbers.  We used green counters for positive numbers, and red for negative.  We use these two colors throughout our unit.  I really putting an emphasis on the fact that adding is PUTTING TOGETHER.  This would later help with differentiating between the rules for adding negative and the rules for subtracting negatives.  After doing a few sample problems, the students were able to come up with their own rules for adding positive and negative numbers.  They quickly caught on and all was well in the world! (Don't worry.....subtracting is next....that's a WHOLE different ball game!)  Here are our notebook entries for adding:

Next, we began working on subtracting.  This time, I used a clear bucket to show what we had in the container, and what we needed to subtract, or take out.  Before giving them any tricks, we practiced many many times with counters.  For example, if the problem read: 8- (-4), I would fill the container with 8 green tiles, or 8 positives.  Then, I would ask if we were able to take out 4 red tiles.  Obviously, there were only green tiles in the container, so I couldn't take any out.  We had discussed earlier how the opposite of every number added together equals zero, so I demonstrated putting in groups of one red and one green tile at a time, until I had 4 reds to take out, all the while emphasizing that I wasn't changing the value of the container since I was just adding zero to it.  Then, I was able to take out the red counters, leaving only green behind.  This was not an easy concept to teach, and there were some frustrating looks around the room, but we kept at it until it slowly started clicking.  That's when I introduced Mr. Minus.  

Who is Mr. Minus, you ask?  It's more of a "what".  Mr. Minus is a poem my mother, Mrs. McHugh, also a 5th grade math teacher, made up years ago.  I owe her A LOT this year!  It goes like this:

Mr. Minus, Mr. Minus

Learn you I must

But I think I’m going to turn you into a plus,

Now change the second number to its opposite sign,

Add them both together and life will be fine!   Yeahhhhh!

It's catchy and the kids love it!  Here is our notebook page on subtracting.  Again, the pictures are of containers holding what we need to take out and the step by step change it goes through.  

Probability: Math Notebooks

Today's Math Notebook was inspired by the WONDERFUL Mrs. Runde over at Runde's Room, although, I catered it to our needs.  I really liked how she had her students write out the learning goals and task before starting, so I decided to try it.  I always let the kids know what our goals for the day are, as well as have our "I Can" statements posted clearly in the room, but this is the first time we really put it in writing as a class.  We have been working with probability, so I decided to make a spinner as well. I also liked how she had the kids figure out a way to divide their circle in ten equal pieces as a nice review on dividing as well as remembering how many degrees are in a circle!  We also love working with protractors, so it was a nice review.   (We also had the chance to review equivalent fractions AND percents!  All in one activity! YAY!) Below is our page:

Because our goal was to compare theoretical results with actual results, and to make predictions, our notebooks ends up steering away from Runde's a little bit.  We have been having trouble creating frequency tables lately, so here was a great opportunity to do so.  I was going to have them do their experiment 20 times instead of just ten, so we based our theoretical results off of 20 (and reviewed equivalent fractions at the same time!)  I also had them come up with a prediction of what they thought would happen, based on their theoretical results.  Then, it was spin time!  They recorded their results in their frequency table and wrote out their conclusions based on their actual results.

Lastly, we met to talk about the reasonableness of certain results.  We used our theoretical results to determine if the actual result was reasonable or not.  Before this lesson, only 50% of my students answered a "reasonable or not reasonable" question correctly.  After this lesson, we were up to 87%!  I would say we succeeded!  :)

Algebraic Expressions and Math Notebooks

We covered Algebra way back in October and did a really great foldable for the four steps to follow when solving a problem with unknowns.  We came back to it today, both as a review, and to extend our thinking to expressions and patterns.  Our learning target for today was to introduce the use of algebraic expressions to represent situations and describe rules.  First, we watched this StudyJam on Function Tables, as well as one on Addition and Subtraction Equations and Multiplication and Division Equations.  We also did a sample problem in our notebooks, as shown below.  The problem reads:

Joe and Maria are 5th graders.  Joe is two inches taller than Maria.  

I asked the students to put this information into a function table and give me some measurements for Joe and Maria.  I reminded them to make them reasonable measurements for 5th grade students.  Of course, everyone had different answers, and all were acceptable as long as Joe was two inches taller.  Next, I recorded some of their measurements on the board and asked them to explain how they came up with Joe's height.  We also noticed that our function table was similar to the In and Out boxes they have become so familiar with using Everyday Math.  I asked them what the rule was for this table.  Of course, they all said "add 2!"  Then, I encouraged them to tell me the rule for finding Joe's height based on Maria's if we use m for Maria.  This was a little tricker, but with a little prompting, they said "m+2!" Now we're talking!

Next, we wrote some sample problems in our notebooks and came up with expressions to describe the problems (EDM 10.3):

After some more practice independently, I gave the students an opportunity to write their own problem situations that would require the use of a variable.  I forgot to capture this in a photo, but they did a really great job!  Being able to write their own, and accurately, made me feel confident in their understanding of expressions.

Capacity: Interactive Math Notebooks

Capacity is always fun!  Unfortunately, when moving to our new building, all the containers I had saved for this day were tossed thinking it was trash!  Can you believe it? Someone thought our old milk cartons and jugs were trash!  Anyway, I'm lucky to have a mom who also teaches 5th grade math (you should hear our phone conversations....very animated!) and because she had taught capacity the week before, she was willing to lend me her collection.  Saved by mom (again!).  After a lot of spilled water and "do you predict this container will hold two or three of these?" we moved into our notebooks to record our thoughts.

First, we wrote down the Big G.  Why didn't we have this when I was a kid??  I never remembered what the Gallon Bot was and it was way too difficult to recreate, but the Big G is EASY!  It's truly how I remember my conversions.  Here is a picture:

Next, I gave them a moment to free write.  I really wanted them to write down everything they knew about capacity, so each student's page is very different, but many chose to write down conversions based on the Big G.  I walked around making sure that they understood how to read it.  Here is one example:

Of course, there's also an anchor chart hanging in our room of the Big G that we refer to often, but I didn't feel the need to post it here because it is the same as the notebook page :)

Measurement and conversions: Interactive Math Notebooks

Again, I'm behind on posting this! Where does the time go?  We did measurement back in March, and while we did a lot with it, I'm just going to outline what we did with our notebooks in this post.  We began by discussing the two systems of measurement.  We decided the best way to organize this information was in a T chart (we REALLY like those around here!) and we wanted to keep all the types of measurement (length, capacity, weight) together as well.  Here is the list we came up with:

Next, we brainstormed some of the basic conversions we remembered for length for each system and wrote them down.  Lastly, we took notes on an easy way to convert between units of measurement.   A BIG "thank you" to my coworker Adrienne, the 4th grade math teacher, for sharing her mnemonic device!  Some of my students remembered it from last year!  She taught us that there are two ways to change units: to multiply or divide.  When converting from a Big unit to a Small unit, you Multiply, so we think "Best Soccer Mom."  When converting from a Small unit to a Big unit, you Divide, so we think "Silly Babies Dancing."  We always giggle at "Silly Babies Dancing!"

Below is the anchor chart hanging in our room to serve as a reminder.  Another "thank you" to Adrienne!  I got this straight from her!  Thank goodness for kind coworkers willing to share and kick around ideas with!

Surface Area: Interactive Math Notebooks

We didn't spend a lot of time on Surface Area after all the time we spent on area, perimeter, and volume.  I really just wanted the kids to understand the concept of it, not necessarily how to find it.  We will get to that later in the year.  So, in order to demonstrate the idea of surface area, I wrapped a box in wrapping paper to show that we needed to know how much paper to use to cover the box.  I also gave each student a net for a rectangular prism and asked them to color just one side to represent wrapping paper.  Then out came the interactive math notebooks!  We creased each net and glued them into our notebooks so that when they flip to that page, they can easily fold it back up to look like a 3-D shape!  Here is an example:

Because we just finished up volume, we made a T chart in order to make comparisons between volume and surface area.  I told the kids to think about a tool box or toy box to tell the difference.  Here is what we came up with as a class:

Add caption

 Well, there ya have it!  A quick and easy day!  Thank goodness for math notebooks!

Measurement: Perimeter and Area

Teaching perimeter and area is one of my favorites because it can be so hands on (oh but just you wait until volume, surface area, and circumference!) Because our 4th grade teachers did such an excellent job in this area (pun completely intended!) I didn't feel the need to dwell on perimeter and area of rectangles for long.  Thanks to Pinterest, I found an easy, hands-on way to review area and perimeter of rectangles that wasn't just a worksheet to complete.  After watching a couple StudyJams videos, we went straight to work with some grid paper, two dice, and our pencils.  Students were to roll the dice and create an array to display the numbers on the dice.  Then, they were to find the area and perimeter of the array.  We discussed either counting the squares, or using the formula they used in 4th grade for area.  I worked with a couple struggling students.  I had them highlight the outside of the array in one color to find perimeter, then color the inside of the array with another color to find the area.  This really seemed to jog their memory!  Below is a picture of Aaliyah's arrays (although, I'll have to remind her that area is measured in square units....or squAREA!)

The next day, we moved on to finding the area of parallelograms and triangles.  This proved to be much more difficult, they concluded.  The triangles and parallelograms were drawn on centimeter grid paper, and at first, some students were trying to shade whole squares, then put together pieces to create more wholes.  When we shared answers and found that we all had something slightly different, we need we needed a better solution.  I asked them why finding the area of a rectangle was so easy for them, but these polygons were so much more difficult.  The conversation went something like this:

Student 1: Because with rectangles you just count the squares and that's the area!
Me: Ok.....so why are triangles and parallelograms so much harder?
Student 2: There are sometimes all these little pieces of squares that you have to put together.  Sometimes they are easy, like if it's just a half of a square.  But other times they are harder.
Me: What is it about rectangles that make it have just whole squares or half squares then? <silence in the room>
Student 3: It....has....right angles?
Me: And....?
Student 3: Right angles are squared off....so they are easier to count!
Me: So what if we made our parallelograms into rectangles so that they are easier to count?

We went on to change each polygon into rectangles and go from there.  I let them struggle with this method for homework: not all of the polygons were easy to change accurately.  On day 3, we went at it again.  Many students came in frustrated and unsure if their answers were correct.  And this is how we led into the discovery of the formula for finding the area of triangles and parallelograms!

We began class by looking at a scalene triangle and trying to find its area.  I had them do what they did before.....divide it into 2 rectangles and find the area of each, then add them together.  This time, however, they had to add up a bunch of half squares and quarter squares.  It took them forever and they were frustrated (yet again).  After all that, they knew they still had to divide it in half because it was only a triangle.  That's when they saw that they were really just finding the area of a rectangle and dividing it in half!  We tried this method, which was much simpler to multiply the length by the width, then divide in half, rather then count up all these pieces of squares.  I actually had one student's face light up as she yelled out "OhMyGosh! That is SO much easier!  Can we ALWAYS do it like that?!" To which I happily replied, "Yes, Ivy, because you just figured out the formula for finding the area of a triangle!"  They were so relieved, and shocked, that it was THAT easy.  We did a similar discovery lesson with parallelograms, cutting them in half and making them rectangles.

The last 15 minutes of class were spent updating our notebooks and reflecting.  We used the right side so show sample problems and made sure to highlight the base and height of each polygon in separate colors to help out our brains.  Then, on the left side, I allowed them to write whatever they wanted on area of parallelograms and triangles.  They were told they could use words, pictures, examples, formulas, or a combination to reflect on what they had learned.  I think they did a great job! They even asked for more time! Here are some examples on the work we've done in our notebooks for area and perimeter:

Each student received two random arrays.  They were asked to find the perimeter of one, use it to define perimeter, and think of examples.  Then, we did the same with area of the other.  Great visual! 

Here is the right side of Allison's notebook.  They used one color highlighter for base, and one for height to help them see that they come together to form a right angle.

Here are a couple of the left side thinking from my students.  Again, they were able to write their own notes on area.  Each are a little different, but reflect how each student prefers to receive their information.

And lastly, as usual, we always make an anchor chart after we have learned something.  Perhaps, if they glance at it enough, the information will stick!!

Area of Triangles and Parallelograms Anchor Chart

Fun with P.E.M.D.A.S!

Teaching the order of operations is fun because the kids really put together all they have learned about multiplying, dividing, and exponents into some really difficult looking math!  They really feel smart and successful!  I took to the Internet to find some really creative ways to teach order of operations and to get my students something to look forward to.  After first reviewing the use of parenthesis in expressions, I taught them all about my dear Aunt Sally.  We also took advice from Runde's Room and created hopscotch squares as a kinesthetic way of learning.  Unfortunately, we couldn't go outside, so we improvised and used painter's tape in our own classroom.  (They even practice at indoor recess!)  Check out our video!

After this, they really got into it.  They even came up with their own mnemonic devices for P.E.M.D.A.S.  Some of them were hilarious!  My personal favorite was: Please Excuse My Dad's Awful Smell!

Please Excuse My Dear (Students') Awful Spelling! ;)

And of course, we just had to update our Interactive Math Notebooks.  These are really turning into something special.  We added a foldable that showed what P.E.M.D.A.S. means and the symbols we can look for for each.  We also used the left side to duplicate our hopscotch squares (we can do finger hopscotch!) and to add our own thoughts about order of operations.  Some students chose to write a letter to their future selves explaining it, while others chose to write out the steps in their own words, and still others chose to show examples.  I'm really trying to embrace the left side thinking into my notebooks more.  Here's to another try!

Check out the I <3 Math add in next to her foldable!  YAY!

Interactive Notebooks: Data Analysis

It was nice to follow our unit on fractions with a unit on data analysis and graphing.  It gave ALL of our brains a nice little break!  Our 4th grade teachers really did an excellent job of teaching our kids all about bar graphs, line graphs, and data landmarks.  In Ohio in 5th grade, we move toward more complicated graphs such as double line graphs, double bar graphs, and circle graphs, as well as being able to look at a set of data and decide which graph would best show that data.  Going into this unit, I knew I didn't want to just show my students a bunch of graphs and have them analyze them.  I wanted to get them really involved and care about their data, which meant that we had a LOT of data to collect!  We began with a survey:

Each student got to pick their survey question and had to survey 20 people

Next, we created a frequency chart to organize the data we collected.  Because we just finished our unit on fractions, we were able to change our totals into fractions, then percents.  This allowed us to display our data in a bar graph as well as a circle graph.  To create the circle graph, we colored a strip of centimeter grid paper in the same fashion we colored our bar graph.  For example: in the picture below, Allison colored 12 squares purple because 12 people voted for math as their favorite subject (YAY!!!!), 6 squares pink for science, and so on.  We cut the strip out and taped the ends together to create a circle, and made our circle graph based on the outer edge of the circle.  That way, the regions in the circle graph were accurate.  It also gave us a great visual for how to create a circle graph.   The result is below:

 Next, we moved into double line graphs.  I had assigned one student to be in charge of taking the morning temperature and afternoon temperature everyday for a week.  Then, we organized our data into a chart. Next, I asked my students what they could tell from the data.  It was difficult for them to come to any definitive conclusions because it was difficult to visualize how the temperature changed from morning to afternoon and day to day.  So, of course, we plotted our data onto a line graph.  Once all the data was in a line graph,  the students were really able to see what happened to our temperature from day to day.  Some conclusions they made were:
"It is more likely to be warmer in the afternoon than the morning."
"The sun doesn't have time to warm up the earth at 8 a.m. in February, but by 3 p.m. it had a chance to warm everything up."
"Ohio weather is really unpredictable, Ms. McHugh!"

Welcome to Ohio, kids!  Allison's notebook is below:

 Next, we collected data on 4th graders who wrote with their left/right hand and 5th graders who wrote with their left/right hand so that we could create a double bar graph.  I left them in charge of coming up with a title, labels  and a scale.  They also dictated how they wanted their chart to look.  Each class had a slightly different title and slightly different charts, but their bar graphs were spot on!  We also decided that while a scale of 10 wasn't optimal, it was our best choice for the size of the grid paper.  The students concluded that the data would probably be pretty similar if we asked the 2nd and 3rd graders too, since the 4th and 5th were so similar.  Poor lefties! 

 We created foldables to show the definitions of our data landmarks.  I let them use a glossary but asked them to use their own words that would make sense to them and create an example.  They really know these well!  (Thanks 4th grade teachers!!!!!)   We will wrap up our unit on graphing with a project this week.  Thanks to Friday's snow day, we had to push it back.  Stay tuned!

Fun With Fractions

Fractions! It's always that time of year that requires the most focus and stamina out of my students.  I really get to see their true colors and get to see just how far I can push them.  We started with a pretty basic understanding of what fractions really are.  We busted out the fraction squares and explored how they worked and what they showed us.  This really helped, not only for my lower kids, but for my higher kids too.  My high kids love to be able to show me they know the numbers, but I also like to see that they really understand what's happening, and what better way than through the use of models?

Allison was kind enough to let me borrow her notebook.  These fraction bars gave us a great visual to start our unit on fractions!

 We really utilized our interactive math notebooks during our fraction unit.  Because this is my first year with notebooks, I haven't done a stellar job implementing the traditional left page/right page setup of notebooks.  That's something I plan to tweak for next year.  You'll notice that in the pics :) Anyway, back to fractions.  We began by using our fraction squares to find equivalent fractions.  In fact, we used fraction squares for just about everything before moving into the numbers of the problems.  Once we had the models down, we talked about what was actually happening with the fractions.  The visuals really helped when making that transition!  Below are some of the pages we entered into our notebooks on simplifying fractions and finding equivalent fractions.  

Step 3 is KEY!  If they can't prove their work with a model, then they don't really 'get it'!  We also learned how to use prime factorization as a way to find the GCF.

I did ask this student to go back and change what she was dividing by to look like a fraction rather than a whole number. !

This is one of my favorite foldables.  It took us a while but it was worth it!

 The fraction squares were perfect for teaching how to add and subtract fractions with unlike denominators.  It really was an 'ah-ha' moment for them when they saw that they could use other fraction pieces and replace them so that the result were squares of all the same size and color.  

Here is our foldable on adding and subtracting fractions with like and unlike denominators.  

Here are some of the other things we did with fractions.  Phew....it was a long unit!

This page was totally independent.  Up until this point, I had written up notes for them.  Starting now, they are coming up with their own notes and steps!  I think they did a pretty good job on their first try!

Here is a sample problem adapted from a former OAA test question.  I wanted to see if they knew to change the fractions to percents and the percents to fractions.  Most got it!

Fractions -->Decimals --> Percents foldable