## Friday, March 29, 2013

### 100 Bead String Decimal Number Line

I have been working hard with my fourth graders on decimals and decimal fractions.  It was time to work more on putting decimals on the number line.  I have been using 100 bead rekenreks with my first and second graders and I decided to grab one to use as a decimal number line.

As you can see, we found 0.5, 0.1, and 0.01 on the number line.  I just took the 100 bead string and clipped it to the white board (I am lucky enough to have a magnetic white board in this room!)

The kids were able to find a lot of equivalent decimal and fraction names as well.  Then someone started pointing out that you can multiply the numerator and denominator of a fraction by 10 and come out with an equivalent fraction.  Here are some of their examples

I think this could lead nicely to the class constructing knowledge about how to find equivalent fractions.  It is something we can build off of in the coming weeks.

My next idea is to use a meter stick as a decimal number line.  Great way to connect metric measurement into decimals and fractions!

I will be posting more next week about how I made the 100 bead strings and how I use them in First and Second grades!  Stay tuned!

## Order of Operations

I have been working on order of operations with both the fifth and sixth graders at my school.  This is something I see students (AND ADULTS!!!) struggle with every year.

Quick Quiz

1 + 1 + 1 + 1 + 1 + 1 x 0 + 1 = ???????

Than you need to work on your order of operations!
The answer you should have gotten was 6!
Order of operations puts multiplication before addition.

A lot of students/teachers use the acronym PEMDAS
P = parenthesis
E = exponents
M = multiplication
D = division
S = subtraction

PEMDAS can cause its own problems
Here is another one that gets people

Quick Quiz 2

48 / 6 x 8 = ?????

Did you get 1?

You made a very common mistake for those who love PEMDAS.  The problem with PEMDAS is that it makes it seem like multiplication comes before division because it comes first in the acronym.  But REALLY with order of operations, multiplication and division are on the same level.  They should be done simultaneously from left to right.  So on Quick Quiz 2 you should have done 48 / 6 = 8 and then done 8 x 8 = 64.  Much different answer!

Try one more.  I am sure I won't be able to get you again!

Quick Quiz 3

25 - 5 + 80 =

If you got - 60, you fell for the PEMDAS trick again.  Addition and subtraction are on the same level for order of operations just like multiplication and division.  You need to do them simultaneously from left to right.  So 25-5 = 20 and then you add 80.  If you got 100, good for you!

How do you help students with order of operations???

## Monday, March 25, 2013

### Frogs and Flowers Addition Fact Doubles FREEBIE!!!!

 Available as a freebie in my TPT store!!!!

This is a great game that is easy to make and play.  It helps kids practice their double facts!  Included are two versions of the game to help you differentiate instruction to meet the needs of all your students.  You can play the game with dice (1-6 die for version 1 and 5-10 die for version 2)  If you don't have the right dice, don't worry!  I also included spinners that you can make with a metal brad and a paperclip.  There is also a quick practice worksheet or little assessment included to check in with your students on their double facts!

 Head over to TPT to grab this freebie!

Hope you enjoy your freebie!  Ready to see more frog and flower products (and a few more freebies!)?  Check out Frog and Flower Equality, Frog and Flower 10 Frames, and I Have Who Has with a frog and flower theme!

## Not Teaching Second graders "How to Subtract"

I have been working hard on a lot of place value up to 1000 concepts with my second graders as well as two digit addition and subtraction.  To ties some of this work together, I wanted to investigate our unifix cubes.

When we purchased the unifix cubes, there were 1000.  I told my students this and dumped them out. I wanted to know how many I still had (it has been at least 7 years!)  I really didn't know how many were there.
We started by trying to estimate how many cubes there were.  This was REALLY hard!  It helped a lot when a student suggested that we take out 100 as a benchmark.  The estimates ranged from 500-1001. The kids organized them into 100's 10's and 1's and we figured out that there were 658 cubes.  We wrote the number in expanded form and talked about how that matched the picture and what it meant.  Then......

A student said, "Wow! You are missing a lot of cubes!"

and another replied "I think it is over 400!"

and then they all wanted to figure it out.  YESSSSS!!!!!

I gave them a few quiet minutes to figure out how many they thought were missing.  They had no pencil/paper etc during this part.  After everyone was giving me the ready signal, I gave them white boards and had them record what they thought the answer was and how they knew they were right.  WOW was I impressed!  It is amazing how many strategies they had and how many correct answers I saw.  And this is all before anyone has "taught" them how to do 1000-658!

Let me show you some of the strategies!

 This student added the 100's, 10's and 1's separately and then knew 900+90+10=1000

 This student started by adding 2 to 658 to get to a friendly 10 (660) and then added 40 more to get up to 700 and then added 300 to get up to 1000.  This reminds me of how we "Teach" kids to use the number line to add up at our school!

 This student used sketching of base 10 pieces along with horizontally written number sentences
 This student (and several others) got the answer wrong.   They added up to 10, 100 and 1000 but forgot to take into consideration the tens and hundreds that get traded up.  They looked at 658 and wanted to add 400 + 50 +2.  Their final answer is 110 off and you can see why!

This lesson really opened my eyes to the importance of letting students create their own strategies and letting them share those strategies with neighbors and the class.  This lets everyone learn and compare and contrast strategies.  I think even I learned a thing or two!

## Measuring to fractions of an inch and making line plots

It was a busy week!  I finally got to tackle one of the new common core standards for grade 4 that I have been waiting to sink my teeth into!

• CCSS.Math.Content.4.MD.B.4 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.
That's it.  That is the represent and interpret data standard for grade 4.  Big change for me.  I used to do an entire unit on representing and interpreting data.  I love that this standard is so focused.  I love that it connects directly to fractions.  In fact it really is a fraction number line!!!!

So... here is how I addressed this standard with my fourth graders

We measured pencils to the nearest quarter inch (I know the standard wants 1/8ths but we were not there yet) I divided the class into 4 groups, gave each group a pile of pencils a bunch of rulers and some scrap paper.  They were told to measure their pencils to the nearest quarter inch and record their results.  Here is one groups results

Several groups got really into making sure they were putting the measurements in numerical order and ended up arranging their pencils from smallest to largest before beginning.

I drew a line plot on the white board and as the groups finished, they came to the board and plotted their results
Here is what it looked like when it was done!  Each group decided to use a different color.

Here is where the fun began!
We gathered around and I started asking questions, focusing on interpreting line plots, fraction equality, scale, and fraction addition and subtraction.  Here is a sampling of some of the questions

Why did I chose to start and end the number line where I did?
What do the small marks between the numbers mean?
What is the most common pencil length?
How many pencils did we measure?
How much longer is the longest pencil than the shortest pencil?
If I put the three shortest pencils end to end, how long would that be?

This led to a GREAT discussion.  I can see lots of applications to this in science class.  We MIGHT do this again and measure something to the nearest 1/8 of an inch but we will probably try to tag team this lesson with science class and make a line plot of measurements and data that are important for scientific purposes.  We shall see

Two more pictures

How do you plan on addressing this new data standard?

To read more about how I took this lesson to the next level, click here to check out my guest post on Minds in Bloom about taking this standard to the nearest eighth of an inch.

## Friday, March 15, 2013

### Multiplication Fact strategies

In grade 3 over the last few weeks, we have been working on developing conceptual understanding of multiplication and division with small numbers.  Now we are at the point where we want to work toward a level of fluency with computing multiplication facts.  We do this by spending a lot of time sharing different strategies.

We gather all the kids together and make sure they are sitting next to someone who will be a good neighbor.  We put a fact on the board and ask kids to think to themselves and give us a signal when they are ready.  (we like to do thumbs up or thumbs up and bring to lips)  When most (or all) kids are ready, we have them turn and talk to their neighbor.  We have them share their answer and how they know they are right (aka how they solved it)  We listen in to these conversations and then bring the entire group back together.

We call on a student to either share their strategy or their neighbors strategy.  Having students share their neighbors strategy is a great way to increase student engagement and have them experiences another way of thinking.  We then ask if anyone else did it that way.  Then we look for a student to share a different strategy.  We then look at how the strategies are the same or different.  By listening in on the conversations when they were sharing with a neighbor, we have a good idea who we will be calling on beforehand.  Sometimes we will share a third or a fourth strategy before moving on.

When we go onto another fact, we repeat the process.  After 2-3 facts, we ask kids to try a different strategy (or their neighbors strategy)

Today we worked on the x 8 facts.  Here are some of the kids strategies

#### I started with 8x8

Some kids knew the rhyme "8x8 fell on the floor, when I picked it up it was 64"

Many other kids did the Double, Double, Double strategy our math program teaches

Several kids said they knew it was two groups of 8 x 4 so they added 32 twice

#### The next fact I put up was 8x7

Right away, many kids were ready.  The majority of kids used the fact that we had just done 8 groups of 8 and 7 groups of 8 would just be 8 less.

Some kids did the double, double, double strategy

Another young man said he knew 5 groups of 8 was 40 and he needed 2 more groups of 8 which was 16 and 40+16=56

I was pleasantly surprised by all the different strategies   Now I have to move these kids towards computational fluency.  I think talking about strategies over and over is a great way to expand a students' thinking and help them develop conceptual understanding and fluency simultaneously.  How do you make sure your students learn multiplication facts?

## Common Core, Place Value and Numbers under 1000

This year I am working with a challenging group of second graders.  Their range of abilities in this one group is incredible.   There are several many students who need a lot of support in terms of visual modeling and other support when dealing with numbers.  The Common Core for second grade says that these kids need to be pretty proficient with a lot of skills with numbers up to 1000.

My second graders have been struggling with this idea, so I created a place value deck for them.  The goal of the place value deck is to support them in some of these skills using a visual model.  I purchased some great clip art from Teachers in Love

The deck I created consists of 54 cards showing numbers ranging from 100 to 1000.  This is what the cards look like
 These are currently my BEST SELLING item in my Tpt store.  Teachers love them and they are getting great reviews:)  Check them out!

I copied the deck on to card stock and chopped it apart.   I think I will end up using this deck in so many ways.  Here is what I have done with it so far:

### Compare Game using Greater Than and Less Than Symbols

The common core for second grade expects that students will use the greater than and less than symbols.  I created this simple (and fun!) game board for kids to use to show inequalities.  The rectangular strips are attached with brads and can move back and forth between a greater than, less than, and equal to sign.
The kids love it.  I walk around and have them read the inequalities to me.  I sometimes have students use whiteboards to record their inequalities after each turn.   This provides a record that lets me check for understanding.  It is formative assessment without feeling like an assessment.

### Ordering Numbers Under 1000

My students need a lot more practice with this!  For the students who struggle the most, when I gave them just numerals, they would focus on the hundreds or the tens or the ones and not consider all parts of the numbers.  When I first began with this place value deck, I would have students just order the picture cards.  Now that they have had more experience  I am having them write the corresponding numeral underneath.  Eventually, I will have them order just the numerals and then they can check with or fall back on the picture cards.

### 10 More/10 less AND 100 More/100 Less

The common core has a heavy emphasis on 10 more and 10 less for second graders.  Beyond that, it is really helpful when you move into trying to develop conceptual understandings of 2 digit addition and subtraction if subtracting 10 and adding 10 is second nature to kids.

I have them create a record sheet on a whiteboard or a piece of scrap paper and they pull a card and record 10 less and 10 more.  The pictures REALLY support kids who need it.  When they have filled in their whiteboard with 10 less/10 more, I have them erase and change it to 100 less/100 more and do the same thing.  While I am sitting here writing about this activity, it doesn't seem like it would be very fun, but my students LOVE it.  I don't know if it is the white boards, or making their own record sheet or what but they think this is super fun and have been asking when we can do it again.

I am getting so much mileage out of this deck already!  I am working on creating a similar deck for my first graders focusing on the numbers 0-120.

What uses can you find for this deck?  Want to grab a copy?  Head over to TPT and grab the deck and activity list.  They may even be on sale ;)  You WILL be glad you did!

Looking for a deck of cards for younger kids?  Check out this post about my numbers to 120 place value deck!

Here is how I use this deck in a place value intervention and to work on expanded notation.

## Monday, March 4, 2013

So today in grade 6, we jumped right into our integer unit.  In the past, the focus of our sixth grade integers unit has been on number line placement and on developing strategies for adding and subtracting integers.  I have always enjoyed teaching integers and have exposing kids to using manipulatives such as chips, virtual chips on the National Library of Virtual Manipulatives  and number line models.  So last Friday, as I was finishing up the school day, I pulled up the common core standards to see what they said about integers.  Here is what I found.

• CCSS.Math.Content.6.NS.C.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
• CCSS.Math.Content.6.NS.C.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
• CCSS.Math.Content.6.NS.C.6a Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.
• CCSS.Math.Content.6.NS.C.6b Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
• CCSS.Math.Content.6.NS.C.6c Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
• CCSS.Math.Content.6.NS.C.7 Understand ordering and absolute value of rational numbers.
• CCSS.Math.Content.6.NS.C.7a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
• CCSS.Math.Content.6.NS.C.7b Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 oC > –7 oC to express the fact that –3 oC is warmer than –7 oC.
• CCSS.Math.Content.6.NS.C.7c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.
• CCSS.Math.Content.6.NS.C.7d Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars

"WOW!  That is a lot!" was my first reaction.  My second was, "where is the adding and subtracting integers?"
I had to look again.  Nope.  Not there.

I opened seventh grade Common Core standards.......There is is!

So...... I don't have to teach integer addition and subtraction?!?!?Hmmmmm..... I guess my self designed unit needs a major overhaul.  I have to design new items to go with these new standards.  But I really want to share my stuff around integer addition and subtraction with seventh grade teachers so I will be posting it as a freebie in my TPT store.  Check it out.

For those of you looking for stuff on integers to help meet Common Core Standards for grade 6... Stay tuned.  I am working on developing materials and will let you know how it is going.

What has been the biggest instructional shift for you as you move towards the Common Core?