Monday Math Literature Volume 12

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The lesson I did on equality this group with my first and second graders got me thinking about how kids perceive the word equal.  Much of their knowledge comes from what they get exposed to in school and it made me realize how important it is to give kids opportunities to be exposed to equations that look different and to have kids get a chance to decide if equations are true or false.

The book I pulled out as a follow up for these students is a great story by Virgina Kroll who is an amazing author who has over 50 published books.  Now that I know the name, I keep seeing books that I own that she wrote!  Quite a collection.

Equal Shmequal

This story has the sweetest illustrations and a story line that really supports students' developing understanding of equality.  Mouse wants to play tug of war but she wants the teams to be equal.  They can't seem to come to a solution until Mouse starts using math to help her.  The story ends with a look at what equal means for math, art, the law and sports.  

I first started thinking deeply about teaching equality when I read Young Mathematicians at Work: Constructing Algebra.  That book made me really think about how intentional we need to be when we teach equality.

Want to read more about how I use math literature in the classroom?  Click here to start at the beginning of the series!

Click here to head to volume 13!

Estimating and Counting Routines Part 3

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Click HERE to start at part 1 or HERE to check out part 2!

Today's estimating and counting routine was done with students who are about 1 month into grade 3.  This is also an excellent routine do with kids from K up to grade 4.  The strategies and efficiency will vary greatly among the grades but there is something different you can bring to each level.

This routine was not done as a whole group lesson but rather in small groups of 2-3 students.  I gave each students a scoop of dominoes and asked them to spread them out on the table in front of them with the dot side up.  I gave them about 15 seconds to look at the dominoes and come up with an estimate about how many dots they think are there.  Then one person in the group covered the dominoes with a student sized white board and each person wrote down their estimate.

The dominoes awaiting estimation

A group of students record their estimates

Once students were done estimating, their job was to figure out how many dots there actually were.  I did not provide parameters about how this had to be done or make suggestions about ways to do it efficiently.  It was late on a Friday afternoon however, and they knew if they had any time left before end of the day pick up, they got to play with the dominoes so they were super motivated to do this task efficiently.

Here is a look at some of their strategies.

This team was grouping by 10's but had several dominoes that did not fit into their schema.  The black and blue one in the upper left corner got pushed together to make a group of 20.  The two black and one yellow one above those they figured out by imagining breaking apart the dots on the center domino and giving one to each of the top and bottom of the domino on the left and 2 to the yellow domino.  then they had 30 and 5 left over.  From 35 they simply counted on by tens touching the group of 20 twice. 

This group had some simple combinations (silly teacher forgot to put some double 9 dominoes into this cup!)  This activity done this way without the double 9's would be great in K or grade 1 as a way to practice combinations of ten and counting by tens.  You could even contrive exactly which dominoes go into a cup and really push on the idea of 10.

This group started by putting some dominoes together to make 10's but had more that could not make a friendly ten and they did not think about going farther and trying to make 20 or imagine moving dots to make tens.  They pulled out the 100 bead strings to help them when they got stuck adding unfriendly numbers. 

Here is a look at a group of 2 kids estimates and the real answer.  As you can see, kids still need more practice with estimating.  My next question as usual when they finish is how far were you off by?  This often leads to a lot of discussion about adding up versus subtracting.  On this particular day we were running behind so this part got a little shortchanged.

I closed this session by having each team share their strategy for figuring out how many dots and telling how well their strategy worked and if they thought it was an efficient way to do it.  I am always amazed at how perceptive kids are about their own struggles and successes.  

Want to make sure you see part 4?  Check the upper right hand corner of this page for ways to follow my blog so you will always stay informed!  

What is your favorite math manipulative to use for estimating?  Respond in the comments below!

Click here to head to part 4!

Big ideas about equality for little kids

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This week I was working with a small group of first and second graders who are having a hard time with this grade 1 Common Core Standard around equality.

CCSS.Math.Content.1.OA.D.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

They are also having a difficult time with addition and subtraction facts so that is what I decided to tackle first.  We were working on fluency of addition facts using a dice game and our 10 bead sticks when the opportunity to work on this idea about equality presented itself.

A pair of students had just rolled a 9 and an 8 and were talking with each other about how they found their answer using the 10 bead sticks to help explain their thinking.

The student whose sticks are in the top of this picture said, "I put 9 on one stick and 8 on the other but then I took one from the 8 stick and gave it to the 9 stick and made it 10 and 7."  I asked this student if he thought 10 and 7 was the same as 8 and 9 and he said yes.

His partner then said that she showed 9 on one stick and 8 on the other and said he saw 8+8+1.  I asked him if he thought that was the same as 9+8 and he said yes.

Here is where I stopped the whole group and had these two students share what they had noticed.  I recorded their ideas as equations on the board.

We talked about one equation at a time and kids shared ideas about whether or not they thought the equation was true and how they could prove it.  There were some kids who were not convinced that these equations were true, so I pulled out my math balance and the kids who thought the equations were true used it to convince the other kids that they were right.

The left side is showing 9 + 8 and the right side is showing 8 + 8 +1 the  bar is parallel to the table which means the equations are equal

After discussing the equations 9 + 8 = 8 + 8 + 1 and 9 + 8 = 10 + 7 a student pointed out that we could write 17 = 9 + 8 so we added that to our list of true equations.

The students got back to playing their game and several times I had groups notice something similar about equivalent expressions and add their equations to the board.

Next time I take this group, we will continue talking about the ideas of equality as well as addition and subtraction facts with my Frog and Flower Equality game.

How do you make sure your students understand equality?

I chose the math literature book Equal Shmequal to follow up this lesson.  Click here to read more about it!

You might also be interested in
Another lesson on equality
Penguin themed equality freebie
Frog and flower equality

Multi-digit Addition and Subtraction and the Common Core part 1

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One of the things I am finding teachers are struggling with as we transition to the common core is how and when to teach multi-digit addition and subtraction.


Here is a look at grade 1

Use place value understanding and properties of operations to add and subtract.

• CCSS.Math.Content.1.NBT.C.4 Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.
• CCSS.Math.Content.1.NBT.C.5 Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.
• CCSS.Math.Content.1.NBT.C.6 Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Grade 2

Use place value understanding and properties of operations to add and subtract.

• CCSS.Math.Content.2.NBT.B.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
• CCSS.Math.Content.2.NBT.B.6 Add up to four two-digit numbers using strategies based on place value and properties of operations.
• CCSS.Math.Content.2.NBT.B.7 Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
• CCSS.Math.Content.2.NBT.B.8 Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.
• CCSS.Math.Content.2.NBT.B.9 Explain why addition and subtraction strategies work, using place value and the properties of operations.^{1 *explainations my be supported by drawings or objects}

Grade 3

Use place value understanding and properties of operations to perform multi-digit arithmetic.¹

• CCSS.Math.Content.3.NBT.A.1 Use place value understanding to round whole numbers to the nearest 10 or 100.
• CCSS.Math.Content.3.NBT.A.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

Grade 4

Use place value understanding and properties of operations to perform multi-digit arithmetic.

• CCSS.Math.Content.4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.

WOW!!!! There are a lot of expectations about how kids are supposed to use multi-digit computation in addition and subtraction.  Notice the STANDARD ALGORITHM is not used in the standards until GRADE 4!!!!!

So what does all this other stuff mean?  

I think it means we need to help kids learn to think about math rather than just doing it.  I have shared some ideas around this on my blog before but I will be taking a closer look at this important idea and providing a lot of examples of how I am attacking this issue in grades 1-4.  

Today we are going to take a peak at some strategies being used by students a few weeks into third grade.  This was done as a warm up to another lesson and took about 10 minutes.  I started kids off with a problem string culminating in the problem I wanted to discuss in further detail.  Here is what the string looked like

195 - 10 =
195 - 30 =
195 - 6 =
195 - 9 =
195 - 68 =

I pose the problems in the string one at a time and give kids a few seconds of quiet think time.  When they are ready they give me a signal and when most if not all kids are ready I ask for an answer and how they did it.  We obviously spend more time discussing different strategies as the problems increase in difficulty.  By the time we got to my culminating problem, I had kids turn and talk to a neighbor when all students were ready.  While kids are talking with a neighbor, I get a chance to listen in on different strategies.  I then know what kids I want to call on during our full group discussion.  When I call on a student, I often have them explain their partner's strategy to me.  This does wonders for student engagement and makes it so that they actually listen when their partner is talking.  As we have the whole group discussion, a student or myself will record the speaker's thinking on the board.  Here is a look at the three most common strategies used by my students.  

The most popular strategy and the one most apt to be suggested by the problem string  was to subtract the tens and then the ones in 2 steps.  Some kids provide further detail into their thinking and describe decomposing the 8 into 5 and 3 more and talking about how they subtracted 60 and than 5 and than 8.

The students who used this strategy decomposed 195 into 190 +5.  They then put the 5 to the side for a minute and subtracted 60 to get 130.  Then they put the 5 back before subtracting the 8.  This strategy led to a great discussion about when to add a number back versus when we need to take more away.

This student thought about adding up.  Because of some hard work on their part last year, these kids are fluent with combinations of 100 and were easily able to follow this students' thinking.  He started at 68 and added 32 to get up to 100 and then added 95 more to get to 195.  Then he added 95 and 32 by adding the tens and then the ones.  

Be sure to follow my blog so you can see part 2!!!  Coming sometime next week:)

How would your students solve these problems?

Want to read more?
Here is a post about fourth graders subtracting money amounts
Here is a post about multi-digit subtraction in second grade
Here is a post about extending fact strategies into multi-digit subtraction in grades 2 and 3

Fall Theme Decimal Task Cards

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Over the last week, I have been working to create a set of task cards to use with my fifth and sixth graders around decimals.  I decided on a fall theme and we were able to get outside and use them for track math!  We are really getting spoiled with all the nice weather and I have to take advantage of it while it lasts.

Here are my fall themed decimal task cards: organized and ready to use!

I created this set as a review of some fourth grade Common Core standards and to really target fifth and sixth grade standards, I added in operations and numbers to the thousandths place.  There are 20 task cards and some blank ones that I also include so that kids who finish early can write their own problems for other early finishers.

Here is a peak at one of my cards

Can you tell how much I love fall?

My favorite part of this experience is looking over the follow up questions.  I use these as a formative assessment to see who is getting it and what types of strategies they are using.  Here is a peak at 3 different answers to one of my follow up questions

Based on this evidence, I can see who needs more support tomorrow and which successful strategies I want to highlight.  

If you are interested in these task cards you can click here to get the free preview or click here to purchase the full set.

Math Intervention Books

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Browsing through the bookstore this summer, I happened to see 2 books that I had never seen before.  The words math interventions, differentiation, games and formative assessment caught my eye.  Of course when I got home, I had to look them up on amazon and find out more about them.  You know how this can lead to buying more books, and sure enough, I decided I needed to buy them both.

Here is one for preK-2


Here is one for grades 3-5

So now that I have had them and used them for a few months, here is what I have discovered.

PeK-2 book

    This book defines 4 goals for math understanding. 

    Accuracy

    Efficiency

    Flexibility

    Fluency

These goals align with what I try to do so I was excited to jump farther into these books.  This book is also almost completely game based which is something I love when doing intervention.  Terrence are games designed to work on counting, cardinality, subsidizing, place value, basic fractions, composing and decomposing numbers and addition and subtraction facts.  All of these things support the big ideas of mathematics at the preK-2 level. Throughout the book there are also problem strings and other things to use as formative assessment to see where your students are and how much they are learning from your interventions.  

Overall, I think this is a great book for someone who teaches at preK-2 level who wants to differentiate instruction for their students or help their students catch up on important math concepts.  For special educators or para professionals who work with kids struggling in math this book would be a step in the right direction towards the type and quality of intervention kids need.  Even the veteran math specialist can learn a new trick or two from this book. 

3-5 book

     The 3-5 book picks up with some of the same games and activities around addition and subtraction facts that the preK-2 book leaves off with.  Even though this is a double of what I had in the last book, I think the placement in the 3-5 book is absolutely appropriate.  I often begin the year with third grade intervention around these very concepts.  It is important that kids have a solid foundation of additive reasoning before moving onto multiplicative reasoning.  This books also talks a great deal about efficiency of strategies which is paramount to my own math teaching.  

     In addition to the addition and subtraction fact activities, this book includes ideas and games around multiplication and division facts as well as multiplying and dividing with larger numbers, fractions, decimals, number theory and place value.  It is a nice sampling of activities that can differentiate learning and help kids catch up in math.

As with the k-2 book, I can see this books being beneficial for classroom teachers, special educators, para professionals and math specialists.  There is a great philosophy behind this book and it is packed with useful and engaging lessons.

Way resources do you use for kids who need intervention in math?

Monday Math Literature Volume 11

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New to my Monday Math Literature posts?  Click here to start at the beginning 

This week's Math Literature Monday is all about BIG NUMBERS!  I use these books with kids from K-6 because I find kids of all ages who are very interested in learning more about big numbers.  In the common core, first graders are responsible for numbers up to 120 and second graders are supposed to be comfortable up to 1000.  Older kids are responsible for more numbers.  That is what the common core says.  What I know is that all kids are interested in learning big numbers.  These books can both help motivate your students to learn more about numbers and help to build a lifetime enjoyment of mathematics.

On Beyond a Million: An Amazing Math Journey

This book is written in comic book format.  In the story, a popcorn machine has gone crazy and a group of kids are trying to count all the kernels and can't keep up.  They are introduced to the idea of power counting, 1, 10, 100, 1000 etc.  This book in is in a fun format and really draws kids in.  It also gets to numbers past one million and probably has more numbers than you know the names of.  It is super engaging and I most often use this book in grades 2 and up

Another story about bigger numbers (although not as big!)  that is great to use with kids from K up is

Count To A Million

   

     I think this book is currently out of print but I found several nice used copies on Amazon.  This book shows what each base 10 number would look like in terms of bugs or people or grains of sand.  It is great at giving kids a frame of reference for just how large these numbers are.  I find myself going to this book over and over again when I am trying to get kids to see how big these numbers are and how much bigger (or smaller!) they are than other numbers. 

Do you have a favorite story that you use for talking with students about really big numbers?

Click here to head to Volume 12!

How to Make 100 Bead Strings

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I have had many requests lately to talk more about 100 bead strings so here we go:

Here are a few shots of me using my 100 bead strings.

Using the 100 bead string as a fraction and decimal number line

Finding pairs of 100

Playing a game with pairs of 100

Playing 100 Take away with Base 10 cards

  For someone who has only been blogging for a few months... you can see that these get used over and over again.  The pictures above were taken in grades 2-5 so there is a wide range of times where these are used.  I also use them in K and 1 for counting and grouping activities and thinking about adding and subtracting multi-digit numbers.  

Materials Needed (To make 20 Bead Strings)
All prices given are for what I paid on Amazon 

Red Pony Beads (1000 beads per package)  $5.14

White Pony Beads (1000 beads per package) $6.13
Colored Cords (42 yards) $5.10.  You will only use about half of this and have the rest left over for another project or to share with another teacher!

Grand total $16.37 which is about 82 cents per bead string.  The shipping was free when I spent $25 which meant I ended up ordering a few other things while I was on Amazon.


How to make
- Cut colored cord into sections about 1 yard long.  Fold over a few inches and tie off one end.  I like to leave a loop on the end so that I can hang them up when I am not using them.

- Once the ends are tied off, I like to have pairs of students make these.  Working together, they alternate 10 reds with 10 white beads until they get to 100.  Once there, I have them re-count to check and/or I have them put their bead string up against one I already have made to see if their sections match up with mine.

- Once we have checked the bead string, it is time to tie off the other end.  Older students can sometimes do this themselves but I often end up tying off all of them myself in younger grades.

Now your bead strings are ready to use!  The possibilities are endless:)

How do you or would you use a 100 bead string in your classroom?

Beautiful fall day? Time for Track Math!

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  Yesterday was one of those days where the weather was perfect and the kids were wiggly.  With the count down to winter getting closer and closer, I have to take advantage of days like this and get outside.  Only I am supposed to be teaching and the kids are supposed to be learning.  Enter track math.  A fun and engaging way to practice a variety of skills.  I have used this idea in grades 1-6 and it has always been very successful.

The sun seemed to be screaming, "come out and play!"

We have a gravel track around one of the soccer fields and it is about an eight of a mile around.  I take 4-6 clipboards and attach some task cards to each one.  I place them on the ground spaced out around the track.  Kids each have their own clipboard with a record sheet and a pencil and simply move around the track answering one task card at each clipboard.  They get the power of choice, some extra movement and a break from being stuck inside.

A look at one students' record sheet

Classroom management is almost unneeded when we do math on the track.  The kids are super engaged and spread out and get quick mini breaks while they are walking (or running) between stations.

A student working on a task card

I like to keep my task cards organized and ready to go so that when a day like this arrives, I don't have to do any prep to get the kids outside and moving.  Check out this post about how I organize task cards!

This is how I space out the clipboards when I choose to use 6.  If I am only using 4 clipboards, I eliminate the spots where you see the x's on the long sides.

These kids needed some work on addition and subtraction number stories and since we are heading into fall, I used my fall themed addition and subtraction number story task cards.

A little visitor showed up on one of the cards!

How do you get outside on these perfect weather days?