Individual student toolkits for CRA instruction

If students are in your classroom in August, it will be worth your time to have some sort of math tool kits prepared for your students because it is not recommended that students share resources, and you don't want to be cleaning them every night. What you are able to put in the kits will depend on your resources.  

We know that CRA instruction is vital for conceptual understanding of mathematical concepts.  Just because we will have some limits in place (for everyone's safety), doesn't mean we should abandon best practices.  We just need to adjust and do what we can for our students.

As I have said before, the use of virtual manipulatives will probably increase in your instruction even if you are seeing students in person.  This is especially true for those manipulatives where students usually need a lot of them to do their work--like pattern blocks and base ten blocks. I recommend that you use the virtual versions some in the classroom so that students are comfortable with the websites if they should need to use them from home without your direct support.

You might also consider paper versions of some of these tools to have in students's toolkits.  Not only can you get them easily and cheaply for the classroom, but you can also send them home if you need to without the worry of losing valuable manipulatives.

You may not start the year with all of the items in the toolkits.  You might add some as students are introduced to new concepts. Here are some things you might consider putting into individual toolkits for your students:

K:

five frame

ten frame

hundred grid from nrich.maths.org

number path  1-20 by Berkeley Everett

ten counters (unifix cubes?)

rekenrek

real or paper pattern blocks by Mathwire

+/- Math Flips cards (+/- 1) by Berkeley Everett
Quantity Cards from Erickson Early Math Collaborative

pair of dice

1:

ten frame

double ten-frame

hundred grid from nrich.maths.org

number path  1-20 by Berkeley Everett

twenty counters (unifix cubes ?)

rekenrek

beaded number line

small clock face paper version

real or paper pattern blocks by Mathwire

+/- Math Flips cards (Count on within 10)  (+/- 1 or 10 within 100) (subtraction within 10)

pair of dice

2:

double ten-frame

hundred grid from nrich.maths.org

number line 1-20 by Math Salamanders

twenty counters (cm cubes?)

rekenrek

beaded number line

small clock face   paper version

play coins, dollar bills

ruler

real or paper pattern blocks by Mathwire

real or paper base ten blocks from Tim Van de Vall

+/- Math Flips cards by Berkeley Everett (Doubles/Near Doubles) (make 10 with 2 addends)                        (make 10 with 3 addends) (2 digit + 1 digit) (2 digit + 2 digit) (subtraction within 20) (subtraction within 100)                                                                       

pair of dice

3:

ten frame

hundred grid 

twenty or more counters (cm cubes?)

beaded number line

small clock face   paper version

play coins, dollar bills

ruler

10 x 10 array (laminate)

real or paper pattern blocks by Mathwire

real or paper base ten blocks from Tim Van de Vall

paper Fraction Strip  from Math Salamander (blank version)

paper Fraction Circle Pieces Page 1   Page 2  Page 3

+/- Math Flips cards by Berkeley Everett (if needed from previous grades)

 x Math Flips cards by Berkeley Everett (2s,5s, and 10s) (4s, 6s,& 8s) (3s & 6s) (9s & 4s)

pair of dice

4:

hundred grid 

thirty-six counters (cm cubes?)

beaded number line

ruler

10 x 10 array (laminate)

real or paper pattern blocks by Mathwire

real or paper base ten blocks from Tim Van de Vall

paper Fraction Strip  from Math Salamander

paper Fraction Circle Pieces Page 1   Page 2  Page 3

+/- Math Flips cards by Berkeley Everett (if needed from previous grades)

 x Math Flips cards by Berkeley Everett

pair of dice

5:

hundred grid 

thirty-six counters (cm cubes?)

beaded number line

ruler

real or paper pattern blocks by Mathwire

real or paper base ten blocks from Tim Van de Vall

paper Fraction strip from Math Salamander

paper Fraction Circle Pieces Page 1   Page 2  Page 3

Math Flips cards by Berkeley Everett (if needed from previous grades)

pair of dice

If you laminate many of the paper tools, students could use a dry erase marker to create representations.

What am I forgetting?

Looking for Virtual Manipulatives?

While we are still unsure how our instruction will be delivered in the fall, there is one thing we are sure of....student's involvement in the concrete phase of learning math is important!

Sending home concrete manipulatives for all of our students may be too costly for most schools.  Purchasing enough to send home is not the only problem: we know it is doubtful that many of these tools will return to our buildings.  With that in mind, here I am going to highlight some of the best virtual manipulatives I have been able to find and use.

I highly recommend using these with students this fall even if we are face to face.  Allowing for guided use now will make it easier should they need to use them again at some point in the year.

I have always been a fan of Math Learning Center Apps, and I have shared these apps with many teachers before! These apps have always been free and easy to access and use.  Because of the pandemic's rise in at-home schooling, they have added an extra feature!  Sharing!  Four of the apps can be set and shared with students and/or students can share with you.  There is a video explaining how this works, and I have had success with it myself.  There are many great apps here.  Some of my favorites include the Number Rack, the Number Line, and the Partial Product Finder.  However, I can think of many great uses of the other apps.  I just have not used them myself in a classroom.

These are the apps that allow sharing at this time.

Toy Theater also has a great selection of virtual manipulatives. I do not have as much experience with these, but it has some great ones such as interactive hundreds and mulitiplication charts, a variety of counters, dice, and some graphing apps.  I think it could meet a lot of needs!

A great variety

Didax also has free online manipulatives. I have only found these while I've been sheltering at home, so I do not have experience in the classroom with them.  You'll find many that are available above, but here you will also find linking cubes and a math balance!  It also has algebra tiles and prime factor tiles that can be used in middle grades. (because manipulatives are just as important there!)

A few other places that you can find specialty links:

MathToybox has a Cuisenaire rods app.

ABCYa has some fraction tiles that are nice.

I hope you can take some time to explore these apps and find the ones that will work best for you and your students' needs!  It might be a good way to start the school year, whether we are virtual or in-person, to spend some time walking students through these tools, how to get to them, and how to use them.  That way, when the time arises, they will be able to use them more independently!  Remind them that they can use them and explore with them whenever they want.  Exploring these tools independently helps students to see relationships and make sense of it all on their own.

The other thing that is great about these virtual tools is that you can use them to create your visuals for your SMART docs, Google slides, or student tasks.  Math is visual, and these provide us with good ways to show the math to our students.

I hope we have the chance to use concrete manipulatives in August with our students, but now is the time to prepare in case we don't.  I hope you can find tools at some of these sites that match the ones your students typically use in the classroom.

Math is Visual

Our students need access to math visually as much as possible.   This can come in different ways.  Drawing, manipulatives and online components are some places to start.

I spend a lot of time in classrooms working around word problems.  One of the things I find myself constantly coming back to with students is "Think like a reader:" use schema, make predictions, infer, VISUALIZE...that is what good thinkers do.  Asking students to draw what they are seeing happening in the problem is an important step.  That drawing can be a true drawing, a number line, a tape diagram, or anything else that helps them make sense.  Are they thinking through the problem or are they just pulling the numbers and computing without making sense? 

I also use numberless word problems a lot. Numberless word problems give students the opportunity to see the problem as a story and to make sense of it without worrying about the numbers.  Once you have practiced this strategy in class, it is a strategy they can refer to independently when they come across a word problem that confuses them.

Manipulatives are important, too.  CRA instruction helps us to build the visual into the child's mind by using tools to represent the math.  This is important for all students and provides equity in instruction.

There are many websites with visual representations.  One of the best is Math Visuals.  Berkeley Everett has created a treasure trove of visuals for the K-5 classroom.  His short videos to accompany skills like counting and computing are a fabulous resource, but they are just the beginning.  He brings visual supports to understanding mathematical properties and making connections through visuals, too.  In addition, his Math Flips cards are well worth the time to cut out and use with all of your students.

Many other awesome websites exist, too, that are very visual.  Math is Visual, Fraction Talks, Same or Different, Number Talk Images, and Slow Reveal Graphs are just a few of the visual resources that may help you and your students make more sense of mathematical concepts.

Visual math is for all students.  We have to be careful not to assume they can see it in their heads.  We have to help them see it in their heads...Visuals help students to make sense.  They help students to look for patterns.  Visuals help students to answer the question, "What does the data tell us?"

In what ways can you provide your students with more visuals?  This is an important area for all of us to grow.

Sorts

Sorts are a great way for students to make sense of things around them.  It could be images or numbers or objects or words....the point is to get them thinking about a subject and have them make sense of it in some way.

Here are some ideas for sorts.  Don't think that these are just for Littles.  Older students enjoy them.  They offer a non-threatening way for students to look at things, and they offer us good insight into our students' thinking.  Sorting is very mathematical even when it is not about numbers.  It is about looking at things carefully, finding patterns, and making sense--that's math!

SORT:
pattern blocks
doors
coins
book characters
drain covers
shapes--2D and 3D
numbers (odd/even) (prime/composite) (square/not square) (multiples)
vehicles
words
patterns
emojis
food
volume/area/perimeter
shoes
states
types of graphs (no numbers or titles are necessary)
expressions
angles

Put Math in their Hands

I often talk to teachers about the importance of concrete representations for our students.  The use of concrete tools helps all students to gain a better understanding of the function at hand.  It also creates a visual representation that will be a reference for students to use as they advance into more advanced mathematical ideas.

In this post, I want to share some simple concrete representations that teachers can help students develop as they work to make sense of a new concept.

Doubles/Doubles +1: Both the rekenrek and ten frames help students to make more sense of what a double is and how we build from that to doubles + 1.

Doubles on ten-frame:  3 + 3 =6

Doubles + 1:  3 + 4 = (3 + 3) +1 =7

Doubles on rekenrek: 6 + 6 = 12

Doubles +1:  6 + 7 = (6 + 6) + 1 = 13

The rekenrek is a great tool for doubles facts between 10 and 20.  Student can see that all of the green beads have been pulled over, which is 10.  They only need to add on the white beads.


Addition/Subtraction within 100:  For this skill to take hold, I find beaded number lines and base ten blocks to be good concrete tools to use because they carry over so nicely into an open number line representation.  Open number lines are important for students to use to build number sense.

Addition on Beaded Number Line:  17 + 35 = 52

Addition with Base Ten:  17 + 35 = 52

Subtraction on Beaded Number Line:  44 - 18 = 26

Open number lines are important, but they are often too abstract for many of our students.  Combining the use of the beaded number line or base ten blocks with the creation of an open number line helps for it all to make more sense for the students.

Multiplication:  For learning multiplication facts, creating arrays is a nice way to start.  This should help students to see how repeated addition is connected to multiplication.  Any small object will work to help students create arrays.  Using the number card templates will allow them to create a visual more quickly at times when they need a visual but don't have time to build an array.

Multiplication with Base Ten Blocks:  22 x 23 = 506

The 22 and 23 were created at the top of this image and on the left side of the problem using base ten blocks.  Multiplying a blue ten and anothe blue ten gives you an orange 100.  The same continues until we end up with 4 hundreds, 10 tens, and 6 ones for a total of 506.

For multi-digit multiplication, building the array using base-ten blocks works well to connect the arrays that student made for facts under 100 with the longer problems that they are ready for now.  Besides--the base-ten blocks transition nicely into graph paper representations and finally into the area model of multiplication for multi-digit numbers.

Decimals:  The beaded number line can be used for addition and subtraction of decimals less than 1, and it can also be used to round and compare decimals.

Rounding on the beaded number line:  0.86 is closer to 0.9 than 0.8

Adding decimals on a beaded numer line:  0.6 + o.14 = 0.74

I hope that you find some of these ideas helpful and that you find ways to incorporate them into your explanation of numbers with your students.  There are many ways that can be used for different problems.  If you have a concrete tool that works well for teaching these skills, don't think you have to switch to one of these--do what works best for you and your students.  The key is to get the concrete math in their hands!  This way it will stick in their heads.  😊

Virtual Concrete Manipulatives

We know how important concrete manipulatives are for our students as they build a deeper understanding of mathematical concepts. Being able to see and manipulate objects enables students to visually represent problems, see patterns, and make connections.

Hopefully, you have a variety of manipulatives in your room that students can easily access at all times.  Virtual manipulatives are important to have easily accessible, too!  Now is a good time to download some of these apps onto your iPad, bookmark on your computer, links in your SMART Notebooks or PowerPoints and/or add to your Symbaloo!

There are a lot of great FREE virtual manipulatives out there, but today I want to focus on the apps by Math Learning Center.  I have good success with many of them. The site has a lot of other good resources, so when you have some time, check them out as well!

Fractions: This app not only allows you to create bar or circle models of fractions, but it also to layer fraction models to see if they how they compare to one when you add them.

3/5 + 1/3 < 1

The app allows you to place number models or write directly on the screen as well. Take some time to explore the capabilities of this app!

Vocabulary Cards: Another great app that you can probably find multiple uses for! There is a large database of words in this app divided into grade bands K-2 and 3-5.  You can choose to see them all or only certain words.  Each card has 3 parts: the word, examples, and a definition.  You can choose which part you want hidden.  You can also choose the language for the card. I can see these being used as individual review for students, but I can also envision one of these on the board as an intro activity or even a quick exit task.

Money pieces:   The money pieces app allows you to display money with or without the accompanying base ten blocks depending on student needs.  It also has a variety of tools similar to the games Bears in the Cave or Pennies in the Hand where you can put your coins up and then hide some.  For example.  I have a pocket in the above screen.  If I told you that I have 35 cents altogether, can you tell what I have in my pocket?

With the click of a button, I can remove the pocket to show that there is a quarter in it.  You can do this similarly with a hand or a bank.  

Money seems to be difficult for children anymore because they have less interaction with it than we did. Our students don't get the same opportunities we did to spend cash, but it is still important to understand, and this app will give them some basic experiences with it.

Number Frames:  This app not only allows the important 5 frame, 10 frame, and Double 10 frame, it allows you to customize the frame you want up to 120. As with many of the apps, it allows you to enter number sentences and to write on the screen.  This app is not just for Littles!  Teachers in 3-5 can use it to explore place value as well as patterns in multiplication.

Number Lines:  I love this app!  It has so many options for you to customize the number line including fractions, decimals, hidden numbers, hidden tick marks... I think it really helps students to represent their work, and it marries nicely with the beaded number line for moving from concrete to representational.

Number Pieces:  There are two versions of this app.  The one I am showing above is more advanced than the version called Number Pieces Basic. It is base-ten blocks, but you can break the large pieces apart to show number relationships. You have choices in color and orientation of the pieces.  Again, you can enter number sentences or write on the screen.

Number Rack:  You know how much I love this one!  It comes in handy as a teacher model on the SMARTBoard as students manipulate their own Rekenreks.  It is customizable by sets of 10 up to 100. It allows for teacher annotations like the others, but it also allows for teachers to hide beads as below.  Well worth your time to explore this one--not just for Littles!

Do you know how many beads I have hidden?

Pattern Shapes: It is so important that we give students time to play with pattern blocks!  This offers a blank mat for students to create their own patterns.  It also has templates (as above) for students to fill in with shapes.  (Very good for sharpening visual skills)  For older students, it has two different grid backgrounds to allow exploration of area and perimeter.

Geoboard:  The geoboard app has different sizes of geoboards and allows for customization in many ways.  Another great one to have up on the SMARTBoard as students manipulate their own geoboards.

Partial Product Finder: While still in Beta form, this is an awesome app to help your students better understand partial products as well as the distributive property.  You can customize the rectangle up to 30 x 30 and then decompose one or both sides.  The matching equation shows up at the bottom of the screen. I have blogged about this one before--it is awesome!

I highly recommend that you take some time to explore these!  Hopefully you will find some that you make available for student use just as you do other manipulatives.  Maybe you'll find ways to use them within your instruction.  Whatever works best for you and your students!  

Let me know if you have other virtual manipulative apps that you would recommend!

Using Counting Collections in the Classroom

Have you tried counting collections? 



Counting Collections: Kindergarten - a common core classroom friendly exercise from Luna Productions on Vimeo

This activity is a great opportunity for our primary students to gain a better understanding of counting and number, and with some modifications, I think it could be used, at some level, in the upper elementary classroom as well.

You will need to begin by creating some collections.  In the link above, they give some examples of objects you can gather for counting, but I'm sure you can find other items around your home or classroom that will work as well.  You will probably want them to be smaller in size so that storing them doesn't become much of an issue.  Hopefully, you can find other teachers in your classroom to join you, and then you can find a common space to share your different collections. This handout will also be able to help guide you as you plan for your collections and fine-tune the activity.

I think that these would be a great way to kick off your math habits to start the year.  You would be able to learn a lot about your students by interviewing and talking to them as they work on organizing and counting their collections.  

For older grades, I have thought that you could have them count objects in multiples or fractions to get a total.  You could also have them count by sets.  Packs of items...Can they count by 24 or 36? What if you offer them decks of cards? Can they count by 52s?  While they may not be fluent as they move through these unusual multiples, it will certainly aid them in developing mental math strategies.  Introducing fresh concepts through counting offers all students an access point, so students can practice new concepts using the math routine of counting.

Here is an example of a Counting Collection in a 3rd grade classroom:
Counting Collections: Third Grade - a common core classroom friendly exercise from Luna Productions on Vimeo.


Counting collections can adjust as your students develop their number sense. It is a routine that will allow students to think about better ways to organize, more efficient ways to count, and concepts of number. It can be used all year long.

What do you think about this?  How can you make it work in your classroom?  Please share your ideas!

I'd love to join in on the fun as your class does a counting collection!  If you are okay with that, let me know when you think you would like to do one, and I will see if I can join you.

Building Understanding and Developing Culture

I have done this problem with first graders for the past couple years.  The first time I offered it to students, I was surprised by how difficult they found it (due to the majority of my experiences being with older students).  When I did it subsequent times, I offered a lot of "up-front" guidance to help them think about it, and I highly encouraged them to use manipulatives and images to help them make sense. However, they still struggled...

I have decided that if  I work in a first grade classroom next year, we will again do this problem. I have been pondering some different ways to approach it.  

I think it will make a great numberless word problem that we can do with a slow reveal in hopes that students work to make sense of it rather than just trying to solve it. Using the slow reveal will offer a great way to directly instruct the students in the process of notice/wonder and making sense of the problem before trying to solve it.  This use of numberless problems should help them as we continue to build understanding throughout the year.

Doing a general overview as a class, offering the students manipulatives and encouraging them to work with a partner to solve it.  Once they think they have solved it, they must find another pair who has a solution, and each group must convince the other that their solution is correct.  I like this idea, too, as it should help to build mathematical culture int the classroom--collaboration, sharing methods, convincing others of our solution... I think that the deliberate teaching of convincing others with math talk will need to come before this lesson, so that both sets of students don't just say the answer they got and move on.  

I could provide a picture of two spotless ladybugs and then have the students work to put the spots on to match the words in the problem. This one seems very direct, but it does emphasize the importance of using visuals to help us make sense of the problem. Again, I think partner work is a great way for them to approach this.

I think the sharing out of this problem is important and recognizing the different ways that students went about solving it. Determining as a class what we think the correct answer is and acknowledging how we worked with our partners to solve it.  Maybe a gallery walk of our thinking?  

A possible follow-up activity could include students writing their own kind of problems like this and then switching problems with others.

This problem is written with 1st grade in mind.  Certainly, this type of problem could be adapted for different grade levels:
* I have six pieces of candy in my purse.  There are four more pieces of gum than mints. How many pieces of gum do I have, and how many mints do I have?
* We have 27 students in our class.  There are 5 more girls than boys in our class.  How many boys and girls are in our class?
* There are 114 vehicles in the parking lot.  There are 72 more vehicles with 4 wheels than vehicles with 2 wheels.  How many vehicles have two wheels?  How many vehicles have four wheels?
* In my closet, there are 18 shoes on the floor. However, there are 2 more right shoes than left shoes.  How many complete pairs of shoes do I have in my closet?
* I found thirty-five coins in my car.  There were four times as many pennies as all of the other coins combined?  How many pennies did I find in my car?

What do you think?  How might you approach a problem like this with your students? Do any of the ideas I shared seem to be better than the others?  I'd love to hear your thoughts.

EQUALITY is SO IMPORTANT

This online tool is a great option if you don't have access to a concrete tool in your classroom

Research has shown that a strong understanding of the equal sign in elementary school has a positive effect on student math success as they progress through middle and high school.  The standard sits in first grade (1.OA.7), but it is up to all grade levels to provide practice with the importance of the equal sign so that students can see the equal sign not as operational but as relational.

This is an important concept that I find myself working to build in our 1st and 2nd grade students quite often.

We typically begin by using the scale to think about missing addend problems.  The fulcrum of the scale represents the equals sign in the number problem. Because this concept is still difficult for most of them, we use cubes to look at the relationship of the two sides.  We always tell a story to go along with the problem. What do I need to add to 3 to make it be equal to 9?  I also draw it so that they see a representational model.

When we add the 6 weight, the students confirm that their answer was correct. 

After they become pretty proficient at solving the problems within 10, we stretch to equalities between different sums.

We again use cubes to help us make sense of the numbers and write number sentences to match our work.  We explore these problems always with context, and we make sure to write our number sentences with the missing addend in a variety of places. When the missing addend is in the spot directly after the equal sign is when the students tend to struggle the most. (4 + 5 = ___ + 1)

Giving students much time to explore and think about the numbers with the scale and other manipulatives is important.  It will also help them to better understand addition and the facts that go with it.

Here is an Equality dice game you can play, too.

The Math Coach's Corner has a good game for practicing equality with two-digit numbers on this post.

Illustrative Mathematics has some other good tasks that you can do with your students to help build their understanding of the equal sign.

What do you do to help your students truly understand the equal sign?