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Number Concepts

Best Results Ever with Kids Learning Math Facts!

I am going to share with you what I did this year to be successful in helping our school be more fluent in math facts than ever before!  We have had Reflex Math for about 2 and a half years now.  The first year everyone didn’t know much about what to expect.  The second year some classes were making progress with it.  Now this year being in a new school with new people (we consolidated with another school), we had a slow start but by the end students’ fluency took off.  Albeit some of the increase has been due to the increase of available technology and the new Reflex Math app on the iPad, but students this year were more motivated to achieve than ever before.  I believe that is due to a few things I added to encourage friendly competition!

1.  I added the 70% and up club.  What child doesn’t want to be a part of a club?!  When children reach the 70% mark, I post their certificate on the wall with their pictures in a central location.   I try to take their pictures close to window light so they look nice.  If students’ certificates are up too long without their picture they make sure they let me know about this.  I print the certificates off weekly.  These are available to easily print off from the home dashboard screen underneath the unprinted milestones link.  I also have these students names called on the announcements with “Welcome to the 70% and up club…” and then I have their names called out.


2.  To encourage students along the way to reach the 70% and up club, I send home their certificates if they are below 70%.  Students get certificates for answering facts and they get certificates for answering a certain amount of facts.  One of these certificates may say something like Joe Bob solved 2,000 facts.  If the certificate is one that is fluency related then I attach a little prize such as a pencil or fake tattoo.  These certificates may say something such as Mary Sue learned 25 new fluent facts.  All certificates that are 70% and up go on the wall.  I replace the 70% with an 80% or 90% certificate so that others walking by can see their achievement.

3.  I have a special place to hang students who achieved 100% fluency during the year.  These students have their picture retaken again for this spot on the wall.  Also, I had a special ceremony for these students at the end of the year in which I invited the principal and guidance counselor to shake the students’ hands and pin them with a special pin I ordered from Jones Awards.  I had wanted to order them all trophies, but I didn’t think I would be able to afford enough trophies with the school budget for those kids who had achieved 100%.  I also have the 100% fluent students’ names called out on the announcements when they achieve fluency.


4.  Each quarter I have parties for all the students who achieved at least 70% fluency.  Once students achieve 100% fluency, they are able to go to all parties for each quarter.  Most students who achieve 70% fluency go ahead and work towards the 100% fluency spot on the wall even though there isn’t much more reward to achieve 100%.

5.  In addition, I have parties for a class winner in 1st-2nd grade and a class winner in 3rd-5th grade.  I didn’t do a party for the 1st quarter.  In the 2nd quarter we did a hot chocolate party with toppings, in the 3rd quarter we did an Easter egg hunt, and in the 4th quarter we did a water play party outside with sprinklers and water squirters.  The kids anticipate these parties with great excitement!

6. Within each grade level I hand out 1st, 2nd, and 3rd place ribbons for the highest percentages of fluency achievement.  These students pictures are also displayed in a central location for everyone to see.  Some students are very competitive about this.

Because of all of these things, more students achieved 100% fluency than ever before.  The excitement built around the achievement of math fact fluency built a positive momentum with the children for something to earn.  When parents visit the building, they can often be found near the wall looking at all of the children’s pictures on display.


Some teachers became very competitive about their classes beating another class.  These teachers built a competition within their own class.  Towards the end of school, some teachers were sneaking their children who were still below 70% to the computer lab just so they could reach 70% fluency to attend the party outside. (I know the paper colors don’t exactly match, but I was just working with what I had :/.  I will make it look better next year :))



I hope this sparks some ideas within your own building!

Using Number Sense with Larger Numbers and a Freebie!

Thanks to the snow day (or shall I say ice day), I finally finished these number charts!  Back in the fall I had the idea for this product because I was working with a group of intervention children and they just weren’t able to tell me what 1,000 more or less than a number was when the number had more than four digits.  After the second grade standards, there are no standards that have children count past 1,000.  I think somewhere, someone who wrote the standards just assumed that children would be able to pick up on these patterns, but many times they aren’t able to see these patterns without explicit teaching.  That is what these number charts are meant to help teachers do.  In celebration of a snow day (2 snow days now), and over 600 fb followers, I made a free product with charts counting by 500.




Here is the main product that has number charts counting by 100, 500, and 10,000.  There are charts that count to 10,000; 100,000; and 1,000,000!  These will help students begin to see the patterns of larger numbers and help give you a basis for discussing rounding.  Not only that, but they increase in difficulty giving you a way to either scaffold or differentiate for your students!  Here is a peek at the complete set of number charts….

Number Charts to Million

Number Chart 1

Number Chart 2

Number Chart 3

You might also be interested in these fill in number charts with smaller numbers:number charts 1200 3.36.36 PM


This Helped a SPED Kid Learn Subtraction Regrouping!

I posted earlier about a strategy that helped a struggling SPED student add with regrouping.  Now, I am sharing the strategy I taught this same child to learn subtraction with and without regrouping.


Thank you to Lovin Lit and Educasong for the clip art!

Above there are two examples.  One of the examples is with regrouping and the last one is without regrouping.  This strategy will work both ways.  I admit I have been using the rhyme that a couple of teachers have gotten from Pinterest…”More on top, no need to stop.  More on the floor, go next door, and get ten more.”  I have kids recite this first.  The rhyme works very well so I use it.  Anyhow, I have the kids always circle the number on the bottom in subtraction.  The circle represents their head.   Then they make dots to count up until they get to the top number.   The dots are like their fingers.  To get the difference they count how many dots they drew.  Simple, easy, and if kids can make the jump to use their fingers, they can go ahead and don’t have to draw dots.  I did this because I found some students don’t know how to effectively use their fingers to count up yet.

You may also like Addition Intervention Strategies:


You’re Kids Aren’t Learning Their Addition Facts? Try This…Part 5


(Thank you Erin Cobb from Lovin’ Lit for the pretty border!)

Now after I have taught everything that I previously blogged about in Parts 1, 2, 3, and 4, which includes tens and tens plus one.  Learning the sums/bonds of 10 is the foundation for this discussion.  One of the tens plus 2 will already have been learned because it is a double, but there is no harm in learning multiple strategies to reach one fact.  Also, doubles plus two facts will be learned later and doubles plus two will also give students a strategy to reach 7+5=12 and 5+7=12.  Allow students to recognize this on their own when you reach that lesson.  The more ownership students can have of the strategies without you telling it to them, the more they will remember the strategies and feel smarter for being able to discuss the strategies.

Again when you introduce these facts write them out of order on the board.  Step back, wait, have children quietly look at the number facts and find relationships or patterns in their head.  I use the Number Talks strategy and have them put their thumb on their chest when they find a pattern.  This keeps everyone attentively looking for more patterns without the dramatic hand raisers flailing their arms in the air.  If students say that they see lots of tens and twelves acknowledge this and then ask students to look for more.  Eventually you will get what you are looking for if you have the foundation built from the previous lessons.  If no students say that one of the addends goes up by 2 and the sum goes up by two, offer a hint by underlining these numbers so that they are focusing their attention there.  Follow this up by fact (flashcards if you prefer) practice over the sums they have just discovered a strategy for and over previously learned facts.

Happy Thanksgiving!

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Your Kids Aren’t Learning Their Addition Facts? Try This…Part 4

So far if you have followed my previous posts, students will have learned their bonds of 10, their +1, +2, +9, +10, and adding one more to their bonds of 10 facts.  Next, I like to focus my students’ attention on learning their doubles.  Most of the time students are already comfortable with their doubles up to 5+5 since they easily see these doubles on their fingers, on dice, and in other real world examples.  At least when working with my intervention groups, this is the case.  The doubles kids most often struggle with are 7+7, 8+8, and 9+9.   When writing the doubles on the board, kids can easily see that the sums of double numbers turn out to be even numbers or the numbers that count by 2’s.

Slide1 I also like to use videos and games to help kids remember their doubles.  Here is one of the videos that I like to use.

This is only a preview of the video.  The other part used to be free but is no longer free.  The video costs $2.49 to download the 6-10 doubles, but is worth the purchase in my opinion.

After kids have learned their doubles, show them these doubles plus one more.  Don’t tell them that they are doubles plus one more, but let them see the pattern and tell you about them.


 Allow the kids to notice the pattern in the doubles and doubles plus one and express to you how the numbers change when one is added.  Kids will excitedly see the relationship between the double and how it goes up by one more.  After discussing the patterns from the previous posts, students will more readily see this pattern and relationship.  Then when using flashcards to follow up, students will sometimes think out loud about their strategy, and you will hear them thinking about the relationships they see to get to a new sum.  When you hear this you know you have taught them well!

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Your Kids Aren’t Learning Their Addition Facts ? Try This…(Part 3)

Up to this point if you have been following my previous posts and tried them with your students, you’re students will have learned their bonds of 10, +1 facts, +2 facts, +9 facts, and +10  facts.  Now it is time to build on some of that foundational material that you have been working on with your students.  With consistent review of what they have already learned students will be ready to move on to using their bonds of ten to find other sums.  While allowing them to sit and think, show students these facts side by side and allow them to comment after a few minutes on what they notice.  I like to use the Number Talks idea and have students sit and think for a while and when they notice something in the patterns to then respond with a thumbs up on their chest.  This allows the other students to think without the over zealous arms dancing in the air with the correct answer.  Here even if students say something that isn’t quite what you are looking for, don’t discourage their contributions.  For example, if someone says that they all have 11’s respond by agreeing but asking for something more.  You might ask, how are the facts on the left like the ones on the right?  What are the only numbers changing?  How much are they changing by?  Only ask these questions if you don’t get much response initially.  Allow students time to think and study what you have written. Slide1

You may also like these earlier  posts about learning addition facts:



*Thank you Erin Cobb: Frames courtesy of Lovin’Lit.

Fill In Decimal Number Charts. Could This be the Answer?

Ok!  So I won’t lie!  I have struggled with the next teacher.  Kids just fumble through decimals like there is a missing link. You try to have them do number lines, and they give you blank stares.  You give them card sorts.  They jumble all the cards up in the wrong order.  They tell you the wrong answers almost always.  There MUST be another way!!  Well, 1 year later, I have finally put the pieces together.

Why can’t kids compare decimals?  They are just numbers that follow a pattern with DOTS in them no less!

Have we ever stopped to look at the patterns that are formed when decimals are put in order.  Have we stopped to reason about why the zeros drop off the ends of the numbers and they have the same value?

In kindergarten, first, and second grade, we have it somewhat figured out.  For three years, students spend time counting and looking at patterns, and building numbers–for THREE YEARS.  THEN BOOM!  All of a sudden, they are supposed to draw their own conclusions about how to compare and round numbers that are abstract to them in 5th grade.  So students  CAN build decimals “reasoning about their size”, but where is the repetition that we give students in primary grades so that they can draw their own conclusions about the patterns.  There is no counting standard that I can find…but maybe I just missed the standard or maybe I am just going on a rant here.

Anyway, I think students struggle with decimals, because we don’t give kids anything to hang their learning on…they have no foundation!  I made some decimal number charts last year, but never really used them in depth.  This year I made some fill in charts thinking this would solve the problem of students’ glassy eyed look when learning about decimals–AND NO…I’m not even talking about the kids on meds!!  I really think that this is the problem…they need the foundation of counting before they can reason about decimals and move on to comparing, rounding, and ordering.

Because you are reading this, you obviously care about your students.  You most likely wouldn’t be on the computer during your down time looking for materials for your kids.  I am going to give you a few of the pages I made for FREE just because you care.

 Decimal Charts Free Download

More charts are included than this single picture below.

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I am also going to tell you about the pack of number charts I made that may help you even further.  There are number charts for each section of decimal numbers counting by hundredths and thousandths.  There is also a decimal number chart that counts by thousandths that is small enough to glue in students’ journals.  Not only that, there are small number charts the same size as a base ten block that will help students put the concrete together with the abstract counting numbers as they place blocks on top of the charts.  You can see a bit more below:

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Screen Shot 2014-01-14 at 10.39.26 PM Screen Shot 2014-01-14 at 10.38.50 PM


How In the World Do You Teach Rectangular Arrays and Division in 4th Grade?

Well, with large numbers this is something that my fellow colleagues  did not feel comfortable teaching.  And when that happens…who steps in??  None other than The Mathemagician…ta-da!  (which is me of course, but shhh don’t tell anyone!!)…I’ve been off for a few days as I write this, which makes me a little sillier than normal–and probably slightly more interesting! So, on with my lesson!  Now just remember when I post these pictures it is not a beautiful, I spent weeks preparing, colorful, lesson.  This is a practical lesson anyone could use whether you are savvy with a computer or not.  (I may turn this lesson into something more aesthetically pleasing later on.) The part that stumped the teachers was the fact that the standard says “up to 4-digit dividends”.

CCSS.Math.Content.4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

I had to think myself about how to teach this before I spouted off a lesson idea.  (I know I am supposed to be the math expert in my building, but, honestly, I have to look up what exactly common core is expecting before I assume I know what a standard is asking students to do.)  I looked at North Carolina Unpacked–my go to document for what common core expects…

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North Carolina shows arrays being broken into smaller arrays similar to ladder division.  Since these students had already been studying ladder division, I thought that they would have an easy time relating arrays to ladder division.  After reading the North Carolina document, I decided to create a large array with base ten blocks and have students find the missing side, but then I realized that students would just count the missing side.  I needed something better, but what?  Then I thought about a training I had been to which had arrays with part of one side covered up.  Students had to figure out a missing side .  However, these type of lessons  were for double digit numbers.
Here is what I decided to do to push students towards solving arrays with large dividends.  I used cm grid paper and cut out different sized arrays.  I cut out a very small one at first to use for modeling and discussion.     I told the kids that the principal had asked me to set up the auditorium with chairs for the 5th grade graduation.  Each cm square represents a chair.  Then I explained that I had spilled coffee all over my seating plans, and I didn’t know what I was going to do.  I told them I needed their help to figure out how many rows there were so I could recreate my plans.
I started  with this one.  Students had no problems figuring out the missing dimension in this array since they knew their facts.Array1
All I did to accomplish this spill effect was to cut out pieces of colored paper in blob shapes and tape them to the array before setting it on the copier.  In this case, the large blob was actually red, but it printed out black on the copier.
Then I showed an array that was larger with a 3-digit dividend.  This array actually had a blob on it, too, but I removed it so that we could discuss different students’ thinking after students figured out the missing dimension.  The blue marker shows my recording of the students’ thinking.
Then I gave students a much larger array to figure out a missing dimension on their own.  Several students tried to count how many squares there were underneath the coffee spill instead of using more efficient methods.  As you can see, this student drew lines on top of the blob to figure out how many rows there were.  She arrived at 22 rows.  (there were 24)
This paper above although wrong, was a valuable piece of thinking to tie everyone else’s work together at the end when I had students share because it was so basic.  ( I typically don’t write on students’  work, but since the students were a bit shy about sharing  I asked her if I could write on the work to help other kids make connections.)
The array above shows a student who is thinking of each row as a group of 12.  Rather than multiply he chose to add 12 repeatedly.  He realized that 12×12 made 144 and that adding 144 twice would give him 288.  Even though this student is oh so close he isn’t explicitly saying that there were 24 rows of 12 chairs.
I’m still looking for someone who is thinking with groups of 10 so that I can relate this to the ladder method for division.LargeArray2
Here is yet another student who is thinking in multiplication with equal rows.  Surprisingly to me, the students are more comfortable with 12×12=144 than thinking with groups of 10 to break off parts of the array.  The above students is a GT (gifted) student and he had 144 + 144 on his paper for the longest time because he recognized the relationship among 288 and 144.  However, it wasn’t until we had closing discussion that he labeled his other dimension with 24 and columns with 24’s.
Let’s pause here for a brief  teacher reflection moment by Ms. K (soft music playing)next time I prepare an activity like this, I will make sure that I cover up enough rows so that the only rows that show are a group of 10 rows.  This will hopefully push students towards thinking about ladder division.
Ok, finally, someone who thought about the ladder method for division!!!!!!!!! (picture me running through a grassy field to meet this paper).  This student’s paper will let me tie all the other students’ papers together in closing discussion and to relate arrays to everything that the class has been discussing for weeks!!!! (woooooohoooo!)
This lesson went successfully.  Now for the next lesson, I plan on giving the students another large 3 digit dividend array that is covered after row 10.   After this the next steps are giving students 4 digit dividends and/or rectangles with no lines.
Just so you know…one of the kids asked me if I really did spill coffee on the seating plans.  (so cute) I said, “What do you think?” 🙂

Counting by 1’s and Counting by 10’s Freebie!

This week, I was helping our kindergarten teachers gather some resources to teach counting by 1’s and counting by 10’s.  I made these simple blank number charts that the kids could use with unifix cubes when counting.  Because the vertical lines on the number chart for counting by 10’s are missing, this prompts students to group their blocks into a stick of ten for counting groups of ten.  The squares are ¾ of an inch so they fit exactly with a snap cube or unifix cube.  The counting by 10’s mat is made to fix ten unifix cubes exactly as well.

I am sharing these sheets for counting by 1’s and counting by 10’s with you (freebies)

counting by 1's mat

counting by 1’s mat


Counting by 10's

Counting by 10’s

Click here for the Counting By 1’s and Counting By 10’s Mat



How Can You Use Real Life to Teach Estimation?

Since some of our students are struggling with estimation, I’m thinking of bringing in a few pictures, that might help put estimation in perspective.  While out today at the grocery store, I came across estimation.

The sign said, “About 15 items.”

Oh, my gosh!  What does that mean?  I have 21 items in my cart!  Can I still go through the line?  The cashier looks haggardly and tired.  Is 15 about 20?  Is 21 about 20?

I think I will be okay going through the line with 21 items and without a fuss :).  Smooth sailing!

Then I found estimation again on a can of almonds.

The can displayed, “About 28 nuts.”

I’m on a diet.  Does that mean I can have 45 for the same calories?  No, I don’t think so.  I think that 45 is close to 50.  I know that 25  doubled would be about 50 and 28 is close to 25.  What about 30?  Would 30 be about 28?  Yes, 28 is only 2 away from 30.  I think I could have about 30 nuts for the same amount of calories!

I’m on a diet to teach estimation!

Having conversations like this reasoning about numbers will be my plan for this week!

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