## Use This Strategy to Easily Multiply Large Numbers and Fractions

You know the guys who easily multiply in their head who leave you picking your jaw up off of the floor?  Well, these folks have special strategies.  I am going to teach you one of these so that you can teach your students!

First, let’s look at this example with fractions.  If you double ½ you get one.  Instead of going through all of the steps it takes to multiply fractions, why not simply double the fraction and multiply?  In the case of one half or any other fraction with an even denominator, this process is simple.  ½ becomes 1.  Then multiplying by 1 is super simple.

In the case of a denominator such as ¼,  in the second example, you can double the number twice and halve the other number until you find a factor that is easy to multiply.  ¼ doubled becomes ½ and ½ doubled becomes 1.  As long as the other factor is easy to halve, this works great!

This may be done with mixed numbers as well.  As long as one of the numbers is even, you can double the other.

Now let’s look at examples with whole numbers.  Again, double one factor and halve the other.  Hmmmm 6 x 24. I don’t know that in my head, but I do know that I can easily double 6 to 12 and halve 24 to 12.  Wow!  I do know 12 x 12! 144!

I will skip the next two examples (12 x15 and 25 x 16) because these are self-explanatory.

Let’s look at 6 x 32.  If we double 6 and halve 32, we get 12 x 16.  Still not an easy fact.  Ok, I will try to double and halve again, and I get 24 x 8.  Hmm, again I don’t know that one.  Let’s try another time.  We get 48 x 4.  Whew!  Still difficult.  One more time.   Ok, 96 x 2.  To solve this problem, I will use a combination of strategies.  First I know that 96 is 4 away from 100.  If I have two groups of 4 away from 100, then I know that I will be 8 away from 200 because 96 is almost 100.  If I take 8 away from 200, this gives me 192.  Teach children through number talks etc. to think flexibly about numbers and ways to solve problems.  By teaching children these strategies, you will become stronger at solving math problems in your head as well!

When students struggle with ladder division, many times it is because they learned a procedure and haven’t made sense of the procedure for themselves.  In this case students haven’t had enough experiences with division problems that are near friendly numbers so that they can reason about the numbers. Try giving students some problems that are near friendly numbers first if you notice that they aren’t using number sense to form partial quotients.  For example, if students continue to subtract only small groups of ten and aren’t able to estimate a larger number for partial quotients, then try giving students numbers that are easier with which to estimate, like in the picture below.  In the above picture, I started with some students in intervention who started solving the problem using groups of ten.  2499 divided by 25 is obviously close to 2500.  Why not start with 100 groups and reason about taking away one group so that the quotient isn’t too large.  When students look at how they could estimate to solve this problem, they have a lightbulb moment.  Give them other examples like this to get students in the habit of solving problems with reasoning and number sense as opposed to a procedure.

## How to Intervene with Children Who Don’t Know Their Addition Facts

I have worked with children from 2nd grade on up to help them learn their addition facts.  One common denominator exists among all of these students.  That is THEY DON”T SEE PATTERNS!  I remember having a difficult time learning my 9’s facts when I was growing up.  To help myself, I just took one off of the number I was adding to 9 in the ones place.  I noticed this pattern.  No one taught me this.  When I was growing up, learning facts was like, “Ok, Class, let’s learn all our 8s facts, let’s learn all our 6s facts and so on.”  This is not effective for students who don’t recognize patterns on their own. Now with the common core mathematical practices, we should be teaching children to explore patterns through thoughtful placement of number facts to help them recognize these patterns.  Giving students opportunities to see the patterns will result in more students who are fluent in their facts.  I have shown examples of this before such as in this post about using 10s to help with adding 9s.

But now I have actually put all of my work with struggling learners into a packet which could be used whole group for grades 1 or 2.  At the 3-5 level this could be used for students in intervention or as part of the RTI process.  Here is a look at the packet  that I have put together to help students become fluent with all of their addition math facts.  It is on TPT !

You can also try out a little sample of this product for FREE here.

Are you teaching your firsties to add ten, subtract ten, add 1 and subtract 1? The week before Christmas we added this game to one of the selections in the students’ math stations. This game is called “Bubble Gum Pop”. The kids absolutely LOVE it!!!

Students move “bubble gum balls” (bingo chips) up and down the 100’s chart mat according to the spinner. The game is differentiated for students who need more of a challenge so that they can use a mat that counts to numbers past 100 or they can use a bubble gum spinner that allows them to even add or subtract multiples of up to 20.

In this photo above, students are tied with both having an equal number of chips on the board.  The one who knocks the other student’s chips off the board first is the winner.  What makes this game fun is that there is an element of chance when students land on the pictures, their chips are out.  Also, the game requires children to know which direction to move on the board to add or subtract 10s and 1’s so they are learning at the same time.

The game is also available in color.  I copied it in second grade however on colored paper, but ended up liking the black and white better because I felt the students could see the chips and numbers better on the board.  The color definitely did make the game happier though.

## What do you see? A Freebie?

I have been missing in action from my blog lately.  Hopefully this will make it up to all of you faithful followers 🙂 !  I have been working on this packet of addition fact lessons that I used with intervention groups all last year with much success.  The lower students really seemed to enjoy the thinking aspect of these lessons.  I have been working on putting this into a format that is cute enough to post.  Because I have been working on the whole packet for months, I thought I would give you a free preview sample in the meantime.  I will be posting the whole packet soon for sale.  Without further adieu, here is the Freebie!  I hope you enjoy using it!

Thanks to Winchester Lambourne for the spooky eyes clip art!

## How Do You Teach Rounding?

To teach rounding I take several approaches.

The first method I use is to teach rounding with a sentence strip number line.  I have students build a number line on sentence strips with whatever numbers we are working on.  If they are working on the nearest 1,000 and nearest 10,000 for example, I may double side the number line sentence strip.  If we are working on nearest 10 and 100 then I would double side the sentence strip counting by those two numbers.  Here is how I have students build their sentence strip without much fuss.

First, I have them put a finger space down with one finger and make a mark.  We put a zero here.  I also have them leave a finger space before the end.  They put the last number here such as 10,000 in this case since we were rounding to the nearest thousand.

Then, I have them fold the strip in half so that students can at least find a mid point.  They put a mark at the  mid point.

I have them put four fingers down to hold the space to make the next mark.  Students repeat this four finger spacing until they get to the midpoint and then repeat the four finger spacing after the midpoint.  This gets a fairly even number line if students do this.  IT ISN’T Perfect, but it’s close enough to reasonable spaces give or take the size of the students hands.

Next, students label the numbers underneath the marks.

This will give students the numbers they need to use when making a number line sketch such as in the rounding roller coaster model I like to use.  Before actually talking about rounding.  I like to pose a number such as 8,456 and ask students where this number would fall on the number line.  I have them place their finger where they think the number would go and I do a quick sweep around the room to look for understanding.

Here is how you can progress to the rounding roller coaster.  Whichever numbers the students’ fingers are pointing between on the number strip go on the end of their roller coaster.  For example with 8,456.  The numbers would be 8,000 and 9,000.

Next, have students put the midpoint number in between the numbers on the two ends of the roller coaster.  Then have them put a dot where the number they are rounding actually is.  Explain that when a roller coaster is on top of the hill at the midpoint it will coast all the way to the end.  If the roller coaster isn’t all the way at the midpoint then it will coast back down to the beginning.  Whichever side it coasts to is the answer.

Now of course in the midst of all this, I have students learn the rhyme “4 or less let it rest, 5 or more raise the score” so that students have another rounding strategy to fall back on.

Now the rhyme and the roller coaster I cannot take credit for.  I either learned it on the internet somewhere or from another teacher.  I can’t remember, but both of these strategies support students’ understanding.  These are my preferred ways of teaching rounding.  Now, of course you will have students who don’t understand the above because they cannot count that high or have understanding of numbers that high.  That is when I give them some counting practice using these number charts:

This is free in my TPT store:

And you can get these which count to larger numbers and they cost \$4:

Or these count by smaller numbers up to 1,200 and are \$7.  I use these a lot with kids at school.  They are great for up to grade 3 or as an intervention for older kids.

Now here is what I do with students who are seriously struggling.  I don’t like teaching rounding this way because it really takes the number sense out of what they are doing, but some students just need to know how to get the right answer and do not have the number sense to build on to be able to round with understanding.

I showed the method above to a group of 5 struggling learners and all were really getting correct answers by the end except a resource student.  Being able to write something down on their paper before they did much thinking really helped the students.  To know that they could go ahead and fill in the zeros and fill in the beginning really helped them.  However, like I said this isn’t the best way to teach for understanding.

## 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….

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

## 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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