Tuesday, January 27, 2015

Addition with Regrouping... The Place Value Way!

A few months ago, I wrote a post on using place value strategies to add two 2-digit numbers.  If you remember, students were decomposing numbers into tens and ones and combining like units. By using base ten blocks, students were able to easily make the connection between the digits in a number and their values.

 <<Here>> if you would like to take a look in more detail. 

I also promised in that post that students WOULD eventually get to the traditional algorithm. You'll also remember that I said that the traditional algorithm was a 4th grade standard so we are still eeking out as much place value understanding as possible as taught through addition.

If you want to skip to the addition strategy jump below to the large letters that say "STRATEGIES START HERE". Otherwise, keep reading to see how we have bridged the gap between these basic skills and larger sums.

Our second graders have been working towards this goal over the past few months. They have done extensive work in the world of place value and also in mental addition strategies. I am condensing here <in a big way> but one aspect of place value work they have studied is grouping and renaming units. 25 is 25 ones but is also 2 tens 5 ones. 345 is 345 ones, 34 tens and 5 ones or 3 hundreds, 4 tens and 5 ones. Etc.

Additionally, they are thinking of addition and subtraction in terms of place value. For example, if you have 46 and want to add 10 more, a student well grounded in place value knows that 46 is also 4 tens and 6 ones. "+10" is really just adding another ten. The result is 5 tens and 6 ones or 56.

Here is a look at some of the place value work we have been doing. 
Students are also able to add numbers such as 59 + 12 using mental strategies. If you were to add 59 + 12 in your head, you likely wouldn't line up the digits in your mind and think  "9+2, put down a 1 and carry a 1 and 5 + 1 + 1 = 7 so 71!"

You would likely solve the problem by thinking something like "59 + 10 is 69. Two more is 71." Or, 50 + 10 = 60. 9 + 2 = 11. 60 + 11 is 71." Students have learned ways to note this mental strategies and, as they have, their mental math has really grown.

Strategies Start Here
*If you are REALLY in a hurry, scroll down to the bold conversation below*

I walked into a 2nd grade class last week ready to show them a new way to organize 2-digit numbers vertically **read: leading to the standard algorithm*. Students would be using place value disks to show addition. If you aren't familiar with place value disks, they are foam disks about the size of a quarter that say "1" "10" "100" etc. They are a non-proportional model that students who already understand the magnitude of 1s, 10s, and 100s can use to organize their learning when working with place value.

To add 59 + 12, students would build 59 with 5 tens disks and 9 ones disks horizontally. Below this number they would build 12 by showing 1 ten and 2 ones. Students would then look at the ones and determine how many there were:
S: 11
T: What is another way to say 11.
S: 1 ten 1 ones.
T: How could we regroup our ones to show 1 ten 1 ones.
S: Trade 10 ones in for a 10 disk

Easy as pie.

So we did. We traded in ones for tens. We found the total number of tens and ones and were pleased to find the answer.

This was the first day of this new instruction on NON-mental addition strategies. We weren't doing any writing or algorithm to accompany the number disks. And something funny happened.

T: Can we build 23 + 72?
S: Do I have to? It's 95. I did it in my head.
T: Alright, I'll give you a harder one. How about 34 + 57.
S: I can do that in my head too!

... thank you place value and mental strategies... No worries, they built them all "because when numbers get larger you will be glad you learned to organize your work!"

So the next day, students were tasked with writing numbers that match their number disks. *Read: The traditional algorithm". Their work in place value again supported this work. Here's what a problem might sound like:

T: Let's build 75 + 18. 
S: [Build 7 tens and 5 ones horizontally. Below, they build 1 ten and 8 ones.]
T: How may ones are there in the two numbers together? 
S: 13 ones. 
T: What is another way to say 13? 
S: 1 ten 3 ones. 
T: Can you regroup your disks to show 1 ten and 3 ones? 
S: [Students trade in 10 ones disks for 1 ten disk] 
T: On our paper we can show 13 ones as "1 ten and 3 ones". Let's put the 1 ten in the tens column above [or below, on the line if you are doing "New groups below"] the 7 and the 1. We'll put the 3 ones below to show the total. 
T: How many tens do we have in all now? 
S: 7 tens plus 1 ten is 8 tens. Add in the new 10 and we have 9 tens. 
T: What is the total? 
S: 93. 

By thinking explicitly about teen numbers as tens and ones, students give meaning to the traditional "carry the 1". Fewer students are making errors like carrying the 1 where no new 10 exists, not carrying a ten when there ARE more than 10 ones and other errors that demonstrate a lack of understanding.

Even better? When a student does make an error, you know that there is a breakdown in place value and addition understanding rather than an error in a procedure. When the algorithm is built on place value and understanding the results will be more consistently successful!

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Monday, December 8, 2014

iHeart Math Holiday Hop

Welcome to day #9 of the iHeart Math Holiday Hop! I am so excited to join 22 other math bloggers to bring you holiday tips and treats through the month of December! If you have never been to my blog before, I am a math interventionist in NYS. I work with students in grades K-5 and have a background in intermediate special education. Image Map
Holiday Tip #1: Giving Back
It is easy to get caught up in the stress of the holiday season. I love the idea of committing to 1 act of kindness towards someone else and 1 act of kindness towards yourself each day of the month to keep yourself and others sane! This may be as simple as committing to leaving within 45 minutes of the end of the school day to get home and spend time with family, treating yourself to a special drink in the morning (Starbucks!), or putting down your pile of grading so that you can go to the gym. Work will always be there, it will NEVER truly be "done" so take time for yourself. An act of kindness towards others could be as easy as leaving a copy of that great lesson you are doing on a colleague's desk, bringing in a muffin for the teacher who comes flying in having already readied their 3 kids and dropped them off at their own respective schools. You know how exciting it is to have an anonymous treat dropped off in the morning- commit to giving someone else a great start! 
Holiday Tip #2: Math Tip
Picture this, it's the day before the holiday break, the kids are excited and bouncing off of the walls and you just want everyone to be safe until dismissal. Learning would be a MAJOR victory but it seems like a far fetched dream... 

Or is it? 

I save my best trick for the day before breaks. Math scavenger hunts :) Ahead of time, I prepare, essentially, task cards and tape them up around the school. Students are broken into teams and they go off and do math review. But because they are in a team and walking around the school it seems super novel and they have no problem at all doing a set of math problems quickly and accurately! 

I would recommend making the tracking sheet novel in some way. At Thanksgiving, I have students collect a turkey feather at each station, when their turkey is complete with all 8 feathers, they are done. At Christmas, they are collecting ornaments as they go. When their tree is fully decorated the hunt is over. Before April break? Easter egg hunt with fake money inside. Get counting kiddos :) 

Also, if your students tend to bring in a CRAZY amount of snacks on the day leading up to a break, it is fun to set up a snack in the classroom and allow students to come back at the 1/2 way point for a special treat. 

I am telling you, I have used this strategy for years and it works. every. time. 
Holiday Gift for YOU!
Want to try the scavenger hunt? I put together a scavenger hunt for you! I have included one set of task cards appropriate for grades 1 and another appropriate for grade 2. Both use the same game board. Click the collage below to grab it. Enjoy! 

Thank you so much for stopping by today! Tomorrow, head over to Math Coach's Corner for the next day of the iHeart Math Holiday Hop!

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Saturday, December 6, 2014

Keywords are NOT the Key to Word Problems

Solving word problems is hard.

Really, really hard.

In a world where reading comprehension, logical thinking, math computation and visualization come together, word problems were born.

As teachers, we are interested in doing everything we can to make instruction make sense, come alive and "click" for our students. And, unfortunately, in the interest of the end goal it can be very easy to try to teach tricks to our students. In the long run, however, this does SUCH a disservice!

Let's look at the list of commonly taught "keywords" and match it up to a progression of addition and subtraction word problems from K-2 to see if learning these key words will serve a student well.

Keywords for addition often include:
in all

Subtraction often sounds like:
take away
gave away
left over

First up: Do the keywords hold up in Kindergarten? 

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

So, the standards alone here really aren't enough to determine what word problems look like at this level. To learn about what the standards "look like" you need to look into the progressions document. Have you seen this chart? I know it's difficult to read, so I have included a link.
Basically, the chart lays out the different type of word problem and then, through the shading, explains which problem types are expected at each grade level.

Back to kindergarten now. The bulk of the problems are put together, take apart, add to and take from problems where the result is the missing piece. Looking at the questions, one by one:

4 bunnies sat on the grass. 5 more bunnies hopped over. How many bunnies are on the grass now?
Keyword indicates addition, addition of 4 + 5 will solve the problem. 

10 apples were sitting on the table. I ate 4. How many are on the table now?
No keyword... you could argue that "ate" means take away so that's subtraction but, in the world of keywords they really offer no help here. 

2 green apples and 4 red apples are sitting on the table. How many apples are on the table?
Keyword indicated addition, addition of 2 + 4 will solve the problem. 

Grandma has 10 apples. How many can she put in the red vase and how many can she put in the blue vase?
This question is open ended and contains no keywords. 

Conclusion? In kindergarten, keywords are not misleading, however, they are not helpful in solving all types of word problems.

Next up: Do the keywords hold up in 1st Grade?

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.1

The "1" at the end of this standard is directing you to the chart that describes the problem types. Too bad they didn't make the same note in the Kindergarten standard.... I digress...

So here you see we have unknowns in all positions in all types of word problems. The only real caveat here is that change problems where the start is missing and comparison problems where the language is intentionally misleading are saved for second.

Let's go through a few problems:

6 bunnies were sitting on the grass.Some more bunnies hopped there.Then there were 12 bunnies. How many bunnies hopped over to the first 6 bunnies?
Keyword indicates addition. 6 + 12 does NOT solve the problem. A student needs to be quite flexible with situation and solution equations for this key word to make sense. 

18 apples were on the table. I ate some apples. Then there were 10 apples. How many apples did I eat?
No keyword. If you were in the camp that said "ate" indicated subtraction before 18-10 will yield the correct answer. 

14 apples are on the table. 7 are red and the rest are green. How many apples are green?
No keyword.

Lucy has 16 apples. Julie has 9 apples. How many more apples does Julie have than Lucy?
Keyword indicates addition. Addition will NOT solve this problem. Without any obvious action in this word problem, even a situation equation is a difficult argument for a 6 year old. 

Lucy has 10 fewer apples than Julie. Julie has 18 apples. How many apples does Lucy have?
Key word indicates subtraction. Subtraction will solve the problem... but wait until 2nd grade... 

Conclusion? In first grade, many problem do not have obvious keywords and those with keywords may be very misleading with a keyword leading to a situation equation rather than a solution equation. But when was the last time that you saw a "keywords" poster that said "Keywords for determining the operation in your situation equation!" I haven't seen one yet. 

If I haven't yet convinced you that teaching keywords is doing a disservice to students, continue on to reading about 2nd grade. 

Last, but CERTAINLY not least: Do the keywords hold up in 2nd Grade?

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1

I am not going to even TOUCH the multi-step nature of word problems at second grade. Obviously with multiple steps there may be multiple keywords for a student to sift through.

Here we add in the last 4 type of word problems.

Some bunnies were sitting on the grass. 19 more bunnies hopped there. Then there were 27 bunnies. How many bunnies were on the grass before?
Keyword "more" indicates addition. 19 + 27 will not solve the problem. Addition is appropriate only in a situation equation. 

Some apples were on the table. I ate 7 apples. Then there were 45 apples. How many apples were on the table before?
No keyword unless we count "ate". In that case, subtraction is appropriate. 

Lucy has 19 fewer apples than Julie. Lucy has 36 apples. How many apples does Julie have?
Keyword indicates subtraction. Subtraction will NOT solve this problem. In fact, a solution equation using subtraction would have to look like J-19=36. How many of your students would have written that? In terms of pure key words most students would write 36 - 19 as Lucy was written directly before the word "fewer". The keyword is very misleading in this problem! 

Julie has 63 more apples than Lucy. Julie has 89 apples. How many apples does Lucy have?
 Keyword indicates addition. Addition will NOT solve this problem. Most students, in an attempt to write a situation equation matching addition will write 89 + 63 = L as Julie's name was written first. The keyword, again, is very misleading. 

In fact, in the problem type diagram for the last 2 problems, the descriptions are "fewer suggests wrong operation" and "more suggests wrong operation".

Overall conclusion: 
In kindergarten keywords don't always help but they don't really hurt either. However, if students in K are taught key words, they are set up for trouble in first grade where key words lead to situation equations but NOT necessarily the answer of the problem. By third grade, key words can be misleading in a way which will cause students to write even a situation equation in the wrong way.

Suggestion? Focus on the action of the problem, what is a part, what is a whole. Are we missing a part? Use subtraction or missing addend addition. Missing the whole? Use addition. In comparison problems, focus on the larger amount as the whole, and the parts being the smaller number and the "more/fewer" piece.

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