Tuesday, May 5, 2015

Attacking Angles

I recently had a conversation with a 4th grade colleague. She had been measuring and drawing angles using a protractor with her student. The trouble was that no matter how much guided practice she provided, how many real life examples, how ever many ways she tried to attack angles, there was still a handful of students who were using the wrong numbers on the protractor. A 150 degree angle was recorded as 30 degrees and when drawing a 20 degree angle the student would draw a seriously obtuse illustration!

If this sounds familiar, and if you teach 4th or even 5th graders I'm sure it does, your students would likely benefit from vocabulary work along with some activities that will develop a sort of number sense around angles. So without any further ado, here are my strategies for Attacking Angles!

Strategy #1: Sort & Discuss Pictures
Students need to begin by developing a visual anchor for what an acute, right and obtuse angle looks like. Begin by talking about a 90 degree angle. Given a set of angles, students should look through and find the right angles. When students are confident, go ahead and relate right angles to acute as any angle which is more closed or smaller than a 90 degree angle. Given a set of angle pictures, have students sort through and find the acute angles. You may wish to stop right there and mix up all of your right and acute angles and ask a student to sort them. When they have developed confidence around these types of angles, you can repeat the activity with obtuse angles and sort the three.

Could this angle be 130 degrees? NO! Any kiddo with a
spatial understanding of angles would say so :) 
As students are sorting all 3 angle types, give them vocabulary to match the pictures.
"That's an acute angle, it's smaller than a right angle so it must be less than 90 degrees."
"That's an obtuse angle, it's larger than a right angle so it must be more than 90 degrees."
"I see you have an acute angle, could it possibly measure 120 degrees? Why or why not?"
etc.








Strategy #2: Sort & Discuss Angle Measures
What might 157 degrees look like? Could it be an acute angle?
No way!! 
Once students have a spatial understanding about angles, move into a sort that has NO pictures. By looking at a measure, a student should have a general idea about whether or not that measure is going to result in an acute, right or obtuse angle. Repeat a sort with only angle measures rather than pictures much in the same manner as you sorted and discussed the angle pictures.






Strategy #3: Synthesizing Our Learning. 
Let's measure, this could be a 50 degree angle. 
Given a set of angle pictures and a set of angle measurements that match up, students can play a game of memory. The trick is that students should NOT measure every single set of cards they flip over, only the cards that they really need to measure. This is a way that you can synthesize the learning of the first two activities. For example, if a student flips a card reading 170 degrees and an acute angle, there is no reason they should need to get out their protractor, they know that these two cards can not possibly be a match.
No need to measure, this obtuse angle could NOT be 23 degrees! 



After completing these activities and having thorough understanding of the vocabulary terms acute, right and obtuse and the associated relationships to 90 degrees, students should no longer be making errors in reading the wrong numbers on a protractor because they will know when a measure doesn't make sense. Likewise, they will know when a measure seems reasonable. 

Good luck attacking angles in your classroom!

If you are looking for a ready made resource to put these strategies into place, click on the cover below and head over to my store where these angle activities (along with additional task cards and word problems) can be found.



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Friday, February 20, 2015

Fly On The Math Teacher's Wall: Fractions!

Thanks for joining in on the Fly on the Math Teacher's Wall Blog Hop. This is definitely one of my favorite blog hops out there because it is CONTENT focused! Just a bunch of math nerds talking about instruction :)



If you look at the CCLS, fraction standards don't start until 3rd grade.

... or do they?

Primary learners are studying the foundations of fractions and the K,1,2 standards support this learning. If you are a primary teacher or an intermediate teacher struggling with how to reach students who are having difficulty with fractions take a walk back into primary math standards.

If you stick with me to the end of this post, you will be rewarded with an activity that can be used with either second grade students or as an intervention for older students who are struggling with the idea of understanding the meaning of a denominator and how denominators can be used to compare.




If you would like to explore unit size through measurement with your second graders OR if you have intermediate students who still struggle to compare fractions with like denominators who need more instruction on the relationship between unit sizes, click HERE to grab a free activity.



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Next stop on the blog hop is The Recovering Traditionalist. Head on over! 


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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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