Cracking the Code

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My third grade students explored different addition and subtraction algorithms this week.  It’s been a challenge.  In the past, students used number lines to add or subtract by the highest place value first and then slowly move towards the lowest.  Moving towards a  standard algorithm has been a process this week.  Students started by using the partial-sums method.  This was similar to what they were used to and mimicked the number line models.  Students seemed fairly comfortable in using this method.  The column addition method was next on the docket.

Many students come into my classroom already knowing how to use this method.  It’s often referred to as the traditional/standard addition algorithm.  Students complete the steps and out comes an answer.  Does it make mathematical sense to my third graders?  In some cases the answer is no. So, in an effort to bring a bit more meaning to why it works I decided to use a code activity to reinforce the idea of base-ten and place value.

Cracking the muffin code is an activity found in the Everyday Day math curriculum. A quick Google search will also bring up many different threads related to this activity.  I’ll be paraphrasing the lesson throughout the post.  Basically, students are given a scenario where they’re in charge of a muffin market.  At the market the muffins are packed into boxes.  The boxes only hold a certain amount of muffins.  When someone asks for muffins, an employee fills out an order form.  That order form contains a code.  The largest box needs to be filled first and the employee needs to send boxes that are full. Here’s an example that I paraphrased from class:

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I had students work in groups to figure out the code.  I gave them around 10 minutes and at the end of time a few groups were fairly confident with their answer.  We discussed the code and students started to notice a pattern.  They used trial-and-error to figure out which column matched the box size.

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There was a lot of excitement in the air as students solved the puzzle. Afterwards, students connected this to the idea to place value for the next problem.  This puzzle was designed differently. Now, students were asked to pack boxes of granola bars. The packages hold 100, 10, or 1 bars.  The employee uses a coding system.  Here’s another example:

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Almost immediately, students were able to see that the first column was designed to package one bar.  The second was for ten, and the third, one hundred.  I gave time for students to look at the similarities and differences between the granola and muffin codes.  Students were then asked if the base-ten system was similar to the granola code.  Students nodded their heads and I even had a student say that when we regroup numbersit’s like adding another package.  Another student stated that sometimes not all the numbers fit in a package so we have to find another place for them.  Students were making connections to how the base-ten number system works and why regrouping is sometimes necessary.

This was an eye-opening experience for some students as they started to look at the place value positions as bins or containers.  This lesson had students talking about how place value can be perceived as “containers” or “boxes” for numbers.  Each box needs to be filled to it’s capacity until a new one can be used.   I’ll be referring to this activity throughout the year as it seemed to help students make connections when exploring the base-ten system.

Afterwards, students used the column addition algorithm with a bit more confidence.  Next week we’ll be discussing multidigit subtraction.

 

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Exploring Subtraction Computation Strategies

During the past few weeks my second grade class has been taking apart and putting together two and three-digit numbers. In the process students have been developing a better understanding of numbers.  They’ve been exposed to using a variety of computation strategies to find the sum and differences of numbers.  Through all of this I’m finding that the students are becoming more confident in their ability to use these different computation strategies more fluently.   Although they’re confident they tend to gravitate towards using one specific strategy for computation.  The traditional algorithm is usually the primary method that they use.  Even though students can add/subtract using that method I found that they weren’t expanding their understanding of other computation strategies.  This was a bit of an issue for me because students started to look at computation as the shortcut and not delve into the understanding of why it works.

After speaking with a few other teachers I decided to use a math task found in this book.  I briefly reviewed the different strategies that we’ve learned this year and gave the students this prompt.

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I wanted to make sure that students showed two different strategies and provided some type of written explanation.  The template I copied also had fields for a number model and explanation boxes.

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The bottom of the sheet was designed for students to be able to check their work using addition.

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I gave the students about 10-15 minutes to complete the formative assessment.  Most of the students tried out the standard subtraction algorithm but had a bit of trouble with the second strategy.  After a few moments students started to dig deep and think of how to take apart numbers using different strategies.  Some of the  students truly had trouble using a different strategy and this was evident in what they produced.  I was impressed with some of the different strategies that students used.

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I wrote feedback on the papers and handed them back to the students the next day. Afterwards, I removed the names off of the papers and shared some of the results with the class.  As a class we decided on the following:

  • Students remembered many of the different computations strategies that were discussed earlier in the year
  • Some of the students invented their own strategies on this particular sheet
  • Students need to strengthen their written explanations
  • Students had some trouble explaining what regrouping means

Next week the class will be setting goals in improving our written responses.  Overall, I feel like this activity helped showcase different computations strategies while bringing awareness to areas the need improvement.  I’d like to use this template with a few other classes later in the year.  Feel free to download and edit this file for your own classroom.

Transitioning to the Standard Algorithm

 

When should the standard algorithm be introduced?
When should the standard algorithm be introduced?

 

Many of my second grade classrooms are in the middle of their addition units. The classes often teach place value and addition strategies during the months of September and October. When introducing addition strategies, teachers rarely start using the standard addition algorithm (see # 4). Manipulatives and visual representations are heavily used during the first month of school.  The process below differs per school, but I’m finding that this is often the case in many of the second grade classes that I’ve observed. Keep in mind that I’m missing other approaches, so perceive the following as a few highlighted strategies that are used during the first few months of school.


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Finding the sum of objects

Students are introduced to some from of unifix cubes or counters. Students are asked to compile the groups of counters to find the sum. For the most part students find this task quickly and are ready to move onto the next portion.  This is also a first grade skill that’s reviewed at the beginning of second grade.

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Identifying numbers on the number line

The number line usually follows the counters. Sometimes the number line makes an appearance before the counters, but it’s usually afterwards. The number line is used extensively. Students are asked to find numbers on the number line. This builds number sense and an understanding of the reasonableness of an answer. Eventually students are asked to add numbers with hops showing the addition involved.

Find the sum of 25 and 15.
Find the sum of 25 and 20.

Base-ten blocks are then introduced to emphasize place value. Students are asked to combine base-ten blocks to find the sum. They are asked to find the sum of the base ten blocks and place them on the number line.

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What is the sum?

The four processes above aren’t necessarily mandates, but it’s found in the sequence of the textbook.  I should mention that the same process is used with subtraction later in the year. What I’m finding is that there’s rarely a mention of using the standard addition algorithm to find the sum. I don’t necessarily think that’s a problem, but it raises the question of when should the algorithm be introduced? In what cases should the algorithm be introduced and is it only used in certain circumstances.

Think of 200 + 198. Would your students use the standard algorithm for this?  Some might, other might prefer to use another method. Regardless of when the standard algorithm is introduced, there will still be students that would prefer to use the algorithm.  Is that the most efficient method?

The topic of this post was tackled during last Thursday’s #ElemMathChat. Most of the questions revolved around when the standard algorithm should be introduced and mistakes that occur when students focus on the steps.  There were many useful answers, but I’m not positive if one right answer climbed its way to the top.  There are many factors at play here. Some teachers feel pressured to move through the curriculum at a high pace because of testing.  They might teach the algorithm sooner, while others might not mention it and scroll through the prescribed lesson sequence.  Most teachers would like students to have a conceptual understanding of numbers and systems before moving towards a standard algorithm. How much time is spent developing that truly depends on the teacher and student. I’m not judging any teacher is these situations as we are all in this together, but I believe having this discussion is important.  I think this also plays a role in how other algorithms are introduced.