It's Not Your Father's Math Anymore!

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The work above shows some more interesting strategies for subtracting and for showing possible combinations. On a recent test, students were asked to subtract $25,000 from $712,000. Many students used the traditional algorithm with mixed results. Today, the class decided that decomposing the numbers, less the 0s, and using positive and negative values made solving the problem easier. Most students realize that using negative values is way easier than it first sounds. Look at the middle photo.

The possible combinations problem involved making sandwiches with three major ingredients, a bread (Wheat or Rye), a meat (turkey, ham, or chicken), and a cheese (Swiss or American). Most students as comfortable using a tree diagram to show the combinations, but many students fail to understand exactly what a combination is. In this case, a combination is a type of bread with one type of meat and one type of cheese. Once that became clear, students could easily see (via their tree diagrams) that each bread type could have six possible combinations of ingredients. Look at the top photo.

Why Do We Do the Things We Do?

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This edition of my blog emphasizes two things that are hard for most 5th graders and adults to do well. The first is subtraction (be honest adults, you know it's easy to make mistakes), and the second is metacognition, or thinking about your thinking.

This task asked that students solve a fairly easy subtraction task two ways and then to give a rationale for the method that they prefer. Many kids wrote that they preferred one method over another, "because it was easier for them to do" or because, "that's the way that they did it last year in class". However, some students were able to make statements that relate to the value of the numbers, the distance between the numbers, a preference for addition over subtraction, a need for a visual strategy, or a like or dislike of negative numbers!

If you are an adult, try to answer that same question. I bet it will not be easy :-}

Rounding Using A Number Line

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Today we worked rounding numbers using a number line to provide "visual evidence" that conclusively indicates which end of the number line the number is closer to. This process helps students see the relative distance of numbers, and leads students to really think about place value. This, in my mind, is much more valuable than the traditional method of looking at digits and playing the "4 or less is down and 5 or more is up" game.

Counting Puzzles Stimulate Great Thinking

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These pictures show two recent student sheets that we have been working on in class. Both are designed to stimulate thinking about counting patterns. Often, students fall back on methods of solving these questions that are far from efficient. For example, a student might simply list all of the multiples of 25 in order to get to 300 and then state, after counting the written numbers, that it takes 12 people counting by 25 to get to 300. However, many students quickly latch onto the idea that it is much easier just to figure out how many 25s are in 100 and then triple the amount to get to 300.

On the "What's in Between" page, students have to sort through a multitude of important concepts in order to find the answers to these puzzles. For example, students must be able to find the mid-point of two numbers in order to find a reference point to know if a number is a number is closer to the numbers on the ends of the number line. Some students get really confused with larger numbers like 7,900 and 8,100. These same students would also have no trouble finding what comes exactly in the middle of 79 and 81. Sometimes, "ignoring the zeros" can be a great strategy. Students also have to be flexible and efficient when they are forced to find numbers that are multiples of two different numbers. If a puzzle said that the mystery numbers were said if you count by 125s and were also multiples of 500, then only certain numbers would qualify. I would try to think about multiples of 500 first, as that means only numbers that end in two or three zeros would qualify. See if you can find the puzzle with answers that are not quite correct.

Basic Operations Voted Most Efficient Week 2!

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Our first homework sheet proved to be of great use, as many simple and very efficient strategies emerged!

Problem number one was a subtraction problem, and the most popular strategy was the use of an open number line to add up from $267 to $323. The class then gave a go at "straight subtraction" which uses positive and negative values. Many in the class found this to be even easier to navigate, and most thought that it was faster too. Fast and accurate; you cannot beat that!

The second problem shows a very efficient execution of left to right adding. Adding the greatest place value first can help reduce large errors in calculations. I'd much rather be off by a dollar than a million dollars!

The final problem was almost universally solved by using a "generic rectangle" to multiply 16 by 12. The partial products are usually accurate, easy to calculate, fast, and easy to add up. Some students used multiplication clusters like 16 X 10 and 16 x 2 to solve the problem, and that is another fast and accurate strategy that is great for this factor combination!

These students ROCK!

Homework With Meaning!

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These two homework problems may seem too easy for a 5th grade student, but on examination of the thinking displayed, it is clear that a deep understanding of important concepts, like place value, are well understood by this student.

The first problem was quickly identified as a "subtraction" problem, but it was solved using simple addition. This young mathematician added up from 267 to 323 on an open number line. This is kind of like what folks did in the good old days before cash registers had built in electronic calculators. They would count back the change using landmarks along the way. This student also used landmarks (easily recognized and "easy to work with numbers"), as she first "jumped from 267 to 270. This allowed her to easily add on to 270 in order to get to 300. From 300, the jump to 323 was a piece of cake. Finally, all of the jumps were totalled, and the distance between $267 and $323 was found correctly.

Also of note are the sentence restating the prompt (question) and the matching equation, These also signify a real sense of math understanding.

On the second problem, the student added from left to right, and to me that is great! Adding the largest place values first makes it less likely to make a mistake of great magnitude. Using the traditional algorithm makes it more likely to make a mistake in the larger place values, and that's a real drawback to sticking with traditional algorithmic thinking, unless you REALLY understand the method well.

EDC and More!

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Today we spent time completing another day of our EDC curriculum, and we began to diagnose the needs of students. It was great to see that many students do remember the strategies that they have learned over the last five years. It was so great to see students paying such close attention to detail on our fourth day of EDC. I wrote, in black pen, some of the important concepts that we discussed during today's session on one student's awesome work :-}