Double-Trouble or Split-It!

Recently we have added a "game" to our EDC notebooks. Double-Trouble and Split-It are games that are designed to get students to strategically think about doubling and splitting numbers. These skills are crucial if a student is to build great mental number sense.

For example:

When I ask kids to split 1,700, many kids balk, because they see 17 groups of 100 as an odd number of hundreds. Most kids are not seeing this as 8 1/2 groups of 100. However, most students can readily replace 1,700 with 1,600 + 100, and they can easily split these numbers to get to 850! This still may not seem important, but since the product of 17 X 10 is so easily calculated, it seems logical that 17 X 5 (half as big as 17 X 10) should be easily found if 1,700 can be easily split.

In a similar way, being able to double numbers with ease leads to being able to double small factor pairs into larger factor pairs. If 7 X 2 = 14, then 7 X 4 is twice as big. So, 7 X 4 = 28. Then, 7 X 8 = 56, and 7 X 16 = 112...likewise, 17 X 2 = 34, 17 X 4 = 68...17 X 4 = 68, 17 X 40 = 680, and 17 X 80 = 1,360 ...all done through doubling.

When doubling a number, many students find it easiest to double the largest place value first and then work through the smaller place values.

Doubling a number like 1, 486 might look like 1,000 + 1,000 = 2,000...400 +400 = 800 ...80 +80 = 160...6 + 6 = 12...added together, the sum is 2,972.

These are VERY important concepts! Please practice. It's the kind of thing you can do in the car.

Division and Multiplication A Match Made in Heaven

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This student shows why a great understanding of basic multiplication is so vital to solving division problems. By knowing the simple factor pairs shown (15 X 1 =15 15 X 2 =30...) and knowing the relationship of these to larger landmark factor pairs (15 X 10=150 15 X 20=300), this division problem becomes an absolute snap to complete correctly. I highly advise students to write the first several multiples, and a few larger landmark multiples, of the divisor when solving almost any division problem.

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Boxes of Markers is a student sheet designed to introduce the idea of portioning items evenly as a form of division. Many students simply skip count by the number of students , in this case 23, in order to find out how many groups of 23 can be made by multiples of 70 markers. It might look like 23, 46, 69;at which point the kids realize that they have run out of markers. Most can see that each "skip count" represents a student with a marker. So, three skip counts would equal 3 markers per student.

The work above is from a student that immediately recognized this as a division problem that could be solved using easy multiples of 23 until the number of markers was exhausted. This really is the goal of this type of exercise, as it makes division and multiplication forever seen as related activities. Once the student that did this work adds sentences that explain the numerical answers, the work will be at standard to say the least!

Revisions Are Vital to Mastery!

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Students in our class are able to complete "revisions" of tasks to insure that concepts that were missed are cleared up or at least made more clear. Students do have to prove that they understand why they were wrong, and they have to show new work that proves that they really have reached an understanding of the concept.

In the top photo, the student explains what went wrong in the problem in the bottom photo. In this case there was a fairly major misconception about the formula used to compute the amount of money to subtract ($1,000 X the date or $25,000) from the original balance ($712,000). This misconception made the problem impossible for the student to solve. However, it would not be correct to assume that the student in question did not know how to subtract. Therefore, this thorough revision provides equitable remediation. Revisions are hard work, but they are well worth the effort!

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!