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Cross Number Puzzles

Course: 
3rd Grade

This term names a format or teaching style that is used in Think Math!. The term, itself, may not be recognized by people who do not use Think Math!.

A way of modeling addition or subtraction so that the values in each place are computed separately before the final answer is found.

See also Cross Number Puzzles PowerPoints for professional development.

A surprise

Write down any two addition sentences, lined up one above the other, like this:

Then add the numbers in the columns:

Your results, like ours, should form a true addition sentence!

If you are teaching second grade, this is a nice opportunity to ask children to see if they can find any examples that don't work! Many children will be quite sure that there is some way to "foil the plan." (And some, of course, will find ones that don't seem to work! Hmm...)

You can also subtract one addition sentence from another, like this and the result will still be a correct addition sentence!

 

You can even add and subtract subtraction sentences: or

Why does this work?

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What is a cross number puzzle?

Like any puzzle, the CNP has “rules.” In Sudoku, every digit from 1 through 9 must appear once and only once in each row, column, and box. In CNPs, the total on one side of the heavy line in any row or column must match the total on the other side of that heavy line in the same row or column. That’s not a rule of math; just a rule of the puzzle.

In this puzzle looking only at the top row, we can make the two sides of the heavy line equal by filling in a 6:

Continuing in this way, we can fill in the other white boxes:

So far, this just illustrates the “rules of the puzzle.” The first thing that is mathematically significant is that the final box comes out the same however we calculate it (adding the 6 and 4 or adding the 7 and 3). That fact makes it possible to generate real puzzles.

or

Think Math! uses this structure, and its logic, for building the addition and subtraction algorithms.

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The logic of cross number puzzles

The reason why this works is important. Imagine, instead of numbers in the original puzzle, buttons sorted according to size and color (or any two attributes, like color and shape, or size and shape, or...[1]):

 



Filling in the right side of the heavy line, then, is like “summarizing the data,” answering how many small buttons (total) and how many large buttons. Likewise, the 7 and 3 on the bottom answer how many blue and how many gray.

 

 

So, whether we are asking how many small and large buttons, or how many blue and gray buttons, the total must be the same: it is all the buttons.

References:

  1. ↑ In Think Math!, see, for example, Grade K, Chapter 5, Lesson 8, or Grade 1, Chapter 4, Lessons 1 and 2.

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Real-life applications of this structure

This property is used in all kinds of bookkeeping situations, on the paper ledgers that people used before computers, and on today's electronic spreadsheets as well. In Think Math! grade 2, students use base ten blocks to keep a record of the stock in the Wonder Wheel Works, as the factory ships out orders during the day. The following record shows that the factory started with 100 wheels in stock in the morning, and shipped 6 during the day.

From this, we can fill out the number of wheels left at the end of the day.

If Wally, the owner of Wonder Wheel Works, opens another business, he can make a similar record for its stock.

Because both businesses are his, he might want to combine the records.

It makes sense that we can summarize the two businesses' transactions by adding their morning inventories to give the stock of the combined businesses, adding the shipments to find out what the two businesses shipped during the day, and adding "down" in the third column to find the combined stock remaining at the end of the day. It also makes sense that the bottom row, itself, "balances" -- the two sides of the heavy line are equal in quantity.

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Using cross number puzzles to build the algorithms of elementary arithmetic

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Addition and subtraction

Outline, to be expanded

  • Any order any grouping
  • Columns in the "body" of the puzzle restricted to place value (100n, 10m, r, where n, m, r vary from 0 through 9)

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Multiplication and division

See developing the multiplication and division algorithms in the article on multiplication and division.

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How cross number puzzles contribute to problem solving and proof

Like any puzzle (see article on puzzles), Cross Number Puzzles help students replace a "What am I supposed to do?" approach with a "What can I do?" stance. The key difference between an exercise and a genuine problem -- at least the way "real life" delivers problems -- is that problems don't ask what chapter you just studied, and don't come with step-by-step instructions for solving them. One must puzzle at them, and figure out where to start. Most puzzles have that feature; the Cross Number Puzzle offers an way of developing that inclination and facility while building age-appropriate arithmetic skill.

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Leading to algebra: the "elimination method" of solving simultaneous equations

In algebra -- in late middle school or beginning high school -- students eventually see "number sentences" that look like this:

If we knew what numbers x and y represented, these would just be number sentences, and we now know that we can add or subtract number sentences to get new ones that are true.

So let's subtract these!

This last sentence says that 3x = 12, so x = 4.

We could fit the number sentences into cross number puzzle formats:

But we could also re-interpret the equations visually. We can represent the top sentence with bags of marbles, five of one kind and three of another: . That top sentence says that in five blue bags (each of which contains the same number of marbles as the other blue bags) and three yellow bags (each of which contains the same number of marbles as the other yellow bags), there is a total of 23 marbles.

The bottom sentence tells us that in a different collection of blue and yellow bags, there are 11 marbles.

If we remove that collection of bags from the collection described in the first sentence, there will be three blue bags left:

Because we have removed 11 marbles in the process, those remaining three blue bags must contain 12 marbles. So each one contains 4 marbles! And now look back at the original picture. If each blue bag contains 4, and that whole collection of bags contains 23, how many must be in each yellow bag?

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Introducing cross number puzzles to 5th grade students who have not had Think Math! before

Because the multiplication and division algorithms are built on the logic of the cross number puzzles, students who are seeing them for the first time in grade 5 should be given the chance to understand the logic. It should take only about 10 minutes to go through that logic, but then, as students return to the fifth grade work, it will still be important that the children find it easy enough to check their work from time to time, so that they are not just trusting that the "corner cell" (the single cell that is isolated by the heavy horizontal and vertical lines) comes out the same regardless of which addition (vertical or horizontal) one does.

For students who struggle with two-digit addition, have some one digit cross-number puzzles to supplement Activity Master 1.

You can make up your own one-digit examples. Alternatively, take some from the 3rd grade and use them. Some of these puzzles use base 10 blocks to ease the computation while children are becoming familiar with the concept of how the puzzle works. You’d have to judge whether or not that’s suitable for your students. Once students see how the puzzle works, then doing the two digit addition, perhaps with the help of the Number Line Hotel, is fine.

Children must be solid enough in the facts they need so that they can understand and enjoy any activity in Think Math!. As a general rule it is important to keep computation relatively easy while children are learning a new task. Brief, lively, daily, focused work on skills builds fact competence and, over time, allows the children to enjoy, explore, and understand the mathematics.

Teaching strategy: At one of the schools that was implementing Think Math! for the first time this year, and that had been failing the state assessment regularly in years past, the math specialist (Cindy Carter) especially focused on the upper grades, with particular attention on 5th. Because we didn’t want teachers replacing lessons with outside content (slowing kids’ progress and assuring that they don’t catch up, and also most likely changing the tone and spirit of the mathematics), she instituted a number of activities that supplemented the work going on, but “stole” the time from what is generally “lost time” anyway during the day. One of her inventions was the "Fact of the day.” One Fact-of-the-day activity was that for a whole day whenever anyone saw a 3rd – 5th grader—at the water fountain, on line, at recess, anywhere—and said any single digit (e.g., 3) to them, that child would be expected to answer with the number (e.g., 7) that would be needed to make 10. Two or three minutes in class in the morning to establish the game, and for the rest of the day this is like a secret handshake, and kids love it. It is a great way to get the pairs to ten solid quickly. The next day there would be a different secret handshake—different enough to be interesting but same enough to build mastery—for example, pairs (only of multiples of 10) to make 100 (e.g., call 30, answer 70). A few months of these activities are in the process of being posted on the web. In this school, where kids were quite weak initially in skills, most of the kids became very competent and quick at mental calculation, which eases the burden of doing all of the more conceptually deep understanding.