# Math Puzzles for the Win!

Who doesn’t love a good puzzle? We engage students in algebraic thinking and productive struggle when we use math puzzles! Read about various types of math puzzles you can use to challenge and engage your students. Be sure to read to the end to download an activity sampler for free!

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Place ValueโDecomposing in More than One Way

Both the CCSSM and the TEKS (Texas standards) now include verbiage for 2nd Grade indicating that students should be decomposing numbers in more than one way, as so many hundreds, tens, and ones. For example, 34 can be composed into 3 tens and 4 ones, but it can also be decomposed into 2 tens and 14 ones. Take a minute to think about why that is important.

If you said that it leads to understanding the standard algorithm for subtraction with regrouping, give yourself a pat on the back!

We can make a puzzle out of practice by asking students to build a number with their base-10 blocks, but we specify the amount of tens (or hundreds) they can use. In the example shown below, they are asked to build 55, but they can only use 4 tens. They fill out the number using the unit cubes, so they decompose 55 into 4 tens and 15 ones.

### Equivalent Fractions

A big problem with fractions is that we often teach them at a very abstract level, often using tricks or by jumping right to algorithms. As a result, many students lack a true understanding of fractions as numbers.ย  We see that when 5th graders confidently tell you that one-eighth is greater than one-fourth. When that happens, it means that those 5th graders can’t conjure up images of those fractions. They are applying whole number thinkingโeight is greater than 4 so one-eighth must be greater than one-fourth.

Another hugely important fraction concept is equivalent fractions. Students are often only taught the algorithmโmultiplying or dividing a fraction by a version of one, in this case two halvesโfor generating equivalent fractions:

What we find, however, is that even when students can accurately generate equivalent fractions using the algorithm, they often don’t realize that the two fractions represent the same part of a whole or the same point on a number line. In other words, they don’t understand equivalence. Again, it’s because they haven’t seen it. So, first and foremost, we need to use manipulatives for our activities. Next, we can turn it into a puzzle by presenting students an equivalency with a part missing. In the example below, the second numerator is missing. But we can move that unknown around. Students use their fraction tiles to build the equivalency.

Notice that when students are building the equivalencies, they see the algorithm. There is a “times 2” relationship between the fourths and eighths. The eighths are half as big (denominator) so you need twice as many parts (numerator) for the fractions to be equivalent!

### Area Models for Multiplication (and Division!)

At the risk of sounding like a broken record, we also tend to teach computation at a very abstract level. When we progress through concrete and representational activities before introducing the standard algorithm, students have a much better understanding of why the algorithm works. What you see below is a progression for teaching multi-digit multiplication from concrete (base-10 blocks) to representational (the area model) to the partial products algorithm and finally to the standard algorithm. The first three steps don’t replace the standard algorithm, they build understanding for it.

Notice that when we use the area model to multiply, we know the two factors, the length and the width, and we’re looking for the product, the area. We can turn an area model into a puzzle by manipulating the known information. In the example shown below, we know one of the factors (234), but not the other. Given a part of the product, however, we can figure out the rest of the puzzle.

While most commonly used for multiplication, the area model can also be used for division. Using the area model puzzles helps make the connection to division.