![]() The first three steps don’t replace the standard algorithm, they build understanding for it. ![]() 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. When we progress through concrete and representational activities before introducing the standard algorithm, students have a much better understanding of why the algorithm works. 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!)Īt the risk of sounding like a broken record, we also tend to teach computation at a very abstract level. There is a “times 2” relationship between the fourths and eighths. Notice that when students are building the equivalencies, they see the algorithm. Students use their fraction tiles to build the equivalency. In the example below, the second numerator is missing. Next, we can turn it into a puzzle by presenting students an equivalency with a part missing. So, first and foremost, we need to use manipulatives for our activities. Again, it’s because they haven’t seen it. In other words, they don’t understand equivalence. 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. ![]() Students are often only taught the algorithm-multiplying or dividing a fraction by a version of one, in this case two halves-for generating equivalent fractions: They are applying whole number thinking-eight is greater than 4 so one-eighth must be greater than one-fourth.Īnother hugely important fraction concept is equivalent fractions. When that happens, it means that those 5th graders can’t conjure up images of those fractions. We see that when 5th graders confidently tell you that one-eighth is greater than one-fourth. As a result, many students lack a true understanding of fractions as numbers. They fill out the number using the unit cubes, so they decompose 55 into 4 tens and 15 ones.Ī big problem with fractions is that we often teach them at a very abstract level, often using tricks or by jumping right to algorithms. In the example shown below, they are asked to build 55, but they can only use 4 tens. 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. If you said that it leads to understanding the standard algorithm for subtraction with regrouping, give yourself a pat on the back! Take a minute to think about why that is important. For example, 34 can be composed into 3 tens and 4 ones, but it can also be decomposed into 2 tens and 14 ones. Place Value-Decomposing in More than One Wayīoth 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. There is no additional cost to you, and I only link to books and products that I personally use and recommend. This post contains a ffiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. Be sure to read to the end to download an activity sampler for free! ![]() ![]() 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. ![]()
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