By adopting an unexpected tool—a multiplication chart—new teachers can assist pupils in understanding the difficulties of fractions.
Data about multiplication. In all the years I’ve taught math, I’ve never been able to figure out why we still refer to the rows and columns of skip-counted integers as multiplication facts. They seem to me to also be division facts. I can easily find 30 on my multiplication table when I multiply 5 by 6, however when I divide 30 by 5, the only possible result is 6. Finding 30 on the chart, finding 5 to the left, and then finding 6 above will all require my attention. Voilà. If I approach it that way, I may use the exact same graphic as a division facts sheet to get the solution.
This chart can also be put to various uses, which can assist new teachers in making more difficult arithmetic ideas simpler for students to understand.
The usage of fractions is one of my favorites. My children and I “evolve” the multiplication chart into a fractions chart throughout the course of the school year. I wish I could claim ownership of all the new applications for this evolved chart, but I am not allowed to. Every year, students find the most amazing applications for it in each aspect of fraction work you can imagine. Here are a few examples from my class that have done well.
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1. IDENTIFY EQUALS USING THE FRACTIONS CHART
It is common practice to find equivalent fractions by multiplying or dividing the numerator and denominator by the same integer. For instance, 2/4 can be obtained by dividing 4/8 by 2. Or, to find 8/16, multiply 4/8 by 2. These two fractions are equivalent to one half. It’s beneficial if brand-new instructors make sure to assist students in creating models of these counterparts.
You may also get equivalent fractions from the multiplication chart by simply aligning the numerator and denominator on the same column. Give it a go. Find the column on the chart where the numbers 4 and 8 are located. Equivalent fractions can be found by moving to the right or the left. Going right results in a larger number of smaller bits in the fraction. Going left results in smaller but larger parts in the fraction. Any fraction can use it. Go right one column for 6/24, two columns for 7/28, and three columns for 8/32. Get 5/20 by moving back one column. All of those fractions add up to 1/4 as you go on.
Utilizing a multiplication chart
2. Simplify fractions by using the fractions chart
To ensure that our replies are in the simplest form, my students and I frequently use the chart. Let’s say your response is 32/40. The greatest common factor, or GCF, is typically easy to find and “split out.” In doing so, we obtain the following results: 32 divided by the GCF of 8, which equals 4, and 40 divided by 8, which is 5. Therefore, in its most basic form, 32/40 equals 4/5. However, not everyone is able to perceive that as clearly, and even models can take a long time to construct. Just look for 32 and 40 in the same column on the chart. When you get to the first column, which provides you 4/5, move left.
You might obtain simpler fractions that aren’t entirely simplified if the fractions you want to simplify aren’t on adjacent rows. Using the same method, for instance, 36/48 will bring you left to 6/8, which is not entirely simplified. Then, if you take 6/8 in a column where they just so happen to be in an adjacent row, you may slide left to get the fraction 3/4, which is much simpler.
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3. COMPARE FRACTIONS USING THE FRACTIONS CHART
Each person has a method for comparing fractions. We construct number lines, create models, and—especially—find common denominators in my class. However, the fractions chart is available once more to assist. Take 1/4 and 1/5, please. As we all know, 1/4 is bigger than 1/5. Perhaps we have a mental image of a model. Perhaps we know there are fewer pieces in 4 than there are in 5, thus 1 out of 4 represents a larger share than 1 out of 5.
Comparing fractions with a multiplication chart
Perhaps by changing the denominator of each fraction to 20, we can determine that 5/20 is larger than 4/20. The fractions chart can be used as a fallback. Find 1 and 5 along with 1 and 4 in the very leftmost column. On the chart, move 1/4 and 1/5 over until you reach the same denominator, 20. You’ll observe that whereas 1/4 continues until it reaches 5/20, 1/5 stops at 4/20 beforehand. This also applies to other fractions. Examples are 2/5 and 2/3: 2/3 continues until 10/15, while 2/5 finishes at 6/15. Therefore, 2/3, or 10/15, is larger than 2/5, or 6/15.
4. ADD AND SUBTRACT FRACTIONS USING THE FRACTION CHART
Because 1/3 and 1/4 are two different sizes of fractions, if a student adds them together and gets 2/7, we know it is the incorrect response. In order to add parts of the same size, we often locate a common denominator to solve the problem. The good news is that our fractions chart provides an easy way to determine the LCM.
Chart for multiplication
The skip-counted numbers on rows 3 and 4 can be traced by students until they reach 12 as the LCM for both numbers. They can then calculate how many columns or right moves it took them to get there. This instructs them on how much to multiply the numerator and denominator by. In this instance, 3 became 12 by moving four columns, so 1/3 is multiplied top and bottom by 4 to become 4/12. In the meantime, we shifted three columns to reach a common denominator of 12 by adding 1/4. Therefore, 1/4 is multiplied by 3 on the top and bottom to get 3/12.
Students can, of course, just write out the multiples for 3 and 4. Even most math texts advise against it. However, not all students have the time or aptitude to do so. In my classroom, I’ve had a lot of students with special needs, and they do a terrific job with the fraction chart. For pupils, having a second source to reinforce these ideas is incredibly beneficial.
I’ve discovered that it’s crucial to view the chart as a resource rather than a scam. My kids appreciate that it’s frequently another tool to examine their work because we all believe in supporting our claims. Students still perform a lot of model work and estimation in my class, but we now learn how to use the multiplication chart to check our fraction calculations.
Why not let students use the chart to look for patterns in fraction equivalence, simplifying and comparing fractions, and adding or subtracting fractions rather than hiding the patterns that can be seen in the chart from them? Children typically struggle with these calculations, but using their fraction chart gives them another method to assess the logic and predictability of their responses.
