Finding the Lowest Common Denominator with the
What's the most challenging math topic to teach/most difficult for your students to ‘get'?
This was my question in a recent Instagram survey. I got a variety of responses, but the one that came up most often was fractions – remembering the ‘rules;’ students finding common denominators when they were multiplying; students (older students) not being able to find a common denominator; and so on.
So, today, I’m going to share how to use the ladder method to find the lowest (least) common denominator, and hopefully, if your students have struggled with this, it will help them (and you!). Before I explain how it works, I want to share that I've used the ladder method for several years, after many years of teaching GCF and LCM the ‘traditional’ way - the way I’d been taught! And during those years, I’d often get frustrated by the fact that students would miss the GCF because they missed factors, or they couldn’t find the LCD because the numbers got too big so they just multiplied the denominators…..or they listed out the multiples, but made a mistake in one list, and so they never found an LCM/LCD. I'm sure you know what I mean!
The ladder method took these issues away, and it also added something I didn’t initially expect – it appeared to improve number sense for many students who struggled with their multiplication facts or with the idea of finding factors and multiples. It helped them understand HOW numbers were related to each other by making the breakdown of the #s more visual (using prime factorization does this as well, but the ladder method provides a little more organization to the process, and I think that’s helpful).
I used this week's problem in class today (6th grade), for early finishers. Because we haven't gotten too "into" a particular topic, I made the problem a mix of operations - mostly division and multiplication, but I saw students using addition as well.
I really enjoy talking with my students about what they are thinking when they try to solve problems, for a few reasons - because 1) they think about problems in a different way than I do; 2) it makes me rethink the wording of the questions I ask (which makes me improve); and 3) I learn that there will be several ideas to share with class.
I noticed a few different things when the students were solving the different parts of this week's problem:
For part A, I multiplied 85 times 3 to get the total number of cookies and then divided by 24 (when I wrote the problem, I wanted the students to have to interpret the quotient, so I approached it with a desire to use division). And most students did the same thing (except for the few that multiplied 24 x 3 - that gave me some good info: -), but one student was just sitting and thinking, so I asked him what he was thinking. He started to say he divided 24 by 3 and then paused - I almost interrupted his thinking to redirect him to my way, but I successfully restrained myself, and asked why. He said he was thinking about how many baggies could be filled with one batch, and since the numbers worked nicely, he could definitely say that one batch would fill 8 baggies. I really liked his thinking process, because it hadn't occurred to me to do it that way. Now, if the numbers hadn't worked out evenly, it might not have been the best approach, but we can expand our class discussion to explore that. After deciding he could fill 8 baggies per batch, he added on sets of 8 until he reached the correct number of batches.
As some students worked on part C (below), I started to think that I should adjust the wording of the problem. When I wrote the problem, I thought it would be clear that the number of cookies for part C was the same as part A, but some students thought of the part C as using 85 baggies of 2 cookies (same number of baggies), instead of using the same number of cookies. As more students worked on it though, other students seemed to understand that the number of cookies should be the same as the original number they were working with, so I haven't changed it yet. If you use the problem, please let me know what you think.
Again, a few students approached this part in a different way than I did - they said that in both cases, the cookies cost 25 cents each. Using this reasoning, some students said the cost was the same, while others did not - again, a great opportunity for discussion, both in small groups and as a whole class.
To see and/or use the entire problem and answer key, click on the link below the picture.
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Have a great week!
Early last week, I was trying to think of a different kind of activity to help students that needed more reinforcement of the order of operations, and I decided to make a sequencing activity. I haven't tried this before, so I wasn't sure about the best way to design this, but it ended up working quite well. Here's how I put this together and used it:
For the activity, I created 8 different
expressions, and then typed out the steps to simplify each expression. I copied the expressions and steps onto different colored papers, so that two expressions would be on the same color. I cut the steps apart into strips, and then put two expressions and their steps (of the same colored paper) into a baggie...I figured if I put only one equation in a baggie, the activity would be too simple. If I put two expressions of two different colors, it would be too easy. So I went with two expressions of the same color.
I decided to put 3 baggies (6 different expressions and their steps) into a manila envelope for each group. Groups were mostly just partners, with an occasional group of 3.
I had typed directions, and when I gave students their envelopes, I asked them to do their best to follow those directions before asking for clarification (some of the students worked on this activity, while others completed different activities, so I needed them to try to work through the directions themselves before I got to each group to discuss with them). Some students did need additional instruction, while others did not.
After students put the steps into the correct sequence (shown in image above), they had to write those steps onto a recording sheet, pictured here.
In each of my classes, students worked on this activity for about 15-20 minutes. Some groups completed all 6 expressions, while others completed only 2-3. A few more minutes would have been helpful for those students who didn't complete as many expressions, but I can revisit the activity with those students this week.
I will definitely use this again next year:)
Hi, I'm Ellie! My mission here is to support teachers as they work to provide engaging, meaningful experiences for their students. I've been in education for 25 years, teaching all subject areas at both the elementary and middle school levels, and am here to share what I've learned through those years, as well as what I continue to learn. I hope you'll find some ideas or resources here to help you out!