Finding the Lowest Common Denominator with the Ladder Method 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). Finding the LCD So, here’s how to find LCD with the ladder method. Let’s use the problem 9/28 + 13/42 Step 1: Put the denominators in the ladder.
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Use Task Cards in a New Way, to Provide
SelfDifferentiation and Promote Discussion
If you're like me (and so many other teachers), you know that task cards can be used in sooo many ways. From centers to Footloose (or Scoot) to exit tickets to entrance tickets to miniquizzes  the list is long!
However, if you're like me in other ways, you're always looking for something new and different. This year, my "new and different" was to start using task cards to play Truth or Dare in math and language arts classes! To use them this way, some of the task/question cards need to be written as True or False questions, which can make the questions just a little trickier and lead to more indepth thinking. I allow students to discuss the answers after the "official" answer is given, and depending on the question, students end up having great discussions! The Dare questions are a little harder, require more calculation or perhaps more verbal explanation than the Truth cards, and so they are worth more points. (Truth cards are worth one point while Dare cards are worth 2 or 3  I've even thrown in a 4pointer here and there.) What makes this game fun? Well, it's a little different  with the "dare" part in there. Students also don't always know how many points they're going to get to try, so that offers a little excitement. I like the fact that students can choose the type of question they want, so it allows for some selfdetermined differentiation...the choice gives the more hesitant students the chance to feel a little more confident. After creating several paper and pencil Truth or Dare games, my wonderful friend Leah (Secondary Resources for Social Studies & English) suggested that I make a Google classroom version, and I'm so glad I did! It's so easy to use and there's little to no copying needed! (A little copying if I want students to write their work/answers on paper; no copying if I want to share the Truth or Dare game in Edit mode and have students type their answers.) Check out the 2minute video below  it shows how the game works in Edit mode (there are one or two "slow to refresh" spots in the video, so please don't think it's not working:)
Check out this video to learn more about the way the game is played with paper/pencil  in any subject!
I hope you can use this game ideait can be used in any subject!
I'm just writing to express (again) how much I love the ladder method! :) If you haven't had the chance to use the ladder method before, I highly recommend it. In addition to helping students find GCF and LCM, I think it helps students start to see the relationships between numbers a little more clearly. It's very easy to see what factors different numbers have in common and how those factors 'contribute' to the LCM or GCF. I have used the ladder method for factoring as well, and let me tell you  students picked up the factoring concept MUCH more quickly than when I hadn't used it.
What I really like about this method is that the process is the same for each use, but the outside numbers are used differently. I like the fact that the continued use of the ladder method (for various reasons) leads to the students making greater connections between numbers.....finding factors seems to come more easily. Last spring, I wrote a guest post about the ladder method on Rachel Lynette's blog, so if you're interested in reading all the details, check it out here. I had shared a ladder method folditup in my guest post, but you can also find it below, if you'd like to download it. The latest ladder method item I've created is the poster/anchor chart. I had a few different ones last year, so I decided to consolidate! I haven't shown the students how to reduce fractions using the ladder method, but they'll see it on the poster next week. Then when we discuss fractions, it will already be there!
I've also created a fun Doodle Notes page to help students with the Ladder Method!
Click on the image, to see it on TPT.
I have never had the opportunity to teach slope before. This year is a first for several topics, among which was graphing functions  we did this last week, by using function tables to generate points, and I mentioned positive and negative slopes in passing. In getting more specific about slope, I knew that I didn't want to just tell the students about slope (and about yintercept)  I wanted them to figure out how the equation of a line can help them understand aspects of the graph of the line. But, I didn't know quite what to do. So, here's what I decided to do: I created a simple worksheet with four equations and their graphs and I simply asked students to find relationships between the numbers (and symbols) in the equations and the graphs of the equations. I didn't give much more direction than that. I had them each think about this, study the equations and graphs, and write their observation on their papers, without discussing with anyone, for about 5 minutes.
Then, I had them choose a partner to discuss their observations with, and to search for more ideas, for about five minutes. As they discussed, I circulated, listened, and asked questions. For the most part, they had written down how the negative/positive sign in front of the x relates to the slope, and many had identified the "added or subtracted number" as the yintercept. Some had noticed that when the coefficient is higher, the slope of the line is steeper. Next, I repaired the students using popsicle sticks, to allow them to share more ideas. At this point, I wrote on the board: "# that is added or subtracted" and "# in front of the x," and asked them to try to figure out what these numbers could tell them about the line (if they hadn't figured it out already). There weren't many students who made the connection that the slope tells how far to move horizontally and vertically between points, but there were several student whose observation was that the "m" is "how far apart" the points on the line were (they identified the points as where the line crossed the intersection of grid lines  I didn't put points on the lines for them). After the second pairing, I asked student to write their observations on the board and then we went through and discussed whether they were correct or not. Then we looked at the same lines graphed on the Smartboard, and we went through what the "m" tells us  we started with the fractional slopes and moved to the whole number slopes. In all, the entire lesson took about 35 minutes. I was really happy with the students' perseverance (for the most part) in trying to find what I wanted them to find:) I enjoyed their "aha" moments!
One "mistake" I made in the equations was that both equations with negative slopes also had negative yintercepts...this led some students to incorrect conclusions, so I changed that for next year. The fixed worksheet is here, if you'd like to use it:)
Today's thinking day is my favorite kind of day:)
Although we're trying to get through quite a bit of material before our state testing, we took some time today to explore triangles. I'm sure many of you may have done this exploration, but it was quick and fun, so I thought I'd share:) We explored the idea that the sum of the two smaller sides of a triangle must be greater than the longest side. I cut straws of three different lengths, and asked students (in groups) to use the straws to make a triangle.
In my first math class, I used straws that were cut to 2 inches, 3 inches, and 5 inches. These lengths, using straws, made it almost possible to make a triangle, even though it shouldn't have been possible. So, I had to insist that their straw ends be lined up perfectly. I wanted to use 3, 5 and 2 inches to show that even these dimensions won't make a triangle, because the sum is equal to the longest side, not longer than it. So, after understanding how precise they had to be and that they couldn't leave segment parts sticking out of the end of the triangle, they came to the conclusion that it couldn't be done. Next I gave the groups a new set of straws that were cut to 3 inches, 3 inches, 5 inches. In this case, they were excited to make their triangles in about 30 seconds! We then discussed why the 3, 2, 5 didn't work and worked our way to "creating" the rule.
For my next classes, I trimmed the 3 inch straws to 2 inches, so that my next classes would have more difficulty getting the ends to meet. It was so funny to hear their comments  "This doesn't work," "Is this a trick question?" "This is impossible!" And then, their excitement when they made the 3, 3, 5 triangle  "We did it first!"
For my next classes, I trimmed the 3 inch straws to 2 inches, so that my next classes would have more difficulty getting the ends to meet. It was so funny to hear their comments  "This doesn't work," "Is this a trick question?" "This is impossible!" And then, their excitement when they made the 3, 3, 5 triangle  "We did it first!" I think (hope!) that they understood the concept....we'll see tomorrow when we go over their homework:)
Remove One is one of my favorite games! It's a great way to teach probability and the students love it. I've been using it nearly every year since I was introduced to it through a program called the Mathline Middle School Math Project, sponsored by PBS (back in 1997?). I was involved in the program through my graduate studies at Allentown College of Saint Francis DeSales (now DeSales University). Anyway, this year, my student teacher is teaching our probability lessons; so she is the one who taught this lesson.
This is how the lesson works: 1. Students use a piece of paper as their "game board" and number the paper from 122 (or 212) . They then place 15 chips next to the numbers. They are told that they can place one chip next to every number and then place the extras next to any number they want. Or, they can leave some numbers with no chips and put several on others. Usually, they place the chips like those in the picture to the right. 2. Once students have their chips set up, the teacher rolls 2 dice and finds the sum of the numbers that are rolled.
3. If students have a chip next to that sum, the students may remove ONE chip from their paper (thus the name of the game Remove One).
4. Play continues, with the teacher rolling the dice and the students removing one chip each time the corresponding sum is rolled. The "winner" is the student who removes all of the chips first.
Without much class discussion, we play the game a second time. Normally, I just ask them to make some quiet observations to themselves before placing their chips again. Students typically notice that the sums of 6, 7, and 8 are rolled the most often and that 2 and 12 are usually rolled the least often, so they arrange their chips differently.
After the second game, we have a discussion about all of the possible outcomes (sums) one can get when rolling 2 dice. We also discuss how many ways there are to roll each of those outcomes, and what the probability is of rolling each sum. We find this probability in fraction form, and then often convert them to decimals and percents.
After this discussion, we play the game for a third time, and students' "game boards" often look a bit different!
This year, since I was observing rather than teaching, I was better able to hear some of the students' quiet comments to each other... "There's a better chance of getting a seven." "I'm not going to put any on 2, because it still hasn't come up."
When I started discussing this lesson with my student teacher, I searched for the lesson online, just in case it was around, and I found it right away. Click HERE to see the full lesson plan from PBS. Have you played this game? What other probability games do your students enjoy?
As I was thinking about school today, I was thinking about one of our next topics: equivalent expressions. (CCSS.6.EE.A3: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.)
Last year, I worked this concept in through the use of my daily math warmups, which brought the idea back time and again, and the students did well with it. This year, even though we will spend more time with direct instruction, I was thinking about other ways to use equivalent expressions, and I thought of using them for partnering cards! We can use them many times throughout the year as quick, random reinforcement. You can download these for free, if you'd like. There are 6 pages, with 3 sets on each page, giving you 18 sets (36 students). The set is in the download twice  once with a background and once without. ** Update: I used these today and they worked well! I printed the set with the background and quickly glued them onto index cards this morning, because I wanted to use them today. I'll laminate them before using again. I hope you can use them! 
AuthorHi, 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! Categories
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