Are students suffering in math because we no longer memorize multiplication tables?Understanding multiplication is much more valuable for students to apply...
The post [Updated Post] Does Memorizing Multiplication Tables Hurt More Than Help? appeared first on Tap Into Teen Minds.
]]>Post update: Saturday May 21st, 2016
This post was originally written two years ago and admittedly took an angle that downplayed the importance of automaticity of multiplication skills which was not the original intention. Please read on as I attempt to clarify the original intent of the article.
As you may have heard in the Globe and Mail or in my recent post, Ontario Education Minister, Liz Sandals recently tossed a comment into the media about the need for students to know their math facts:
That’s actually a great homework assignment: Learn your multiplication tables.
Liz Sandals – Ontario Minister of Education
It seems that whenever things aren’t going well in the world of math education – or more poorly than usual – people are quick to claim that it is because students don’t know their basic math facts; namely, memorizing multiplication tables. There is an obvious benefit to knowing your multiplication tables when solving problems, but is it only the memorization of basic math that students lack?
If we recall what memorizing multiplication tables looked like when we were in school, you might picture flash cards, repeating products aloud or writing out your 7’s times tables repeatedly until it was engrained in your mind. There is no doubt in my mind that having multiplication tables memorized allows for making calculations without taxing working memory.
Although the debate between traditional and reformed mathematics has been going on for decades, I think both groups ultimately want the same thing: students to be proficient in mathematics. However when people make statements about memorizing math facts or “going back to the basics”, people likely have very different interpretations as to how we should get there.
There are no silver bullets in math education and the memorization of multiplication tables alone will not solve all of the problems our students face. That said, many suggest that Ontario needs to “go back to the basics”, but did they ever leave? While the memorization of multiplication tables is not explicitly stated in the Ontario math curriculum, there is a huge push to develop a deep understanding of what it really means to multiply and in time, this should promote automaticity. As I mentioned in this post, shortcuts in math are only effective when you know how to take the long way; learning multiplication is no exception.
Check out the wording of an overall expectation in the Grade 2 Number Sense and Numeration Strand of the curriculum:
solve problems involving the addition and subtraction of one- and two-digit whole numbers using a variety of strategies, and investigate multiplication and division.
We could easily change that overall expectation to say:
know multiplication tables up to 5 times 5 without a calculator.
Which one would be more valuable to the student?
The grade 1-8 Ontario math curriculum contains the word multiplication a total of 47 times, which hardly suggests that multiplication is not a focus throughout elementary.
The curriculum doesn’t suggest that students should not know their multiplication tables, but instead has a greater focus on developing multiplicative skills and understanding. Digging deeper into the document, the words “variety of” occurs in the curriculum a total of 179 times and suggests that much more attention should be spent on exposing students to a variety of strategies, using a variety of tools, to show understanding a variety of ways. If done well, I would have to believe students would have a very deep understanding of math concepts including multiplication.
Over time, I’d like to think that an effective implementation of the Ontario Grade 1-8 Math Curriculum would allow for committing multiplication tables to memory over time.
The curriculum document doesn’t explicitly state the memorization of multiplication tables as an expectation, but it does require that concepts be delivered using a variety of strategies and tools/manipulatives. Where the problem may lie is how the curriculum expectations around multiplication are interpreted by the teacher.
What if the teacher isn’t comfortable teaching math? What if they only really understand one way to multiply? Does “a variety” mean three strategies? Four strategies? … Ten?
These are just a few of the questions that pop into my head when I read the curriculum and ponder some of the challenges we continue to experience in mathematics education.
Tom is a teacher, who admits not having a real passion for teaching math. When he teaches multiplication, it might look like this:
Great for patterns and provides a strategy for students to “get there” if they are stuck, but might not be enough for students to build a deep conceptual understanding.
We all know the algorithm and it is likely that teachers will use it moving forward as students begin working with larger numbers. Works like a charm if you do each step correctly. Something very interesting about the algorithm is that it can often be taught as a procedure and nothing more. Knowing how to use the algorithm without a conceptual understanding of how it works can be useful, but possibly only slightly more useful than typing the expression into a calculator. Miss a step or press the wrong button without a deep understanding of how the algorithm works and you might be out of luck.
Rather than debating over whether memorizing multiplication tables is necessary, maybe a better question might be:
What is the most effective way to have our students build automaticity with multiplication and other math facts?
Can we promote the automaticity of multiplication facts by building a deep conceptual understanding and spacing the practice meaningfully throughout our math curriculum? Would helping students visualize how multiplication works allow for this memorization to build over time?
Here are just a few visual strategies that might help students build a deeper conceptual understanding than simply learning their times tables through mass practice.
Chunking through the use of an area model is a skill that can be used not only to lower the bar for all learners at every level of readiness to begin multiplying with confidence.
If we promote the use of multiple strategies to complete mental math rather than turning to the calculator, then I believe students will value the skill of multiplication automaticity and feel the need to use it as an efficient strategy. I must be clear in saying that this will not happen unless we promote the use of mental math and multiplication strategies throughout the entirety of our math courses.
If multiplication strategies are used consistently and with purpose, students can build their multiplication automaticity in order to create “friendly chunks”. The model above would suggest that this student may be comfortable multiplying by groups of 10 and thus can save a ton of work by chunking in such a manner until they feel comfortable with multiplying by groups of 12.
While I think there are huge advantages to having multiplication tables memorized, I think we need to be cautious about how we plan to get to that end goal. If we do not take time to clarify the how, we risk promoting the use of repetition through mass practice without promoting a deep conceptual understanding of what multiplication really represents. Even worse, promoting memorization without meaning could push students to dislike math and deter them from building a productive disposition towards the subject area we enjoy so much.
Should We Stop Making Kids Memorize Times Tables? – Jo Boaler – US News
Automaticity and why it’s important to learn your ‘times tables’ – Dr Audrey Tan
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]]>Angle geometry including the Opposite Angle Theorem and patterns involving parallel lines cut by a transversal can be boring. Spice it up with some context!
The post Railing Reconstruction appeared first on Tap Into Teen Minds.
]]>From Grade 7 to 9 in the Ontario Math Curriculum, student understanding of angle geometry is extended to include (but not limited to) the following specific expectations:
Grade 7
Grade 8
Grade 9 Applied
The expectations extend nicely through these three courses, but the topics alone can often be a bit of a drag. While I’m still going to approach finding missing angles like a “puzzle” for my students as they seem to enjoy it, I have felt pretty uninspired when trying to introduce the idea of finding what would seem to be completely random missing angles.
When I was asked by Keri K. from Kingsville PS to join her class as they prepared to introduce some angle geometry, I figured I had better put my thinking cap on to attempt making the intro to this unit of study more meaningful for students than I might have in the past. So, I thought we would try to get kids thinking about what they notice and wonder via the 3 act math task approach.
Here we go.
Before we started, I ask students to think about what they notice and what they wonder when they watch the following video.
I played the video a second time. I then asked students to take 60 seconds to create a list of what they noticed and wondered on a piece of paper.
Depending on the task, sometimes students nail the question you’re looking to focus on that day, but other times they won’t. Either way, we can always try to answer some of the questions they have shared and nudge them towards the question we are looking to tackle for the day.
After some good discussion, I move on to the act 2 video.
Or, here’s a screenshot:
Students are now fully aware of the question to ponder:
How big is the angle?
You can frame this as a challenge or possibly play up a story involving the need to reconstruct the railing in your home. I think I prefer posing it as more of a challenge since students have not yet been exposed to any angle theorems involving parallel lines with an intersecting transversal line.
Because this was not my own classroom, I thought I should start with a low floor and ensure students were comfortable with benchmark angles prior to moving on. So, we did a few warm-up questions in Knowledgehook Gameshow prior to attempting to tackle our main question from the video.
Here’s the gameshow I used: [play as student | view/clone as a teacher]
Note that I didn’t do the KH Gameshow prior to introducing the task because that would have got them thinking immediately about angles and the whole notice/wonder piece would probably be a dud. If your students have some knowledge of benchmark angles, then you might consider skipping the warm-up.
Already on the desks of the students were bins including paper, scissors, markers, etc.
I said the following:
Friends:
Take a sheet of paper and fold it twice to create an “X” with the folds.[I held up a piece of paper and folded it over twice to create an “X”]
It doesn’t matter how wide or thin your “X” is and it will probably look different than your neighbour.
Now, take a marker, start at any angle you’d like and number the angles in order, clockwise.
[I modelled this.]
Cut out your four angles and piece them together on the desk.
Take one minute on your own to ‘play’ with the pieces. Jot down anything you notice. Do you notice any relationships?
Now, have a conversation with your neighbour. Ensure both of you have a turn to speak.
After sharing out with the group, students noticed that the sum of the angles is 360 degrees and that the opposite angles were equal. And so, the Opposite Angle Theorem (OAT) was born.
Then, I asked students to do the following:
Friends:
Take another sheet of paper and fold it twice to create two parallel lines with the folds.[I held up a piece of paper and folded it over twice.]
It doesn’t matter how far apart your parallel lines are and they may look a bit different than your neighbour.
Now, fold the paper once so a single fold cuts through both of the parallel lines any way you’d like.
Now, take a marker, start at any angle you’d like and number the angles in order, clockwise.
[I modelled this.]
Cut out your four angles and piece them together on the desk.
Take one minute on your own to ‘play’ with the pieces. Jot down anything you notice. Do you notice any relationships?
Now, have a conversation with your neighbour. Ensure both of you have a turn to speak.
Similar discoveries to the previous paper fold activity were discussed. We consolidated as a group and then determined that if we have parallel lines cut by a transversal, we basically have two “groups” of opposite angles.
Then, I showed the act 2 video again. Students were now ready to head out on their way to solve for the missing angle.
After students solved and shared out their work, we could consolidate the learning from this task via the Act 3 Video.
Or the animated gif:
If you try this out, let me know what you did differently in the comments!
Click on the button below to grab all the media files for use in your own classroom:
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]]>When delivering teacher workshops, I almost always include a 3 Act Math Task as a way to model the 4-part math lesson framework and for teachers to experience the power of introducing new concepts by leveraging curiosity. Although my workshops spend a significant amount of time exploring interesting lessons, teachers often want a formal description […]
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]]>When delivering teacher workshops, I almost always include a 3 Act Math Task as a way to model the 4-part math lesson framework and for teachers to experience the power of introducing new concepts by leveraging curiosity. Although my workshops spend a significant amount of time exploring interesting lessons, teachers often want a formal description of what defines a 3 Act Math Task from any other effective task. I thought it was about time that I pull apart what I believe makes an effective 3 act math task and why I have found them so useful in my classroom.
As many are aware, the Three Acts of a Mathematical Story approach shared by Dan Meyer when he began creating media rich math tasks that were structured using the effective storytelling technique of “acts” where:
The hook that introduces the storyline, often leaving you curious with questions you are interested in answering and rising tension.
Tension continues to rise to its highest point as more clues are revealed to help lead you to the climax of the story.
Curiosity is satisfied with answers to your questions and tension is restored to its original state prior to the first act.
Click here for an example of how we can apply the three acts storytelling technique to a math task.
It’s no secret that math isn’t high up on most people’s list of things they enjoy to do in their spare time. Worse yet is trying to teach new concepts to students when more often than not, there are no immediately apparent reasons why we need them. As students move from junior grades into intermediate (or from elementary to middle school) where concepts become more abstract, the purpose of mathematics is less obvious to the learner and interest in the subject often decreases. As student engagement in mathematics falls, teachers are left wondering how they can recapture that abundance of natural curiosity younger children seem to possess.
When discussing the topic of student engagement, I often hear math teachers firmly state: “teachers are paid to teach, not to entertain” – something I’d firmly agree with. However, I do believe that it is our duty as teachers to seek out ways to capture the attention of our students by making math class a compelling environment. Just as storytellers and filmmakers are forced to work hard to hook in their audience, it would make sense that math teachers, especially in the middle to high school grades, would have to work even harder to hook in their respective audience. It is easy to forget that many students never willingly signed up to sit in our classroom – they had to. I think educators need to work just as hard to engage our audience as those in other industries do – some for good, others not so much – if we want to have any sort of influence on our learners.
While there are many beliefs as to why the 3 act math task approach is beneficial, here are just a few of many reasons I’ve compiled over the past couple years as a frequent user of this approach.
While I believe the 3 act approach could be used to introduce any question in any subject area, they are best known as a way to deliver media-rich questions in the math classroom. By showing an image or short video, you can quickly spark curiosity in all learners regardless of their mathematical ability.
Dan Meyer and others using this framework typically encourage students to share their thinking throughout the process. Sharing what they notice or wonder, what questions come to mind, predictions and extraneous variables that could affect outcomes are just some of the common discussion starters that can be used.
Because most 3 act math tasks model math in the world around us, often times applying a standard formula or algorithm may not accurately calculate what will actually happen in the given scenario. Encouraging students to think outside of the box and consider what would happen in the real world vs. in the “fake world” math questions often have us living in.
Those who are lucky enough to have the sense of sight are constantly processing images of the world around them. It seems logical that providing students with a visual to better understand mathematics, especially when the content becomes increasingly abstract over each grade level.
Most tasks provide an opportunity for all students to participate and can be extended to more open ended (or open middle) questioning via “sequels” to the original question.
While there are many different ways to engage students in math class, it is clear to me that using the art of storytelling such as Dan Meyer‘s 3 act math approach is a solid way to raise student interest, curiosity and engagement in my mathematics classroom.
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]]>End the ongoing debate; inquiry and explicit instruction both serve an important role in the math classroom to build both conceptual and procedural fluency.
The post Is There a Best Way to Learn Mathematics? appeared first on Tap Into Teen Minds.
]]>The ongoing debate over whether math concepts should be taught using a discovery approach versus explicit instruction has been going on for years. These “math wars” (see examples here and here) typically paint effective instruction in mathematics as an either/or situation between the two extremes. Some might interpret inquiry based instruction as necessary to construct conceptual understanding prior to committing algorithms to memory, whereas others might conversely interpret explicit instruction as the requirement to first build procedural fluency necessary to enable a student to better understand why a formula works later. Regardless of where you stand on the continuum, Dr. Daniel Ansari does an excellent job in this video discussing why these math wars between inquiry based learning and explicit instruction should end.
While I have personally stood on both sides of this war – early in my career using heavy doses of explicit instruction and most recently serving up an almost complete inquiry-based approach – only now am I able to see that neither extreme is appropriate. Some mathematical concepts are likely best suited for a direct instruction delivery, while others may seem more natural to introduce from a discovery standpoint. More importantly, I think the decision as to which topics are taught from different teaching styles could and should be different based on the unique characteristics of the educator such as: personality, teaching style, and their own interpretation or understanding of the specific learning goal.
With that said, I will be honest in saying that I almost always prefer when I can find an interesting way to introduce a task from an inquiry approach. Unfortunately, teacher interpretation of what an inquiry math lesson looks like varies drastically. According to E. Lee May, from Salisbury State University, the definition of an inquiry based math lesson is:
…a method of instruction that places the student, the subject, and their interaction at the center of the learning experience. At the same time, it transforms the role of the teacher from that of dispensing knowledge to one of facilitating learning. It repositions him or her, physically, from the front and center of the classroom to someplace in the middle or back of it, as it subtly yet significantly increases his or her involvement in the thought-processes of the students.
As cited on inquirybasedlearning.org
This leaves many different possibilities for lesson ideas to be defined as “inquiry”. For me, I keep a 4-part math lesson in mind when planning.
While I always attempt using the 4-part math lesson framework when possible, it is not always easily achievable, necessary, or appropriate for every concept. Currently, I do not have every lesson planned using this framework, but I am always trying to think of ways to use this approach to improve my lessons, where possible.
So regardless of whether you feel that you sit more on one side than the other, consider exploring ways to use inquiry and explicit instruction where appropriate in your classroom. A good place to start might be trying to structure a 4-part math lesson or just exploring some 3 act math tasks by Dan Meyer and others from around the web. Here are a couple tasks I’ve created that do some of the heavy lifting for the first two parts of the lesson such as Walk-Out, Stacking Paper Tasks, and Tech Weigh In.
Introducing Distance-Time Graphs & Graphing Stories This 3 act math task was created as a simple, yet powerful way to introduce distance-time graphs and other various graphs of linear and non-linear relationships between two variables. In particular, I'm looking to address the following specific expectations from the grade 9 math courses in Ontario: Grade 9 Applied LR4.02 & Grade 9 Acade...
If you have explored using a 4-part math lesson, please feel free to share your reflections in the comments section.
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]]>Exploring fundamentals that help investors maximizing returns in the stock market and how they apply to professional development and teaching.
The post Earning Versus Learning appeared first on Tap Into Teen Minds.
]]>In recent months, I’ve been doing quite a bit of reading about investing in the stock market. I’ve always had an interest in finance and saving money, but I have never really had a deep understanding of how to be a successful investor beyond handing my money over to a financial advisor or picking random mutual funds. As I dug deeper into the investing world, I began to notice that some popular keys to investing seemed to mirror a reasonable approach to implementing effective teaching strategies.
Regardless of whether you feel like you are a pro-investor or a newbie, my guess is that many have at least heard the popular investing saying: “buy low, sell high!”
Seems like a pretty logical concept; in order to make money, you need to buy something at a price and be able to sell it for a higher price.
This concept parallels what an educator is trying to achieve when introducing a new teaching strategy in the classroom:
If I invest time and effort into a new teaching approach, then learning outcomes will improve for my students.
Despite how simple this may sound, things are rarely as easy as they seem. When investing, many are not comfortable making their own investment decisions. A similar lack of comfort is common amongst many educators when conversations about making changes to their teaching practice are explored. My hunch is that while the concept of buying low and selling high is fundamentally straightforward in both cases, determining what to buy when investing or which strategy to use when teaching becomes much more convoluted once you begin to dig in.
If you’re thinking it can’t be that difficult to find a good company to invest in or an effective teaching strategy to use in the classroom, then you might want to think again. There are over 2,800 different companies listed on the New York Stock Exchange and over 150 teaching strategies listed on The University of North Carolina at Charlotte’s learning resource page. Don’t forget that there are a large number of stock markets around the globe and a wide range of other influences on student achievement that we may not even be aware of.
In the investing world, one could invest in Apple (APPL) at about $100 a share or Microsoft (MSFT) at $50 per share, but which is the better pick? Does Apple having a value double that of Microsoft make it a better pick or does Microsoft seem like a deal you can’t refuse?
In the learning world, a teacher could invest all of their time and effort into direct instruction or an inquiry-based approach, but may still feel uncertain as to whether or not they are maximizing learning outcomes. Is direct instruction a better teaching strategy due to the highly structured and guided approach or does inquiry-based strategies provide a better opportunity for students to construct their understanding of content?
Making a decision based on speculation to invest in a company or in a strategy without specific and logically sound reasoning increases risk. If we allow our decision making process to be influenced without an appropriate amount of evidence, we are ultimately taking a gamble in hope that probability will be on our side. What constitutes an appropriate amount of evidence could look very different when comparing investing and teaching. For example, I think that while many teaching strategies are research-based, there are many other strategies teachers may explore based on their own creativity and an intuition that the strategy could have positive effects on student achievement. On the other hand, some in the financial world believe that intuition can be useful to identify a good buy, but I don’t believe there are any investors claiming to make decisions without thoroughly conducting the necessary research to confirm that their thinking is more than just a hunch.
With that said, my hypothesis is that investors would be less likely to gamble on an investment based purely on intuition than an educator would on a teaching strategy because the investor has more to lose in a very short amount of time. We as educators are protected because it is almost impossible to go “all in” on a strategy and lose it all. A particular teaching strategy may yield no academic gains, but it would seem highly improbable that there are too many (any?) strategies that would cause students to regress.
Despite the seemingly limited risk educators take on when exploring new teaching strategies, teacher resistance to change is common across many school districts. This seems logical since every teacher believes they are doing the best they can by using effective strategies they feel will be most beneficial to student learning. Conversely, I have never met a colleague who openly admitted to doing less than their best by intentionally using ineffective strategies to limit learning outcomes.
So what do education policy makers, districts, consultants and instructional coaches like myself do? We bombard teachers with a huge list of “research-based teaching strategies” to diversify the lesson delivery in classrooms. Just as an investment advisor would tell you to diversify your investment portfolio to limit risk, it would seem that we promote the same logic in education reform.
If exploring new teaching strategies produces minimal risk, why do we need diversification?
Rather than sharing a variety of teaching strategies that educators can use at their discretion, all too often system frameworks are created with an expectation that teachers will follow specific lesson formats regardless of whether the approach aligns with their current (hopefully, ever-changing) beliefs. While research indicates the benefits of using such teaching strategies, the risks may outweigh the rewards when teachers are forced to use them against their will. To make matters worse, we often exacerbate a group of already jaded educators when we forget to celebrate the many great things they are already doing in their classrooms on a regular basis.
This self-induced need for wide diversification in the classroom also arises on Wall Street. The inherent need for diversification when building an investment portfolio has been rejected by some of the most successful stockholders of all time as a “recipe for mediocrity“.
Wide diversification is only required when investors do not understand what they are doing.
What Buffet and other successful investors like him advocate for is focus. Rather than investing into a large list of companies – some that will go up in value, some that will go down – do the necessary research to understand the difference and invest in only those that will maximize the return on your investment.
This thinking can be applied to enhance our teaching practice. Rather than trying to do too many research based strategies without understanding their expected outcomes or how to maximize those outcomes, it seems more reasonable to focus on building our portfolio of teacher moves gradually and with intent.
While the word “formula” is probably more common to us math teachers than those in other subject areas, there is no denying the human tendency to want a simple list of steps to follow in order to achieve a desired result. Whether it is The Formula for Losing Weight, 10 Steps to Getting Noticed by Your Crush, or Five Steps to Make Your House Hunt a Happy One, people are always looking for a quick-fix to their problems. The world of finance and education are certainly not exceptions to this rule.
Unfortunately, there is no formula to getting rich or to being the best classroom teacher. If there was, we’d all be rich and our problems in education would be solved.
Although there is no exact formula that will lead to successful investing or teaching, we can use the following fundamentals to help guide our work:
Avoid widely diversifying without having specific intent or purpose for each of your selected investments/teaching strategies.
The only way you can maximize reward and minimize risk is to deeply understand the investments/teaching strategies you use.
The formula is knowing that there is no one formula for success.
Whether from the perspective of a classroom teacher or an instructional coach, being mindful of these fundamentals can help to ensure that our professional development models are a continuum rather than a teaching strategy checklist.
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]]>Introduce exponential notation with a question: Which would you rather - $1 Million or a Penny a Day, Doubled for a Month? A 3 act math task for exponents.
The post Penny a Day appeared first on Tap Into Teen Minds.
]]>In Ontario, exponential notation is introduced in grade 8 in order to pave way for topics such as area of a cirlce and Pythagorean Theorem. As students reach secondary, exponential notation becomes increasingly more important as more complex functions are introduced in each course. Recently, I had a number of teachers emailing me for an interesting 3 act style problem to introduce exponents. While Double Sunglasses by Dan Meyer and Dark Side by Jon Orr are two great tasks involving exponents, I’ve found that they can be pretty difficult for students in grade 8 to wrap their heads around when they are just being introduced to exponential notation.
I think many are familiar with the Wheat and Chessboard Problem and some have even taken their own spin on that to make it more relevant to current times. Dave Bracken from my previous school, Belle River District High School used to tell students a story that went something like this:
I was speaking with your parents last night and they were all complaining that you weren’t doing your chores around the house. After some brainstorming, we came up with an offer for you. If you do your chores every day for the next month, your parents will give you $10,000 or a penny on the first day total, two pennies total on the second, four on the third, and so on. Which would you pick?
So while the story alone will lead to a pretty cool introductory task, I thought I would try to put something together that might help with the delivery and hook.
Here’s a good starter to lower the floor and get kids talking. My expectation is you shouldn’t require more than 10 minutes for this quick minds on.
Show students this video.
This can be some interesting discussion because the video does not give you details for how long. When would one be better than another? What if this was for the rest of your life? Does that change things?
Show students this video.
Or, here’s a screenshot:
Now, students are given the question:
How long will it take before the penny a day is your best option?
My expectation is that it won’t take students too long to realize that there are 1,000 pennies in $10, so you may not even need to show the Act 3 Video.
Or the screenshot:
Now, the scenario gets a bit more complex. However, I would suggest using this task again more for discussion and not spend much time on it.
Even though there is an Act 1 video (here), I might recommend jumping straight to the question by showing the Act 2 video.
Or the screenshot:
So students are now confronted with this question:
How long before the “penny doubled” option is best?
*You may want to reiterate that students are getting one penny on the first day, then the original one penny turns into two pennies on the second day, it turns into four pennies on the 3rd day, etc.*
Again, students will probably figure out pretty quickly which option is best.
Now, show the Act 3 Video.
Screenshot:
In just over a week, students will begin to benefit from the penny doubling option.
As mentioned previously, tasks 1 and 2 were created as a way to get kids thinking, talking and discussing. Now, we move on to the new learning goal we intend to hit: introducing the need for exponential notation.
Start off by showing the Act 1 video.
Screenshot:
In Act 1, students are asked:
Would you rather take $1 Million or the “penny doubled” for a month?
*Again, reiterating that students are getting one penny on the first day, then the original one penny turns into two pennies on the second day, it turns into four pennies on the 3rd day, etc.*
I’d try to let kids get going on their own at this point, even without showing Act 2.
If some students are struggling to get started, you may choose to give students the visual provided in the Act 2 video.
Check out the answer here.
Screenshot:
Show students this video.
Or, the screenshot:
After a short amount of thinking time, we introduce the idea of exponents and exponential notation for the first time using a direct instruction approach starting with this act 3 video.
Screenshot:
So, give it a shot and let me know how it goes in the comments!
Click on the button below to grab all the media files for use in your own classroom:
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]]>Man. Time FLIES! I feel like it was just yesterday when I made a commitment to myself to do a “Week In Review” series that summarized what I had been up to in the classroom each week. I had all the momentum on my side up until week 6, but then I was off to […]
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]]>Man. Time FLIES!
I feel like it was just yesterday when I made a commitment to myself to do a “Week In Review” series that summarized what I had been up to in the classroom each week. I had all the momentum on my side up until week 6, but then I was off to Norway for the NORCAN Project for over a week. Talk about falling off the wagon hard!
Then, we hit the busy “before Christmas Break” stretch. I spent a much enjoyed chunk of time with my family over the holidays, but next thing you know and we’re at the end of our first semester with the EQAO Assessment of Mathematics standardized test wrapping things up. Can’t believe it.
If that isn’t enough, I’m just now finally getting around to posting my first blog post for the 2016 #MTBoS Blogging Initiative when I should already be wrapping up my second. Yikes!
Well, enough excuses. Let’s get to it!
In the first blogging challenge, we were able to pick between two options:
I’ve decided to talk about One Good Thing.
I picked the first option because it is so easy to focus on one or two negative experiences through each day and completely miss all of the positive we could be enjoying instead. This is especially true for me at this time of year when I’m completely exhausted physically and mentally as I try to “fix” mathematics education with another group of students. Many who know me are well aware of my passion (addiction) to creating math tasks, trying new ideas and sharing innovative strategies for use in the math classroom. However, only a handful of people know how I get when I haven’t reached every learner in my classroom. My wife knows when it is EQAO time because I get extremely down on myself when I haven’t found a way to inspire a student to want to maximize their potential in math class. I think all teachers feel this same sense of disappointment. But for me, I struggle letting it go. As one who constantly promotes the idea of learning from failure and celebrating that experience, I seem to be the one who needs the advice most.
I know what you’re thinking: “I thought this was supposed to be One GOOD Thing?”
Don’t worry. It is.
For the first time in my 10 year teaching career, I feel like I have finally identified that this negativity is a problem and I am working to confront it as such.
In order to use my own advice and learn from failure, I spent a lot of time over the past couple of weeks thinking about some of the good things that happened this semester as well as some of the things I’d like to improve. I could probably write a book about all of the tweaks, modifications and complete revamps I would like to make as I prepare to enter second semester, I think my One Good Thing could be summed up in one word: balance.
Everything in this universe comes down to balance and I don’t think learning (or teaching) mathematics is any different. These past few years I have really been stretching my thinking beyond direct instruction and a largely lecture-style approach. While my assessment would say that my students and I both enjoy math class much more, I also think that it is easy to shift too far to one side. For the first 8 years of my career, I was of the belief that direct instruction was best. After some great learning at conferences, these past two years managed to shift my beliefs to include inquiry/discovery learning. However, my attempts to get better at teaching with an inquiry/discovery approach has led me to almost banish direct instruction from my classroom completely. It wasn’t something I intended to do, but it happened. However, there is a place for both direct instruction and inquiry based learning and I realize this is an area I need to work on.
One of the most challenging questions we as math teachers get to wrestle with is when to use each approach for the greatest effect on learning. After much thought in this area, I believe that there is no right answer.
In my case, this shift in thinking has made me try to envision a math classroom where I hope to better combine the use of the inquiry process when introducing a topic and direct instruction when consolidating a topic.
I’m excited to dive in and share how things go by the end of next semester.
I just hope someone is nice enough to remind me of what I always say about failure.
The post #MTBoS Blogging Initiative Week 1: One Good Thing appeared first on Tap Into Teen Minds.
]]>Convenient & FREE Practice Options For Your Students A few folks were interested in the Free EQAO Practice via Knowledgehook’s Gameshow tool, but felt they didn’t have the time to experiment with a new tool during “crunch time” in the semester. I know the feeling! Now, Knowledgehook has made it easy for students to practice […]
The post Grade 9 EQAO Math Practice Links for @Knowledgehook Gameshow! appeared first on Tap Into Teen Minds.
]]>A few folks were interested in the Free EQAO Practice via Knowledgehook’s Gameshow tool, but felt they didn’t have the time to experiment with a new tool during “crunch time” in the semester. I know the feeling!
Now, Knowledgehook has made it easy for students to practice any Gameshow independently without any setup, registration, or other hassles.
This is a great way to give students the option to practice what they need to work on in a fun, interactive way.
When students access Knowledgehook Gameshow activities via a link instead of through the website, there are some drawbacks including:
So, in my opinion, I would go with teacher and student accounts, however if you are in a time-crunch at the end of the semester, this is still a great way to offer students an additional way to prepare for the EQAO Grade 9 Assessment of Mathematics.
Share the following useful links with your students. Jump to grade 9 academic or grade 9 applied content.
EQAO Release Material From Previous Years:
Content By Course Expectation/Learning Goal:
EQAO Release Material From Previous Years:
Content By Course Expectation/Learning Goal:
Need more practice? Let me know in the comments and I’ll add more!
The post Grade 9 EQAO Math Practice Links for @Knowledgehook Gameshow! appeared first on Tap Into Teen Minds.
]]>This 3 act math task is a great way to introduce the concepts of parallel and perpendicular lines with context by using two graphing stories that produce distance-time graphs.
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]]>This 3 act math task was created using video clips from the Walk Out 3 Act Math Task as a way to get students wondering about parallel and perpendicular lines. Here’s the expectation from the grade 9 academic course in Ontario:
After spending some time asking folks on Twitter about how they introduce the concept of parallel and perpendicular lines, I found that we were all doing something pretty similar.
To investigate parallel lines:
The process was pretty similar for perpendicular lines:
After students complete the two investigations, a consolidation would take place. I’ve been doing something similar to this for about 10 years and while it does appropriately address the parallel and perpendicular portions of the expectation through investigation, it felt more like I spoon fed the investigation a bit too much.
While it is pretty easy to find a contextual situation where comparing two parallel lines makes logical sense, I really struggled to find a useful case for comparing two perpendicular lines. You can compare the linear relations representing the cost of two different taxi companies as parallel lines, but it makes absolutely no sense if you try to make a case where the linear relations are perpendicular.
When distance-time graphs came up in my course, I created the Walk Out tasks and finally found a way to at least spark the conversation about parallel and perpendicular lines.
Here it is:
Show this video.
If you have already used the Walk Out 3 Act Math Task with your class, you could probably just show this image:
Ask students what they wonder? What questions come to mind?
Write down some of the questions/ideas that students share.
Hopefully one or more students come up with something like “are they perpendicular or not?”
Once sharing out those questions that come up as a group, I ask them:
Are the lines parallel, perpendicular, or neither? How do you know?
I give them a minute to discuss with their groups.
In grade 9 academic, it might be a good idea to ask the group what it means for two lines to be parallel or perpendicular. What does it look like? What are some examples? This is the first time those concepts come up in the math curriculum in Ontario, so I never assume everyone has an understanding.
Teacher Move: It should be noted that my intention with the video/image is that the lines do appear to be perpendicular, but the scale on the x- and y-axis’ are not equivalent. This gives the viewer the impression that the lines are parallel, but as you’ll see in the next few videos/images, they are not.
After allowing students to argue it out a bit, I play this video.
Alternatively, here’s the image:
Then, I show them this video to give a bit more information.
Give students more time to discuss and share out. I usually ask students to share out who feels confident in their original guess and who wants to update their prediction.
Now, I play this video.
Alternatively, here’s the image:
Here are a couple more videos you might want to show as they change the scale to 1:1, but aren’t necessary:
While many math teachers have been using the very powerful Desmos Graphing Calculator, the new Desmos Activity Builder is an awesome way to leverage the power of Desmos in a more structured activity. In the past, creating an activity using multiple Desmos graphs involved linking a series of graphs from one to the other. Took quite a bit of time and planning. Now, creating an interactive Desmos Activity takes only a few minutes and it is easy to share with your students and colleagues.
Or, make a copy of the activity so you can edit to your liking!
Here’s some screenshots from the Desmos Parallel and Perpendicular Lines Activity Builder Activity:
After students finish the Desmos Activity Builder Activity and you’ve taken some time to consolidate the new learning about parallel and perpendicular lines, students can now work on some practice to reenforce the new learning. Check out a Knowledgehook Custom Gameshow that I created with some practice questions related to parallel and perpendicular lines:
Or, just try out the gameshow yourself.
*Note that if you run this Knowledgehook Gameshow with a class account, students have the ability to upload solutions which you can then share via the projector in real-time. The only way to go if you choose to run the activity in class.*
Here’s all the links in one convenient spot for this activity to go off without a hitch:
Act 1 [video]
Act 2 [scene 1 video | scene 2 video]
Act 3 [scene 1 video | scene 2 video (optional) | scene 3 video (optional)]
YouTube Playlist [All Videos]
Desmos Activity Builder [Try It | Copy It]
Knowledgehook Custom Gameshow [Try It | Copy It]
Make sure to leave a comment if you try out the task! Would love to get some tips on how to make this better.
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]]>Extending the concept of solving systems of linear equations, here are contextual tasks for practicing solving algebraically with substitution & elimination
The post Piling Up Systems of Linear Equations appeared first on Tap Into Teen Minds.
]]>There has been a lot of requests for more contextual and/or 3 act math tasks related to solving systems of linear equations when I work with teachers. A big realization I’ve come to this past year is that making students curious with contextual tasks doesn’t necessarily have to involve a relevant question. Dan Meyer has discussed it multiple times (like here and here), but it still takes a lot of reflection to start identifying what is important when trying to hook your students in.
Continuing the series of Tech Weight and the Tech Weight Sequel, we take the same context of weighing random items to give us some additional practice. You’ll quickly notice that these tasks are definitely contrived, but I still find tasks like these useful if introduced as a challenge to my students. Even though they REALLY don’t care how much each item weighs, they DO enjoy the challenge of solving the problem. The media and 3 act format gives them an easy entry into the task. After allowing students to intuitively solve simple systems of linear equations like in the Counting Candy Sequel or the Tech Weight Sequel, you can use these tasks as a way to practice substitution and/or elimination as a way to solve algebraically.
Show students this video.
Ask students what they are wondering? What questions come to mind?
After discussing with a group, help direct the focus to the question:
How much does each weigh?
Students can make predictions and argue why they believe their guess is best.
Show students scene 1.
Alternatively, you can show students the following images:
The weight of 4 bottles of glue and 5 glue sticks is 679 grams.
The Weight of 3 bottles of glue and 12 glue sticks is 680 grams.
Now, students can try to solve this system intuitively, by graphing or algebraically with substitution or elimination.
Show students the solution video.
Alternatively, you can show students these images:
The weight of one glue bottle is 145 grams:
The weight of one glue stick is 21 grams:
At this point, your students should notice that their answers are close to these weights, but not exact.
Why might that be?
What extraneous variables could influence the results?
Show students this video.
They will already know the question, so move on to Act 2…
Show students scene 1.
Alternatively, you can show students the following images:
The weight of 13 pens and 5 packs of correction tape is 166 grams:
The weight of 6 pens and 3 packs of correction tape is 88 grams:
Now, students can try to solve this system intuitively, by graphing or algebraically with substitution or elimination.
Show students the solution video.
Alternatively, you can show students these images:
The weight of one pen is 4 grams:
The weight of one package of correction tape is 20 grams:
Again, the actual weight may differ from what students found algebraically.
I find that sequel tasks like these often don’t require the “hook” as we’ve already engaged the learners to come towards the math. This task is simply additional practice prior to moving on to more traditional word problems and/or abstract problems. Thus, there is no act 1 video for this extension task.
Show students scene 1.
Alternatively, you can show students the following images:
The weight of 5 large notepads and 7 small notepads is 882 grams:
The weight of 3 large notepads and 5 small notepads is 563 grams:
Now, students can try to solve this system intuitively, by graphing or algebraically with substitution or elimination.
Show students the solution video.
Alternatively, you can show students these images:
The weight of one large notepad is 116 grams:
The weight of one small notepad is 43 grams:
Again, the actual weight may differ from what students found algebraically.
Try it out and let us know how it goes in the comments!
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