A Free Algebra Exploration to Deepen Understanding & Increase Retention - Slope Formula
As math teachers we know the frustration of having kids who seemed to understand something during the lessons but then suddenly forget the formula on the test, or get things backwards.
For slope, it tends to be kids putting it upside-down or messing up the negatives. They may also be lacking the basic concepts of greater slopes being “steeper” and are then unable to make sense of their errors or catch simple mistakes.
There tends to be a lack of retention in a lot of basic concepts like this. It can be aggravating as a teacher to grade work that should be relatively “simple” like finding slope between two points, and see so many little errors.
Well, I have some great news. If you see this with your class, you may just be missing the simple element of inquiry! Inquiry provides students with the opportunity to discover concepts themselves, which makes the learning more meaningful and powerful.
Teaching through guided explorations allows kids to develop a formula themselves, leading to a deeper understanding and stronger retention.
Inquiry - Based Learning
Inquiry (or discovery) learning has so many awesome benefits. This method is sure to help your students learn AND retain new concepts. When students learn through inquiry they are asking the questions; they are discovering the answers!
When students learn through investigation,
Introducing Slope with Inquiry
The next time you introduce slope, you definitely want to try out this inquiry activity.
1. First, download this “Discovering Slope” sheet (within the Free Guide to Teaching Slope)
2. Group your students into pairs and give each pair one sheet. Partner work is great for inquiry activities, so students can talk through their discoveries and make progress together. Their discussions help solidify concepts, while helping them get unstuck.
On the sheet there are 5 sets of staircases & ramps that the students will need to put in order of “ steepness”.
3. Tell them to read the directions and complete the page. The next part is tricky- sit back, relax, and try not to help!
This will probably go against your instincts as a teacher, but you need to stay strong! Your students learn through their struggle.
Try to avoid giving hints. Your students need this practice too; it will help them build up persistence.
Don’t let them give up. They will end up making great observations once they jump in and embrace the uncertainty!
4. Walk around and listen to the conversations that are happening. Take note of any interesting observations that you may want students to share in the discussion afterward.
They’ll discover that they need to use both height and length… but how? Some groups may be stumped for a few minutes. There may be frustration they have to work through. Then, they will realize they need a kind of a ratio between the two.
They may try it upside-down at first (run / rise), but then eventually they should get to the formula for rise / run themselves!
They’ll work to count the grid spaces, and later can understand the subtraction model to develop the formula for when they are given just coordinates and no graph.
They will now always remember and understand much more deeply that slope is "RISE OVER RUN" because they figured it out for themselves. This is the goal of the guided discovery / inquiry process!! :)
5. Give encouragement to groups that are truly stuck and frustrated, but try not to give hints about where to begin. Train your class to try something that makes sense, and see where they get.
That is the purpose of the discovery process. Each time you let them struggle on their own, they build those skills and will do better next time.
6. Gather back together as a class for a follow-up discussion. In the discussion, clear up any misconceptions that popped up. Allow students to share their approaches and discoveries.
Try to direct the conversation to noticing that, once simplified, two of the slopes had the same number. Or did any of your students disagree that they were equivalent slopes because the directions were different?
Take all observations. This will help later when you teach the differences between positive and negative slopes.
Once this activity is done, you’ll be ready to lead your class into a lecture with guided notes on the Algebra notation, the definition of slope and practice problems. Click here for my Slope Doodle Notes!
I hope these materials make this your most successful year of slope lessons yet! Enjoy the journey, and have a great year of Algebra! :)
Feel free to share other ideas for exploring slope in the comments! I bet doing a life-size measuring layout with actual step-stools would be incredible if you have the time and can set it up!
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What's the Deal with Math Anxiety?
Math anxiety is anxiety when asked to do math; it’s more than just a dislike for the subject. Edutopia tells us it can start as early as kindergarten and affects nearly half of elementary students! Unfortunately, the number of occurrences increases as children get older.
According to Frontiers in Psychology, “People have been expressing mathematics anxiety for centuries: the verse “Multiplication is vexation … and practice drives me mad” goes back at least to the sixteenth century. From a research perspective, the construct has been an important topic of study at least since the concept of “number anxiety” was introduced by Dreger and Aiken (1957), and has received increasing attention in recent years, in conjunction with the generally increased focus on mathematical performance.”
Math Anxiety Robs Working Memory
According to a fascinating article from the University of Chicago in their series “Ask a Cogntive Scientist”, a growing amount of research shows that math anxiety robs people of their mental working memory. Working memory is kind of like a scratch pad- it’s where you keep several things in mind simultaneously. Without it, we couldn’t problem solve and complete multiple tasks.
Working memory “space” varies from person to person, and it is required for solving math problems. So, since this anxiety creeps in when doing math, it makes sense that people with math anxiety have less room for working memory. “Anxious thoughts consume valuable working memory space...As a result, [students with math anxiety] have less working memory space to devote to the math, and their math performance suffers.”
Signs of Math Anxiety
Are you wondering which of your students experience math anxiety? It can sometimes be obvious, but Edutopia gives us some signs to look out for:
How to Help?
Luckily, there are plenty of ways to help our students overcome math anxiety!
The first step is noting these common causes. Actually, there are quite a few common causes of math anxiety. It can help to narrow it down to a source so we can guide our students to overcome it. According to Oxford Learning, here are the top common causes:
Once you’ve narrowed down the common causes, try these ideas to help your students:
Since a common cause of math anxiety is public embarrassment, try eliminating disapproval or negative feedback in math class. Reward students for excellent behavior, well-thought out answers, or even extraordinary effort!
Provide students enough time to truly understand math concepts, instead of drilling procedures. Students need a chance to develop conceptual understandings to develop joy in math!
Try having a student write down his or her worries about math before doing it; by thinking critically students can realize their fears aren’t accurate. After jotting down the feeling on a sticky note, they can physically relax the part of the body where they are holding tension while they put it aside, take a deep breath, and jump in to try a problem.
Help your students see tests and quizzes as challenges, rather than a threat to their grade. Reframing how they see math can improve their fears. Offering test corrections can relieve some stress, because students know they can move on and have success.
Swap out your active brain breaks for more meditative brain breaks. When I taught high school Geometry in a Catholic school, I loved beginning each class period with these 3 minute retreats.
Students also enjoy chances to color. I believe in fitting this in alongside the learning goals that are already in place for each day’s lesson. Doodle notes offer a way for students to get the therapeutic benefits of coloring and doodling while still learning the material. This has been proven to activate the parts of the brain that help a student relax, focus, and retain the material. It will take their minds off of the stress and physically relax them! There is a reason that coloring books are becoming a therapeutic trend. The brain benefits are so easy to build into a class period!
I’ve collected some ideas to help you blend fun and rigor in creative ways. My goal is to overcome the negativity that can pop up in math class, while preserving high standards for content learning! (essentially fun without "fluff!") Check out some of these ideas to get started! Click the images to read each post that fits your grade level / course work:
Additional downloads to Help: All of these fun lessons are absolutely free!
Last, but certainly not least, teachers need to show excitement and positivity when it comes to math. Our attitudes rub off on our students, so even if math isn’t your favorite subject, keep your words and actions in mind!
I know I’ve definitely seen the tell-tale signs of math anxiety in some of my students! Have you?
What are you going to do to help them rise above math anxiety? Start by subscribing to Math Giraffe through email! (below) I’ll keep you up to date and send you materials and inspiration to keep the creative blend of fun and rigor flowing in your classroom!
Are some of us "math people" and some just aren't??
“I’m just not a math person!” is an all too common phrase proclaimed by many adults. You tend to hear this when someone is stuck on an everyday math problem, and some embarrassment has crept in. When adults say this so frequently, it can slowly get soaked up by the younger crowd and start to sound normal, and like an easy escape as something to say to avoid real world math or problem solving.
Here’s the real deal: It’s just not true- there’s no such thing as not a math person! There’s psychology and neuroscience to prove it!
What’s worse is the harmful effects these statements can have on children’s view of math. As adults we have so much influence in their mindset. Think about it: how often do you hear ‘I’m just not a science person!’ or “I’m just not a reading person!’?
It is unfortunately much more common to hate on math. But there’s good news, too! We can help to put an end to this myth! As math teachers, we have both the opportunity and the responsibility to help our students form positive math opinions. It’s all about mindset!
Why It’s Incorrect
Psychologist Patricia Linehan, from Purdue University writes, ”A body of research on conceptions of ability has shown two orientations toward ability. Students with an Incremental orientation believe ability (intelligence) to be malleable, a quality that increases with effort. Students with an Entity orientation believe ability to be nonmalleable, a fixed quality of self that does not increase with effort.”
Basically, some students believe their intelligence increases with their effort, and some students believe it’s all in their genes. It is common for individuals to believe their math ability is strictly genetic, or nonmalleable.
There were two studies done with large experimental and control groups of 7th graders in math; the students who believed intelligence can grow with effort received much higher grades.
Evidently, convincing students that they could make themselves smarter by hard work led them to work harder and get higher grades.
Neuroscience to Back it Up
According to KQED, Neuroscientists did MRI scans of students taking math tests and saw that when a student made a mistake a synapse fired, even if the student wasn’t aware of the mistake. Then if the student recognizes their mistake, a second synapse is fired. This leads to creating new brain pathways.
With this and other studies, we have found that the brain has the ability to grow and shrink; intelligence is not fixed.
A study focusing on cognitive tutoring students with Math Learning Disabilities (MLD) showed promising results. MLD students received 1:1 cognitive tutoring for eight weeks. Prior to the study, functional brain mechanisms showed children with MLD had very different brain patterns when learning math than their peers. This was not the case after the effective intervention, however. The study proves that brains have the ability to grow in math ability.
Why the Myth is Harmful
Adults, especially parents and teachers, have a profound influence in any subject. However, since math tends to get a reputation for being difficult, adults’ words and actions centering around math are especially important.
Comments such as, “I’m just bad at math!” or “I’m not a math person.”, etc. are misleading and perpetuate a dangerous rhetoric. Children hear them, and over time they believe it’s true.
What We Can Do
To expose this myth, there are a few things we need to do:
1. Help Students Develop a Growth Mindset
Chances are, you probably know a lot about this topic. In case you don’t, here’s a quick refresher: People who have a growth mindset believe that success is based on learning and hard work.
Opposite to this idea is a “fixed mindset”, or the mindset that an outcome cannot be changed.
Edutopia gives some ideas on how to encourage a growth mindset, but says you should really focus on feedback. When we praise students for how clever or "smart" they are, we might actually be encouraging them to develop a fixed mindset. On the other hand, if we praise students for the hard work and the process that they’ve engaged in, then that helps to develop a growth potential.
2. Educate Parents
To help form completely positive math opinions, we need to make sure everyone in our students’ lives is on the same page. Just as we need to understand that intelligence isn’t fixed, it is also our responsibility to educate the parents of our students in this.
Meet the Teacher Night or Parent-teacher conferences may be a good time to talk to parents about the importance of instilling a positive outlook on mathematics and encouraging a growth mindset.
We need to help parents realize they shouldn’t discourage their child from making any mistakes, and should encourage perseverance in challenging situations.
3. Make Math Fun, Accessible, and Engaging at EVERY Level
For students of any age, making math approachable is the proactive way to tackle the rumor. Once students have fun and, even better, realize they are having fun in math class, then developing a positive opinion and developing confidence in math just comes naturally!
If you’re searching for an easy way to make math class more enjoyable and accessible for everyone, don't miss taking a quick peek to check out the creative resources available here!
Stay up to date with Math Giraffe to keep teaching math with creativity! --
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