Graphing algebraic function transformations with cut paper videos

In this post are a bunch of function transformations videos showing how vertex form functions all transform using the same pattern. The way that functions transform in the coordinate plane can feel pretty abstract to algebra 2 students just learning about nonlinear functions. But every algebraic function in vertex form transforms the exact same way.

Functions can translate vertically and horizontally, and even reflect over the x and y axes just like geometric shapes. The one quirk is that horizontal transformations feel opposite from expected. 

Why are horizontal transformations opposite? It feels backwards for the vertex of y = |x - 3| to translate right 3 units. With all horizontal shifts, we're looking for the value of x that will "zero out" the inside expression. For x - 3 = 0, x would need to be 3. With x gone, we can find the function's lowest or highest y value, i.e. the vertex's y value.

Here is how we can find the horizontal transformation of a quadratic function in vertex form and why the inside shift is opposite. 

function transformations cheat sheet

This function transformations cheat sheet in my Google Drive has graphics from an algebra 2 word wall for absolute value, quadratic and square root graphs. There's also a free set of dancing skeleton functions posters here.

Below are a bunch of function transformation video shorts using cut paper. In each video you'll see familiar nonlinear algebraic functions transformed in the coordinate plane. 

Sums of angles in polygons hands-on discovery

Ripping the corners off a triangle to prove the sum of its interior angles is 180 degree is my favorite visual proof. Over the weekend, I posted a couple short videos for finding the sum of interior angles in a polygon, so here I wanted to share the videos and the worksheets from the videos.

Geometric transformations in the coordinate plane with a hole punch

Last week, I wrote a post about using a hole punch to find function inverses in algebra. A few people asked on Facebook if the process would also work for geometric reflections, and it absolutely does! 

In this post I share an easy, hands-on method for demonstrating reflections and rotations of geometric shapes and their coordinates in the coordinate plane. The video included in the post covers reflecting over the x-axis, over the y-axis and over the line y = x. This same method will work for reflecting over any line of symmetry in the coordinate plane, even linear equations. I then share an idea for showing geometric rotations with a hole punch (or sharp pencil!).

How to find inverse functions with a hole punch

Are your algebra students learning how to find inverse functions? Here's how to make the process of finding function inverses easy, visual and hands-on-- with a hole punch! This same process can also be used for reflecting any graph or geometric shape over the x-axis, y-axis, y = x or any other line of symmetry on the coordinate plane.

Why a zero exponent = 1 (and negative exponents, too)

Why is a number raised to the zero power equal to 1? And why do terms with negative exponents become fractions? Are we able to see this through visual models?

Yes! 

In this short video, you'll see how exponents take on a pattern and can be modeled concretely with cut paper. We'll start with 3 raised to the 2nd power and work our way to 3 to the -2. 

How to Find the Domain and Range of a Graph (video + sheet)

This is a super quick post sharing a domain & range of graphs poster, a set of domain and range cards and a video showing how to use the poster to find domain and range from a graph.

Simplifying fractions using visual models and primes video

My favorite way to simplify fractions is by working through a list or prime numbers, checking if both numerator and denominator divide by each larger prime. I like this method because it makes simplifying fractions super concrete for students who need it. It also helps eliminate errors that can occur when a factor is missed. Students who aren't as confident with their division facts can even use a calculator (or divisibility rules) to check each prime.

Estimating square roots using visual models video

A teacher messaged asking if I had a video showing how to estimate square roots using manipulatives. I didn't, but it sounded like a lot of fun to make a video on this topic. 

To prep for the video, I typed up and printed a sheet of visual models. If you want to show this method in class, you don't need anything more than paper, but here is the sheet of visuals that I use in the video if you'd like them.

There are 3 examples in the video. The first example shows how we can approximate the square root of 19 and, most importantly, why it works. As it turns out, what we're really doing is finding a square with a fractional side length that is close to the benchmark square root of 16.

Middle School Math Algebra Tiles Tutorial Video

In the video, you will learn how to use algebra tiles to teach solving equations, simplifying expressions, integer operations, multiplying polynomials and factoring quadratics.  
Read more

Multiplying fractions using area models

In this post, I want to share a few multiplying fractions activities and resources for your classroom, along with a video showing how to multiply fractions using the area model. There's a similar video for dividing fractions here.

I've been building a visual math playlist on my YouTube channel with videos for fraction multiplication and division, solving equations, ways to use algebra tiles, strategies for tackling integers, cut paper math, hole punch math etc. This one here is a longer video for how to multiply fractions using visual models.

I got a question about the denominator, and why it's 15 in 2/3 times 4/5 when using the area model to show fraction multiplication.

If we were just multiplying 3 x 5, we'd get 15 without much thought. 

It's the same when multiplying fractions, except instead of multiplying 3 x 5, we're multiplying a fraction of 3 times a fraction of 5. So the whole area is still 15, but we end up with a fraction of it as our product.

The same is true when multiplying fractions greater than 1 (improper fractions). If we multiply 7/5 x 2/3, we're still multiplying a fraction of 5 by a fraction of 3 so the whole is still 15. One way to think of this is (5/5 + 2/5) x 2/3.

multiplying fractions escape room

For a review of multiplying fractions by fractions, by whole numbers and by mixed numbers, this multiplying fractions escape room comes as both self-checking digital and PDF printable.

The printable version of the escape room comes with an answer sheet where students can record their 4-letter answer codes.

Fraction operations - 6th grade math word wall

The two fractions posters above have been added to the 6th grade math word wall. I colored the posters in this image but they also come in color and in an interactive Google Slides version in the file. Fraction multiplication and division are shown with area models on the posters.

Fraction review digital math escape room

Above is puzzle #4 in a Fraction Review Digital Math Escape Room. It also comes in printable b&w, if you'd rather students work on paper. Questions cover: simplifying fractions, converting from mixed number to improper fraction, converting from fraction to decimal, adding and subtracting fractions, and multiplying and dividing fractions.

Multiplying fractions math pennants

And here is a math pennant from a multiplying fractions math pennant activity that doubles as classroom décor when the students are finished with their work. Students show fraction multiplication through the area model, then mark down their final answers.

> Browse all fraction activities here

free math resource library

Graphing Standard Form Quadratics step-by-step Video and Cheat Sheet

I had stalled a bit on my mission to make videos to go along with all of my math cheat sheets. It feels good to be back at it! The graphing quadratics in standard form quadratics cheat sheet in this post was a teacher request. It's a brand new step-by-step on graphing standard form quadratics.

Desmos Math Art Project

While you're stuck at home with your own teenagers because of COVID-19 social distancing, or have been given the monumental task of teaching your math students remotely, what better way to engage than with a little math+art? Desmos has extended the deadline of their Global Math Art Contest to April 30, 2020 so that more students can participate as part of their distance learning. 

But first! Desmos Activities

Before we get into the Global Art Project, I wanted to link you to a Desmos Activities YouTube video that explains just about everything there is to know about Desmos Activities:

All About Desmos Activities

Graphing Log Functions Step by Step Video

Log functions are kind of cool in that they force us to think backwards. Logarithms are inverses of exponentials (which I covered in this exponential functions video) so when we create our parent table we're thinking y first, then x. This can get a little confusing, so I wanted to make a video to go along with the free logs cheat sheet I posted a while ago on my blog (and that I re-linked here).


In this post is a video and a free graphing log functions cheat sheet. The cheat sheet can be given to students for their notebooks or enlarged to create an anchor chart (link to directions on how to do this easily is below).

What's the deal with extraneous solutions when solving radical equations?

What's the deal with those extraneous solutions we get when solving radical equations? Why do we get two answers but sometimes have to throw out one or even both? In this post I share a video explaining through graphs the answers we get when solving radical equations.

How to Graph Exponential Functions by Hand - free cheat sheet and step-by-step video

It's possible that my favorite functions to graph are exponential functions. They look super intimidating at first with their parenthesis and exponents in the air, but by taking them step-by-step, first creating a parent table then shifting this table to find coordinates to graph, they come together before you know it.

Graphing exponentials is one of my favorite things to teach. I say this about most of the topics we cover in the course, but I see so much growth in students during our exponentials unit that it's hard not to love it. At first, students aren't even sure where to start. By the end of the week, they're graphing like champs.

In this post is a video and a free graphing exponential functions cheat sheet. The cheat sheet can be given to students for their notebooks or enlarged to create an anchor chart (link to directions on how to do this easily is below).

How to Graph Radical Functions -- video and free printable cheat sheet

There's more than one way to do everything, including graphing nonlinear functions. The method for graphing radical functions in this post is by far my favorite and one that I learned from my superhero co-teacher Ms. Sullivan. She was the best. Inclusion is the best.

In this post is a step-by-step video for how to graph radical functions and a free printable radical functions cheat sheet (the one I'm holding in the photo).

How to Graph Quadratic Functions in Vertex Form - video, free cheat sheet and free task card activity

My wonderful husband got me a whiteboard for Christmas so that I could start making math teaching videos. After taking a few days to get everything set up, all the while wondering where to start, I decided to work my way through my math cheat sheets and make companion videos, starting with Algebra 2.

In this post is a video example of graphing quadratic functions in vertex form, a link to a free math reference sheet to go along with the video, and a link to a free vertex form quadratics task card activity.

Towards the end of the video is a shortcut for graphing vertex form quadratics with a pattern instead of a table.

Dividing Fractions by Fractions using Visual Models - 3 examples

Explaining how to divide fractions conceptually is one of those topics I find myself needing a refresher on every so often. For me, showing how to divide fractions using visual models is definitely one of those "don't use, lose" situations. So this post came from me recently refreshing my memory of fraction division. But why now? This post:

After sharing Howie Hua's post in our Visual Math Facebook group, it got me reminiscing on fraction division. And full disclosure, I had never considered dividing across as a thing to do. I'm 42 with a Masters in Math for Teaching and this never crossed my mind. I'm not usually too worried about how I come across, but I did appreciate other teachers in the group admitting they never thought of this, either. It allowed us all to learn something new together, and I love when this happens.

In one graduate class we had to write papers on the various algorithms taught in math and why they work. Fraction division was one of those algorithms, and after more than a few "redo"s written at the top of my paper, I finally handed in a paper that could actually be graded. Needless to say that was a taxing class.

Keep, change, flip. Why? I hope to answer that with the 3 examples in this post.


Example 1: (2/3)÷(1/2)

Division asks, "How many of these fit into that?" For 10 divided by 2, for example, we're asking, "How many 2's fit into 10?" We ask the same question when we divide fractions, it's just a little harder to see.

In our first example (2/3)÷(1/2), we're asking, "How many 1/2s fit into 2/3?" It would be easier to answer this question if our fractions were broken into the same number of parts. By creating a common denominator, it will be easier to see how many will fit.

By creating the common denominator 6, we can then see that all 3 of our green bars will fit into the space taken up by our blue bars. There is even room for one more! So all of the green bars (1 whole) can fit into the blue space and 1 more bar out of the 3 (1/3). So (2/3)÷(1/2) = 1 and 1/3. 

Here is a video explaining:



The connection to Keep Change Flip:

When I posted this video on my Instagram feed, one teacher pressed me for the connection to the keep change flip (aka: multiplying by the reciprocal) standard algorithm. Like anything in math, there is more than one good way to make a connection. I personally like bringing it back to whole numbers when explaining why keep change flip works:

If 10 divided by 2 is 5,

Then half of 10 is 5.

So 10 divided by 2 is 1/2 of 10.

We can multiply by a reciprocal for division.

If you'd like to link the keep change flip algorithm straight to this fraction example, here is how that can be done:

We can see that the numerator of the second fraction becomes the denominator of our answer. This is because we are asking, "How many of the second fraction's numerator (bars) can we fit into the first fraction's numerator [after we create a common denominator]?" 

Example 2: (1/2)÷(2/3)

This example is just like example 1, we just switched the fractions' placements, which is fun because it reinforces that division is not commutative.

In our second example (1/2)÷(2/3), we're asking, "How many 2/3s fit into 1/2?" By creating a common denominator, we can easily see that 3 of the 4 blue bars will fit into the space occupied by the green bars. So (1/2)÷(2/3)=(3/4).

Here is a video:



Example 3: (4/5)÷(2/3)  

Sometimes creating a common denominator with just columns would get too messy to be helpful, so we can create a grid to show it instead. We can always do this, I just prefer columns when there is a chance to use them because I feel they are easier to see.

With (4/5)÷(2/3), our common denominator is 15, so we can create a grid of 15 spaces. 4/5 takes up 12 of these spaces and 2/3 takes up 10 of these spaces. So all of our 2/3 can fit into 4/5, plus an additional 2. We can then see that (4/5)÷(2/3) = 1 and 2/10. 

Here is a video explaining this example:



Summary:

All three fraction by fraction division examples from this post are in this video:

These fraction multiplication and division references are included in my 6th Grade Math Word Wall.

I also just created this set of fraction division task cards to go along with this post. The cards can be laminated and used with a dry erase marker so that they can be reused.

Dividing fractions using models tasks

And for a fun review, this fraction review digital math escape room. In puzzle #5, students are asked to multiply and divide fractions. Students figure out the 4 answers then type their 4-letter code into the answer-validated Google Form to unlock the puzzle.

Fraction review digital math escape room

I hope this post has been helpful! 


-Shana McKay, Scaffolded Math and Science

Browse fraction activities here.

More posts:

Multiplying fractions using models

Math Word Walls

Free math resource library

Solving equations using algebra tiles: with pictures!

There are so many cool ways to use algebra tiles in math. Last week I wrote a post about using them to factor. In this post I wanted to show 3 examples for using algebra tiles to solve equations.

New to algebra tiles?

ALGEBRA TILES TUTORIAL VIDEO (click here)

I'm using a free set of paper algebra tiles (linked below), mainly so that I could cut one in half for example 2 that involves a fraction. So let's get into it!

Polynomial Synthetic Division in Algebra 2

I originally wrote this synthetic division post years ago after making a cheat sheet to help our inclusion algebra 2 students with synthetic division. All these years later I am back to update the post with a synthetic division video to go along with the free math cheat sheet download.