The Four 4s - an activity from Jo Boaler's book

Today I spent most of my day in the computer lab where my Algebra IB students took a "mock" end-of-course exam. Those days are so draining for some reason! I am happy that it gave me the opportunity to finish reading Jo Boaler's What's Math Got To Do With It? which I believe every math teacher should read! Thankfully one of the last chapters discusses some puzzles and number talk activities that are good for students. I was so tired I could not imagine having a "normal" class with my 5th period today. Instead I gave them The Four 4s to do as an activity. The task asks them to try to make every number between 0 and 20 using only four 4s and any mathematical operation. The directions for the task gives one example and then asks them how many of the numbers between 0 and 20 they can find.

At first my students wanted me to give them more examples but I refused by telling them I didn't want to rob them of the opportunity to get them on their own (haha!). I finally got a few of them moving by telling them to just write four 4s on their papers and then put some operation signs in between them. Once they did this I told them to evaluate the expression making sure they used the order of operations correctly. This really got them rolling. A cool thing about this task was that everyone worked on it. At the beginning I had to seperate a few that were totally off task but once they saw that they could get some of the numbers they worked on their own.

I think that you need to have variety in your classes. I have many students who do not enjoy an activity when they view it as purely mathematical but when you give them a puzzle to solve they engage. I intend to use activities like this more often. One of the discussions on Boaler's website (youcubed.org) talked about how they just listed the numbers 0 thru 20 on a board in the back of the room and allowed students to write their expressions and put their name beside it. That way all of the classes throughout the day could contribute until all of the numbers are found. This task will engage some students that are bored with the normal daily routine!

Just in case you read this blog because you are using the IMP Meaningful Math textbooks, Jo Boaler has a list of 3 curricula that she recommends for use in 9-12 and the Interactive Mathematics Program from It's About Time is on that list. After I started reading her book I realized that she was one of the keynote speakers at NCTM this year. I learned so much about what the research says about the best ways to teach math for student success!

IMP Fireworks - Distributing the Area II and Square It!

Michael Reitemeyer shared with me that he uses the video above as an introduction to his class. Today, as my students were working on the Distributing the Area II activity, I was reminded of the video. The very first problem is one where the students need to guess and check in order to get the correct answer. There are several students who just stared at the problem. I gave them a hint and then gave them a few more minutes. Afterwards I had a student share HOW she got her answer. I asked her if the first numbers she tried worked and she said no. She tried some numbers and then made some adjustments until she got the result she needed. I reminded the class of the video. I told them that Jada got to an answer faster than most of the rest of them because she was willing to TRY some numbers. After she tried some numbers and saw that they didn't work she was able to make a revision and find the solution.

One of the questions in this activity actually has the students to factor a trinomial but they don't realize that is what they are doing. I stressed to the students that they are being asked to try both the vertical method and the area model to multiply polynomials but afterwards they can choose the method they like best.

Square It!  introduces the standard form of a quadratic equation and leads the students to convert quadratic equations from vertex to standard form. Number 1 has the students practice squaring a binomial. Then they are given problems in vertex form and told to put them in standard form. It amazed me that some of the students in one of my classes didn't make the connection between number 1 and number 2. I even told them to square the binomial first just like we did in number 1. (I try to help them see the connection between the order of operations and the idea of squaring before you distribute.) I kind of  "got onto" them about not trying to make connections from number 1 to number 2. They just kind of shut down on me and started whining. I noticed that there are several more places in the next few activities where the students will practice this skill again and I am glad. My other algebra class did fine with the activity - no whining or complaining. I wonder if my experience with the first class helped me to put more emphasis on the connection between numbers 1 and 2?

**While planning for the next day I realized that the students will be given a couple more chances in the next few activities to work on changing from vertex to standard form.

Squares and Expansions is an activity where the students are first introduced to the concept of completing the square. Then they practice converting from vertex to standard form.

IMP Fireworks - Distributing the Area I and Views of the Distributive Property

I love that Mrs. New has already been through this unit a couple of times. She give me great tips! For "Distributing the Area I" she advised that we make a handout with the 6 area model rectangles so the students could just write inside them. This helped the activity to move along faster than if they had drawn the area models. Some of my students are perfectionists so they would take forever to draw a perfect rectangle and then have little time to work on the actual task in the lesson. My students seemed to really enjoy this activity. After I showed them #1 as an example they pretty much took off with it. I even had one boy who I have to get on to often for sitting and doing nothing to be the first one to get the answers for #6. I was proud of him!

"Views of the Distributive Property" was an activity that made them think about the way that they multiply 2 digit numbers. They are shown through the activity that they have really been using the distributive property and partial products all along. My students whined a little bit about having to do the problem involving the long form but I tried to explain to them that this was just a foundation on which we were going to build as we started multiplying algebraic expressions.

I have taught multiplying and factoring polynomials using the area model for a couple of years. I have called it "the box method" but it is the same thing. Even though I feel like it gave the students a more visual way to work through the problem and it also helped them to organize their work I still had some students who would take forever to get the idea that the product inside the box comes from multiplying the 2 values on the outside of the boxes. I am excited about the way Fireworks has built the idea of using the area model. My students will have the idea of the "lots" changing size from the "A Lot of Changing Sides" activity so hopefully they will feel like the answer is just like adding up all the smaller areas even when we are multiplying polynomials.

Another thing that excited me today is that I got through these activities in the time that the teacher's guide recommends. That is an accomplishment for me!

IMP Fireworks - A Lot of Changing Sides

I really enjoyed doing this activity with my students. It starts with a background story of a housing developer wanting to change the lot sizes for a new housing development. Instead of all the lots being square the city planner wants some of other types of rectangles. The questions tell the students exactly how to change the dimensions of the lots. The first 4 are situations where they increase the size of the lots and the last 2 involve a decrease in the length of at least one side.

Mrs. New had already taught this lesson and showed me the way the teacher's guide recommended the sketch of the lots be drawn. These diagrams will look like "the box method" that they will use to multiply and factor polynomials. The activity asks the students to express the area as the product of the length and the width (which will be binomials) or as a sum of smaller areas. Since the 2 areas are equivalent the students are led to realize that the 2 expressions are equal. I aksed them to look for connections between the sum and the product and a few of them saw it.  I love that the authors have once again provided a context for the formal math to make sense to them!

I led the students through #1 so I could model how to sketch the diagram with the original side length of X. In #5 and #6 I let them come up with their own diagrams to represent the situation. Also, just for the sake of organizing, I labeled the bullets as A, B, and C so that it would be a little easier to organize and discuss.

IMP Fireworks - Using Vertex Form, Crossing the Axis, and Is It a Homer?

In Using Vertex Form, the students have another picture to create with their graphing calculators. Then they are given equations in vertex form and asked to give the vertex. Most of the students could do this without using the graphing calculators but a few still depended on them. I advised them to use the "Trace" feature of the calculator to find the coordinates of the highest point and then compare the coordinates to the equation. This improved their confidence in finding the vertex.

I gave my students a quiz where they had 6 questions in which they matched quadratic equations to their graphs. Then they had a couple where they had the equation and had to list 3 things they know about the equation. The last 2 questions had them describe how to flip a graph so that it was concave down and then sketch a graph with 3 different parabolas and give their equations. I was so excited about the quiz results. I wish I could say that all my students aced it but that is so not true! However, the large majority of my students passed the quiz and many made As and Bs. I have never expected my students to be able to do so much with graphing quadratic equations. The way the activities led the students through the process was so thorough it made the quiz seem easy.

The Crossing the Axis lesson gets students to start thinking about how many x-intercepts the graph of each quadratic equation will have based on the phase shifts. The activity also has students to write the equation given the vertex and another point on the parabola. They have to use the information to solve for the value of a. Numbers 5 and 6 are very important to complete because they give the students the tools they will need to complete the Is It a Homer activity.

Is It a Homer is an awesome activity to me as a former softball player and coach. It was also fun to the students. Mr. Webb shared a link with me of a Youtube video of a "dramatic reading" of a poem about the "Mighty Casey." We watched it before we read the activity. They are challenged to figure out if the ball clears the fence and they must prove it mathematically. I had the students sketch the graph with height on the y-axis and distance from home plate on the x-axis (which was advised in the teacher's guide). I gave the students the hint that they will be using the same process they did on 5 and 6 of the previous activity. After giving the students some time to think I went to the board and sketched the graph and labeled the vertex. I asked the students if there was another time when we knew what the height of the ball was. My 2nd block students chose to use (0,0) and but my 4th block students pointed out that the ball was not hit off the ground so they used (0,3). They struggled some with the computation of this problem but a few students in each class got the answer correct based on the height of the ball at contact. I had one student who told me that he estimated that the ball would fly 400 feet because the maximum height was after the ball flew 200 feet. Even though we have not yet talked about the symmetry of the graphs he had recognized it and used it for his reasoning. Unfortunately he didn't tell me his thoughts until AFTER class. I will be sharing it with my classes tomorrow.

Teaching equations using the cover-up or blob method

I have a group of struggling algebra students. They were taught to solve equations last year in 8th grade. When we approached the Mystery Bags activity (which covers solving equations with variables on both sides) I gave them some simple one-step equations and realized that they needed some more practice - especially with equations involving integers. The first day I "retaught" solving equations using inverse operations it did not compute with the majority of the class. I discussed with Mrs. New whether or not I should try to find a creative way to teach equations. After teaching using our Meaningful Math books in which we have a context and discovery/inquiry based activities it was so hard to just teach by giving notes and examples. I had seen a cover-up method for solving equations but the worksheet I had was a little confusing. Mrs. New had learned about a Cover Up Math app when she was at ISTE last Summer. I decided to let the students work with the app and see their reactions. When they told me that they understood it better using this method I did a little more searching. I found this worksheet along with this video which teaches solving equations with a method very similar to the Cover Up Math app. I had some students really grasp hold of this method because it made more sense to them. I still would kind of go to the side a be a "real" math teacher and talk about inverse operations. I had one student who did great using the traditional inverse operations.

Now...fast forward a week or so and we are now working on solving inequalities. I have realized that since I have so many students in the class that did not solve by inverse operations it is difficult to explain to them about when they need to change the direction of the inequality. We did an exploratory activity in Cookies where they discovered that when you multiply or divide by a negative (when solving inequalities) that we have to flip the inequality for the statement to remain true. However, when we looked at solving inequalities and we "reached back" to the blob/cover up method we do not  talk about multiplying or dividing by a negative. So...in order to modify for the students to get the answers correct I told them to do these steps:

1. Replace the inequality symbol with an equal sign and solve the equation. (It is amazing how happy some of my students were when they didn't have an inequality symbol anymore.)
2. Draw a number line and decide whether or not to use an open or closed circle based on the inequality symbol (we had already discussed this)
3. Test a value on either side of the circled number to see if it makes the original inequality true. If it does shade on that side and if it doesn't shade the opposite side.
4. Lastly, make sure your "solution" matches the graph. This helps them to write the inequality with the sign going in the correct direction.

I did not write these steps on the board. We just worked through several together. I can't help but stand there thinking that it would be so much easier to just use the inverse operations with the "flip if you divide or multiply by a negative" rule. I had a few students tell me today that it all made sense to them now. I just wanted to say, "Really??"

So, I write this entry with a conflict brewing in my head over whether or not I have done the right thing. My reasoning for using the other method was that the students had been taught last year and  this year with the inverse operation approach and it just did not seem to work for them. The math teacher in me tells me that it is so important for them to understand how to solve using inverse operations. The common sense portion of my brain says if I can get these students to improve their ability to solve 1 and 2-step equations then I am doing good. These are students whose math confidence level is so low.

As we go through these problems I often show the inverse operations method beside the cover up/blob method in order to show them the connection between the two. I want them to get to where they can understand the formal mathematics of what they are doing.

IMP Fireworks - Parabolas and Equations I, II, and III and The Vertex Form of a Parabola

I am absolutely loving the way Meaningful Math develops the Fireworks unit. These graphing activities go through each of the ways that parabolas are transformed using phase shifts. The students explore each of the phase shifts using graphing calculators. They start with different equations to enter and analyze and then they are given a picture of multiple graphs on the same coordinate plane. They areasked to create the picture by typing equations into the calculator.

The first few activities I gave each student a TI-83 and each group also had an Ipad. Once they figured out the correct combination of equations the group displayed their graphs on Desmos which I projected on the board using Airplay. This was a neat way to have the students to "show off" their work. However, as we were going through the activities I noticed that many of the students were just waiting for the person with the Ipad to type in the equations. They were not using their calculators like I had intended. I wanted them to use the calculators first and then display their findings using Desmos. Therefore, today I had the students to let me know when they got each of the pictures correct. I initialed their papers where they had written down the equations that gave the picture. We did not use the Ipads today. I liked the change - especially since we were working on the activity that pulls all the shifts together. By requiring initials for each of the problems I was able to formatively assess each individual instead of each group. Since each graph took a good bit of  time to create it was feasible for me to initial.

Today's activity introduced the students to the vertex form of a parabola. It combined all the pieces that we have been working on in the last few days. I feel that after working on getting the equations that generate the pictures the students were beginning to demonstrate a firm understanding of the phase shifts. 

On a side note...Thursday Tom Laster and Laura Murphy from It's About Time came to our school to observe our classes and discuss plans for next year. Observe is really the wrong word...they actually came into my class and sat with my students and participated in the graphing activity. My students really enjoyed this and I was honored that they came to visit!

IMP Fireworks Day 1 - Victory Celebration

I am excited to be "switching gears" and starting a new unit today. We did the Victory Celebration activity which introduces students to the unit problem for Fireworks. I love that the activity asks students to sketch the situation - this gives students with an artistic flair the chance to "show off" in class! There are 4 questions in the activity so I had each group split up. In my 2nd block class I had groups of 4 so I told them to let 2 people sketch the situation while the other 2 started working on the other questions. This worked out well because there was not more than 1 or 2 students in each group who were interested in drawing.

Some groups took longer than others on their sketches so I had everyone who was finished to get a graphing calculator and showed them how to enter the height equation into the calculators. It was necessary to also talk about how to adjust the window for the graph. We played with the tracing features on the calculators in order to look at approximations for the maximum height and the time the rocket was in the air.

I pulled a piece of chart paper and started talking about sketching the graph on the chart paper so we could refer to it throughout the unit. This brought up a discussion about which quadrants were needed. We also talked about labeling the axes and how we needed to be careful about drawing the graph because we didn't know how to scale the axes until we knew the maximum height and the amount of time it took for the rocket to land. (This is when we started playing with the tracing feature on the calculator but we ran out of time.)

Also, we discussed what the height of the rocket was when time was 0 seconds and what the height of the rocket was when the rocket hit the ground. These concepts are common sense really but it takes a few seconds for the answers to "hit" them.

Another cool thing that happened today was that my 4th block came in excited about getting to draw in math class. They took so long with their sketches that we didn't get as far with our discussions...but it was worth seeing them so invested in the activity.

After writing this post today I had an afterthought. I know that many people may ask, "Why are you drawing in a math class?" I told the students today that when people in the "real world" have large problems or projects to solve they often draw sketches or models in order to visualize what is happening. I think it is neat that this activity leads the students to start with a sketch!

Reviewing Exponent Rules with "The Zombie Game" and the Alice Portfolio

I was looking for some ideas over the weekend and ran across this blog post by Sarah Hagan (Math=Love). The post includes alot of great ideas but the one that I borrowed came from yet another blog post by Nathan Kraft. I have really gotten so many great ideas from the teachers I follow on Twitter and through my Feedly blog reader! I am so thankful for teachers who are so willing to share their resources and ideas with others.

So, today was the first day back after Spring Break and I wanted to review problems using exponent rules. I borrowed some individual white boards from Ms. Whitt and gave one to each student. I had my students go write their names on my board and put 4 x's under each name. The last student with an x under his/her name wins. I gave the class a problem to work and had them put their markers down after a certain period of time. Any student who gets an answer correct gets to go erase an 'x' from any person's name. The "zombie" part comes when students have all 4 of their x's erased. Even though they can no longer win the game, they can still erase x's from others.  I will give the winners 5 bonus points on the next quiz grade. Ms. Whitt and I did go over how to work each problem afterwards.

I altered what my students will be doing for the Alice portfolio. Due to time constraints we chose to skip some of the activities that were mentioned in the portfolio list in the book. I created my own Alice Portfolio assignment. I borrowed some items from the Meaningful Math version of the portfolio. I do not necessarily think that mine is better; it just was a better fit for me this year since we had to cut some of the activities.