You can download the file for the comparing-contrasting-changing-forms-of-equations foldable by clicking on the highlighted text.

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Since I think math should be fun, I have to say that I love Sarah Carter’s blog. This summer, I read her post on Function Auctions and thought that it sounded like so much fun that I spent the next two days making a Proportional Relationship Auction ,some proportional relationship anecdotal records and a proportional relationship card set that I can use7 different ways. Sarah’s blog makes you feel like you have this friendly, creative teacher down the hall.

In addition to being fun, math should be meaningful. Students should be making sense and teachers should be thinking deeply to ensure that is happening. The desire to think more deeply about my work and about my students’ work is part of why I have chosen to be part of MTBoS. My students deserve the best that I can give them and Mark regularly poses questions that make me think. His post on exit cards was the first time I really considered whether my exit cards were addressing the spectrum of the things that I want to know about my students. As I read his post, I realized I was doing a fairly good job of incorporating conceptual and procedural assessments, I really needed to incorporate more metacognition into my assessments. I appreciate that his posts don’t allow me to be complacent.

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Student 1: Should the graph be continuous?

Student 2: No, you can’t have half a person.

Student 3: Can it go into the negatives?

Student 1: No, you can’t have negative people.

Student 4: Yes you can. Not everyone is an optimist.

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I love card sets because I can use them so many ways. I can use them when I first teach the lesson one way. I can then use them again a different way for additional practice or as a quick review leading up to a quiz or a test. I can also use them as part of a differentiated instructional piece of a lesson to work with students who still haven’t mastered the concept. Finally, I can bring them back out in a month in yet another way as a quick refresher. (In Make It Stick, the author articulates the power of revisiting a concept after some time has lapsed in order to “make it stick”).

Option 1 – Card Sort

Students work in table groups or in pairs to sort the cards. They need to match the four different forms of the same equation.

Option 2 – Rummy

Students use the cards to play Rummy. Students play in pairs or triads. Each player is dealt six cards. A single card is face up and the remaining cards are face down in a draw pile. The player to the left of the dealer begins play. He or she takes the top card in the draw pile or takes the top face-up card in the discard pile. The player may lay down any matched sets he or she has face up. A player may add any one or two cards from his or her hand to a matched set already laying face up on the table. If he or she does not play any matched sets, he or she must play a card in the discard pile (it can’t be the same card he or she drew from the face up discard pile). Matches consist of cards that are different forms of the same equation. The first player to play all of his or her cards is the winner.

Option 3 – Odd One Out

Remove one equation card from the set. This leaves one equation with only 3 equations. Since the game play involves making matched pairs, the remaining equation is the “Odd One Out” Students play in groups of 3 or 4. The cards are dealt out. Players look at their cards and lay down their matches (in pairs). When it is a player’s turn, he or she draws a card from another player and tries to make matches with the cards in his or her hand. At the end of the game, the player with the Odd One Out card loses the game. Of the remaining players, the one with the most matches is the winner. (This is essentially Old Maid with equations.)

Option 4 – Concentration

Students work with a partner or play individually. All the cards are laid face down. The player turns over two cards. If the two cards represent the same equation, it is a match and the player keeps them. If they do not represent the same equation, the player turns them face down again. If two players are playing, the next player takes a turn The player with the most matches at the end of the game is the winner.

Option 5 – Spoons

You need a set of teaspoons in addition to the cards. You should have one fewer spoon than the number of players. Put the spoons in small circle in the middle of the table. Deal four cards to each player Each player tries to make four of a kind. The dealer takes a card off the top of the deck, removes one of his/her cards and passes it facedown to the left. Each player discards to the person on his/her left. The last player discards into the trash pile. This continues until someone gets four of kind and takes a spoon from the center. Once the player with four of a kind takes a spoon, anyone can take a spoon. The player without a spoon gets a letter in the word SPOON. When a player has spelled SPOON, he or she is out of the game. If the cards run out, reshuffle the trash pile and continue play.

I didn’t have enough spoons on hand, so we played spoons with forks. Spoons took a far amount of time today, but the students had fun and liked the game. This option is probably best for a review activity rather than on the first day of instruction.

Option 6 – QuizQuizTrade

Each student is given a card of a particular form. I direct them to change it into a different form (e.g., they are given a card in standard form and I tell them to change it to slope intercept form). They solve the problem and then do a Quiz Quiz Trade. Quiz Quiz Trade is a cooperative learning structure. The first partner quizzes the second partner to turn his/her equation into a different form. If he or she struggles, the first partner can give him or her a hint. If he or she continues to struggle, the first partner can give a second tip. If he or she continues to struggle, the first partner can tell the second partner how to solve the problem (Tip Tip Tell). Partners then reverse the process and repeat. Finally, they trade cards and find new partners. I have students continue for whatever amount of time I want to allocate to the task.

Option 7 – Concept Attainment Cards

When I make the card activity, I use a different color of card stock for each set. With this activity, I give each group a set of cards that is all the same color for the desired concept (all the pink cards are in slope intercept form) and a different color for the non-examples (purple cards with equations in standard form or in x=kx+d form). Students then have to look at the examples and non-examples in order to define the concept illustrated by the examples.

Option 8 – Give One Get One

Students line up in two lines, facing each other. I give each student in one line a card with an equation in a specified form of an equation (e.g., everyone in the first line has a card in y=mx+b form). I give each student in the second line a card with an equation in a different form (e.g., everyone in the second line has a card with an equation in x=ky+d form). I then tell everyone to transform their equation into another form (e.g., standard form). When they have done so, they step forward into the space between the lines. They trade cards and solve the other card, discuss their solutions, check each other’s work, and then step back into the lines. When all the partners are done, one line passes their card down and the other line passes it up. The second line shifts up by one (the person at the head of the line goes to the end of it and everyone moves forward) and the first line shifts down by one. Now everyone has a new problem and a new partner. I repeat the process for whatever amount of time I want to allocate to the task.

You can download the rewriting-equations-card-sort file by clicking on the highlighted text.

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Today, they worked through a Connected Math lab exploring how to change between forms. I really like the way Connected Math introduces this topic. Students are presented with four pieces of student work in which someone has rewritten an equation in a different form that they must analyze. Students have to determine which pieces of student work are correct, justify the steps taken for those that are correct, and identify the errors for those that are incorrect. Each of the pieces of student work tackles the task in a different way, so students have to really think about things a little more deeply. The lesson follows up with a task in which they move equations in slope intercept form into standard form and equations in standard form into slope intercept form.

Tomorrow, I am going to spend a second day on the concept before moving onto the next lesson in the investigation. I want to take a little more time to solidify the concept. I I plan to use these Interactive Notebook pages with my students to practice moving between different forms of equations

You can download the changing-forms files by clicking on the highlighted text.

I will also have them practice with a game. I will write about that tomorrow after I see how it goes.

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Earlier this week, I contacted a parent because I wanted to nominate her daughter for a STEM camp. I received this in return.

*“Thank you for seeing this talent in …. . She’s not been very confident in her abilities to the point that she came home once several years ago saying that she wasn’t good in math… I could not believe my ears and I am glad she persevered and she was so fortunate to have you as her teacher.”*

I find the fact that this young woman ever could have thought that she was not good at math absolutely stunning and I’m not really sure what I did to change that. I didn’t do anything special. There were no long pep talks. There was no special “cheerleading”. There was just normal math class.

After some thought, I think there might be a few things in normal math class that make a little bit of difference for her (and for other girls as well).

- I genuinely believe that she is good at math. (I believe this of all my students. Everyone can be good at math) I don’t know if this mattered, but I find that having someone believe in me can be very powerful.
- Because I believe she is good at math, there are no softball questions. I ask her hard questions and she rises to them every time. That doesn’t mean she gets every one of them right, but she thinks deeply about every one of them and has learned that she can think deeply.
- I am lucky enough to use a really good, research-based curriculum (Connected Math). It’s not a perfect curriculum (nothing is), but it is designed to allow students to construct meaning. Students leave with a fairly good conceptual understanding of the math rather than a bunch of rules that they blindly follow. Having the chance to make sense of something means she gets to make it her own, to know that she knows it, to believe in her abilities.
- I use Kagan Cooperative Learning Structures a lot. These structures provide this beautiful interdependence that helps everyone rise. Students start out working independently (so they have to figure things out, not just ride on someone else’s coat tails). Then, they work with one or more classmate (depending on the structure), discussing their work and coming to a consensus (the mathematical discourse is so much richer since I started using these structures). They have a stake in each others’ success because they don’t know who will speak for the group. I think this interdependence is good for all students, but girls seem to really thrive on it.
- I make time for regular review (5-10 minutes every day) of previously learned concepts. This delayed recall helps students to retain concepts that they’ve learned. I also use this time to differentiate instruction. Sometimes students are assigned to work on a station based on their mastery level. Sometimes students are paired so that someone with mastery is working with someone who is still working on mastery (providing an opportunity for some extra kid to kid discussion that helps to scaffold the concept). The groupings are ever changing so every student knows the reality that everyone is good at some things and everyone has some things on which they still need to work. I think knowing that no one is good at everything helps kids see that not being perfect at everything does not equate to not being good at math.
- Both of the years that I have had this young woman in my math class, she has been in a class dominated by girls. It is just a strange fluke in the way that schedules were run, but both years I have had a period in which 3/4 of the class was girls and a period in which all of the class was boys .or in which there was only one girl. (I try to get a schedule change for the girl when that happens so that she is not alone). There is a lot of research out there that shows that girls in math classes do better when at least 40% of the population is female because they don’t have to expend energy fighting a stereotype. My anecdotal evidence is that both years that I have had these girl-dominant classes, the girls have grown mathematically in degrees that I have never seen in any of my other classes. It’s not that these classes are filled with perfect, compliant students. It’s more that these classes are filled with strong, smart girls who interact differently than they do when there aren’t so many girls. .

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Quiz/Quiz/Trade is one of my students/ favorite review activities. When we begin, I give each student a card. Each student works independently to solve his or her card. I laminate the cards before using them so that they can use dry erase markers on the cards as they solve them. After everyone has solved his or her problem, they erase their answer and then walk around the room to find a partner. The first student quizzes the second by showing him or her the card. The second student then solves the problem. If he or she has difficulty, the first student can give him a tip. If he or she still has difficulty, the first student gives a second tip. If the second student still has difficulty, the first student tells the second student how to do the problem. They then reverse roles and repeat the process Finally, they trade cards and find new partners. (Quiz/Quiz/Trade with Tip/Tip/Tell if needed). My students love the chance to move around the room and to interact with different peers. I love it because it works. Marzano’s research has shown that movement changes brain chemistry and increases levels of engagement. His research has also shown that the use of cooperative learning structures increase student outcomes.

You can download the solving-inequalities-quizquiztrade-cards file by clicking here.

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My students came into this project with a lot of background knowledge. This is their second year with me. Last year, I introduced them to some basic coding by having them create video games using SCRATCH. The Ozobots use Ozoblockly, which is very similar in structure to SCRATCH. (Both languages are object-oriented visual programming language. I think this is absolutely the best way to introduce coding. They allow the user to focus on the big ideas of coding without getting bogged down in the syntax of the language.) The added challenge this year is having the code control a miniature robot. This required students to work through a few challenges that had nothing to do with coding, but everything to do with problem solving and making things work. I documented the journey on my twitter account . There are videos there showing each step of the projects (posts dated Dec 8 – 15). Unfortunately, you will have to go to the twitter account to see the videos if are interested, because my blog doesn’t support videos.

**Building Background Knowledge **

InitiallIy, I focused on building knowledge. Students had to learn how to calibrate the bots, how to write code, and how to load the code onto the bots.

Next, they worked on getting the bots to do what they wanted. They explored the different capabilities: getting the bot to move in specific ways, playing with the light features, etc. During this early phase, I had students work in pairs. Because there is a certain amount of frustration inherent in the process of figuring out how to make things work, it helps to have someone to share the journey.

**The Project**

Once I was confident that everyone had a little experience, I assigned a two-part project that would require them to review the work we had done earlier in the year with linear equations and also act as a launch into the work solving systems of linear equations that we will do at the start of the spring semester.

**Linear Relationships in Tables and Graphs**

For the first part of the project, I assigned each pair of students a different linear relationship. Each pair was given two ordered pairs, an ordered pair and the y-intercept, or an ordered pair and the slope. They had to find the linear equation representing their relationship and create a table for the relationship. Then, they had to create a large scale coordinate plane and write code so that their Ozobot would graph the linear relationship. The Ozobot had to flash lights on three ordered pairs and perform a different action at the y-intercept (flash different lights or spin on the spot).

**Solving a System of Equations Graphically**

Next, each pair had to use a second Ozobot to graph a second line intersecting the first line on their coordinate axis. They had to calculate the point of intersection and time the two Ozobots so that they would not collide at the point of intersection.

While there was plenty of math in the project, the most important piece of math was not what one usually thinks of as math. It was learning how to make something real actually work. Students had to learn how to frame a task and then to persevere in the face of challenge (one of the math practice standards).

**My Favorite Thing**

My biggest measure of success is that the kids dinner time conversations were all about math for the entirety of the project (a number of them ended up getting their own bots for Christmas as a result).

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I start helping my students make sense of the process using a chip model. They use black chips to represent positive numbers and red chips to represent negative numbers. They combine the chips, making zero pairs (one red and one black chip) to find the sum. I start with the chip model because most students think of addition as combining sets of objects.

After students have mastered the chip model, I move on to a number line model. The curriculum that I use (Connected Math) does a really nice job introducing the number line model. It talks about addition as combining sets by providing a context in which two kids each have a number of video games and then talking about the combined number of video games. It illustrates the problem using a chip model. Then, it goes on to talk about addition also being representative of a context in which one “adds on”. It provides a context in which there is a temperature at sunrise and that is “added on” to as the temperature rises over the course of a day. They model this problem using a number line.

After introducing students to the number line model, I take them out into the hall to walk the number line. Prior to this, I have had my student aides create a number line for each table group. Each number line is created using painters tape (for ease of removal when the time comes). The numbers on each number line range from -10 to 10. I begin by modeling a couple of integer addition problems on the line. I walk forward for positive numbers and backward for negative numbers.

After modeling several problems, I have each group complete a set of problems by walking on the number line. Each group gets a laminated index card with the problems to be completed and a dry erase marker to record their answer. The first group member walks the number line to solve the first problem. The other members of the group check his or her work. The next person in the group walks the number line for the second problem and the others check the work. Each problem is completed by a new student until everyone in the group has walked the number line. At that point, the group rotates through again until all the problems have been completed.

Once students have mastered integer addition on the number line, they use models to construct an algorithm for integer addition. They begin with a group of four problems. They solve these problems using a model. Then, they identify what is the same about the problems and create two more problems to fit the group. (All of the problems in the group have addends in which the signs match.) Finally, they come up with an algorithm for the adding problems within the group. They repeat this process for a second group of problems in which the signs of the addend pairs do not match. If students have trouble figuring out the algorithm, I remind them that each number has a magnitude (size) and a direction (sign) and suggest that they consider the two parts separately. This usually helps them get to the algorithm.

After creating the algorithm, students test it on rational numbers (mixed numbers) and verify its efficacy using a number line model. Finally, they test rational number addition for commutativity.

After students have created an algorithm, I use the foldable shown below to summarize the lesson.

I follow up with this Always, Sometimes, Never exit card.

integer-addition-always-sometimes-never-v2

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