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Number Patterns
My students will be given input-output charts and asked to correctly figure out the patterns and fille out the charts correctly. For example the first problem has a input of 2 and output of 11 and then an input of 5 and a output of 14. The students are then given the input numbers of 8 and 6 and asked to provide the output numbers by figuring out the correct patterns. I believe my students will use subtraction and addition to figure out the pattern and thus the correct output number. The students may struggle with this as I find my special education students often struggle with number patterns and their sequencing.

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Comparing Fractions
The learning goal is for students to use a strategy to compare two different fractions using the greater than, less than and equal symbol.
The students will be placed in partners (strategically) and given a two fractions to compare. They may use various strategies, draw a model and then discuss with their partner how they decided to compare the fractions.
Productive struggle would be seen by students drawing a model or using a strategy but possibly getting stuck and not knowing what to do next. This is where their partner will help first and then I will step in and ask questions to get them thinking but not lead them to an answer. I want them to be able to work through the problems on their own and feel successful in their productive struggle.

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Learning goal: adding 2 digit numbers
Students are broken up into pairs (L/M and M/H.)
Students take turns rolling dice and solving the problem.
Students roll 2 dice twice that give them their 2 digit numbers to add.
Students may choose which strategy to solve the problem.
Students both check the work and answer with discussion and explanations.
Evidence of Productive Struggle:
Choosing which strategy works best for them to solve the problem.
Using drawings, manipulatives, discussion with partner on best strategy/approach.
Discussions/explanations on why they used that strategy, student reasoning and work.  Evidence of No productive struggle:
Students shut down and do not even attempt.
Students do not discuss strategies or explanations.
Students do not try different strategies if the first one does not work for them.
Teacher strategies:
Pairing students to facilitate learning/growth.
Checking on pairs as they work with probing questions to encourage discussion.
If a pair is completely stuck or perhaps discussion/explanation is not being received - facilitate the discussion.

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One of the geometry objectives for 6th grade is to be able to correctly identify the net of a cube. I introduce this by dividing students into groups and giving out a number of "nets" with multiple square faces. Some of these nets are actual nets of cubes, while others are not. Through this discovery activity I tell the students to fold them and play with them to determine which are actual nets of cubes and which are not. As they fold and discover, they are to list things/criteria that they notice about the nets that actually do form cubes. Students learn that the net must be comprised of 6 square faces. However, with some of the nets that they have been given, they realize that just because it has 6 square faces, it may not form a cube. They then begin to determine the placement of the square faces relative to the others. They discover things like, 6 square faces in a straight line will NOT form a cube, nor will 5 square faces or 7. Four in the middle and one square "wing" on opposite sides WILL form a cube. However, with the same 4 in a row and both "wings" on the same side will result in overlap and NOT a cube. Ultimately, we will discuss as a class and each group will share a net that did form a cube. We will then move to a classroom game where I present on the Smart board a net. They must determine cube or no. When they make their vote, the image will, in 3d, fold and show them if it is a cube or not. Once we have gone through these, we realize that there are actually 11 different nets that form cubes. Finally, we move to an independent activity where we attempt to identify the cube nets by simply looking at them in 2 d on paper. Students are given card stock to cut and fold if they need it to assist them in answering each question. Finally, I given them links to games online where they can determine if a net is a net of a cube or not.

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3 Blocks Tower
Learning goal: Students use mathematical reasoning to reflect on multiple ways to solve a problem systematically.
Students will work individually to solve how many ways they can stack 3 blocks.  Students will draw 3 different colored blocks, in this case we will say red, yellow, and blue.  Students will work to make as many towers as possible that are each in different order by color.  Students who are struggling will be given the three blocks to manipulate.  Some students may choose to draw the colored blocks on a sheet of paper. Students might think they are finished after switching just two of the blocks around.  Students may also be struggling with keeping track of how many different ways they have already found and what they were.  Something I would ask my students is, "Is there a way you can keep track of your towers so you don't forget which ones you have used?"  I would hope students would either draw what they have created, or possibly just write the order of the colors in a "tower" of words.  Students who are catching on quickly and need to be  challenged will be asked, "What if we add a fourth block that is a different color?" Then, "What if we add a fourth block that is also red?"  Some other questions I may ask are:
How will we make sure we remember all of the ways we have found so far?
How can we be sure we have the different solutions?
If we cut out each individual solution, how would we order them to help?

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Learning Goal: Students will demonstrate their understanding of the order of operations.
Task: In Bowling for Facts, students are given a sheet that has bowling pins set up with numbers 1-10 on them. The pins are set up like they would be in a bowling alley. Each student or pair of students gets 3 dice. They roll all three dice to get their 3 numbers. With those numbers, they have to create equations that equal a number 1-10. They can use addition, subtraction, multiplication, and division, they just have to remember the order of operations. Their end goal is to find equations that would equal all 10 numbers. As they find an equation that equals the number on a pin, they color that pin, aiming to have all ten pins colored in the end.
Evidence of Productive Struggle: Students will show struggle when they are trying to determine which operations to use and which order to put them in. Some students may try to be simple, using only addition and subtraction, while others may try to challenge themselves to use all 4 operations in some way. Also, students may forget the order of operations. They will be allowed to work with a partner if they choose, so I may hear them talking about the order of operations and defending their choices of equations.
There is an unlimited number of correct answers for this task, so some students may struggle with the "high ceiling" end of it. They may also get confused as to where to start (Should they start with the number they want for the answer in mind, or use the numbers for the equation and see what they get?)
Teacher Support: To support students, I would use questions that would guide them in the right direction, but still make them think about the task at hand. Some questions I may ask are:
Why did you choose that operation?
What would happen if you chose to switch the numbers around?
If you are wanting to get a larger answer, what operations would help you get there? What about a smaller answer?

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Students are given 40 small cubes.  They are given the task of determining how many factors there are for 40 by sorting the cubes into different size groups.  Students can begin by identifying the size of equal groups (for example, you can make groups of 4 out of 40 cubes without having any remaining).  Students also can count the number of groups they have (you can make 10 groups of 4, so 4 is a factor of 40).   Students who struggle can be encouraged to make rows of cubes and see how many cubes are in each equal row.  Review of vocabulary will help students succeed as well.  Productive struggle will involve considering what will make a factor in the manipulatives.  Students will have opportunities to discuss with other students to promote using mathematical reasoning.

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Students work in pairs to create three digit numbers and compare them. First, students must choose if they want to compare numbers based on a less than or greater than. Then students take turns spinning on a spinner with the numbers 1-9. When a student spins a number they choose whether they want to place that number in the ones, tens or hundreds place and record it where they have chosen. When both students have created a three digit number they place those numbers in their greater than or less than equation.
Purpose: The purpose of this task is for students to understand that when a number has larger digits in the higher place value spots the number will be larger. Students will be able to compare three digit numbers and explain why one is greater or smaller than another.
Unproductive struggle: Students spin numbers and place them randomly without thought about which number will make the value of their number greater or less. There is little discussion and they give up when they get to the comparison equation.
Productive struggle: Students are manipulating digits in various place values to try to reach the largest or smallest number. Students are discussing and creating representations (base ten blocks, expanded form, etc.) to compare their numbers. Students may reach points where they are unsure, but are making attempts to make meaning of numbers.
Teacher support: Teacher questioning- How did you choose which digit you put in each place value?
How do you know which number is greater or smaller?
Can you show me some type of model of these numbers to explain which one is larger or smaller?
Students would need think time after each of these questions to ponder, work and discuss.

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How can they both be equal? - Ronda Herrington

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How can they both be equal? - Ronda Herrington

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First graders are working with numbers 1-9.  The question: Walter chose two numbers and added them.  Kim chose two numbers and subtracted them. They both ended with the same amount.  What numbers could Kim and Walter has used? Show your thinking using pictures, numbers, and/or words.
The purpose of this task is for students to explore and deepen their understanding of the concept of equality.
Students may add numbers randomly or use a strategic method for solving, may add both sets of numbers to make a single number, may subtract for both problems.
·  How did you decide which numbers to try?

·       How did you find numbers that gave Kim and Walter an equal amount?

What did you learn from trying out different numbers?
Students can act out problem. Students can start with an equal number and work backwards. Students can have a variety of answers.

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Students choose eight cards and arrange six of them to create two 3-digit numbers which are then added together to get as close to 1000 as possible, either over or under.  To modify, the teacher can easily have students choose 6 cards and create two 2-digit numbers that get as close to 100 as possible, or have them choose 10 cards to create two 4-digit numbers that get as close to 10,000 as possible.

**Evidence of student struggle- Once cards are chose, student cannot write them down correctly, in place value order.  Student cannot carry if need be.
**Productive struggle- Attempting the task.  Drawing cards, writing the problem down.  But maybe not sure where to go from there.
**Teacher support- I feel for this task probing guidance and affordance would be easily attainable and definitely beneficial. For students who need more support the teach can probe and ask for explanations and justifications. They can prompt student thinking and lead to new insights. To show affordance in this task, teachers need to encourage student thinking and their continued efforts. While providing time for them to linger in their own thinking. In return this would reassure and motivate students to be persistent in their own ideas and not dependent on their teacher.

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Rolling dice- comparing numbers with place value.
In pairs students will roll a dice 3 times. Each number they roll will represent a decimal. For example if they roll a 4 5 7 it would be .457
They then have to write their numbers on their white boards and compare it with their partners to see who has the greater, less than, or if they are equal numbers. To challenge students further, I would have them round each number to the tenths, hundredths, and thousandths after they compared the numbers in their pairs. Productive struggle: Discussion between the pairs of students.
Unproductive struggle: No communication between the students-saying the task is to hard and they cannot complete. I rotate group to group when I do this activity in class. I knew last year what students I needed to ask what questions to in order to get them going in the right direction. I think questioning is vital for a teacher when it comes to productive struggle.

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How many Ways Can You Make 121 Using a Combination of These Blocks?

Students will use given blocks to determine different ways to make the sum 121.
Students will need to understand that numbers can be demonstrated in a variety of different ways.  The 100 flat, the tens rod and the 1 cube will be used by my students to show how many different ways they can make 121.
To challenge the students I would ask if they see any patterns develop as they work on creating the sum of 121. The openness of this problem allows students to see many things occur (the pattern of increasing number of blocks by 9 and whether or not the number of blocks is even or odd).

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The game "Counting Counters"
Player 1: Place up to 20 red counters on a surface and screen. Place 2-6 blue counters on a surface and screen. Player 2: Tells the total number of counters by counting on and write the matching addition sentence. Player 1: uncover the screen and check player 2's work. Player 1 and player 2 switch jobs.

Learning focus:
Students will determine tell the total within 20 of two collections when screened.

Productive Struggle Indicators:
*Attempting to use strategies (counting on, using tools (dry erase/pencils, etc) to keep track of thinking), questioning and discussions between the 2 players

Unproductive Struggle Indicators:
*Defeatest talk ("I can't do this", "It's too hard") or no talk at all (staring contest) between the partners

Teacher Strategies:
I typically work in small group, so leaving the general area of students working is not likely. Proximity is the only thing that keeps some of my friends in my classroom at any given moment. I'd like to think that I would use questioning strategies and acknowledgement of student contributions with regularity. I am also a master of wait time. I also realize after taking this class, I have a bad habit of using "telling". I presumably justified this because my kids are all special and need direct instruction; too much struggle leads to shutting down. I need to find the right balance between probing, telling, and wait time to be a more effective teacher, all around.