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According to the Progressions for the CCSS Math, "Students expand their work in addition and
subtraction from within 5 to within 10. They use the Level 1 methods
developed for smaller totals as they represent and solve problems
with objects, their fingers, and math drawings" (p. 10). Students entering Kindergarten may or may not be comfortable counting to 10 and adding within 10.  The misconception is that all Kindergarten students can add and subtract fluently when entering Kindergarten.  Some students will need more practice than others. The “Five Frames” applet from NCTM allows Kindergarten students to work with five frames in order to use objects such as counters, stars, bugs and apples in order to add within 10.

I would allow students time with a partner to practice with the four games on this applet: How many squares are empty? Move 3 bugs to the frame. How many more bugs to fill the frame? Use frames to find 5 + 1.  Peer tutoring is a solid way to motivate and engage learners so carefully pairing students with one strong student and one struggling student will assist with this setup.  In order to further assess student understanding, I would ask students a series of questions.  Top students will be assigned five random questions from the "play all" section while struggling students will be assigned five random questions from the "How many?" section.  I will make the questions easier or harder based on each student's performance.

Applet: https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Five-Frame/

CCSS standard citations: http://www.corestandards.org/Math/Content/K/OA/

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

CCSS.MATH.CONTENT.K.OA.A.1

Represent addition and subtraction with objects, fingers, mental images, drawings 1, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

CCSS.MATH.CONTENT.K.OA.A.2

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

CCSS.MATH.CONTENT.K.OA.A.3

Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

CCSS.MATH.CONTENT.K.OA.A.4

For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

CCSS.MATH.CONTENT.K.OA.A.5

Fluently add and subtract within 5

The Common Core Standards Writing Team (29 May 2011). Progressions for the Common Core State Standards in Mathematics. Retrieved from https://achievethecore.org/content/upload/Draft-K-5%20Progression%20on%20Counting%20and%20Cardinality%20and%20Operations%20and%20Algebraic%20Thinking.pdf

Operations & Algebraic Thinking. Retrieved from http://www.corestandards.org/Math/Content/OA/

Episode 2 – What is UDL in the math classroom?

Retrieved from https://theothermath.com/index.php/2017/09/03/what-is-udl-in-the-math-classroom/

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According to the Progressions for CCSS Mathematics, "Using Level 2 and Level 3 methods to extend addition and subtraction problem solving beyond 10, to problems within 20. In particular, the OA progression in Grade 1 deals with adding two single-digit addends, and related subtractions" (Progressions p. 12). The common misunderstanding by students are how and when to use various strategies such as counting on, making ten, decomposing numbers, using relationships between addition and subtraction, and creating equivalent and easier sums.  The “Ten Frame” applet from the NCTM will provide practice for students and students will be able to use these strategies and become fluent when adding numbers within 20.

I will give students time with a partner to explorer the four modes including how many, build, fill and add.  At the end of class, I will give students an “exit ticket” with one question from each mode.  I will then ask students to create their own addition problem.  The “exit ticket” will allow me to check for individual student understanding as well as whole class understanding.  I will also be able to provide additional instruction and practice on specific types of problems based on the results.

Applet: https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Ten-Frame/

CCSS standard citations: http://www.corestandards.org/Math/Content/1/OA/

Add and subtract within 20.

CCSS.MATH.CONTENT.1.OA.C.5
Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

CCSS.MATH.CONTENT.1.OA.C.6
Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Work with addition and subtraction equations.

CCSS.MATH.CONTENT.1.OA.D.7
Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

The Common Core Standards Writing Team (29 May 2011). Progressions for the Common Core State Standards in Mathematics. Retrieved from https://achievethecore.org/content/upload/Draft-K-5%20Progression%20on%20Counting%20and%20Cardinality%20and%20Operations%20and%20Algebraic%20Thinking.pdf

Operations & Algebraic Thinking. Retrieved from http://www.corestandards.org/Math/Content/OA/

Episode 2 – What is UDL in the math classroom? Retrieved from https://theothermath.com/index.php/2017/09/03/what-is-udl-in-the-math-classroom/

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According to the Progressions of the CCSS Mathematics, "Grade 2 students build upon their work in Grade 1 in two major ways.2.OA.1 They represent and solve situational problems of all three types which involve addition and subtraction within 100 rather
than within 20, and they represent and solve two-step situational
problems of all three types" (p. 18). Thus, the common misunderstanding in 2nd grade is adding and subtracting within 100 especially when dealing with two-step word problems.  The applet from Khan Academy allows students to watch videos then practice word problems.  During the practice section, a word problem is given along with a picture in order to solve the problem.  Students have to decide if the problem is addition or subtraction based on context clues from the word problem.  Struggling students who are stuck can "watch a video or use a hint" in order to solve the problem.

I would first have students watch the videos and practice with the first problem set.  As students are practicing, I would circulate around the class to help students with their thinking and understanding.  This type of informal assessment will also give me insights to what each student can and cannot do.  I would then review the questions and answers with the students as a whole class. During the course of the lesson, I would also ask the class to pause and I would pose a question to the class so that students can think, pair and share.  I would randomly select students to share their work with the class so that students learn from each other.  At a later date - maybe for homework or maybe for a warm up the next day - I would ask students to solve the second set of problems from the Khan Academy website.  For this, I would print the assignment out and ask students to write down their work in order to support their answers.

Applet:

https://www.khanacademy.org/math/cc-2nd-grade-math/cc-2nd-add-subtract-100/cc-2nd-add-sub-100-word-problems/e/addition-and-subtraction-word-problems-within-100--level-1

CCSS standard citations:

Represent and solve problems involving addition and subtraction.

CCSS.MATH.CONTENT.2.OA.A.1
Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Add and subtract within 20.

CCSS.MATH.CONTENT.2.OA.B.2
Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers.

The Common Core Standards Writing Team (29 May 2011). Progressions for the Common Core State Standards in Mathematics. Retrieved from https://achievethecore.org/content/upload/Draft-K-5%20Progression%20on%20Counting%20and%20Cardinality%20and%20Operations%20and%20Algebraic%20Thinking.pdf

Operations & Algebraic Thinking. Retrieved from http://www.corestandards.org/Math/Content/OA/

Episode 2 – What is UDL in the math classroom? Retrieved from https://theothermath.com/index.php/2017/09/03/what-is-udl-in-the-math-classroom/

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One area of common misunderstanding is the difference between composite and prime numbers.  According to the Progressions of the CCSS Mathematics, " A prime number has only one and itself as factors. A composite number has two or more factor pairs. Students examine various patterns in factor pairs by finding factor pairs for all numbers 1 to 10" (p. 30). The applet from Math Playground, Match 10 Multiples, allows students to better understand the difference between prime and composite numbers, and become more fluent with factors and multiples.

I would allow students to play this game individually and with a partner.  I like how the game has a timer and a point system.  This will make the activity more competitive between students and students will concentrate and focus more as a result.  I also like this game because the more multiples you get right, the faster the game goes which makes it harder and students will have to think quickly on their feet.  The game will also slow down if students click numbers that are not multiples of the given number.  As a result, this applet will create a high level of engagement.

Gain familiarity with factors and multiples.

CCSS.MATH.CONTENT.4.OA.B.4

Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.

Applet: https://www.mathplayground.com/mach_10_multiples.html

The Common Core Standards Writing Team (29 May 2011). Progressions for the Common Core State Standards in Mathematics. Retrieved from https://achievethecore.org/content/upload/Draft-K-5%20Progression%20on%20Counting%20and%20Cardinality%20and%20Operations%20and%20Algebraic%20Thinking.pdf

Operations & Algebraic Thinking. Retrieved from http://www.corestandards.org/Math/Content/OA/

Episode 2 – What is UDL in the math classroom?

Retrieved from https://theothermath.com/index.php/2017/09/03/what-is-udl-in-the-math-classroom/

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In third grade, students spend a lot of time learning and practicing how to multiply and divide.  One area that is often difficult for students to grasp is knowing when to use the four major operations including addition, subtraction, multiplication and division.  According to the Progressions of the CCSS Mathematics, "Use of two-step problems involving easy or middle
difficulty adding and subtracting within 1,000 or one such adding
or subtracting with one step of multiplication or division can help
to maintain fluency with addition and subtraction while giving the
needed time to the major Grade 3 multiplication and division standard" (p. 28).  The applet that I chose to address this was called "Make a number..." from the Math Playground website.  In this applet, students are given three or four numbers and four operations.  Students need to use the four operations in order to create the target number.

I would expand on this applet and I would have students write an expression to represent their thinking. Students would need to consider the order of operations, the commutative property, the associative property and the distributive property when creating their expression. Some students might create different, but equivalent expressions.  A lot can come out of this applet.  I would give students time to both practice independently and with a partner.  I would ask students to create their own problems and create an answer key for their work.  Students can exchange their questions with each other.  They can check their partners work as they are the experts of their work. As a result, students will learn from each other.

Applet: https://www.mathplayground.com/make_a_number.html

CCSS:

Represent and solve problems

Understand properties of multiplication and the relationship between multiplication and division.

CCSS.MATH.CONTENT.3.OA.B.5
Apply properties of operations as strategies to multiply and divide.² Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

Multiply and divide within 100.

CCSS.MATH.CONTENT.3.OA.C.7
Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

CCSS.MATH.CONTENT.3.OA.D.8
Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

The Common Core Standards Writing Team (29 May 2011). Progressions for the Common Core State Standards in Mathematics. Retrieved from https://achievethecore.org/content/upload/Draft-K-5%20Progression%20on%20Counting%20and%20Cardinality%20and%20Operations%20and%20Algebraic%20Thinking.pdf

Operations & Algebraic Thinking. Retrieved from http://www.corestandards.org/Math/Content/OA/

Episode 2 – What is UDL in the math classroom?

Retrieved from https://theothermath.com/index.php/2017/09/03/what-is-udl-in-the-math-classroom/

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Simplifying algebraic expressions using the order of operations is not always easy for students.  PEMDAS often can be difficult to maneuver.  Students must first handle grouping symbols then exponents then multiplication or division from left to right then addition or subtraction from left to right.  Working left to right can also be confusing to students.  The applet, PEMDAS Exhibit by Math Playground allows students to practice with the order of operations.  Some questions are easy and some questions are hard.  What I like about this applet is that there is a story to go with the problems.  You are helping the museum uncover missing numbers.  I also like that you have to click the appropriate operation before solving.  You cannot go on unless you choose the correct operation.  Lastly, I like that you get instant feedback from the computer when you get the question right.

I would allow students to practice order of operations using this applet.  I would then give an "exit ticket" at the end of class.  For the "exit ticket," I would give students 3 questions to solve.  One question would be easy involving one addition/subtraction and one multiplication/division, a second question would be medium involving one exponent, one addition/subtraction and one multiplication/division, and the last question would be hard involving grouping symbols along with exponents, addition, subtraction, multiplication and division.  I will then collect the "exit ticket," grade it and use the results to help me plan class the following day.

Applet: https://www.mathplayground.com/pemdas_exhibit.html

The Common Core Standards Writing Team (29 May 2011). Progressions for the Common Core State Standards in Mathematics. Retrieved from https://achievethecore.org/content/upload/Draft-K-5%20Progression%20on%20Counting%20and%20Cardinality%20and%20Operations%20and%20Algebraic%20Thinking.pdf

Operations & Algebraic Thinking. Retrieved from http://www.corestandards.org/Math/Content/OA/

Episode 2 – What is UDL in the math classroom?

Retrieved from https://theothermath.com/index.php/2017/09/03/what-is-udl-in-the-math-classroom/

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In 6th grade, students are required to solve one step equations.  Knowing when to add, subtract, multiply and divide is not easy at first.  The applet that can help with balancing equations is from the Math Playground website called "Algebra Equations...Build a model then find the value of x.  Can you solve it?"  First, students need to setup and model the equation on a balance scale.  Then, students need to remove pieces from the scale in order to balance the equation and solve for the unknown variable.  Students will not be able to move on unless they get the step correct.

According to Tomlinson, "Assessment routinely takes place as a unit begins to determine the particular needs of individuals in relation to the unit’s goals" (Tomlinson p. 8). I would assess students at the beginning of the unit, during the unit and at the end of the unit. These mini assessments would help me determine how I should instruct each student. Assessments during the unit would allow me to see if the individual or class as a whole is understanding and what they are not understanding.  The final assessment will be an opportunity for students to present what they now know.  Furthermore, the article "Episode 2 – What is UDL in the math classroom?" from the othermath.com explained that UDL and differentiated instruction can thought of together through assessments.  The article stated, “Assess student progress during learning and adjust as needed.”  This way student understanding will dictate how lessons should be taught.

Applet: https://www.mathplayground.com/AlgebraEquations.html

CCSS standard citations: http://www.corestandards.org/Math/Content/6/EE/

Reason about and solve one-variable equations and inequalities.

CCSS.MATH.CONTENT.6.EE.B.5
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

CCSS.MATH.CONTENT.6.EE.B.6
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

CCSS.MATH.CONTENT.6.EE.B.7
Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

Tomlinson, C. A. (2005). How to differentiate instruction in mixed-ability classrooms (2nd ed.). Retrieved April 16, 2001, from http://www.teachersity.org/resources/instruction.pdf

Grade 6: Expressions and Equations Retrieved from
http://www.corestandards.org/Math/Content/6/EE/

Episode 3 - What is UDL in the math classroom?

Retrieved from https://theothermath.com/index.php/2017/09/03/what-is-udl-in-the-math-classroom/

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Building on 6th grade, students in 7th grade are required to solve two step equations.  This is often difficult for students and are unsure of what to do next when solving an equation.  I believe that the applet from hoodamath titled "Algebra Balance Equations" can help students practice solving two step equations.  I like this applet because students have to first move pieces to the balance beam in order to balance it.  Next, students have to click the correct operation and type the correct number in order to manipulate each side of the equation.  Students will not be able to move on to the next step unless the step is done correctly.  Furthermore, I like this applet because students have to consider variables on both sides, the distributive property, combining like terms and the four operations addition, subtraction, multiplication and division.

I would assess students at the beginning of the unit with a K-W-L chart geared towards math.  I would present students with a couple one step and two step equations to think about.  I would have students solve each of them (or try to solve each of them) and write down what they Know and what they Want to know.  I would then review the equations and students would self-grade their work.  Students would then write down what they Learned at the end of class.  As mentioned in the NEA article, “K-W-L (Know, Want to Know, Learned),” the purpose is to elicit students’ prior knowledge of the topic of the text, sets a purpose for reading, and helps students to monitor their comprehension.”  In this case, the focus is on math and solving equations not reading and reading comprehension.

Applet: http://www.hoodamath.com/mobile/games/algebrabalanceequations.html

Grade 7: Expressions and Equations http://www.corestandards.org/Math/Content/7/EE/

Use properties of operations to generate equivalent expressions.

CCSS.MATH.CONTENT.7.EE.A.1
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

CCSS.MATH.CONTENT.7.EE.A.2
Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that "increase by 5%" is the same as "multiply by 1.05."

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

CCSS.MATH.CONTENT.7.EE.B.3
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

NEA K-W-L (Know, Want to Know, Learned) Retrieved from http://www.nea.org/tools/k-w-l-know-want-to-know-learned.html

Tomlinson, C. A. (2005). How to differentiate instruction in mixed-ability classrooms (2nd ed.). Retrieved April 16, 2001, from http://www.teachersity.org/resources/instruction.pdf

Episode 3 - What is UDL in the math classroom?

Retrieved from https://theothermath.com/index.php/2017/09/03/what-is-udl-in-the-math-classroom/

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In 8th grade, find a solution to a system of linear equations is not always easy to find.  Using an online graphing calculator can quickly and efficiently help students find the solution or in some cases no solution.  There are many options online.  Personally, I like the desmos calculator.  Type in the equation of line 1.  Type in the equation for line 2.  Find the point of intersection by hovering over the ordered pair where the two lines intersect.

In terms of assessing students, I would give them a pre-test on the following: graphing a line given an equation, graphing a line given a word problem or situation, graphing two lines and finding a point of intersection and graphing two lines give a word problem or situation. Students need to know how to graph lines, but also need to make sense of their solution. The ultimate goal is for students to conceptualize what the point of intersection means in context of the problem. Based on how students do, I will develop practice to help students understand and meet the overall learning objective. Take this proactive approach to learning will lead to student growth and understanding.

Applet: https://www.desmos.com/calculator

8th Grade: Expressions and Equations http://www.corestandards.org/Math/Content/8/EE/

Analyze and solve linear equations and pairs of simultaneous linear equations.

CCSS.MATH.CONTENT.8.EE.C.7
Solve linear equations in one variable.

CCSS.MATH.CONTENT.8.EE.C.7.A
Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

CCSS.MATH.CONTENT.8.EE.C.7.B
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

CCSS.MATH.CONTENT.8.EE.C.8
Analyze and solve pairs of simultaneous linear equations.

CCSS.MATH.CONTENT.8.EE.C.8.A
Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

CCSS.MATH.CONTENT.8.EE.C.8.B
Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

CCSS.MATH.CONTENT.8.EE.C.8.C
Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

Episode 2 - What is UDL in the classroom?

Retrieved from https://theothermath.com/index.php/2017/09/03/what-is-udl-in-the-math-classroom/