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Subtraction: Adding Up

Two ideas to consider when crafting number talks to encourage the Adding Up strategy for subtraction are 1) keep the minuend (the whole) and the subtrahend (the part to be subtracted from the whole) far apart, and 2) frame the problem in a context that implies distance.

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As a result of more effective ESL instruction, students:

understand the content better and are working together to find creative ways to learn

discuss math more and know how to use the instructional supports their teacher has in place

are comfortable with math and asking questions to get the help they need.

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My dad gave me one dollar bill
'Cause I'm his smartest son,
And I swapped it for two shiny quarters
'Cause two is more than one!

- Excerpt from "Smart," by Shel Silverstein

As a student, I struggled with math. I didn't understand why it came so naturally to some students, but not to me. Looking back, however, I realize that I had an advantage that I wasn't even aware of - I understood the language in which the problems were written, even if I didn't understand how to solve them! Although it is easy to assume that many English language learners (ELLs) will excel in math because math is a "universal language" and students may have had prior educational experience that included mathematical instruction, that assumption can lead educators astray.

As I spoke with teachers and did research for this article, it became very clear that making sure that students understand math vocabulary and have ample opportunities to use it are very important. Solving word problems, following instructions, understanding and using mathematical vocabulary correctly - all of these skills require a language proficiency that sometimes exceeds our expectations. We tend to think of mathematics as a subject that does not require a strong command of language. In reality, however, mathematical reasoning and problem solving are closely linked to language and rely upon a firm understanding of basic math vocabulary (Dale & Cuevas, 1992; Jarret, 1999).

For many educators, the challenge of bringing language and math instruction together is a relatively new one. ELL teachers who hadn't taught content areas previously are now being asked to lead or support instruction in the math classroom, and many math teachers who don't see themselves as language instructors are now responsible for providing effective math instruction to ELLs.

High school math teacher Hillary Hansen learned just how big a role language plays in math instruction when she taught her first Basic Math course for ELLs last year. She wanted so much to provide the students with the good foundation they needed, but she felt unable to reach the students or engage them in her lessons, and by the end of the year she was exhausted and frustrated.

That summer she had an opportunity to join a district Sheltered Instruction Observation Protocol (SIOP) cohort to receive professional development and support to meet the needs of ELLs in content classes. She learned about the importance of language acquisition, building background knowledge, increasing student language production, and explicitly teaching academic language. She began this school year with a new set of tools and a deeper understanding of the instructional scaffolding ELLs need in order to learn the content while also learning English. I am happy to report that while Hillary still feels challenged and is working very hard, this year has been much more successful for her and her students.

As a result of more effective instruction, her students:

understand the content better and are working together to find creative ways to learn

discuss math more and know how to use the instructional supports their teacher has in place

are comfortable with math and asking questions to get the help they need.

Hillary feels that she is providing them with the foundation they need not only to understand the mathematical concepts, but also to successfully interact within a math classroom in order to continue learning more advanced concepts.

Following are some strategies that Hillary and some of the other teachers I spoke with found helpful this year, and that they recommend as best practices when teaching math to ELLs.

The importance of teaching academic vocabulary

Vocabulary instruction is essential to effective math instruction. Not only does it include teaching math-specific terms such as "percent" or "decimal," but it also includes understanding the difference between the mathematical definition of a word and other definitions of that word.

The following example, used in a presentation by Dr. Judit Moschkovich of the University of California at Santa Cruz, underscores why vocabulary must be introduced within the context of the content (Moschkovich, 2008):

In this problem, the student is instructed to "find x." The student obviously knew the meaning of the word "find" because he/she "found" it on the page and circled it. The student even put a note on the page to help the teacher in locating the lost "x". The student understood the meaning of "find" in one context, but not in the appropriate mathematical context.

I recently helped a math teacher create a Sheltered Lesson, and I was surprised to find that there were some vocabulary words that I didn't understand. My lack of familiarity with the words hindered my ability to do the math problem and gave me a deeper empathy for ELLs who struggle in the same way with vocabulary and comprehending math assignments. Following is a list of tips for explicitly teaching mathematical academic vocabulary:

Demonstrate that vocabulary can have multiple meanings. Help students understand the different meanings of words such as "table" and "quarter," as well as how to use them correctly in a mathematical context.

Encourage students to offer bilingual support to each other. Students will understand material better if they explain it to another student, and the new student will benefit from hearing the explanation in their first language. (Check the hotlinks for a list of bilingual translations of math vocabulary in multiple languages).

Provide visual cues, graphic representations, gestures, realia, and pictures. Offer students the chance to work with objects and images in order to master vocabulary. If there aren't enough items for each student, use manipulatives on the overhead or posted throughout the classroom, and demonstrate the vocabulary in front of the students. For example, Hillary created a Math Word Wall that has three parts: key vocabulary, "in your own words" definitions, and a variety of ways to portray a function. For example, multiplication is portrayed by the following symbols: x, *, and ( ).

Identify key phrases or new vocabulary to pre-teach. This strategy will help students decide which math function they should apply. Example: "more than" means "add." (See hotlinks for more references about math vocabulary.)

The importance of reading and understanding written math problems

Written word problems present a unique challenge to ELL students and teachers alike. In Reading and Understanding Written Math Problems, Brenda Krick-Morales writes, "Word problems in mathematics often pose a challenge because they require that students read and comprehend the text of the problem, identify the question that needs to be answered, and finally create and solve a numerical equation - ELLs who have had formal education in their home countries generally do not have mathematical difficulties; hence, their struggles begin when they encounter word problems in a second language that they have not yet mastered" (Bernardo, 2005).

Teacher Xiao-lin Yin-Croft has encountered this pattern in her classroom of bilingual Chinese students in San Francisco. She has developed a very creative way to use her students' background knowledge of math as a stepping stone for other language learning. She does this by accelerating math instruction at the beginning of the school year and then building on what students have learned in math in reading and other content areas. In Building Bridges for the Future, a Colorín Colorado "From the Heart" article, Xiao-lin explains her strategy:

First, we read math word problems; I demonstrate the logical thinking process while translating words into pictures and, finally, into number sentences. Soon, they start to explain their own thinking after reading complicated word problems that involve several steps. They correct each other, and argue about which number sentences they should use to arrive at the correct final results. As they sharpen their math skills, I capitalize on their enthusiasm to teach them how to extract the most important information from texts, and move them toward the oral and reading fluency they need to understand and discuss more challenging texts.

Even if you aren't accelerating math instruction, however, there are a number of ways to help students master word problems. Krick-Morales offers suggestions in the previously mentioned article, such as explicit instruction of key vocabulary, daily practice of problem solving, repeated readings of the word problem together as a class, and hands-on activities such as movement, experiments, or drawing to help students comprehend the problem. As students become more familiar with math vocabulary, they will be able to solve problems more easily.

The importance of building background knowledge

As the opening quote from the Shel Silverstein poem reminds us, background knowledge plays a critical role in math class! My colleague Hillary found that sometimes her students would get "lost" in a problem simply because they didn't understand the context. Following are some tips to help in building background knowledge of students.

Modify the linguistic complexity of language and rephrase math problems. Students will understand the problem better if it is stated in shorter sentences and in language they understand.

Guide students to cross out the unnecessary vocabulary in word problems. Doing so allows students to focus on the math function required. For example, one problem Hillary's students came across referred to a "school assembly." Even though the meaning of that phrase wasn't important in the solving of the math problem, students didn't know it wasn't important, and the lack of understanding contributed to their confusion.

Build knowledge from real world examples. Try to reinforce concepts with examples that students can picture and talk students through the situation. For example, if you need to paint a room, you need to know how much area will be covered so that you know how much paint to buy. Look for familiar ideas or props that can be used to engage students such as recipes, news stories about the economy, or discussion of personal spending habits.

Use manipulatives purposefully. This is important at all grade levels. Hillary has found math cubes to be very useful in having students represent the numbers in the problems and then manipulate the cubes to get the answer. She used the cubes and the terms "hot" and "cold" numbers when teaching with the concept of negative numbers. Students would use the red cubes as "hot" or positive numbers and the blue cubes as "cold" or negative numbers. As students laid out the number of hot cubes and cold cubes represented, they could easily see if the answer would be a positive or negative number by which color had the most cubes. A problem such as -2 + 1 = -1 would look like this:

The student then removed pairs of cubes - one red, one blue - until they could no longer remove any blocks. The remaining blocks represent the answer.

The importance of increasing student language production in the content area

As I've worked with content area teachers in my district to develop Sheltered Instruction lessons and activities to enhance ELL learning, I've told them, "If a student doesn't say it in your class, they're never going to say it." This is a bit dramatic, but it's true to some extent. When students learn new vocabulary, the opportunity to use it must be presented in class, because students are unlikely to try it out on their own - especially academic words like "parallelogram" or "function"!

Here are some tips to increase student-to-student interaction with academic language in the math classroom:

Have students translate symbols into words, and write the sentence out. Hillary used this strategy to check students' comprehension of problems before they solved them. For example, 3x + 4 = 16 would be written out, "Three times X plus four equals sixteen." This helps students process the operations involved in the question and gives them an opportunity to think through how to solve it. It also gives students a chance to familiarize themselves with important vocabulary words.

Create a "sentence frame" and post it on the board. Write the format of the sentence you would like students to use in discussion and then hold them accountable for using it. For example, "The answer is _______ degrees because it is a _________ triangle.

Have students share problem-solving strategies. This involves asking a simple question such as, "Did anyone else get the answer in a different way?" Then allow enough wait time so students can think through how their problem-solving process was similar or different to the one offered.

Allow students to discuss how they are thinking about math. This is a way of redirecting the lesson from teacher-to-student to student-to-student. For example a student might ask a question, "How do you know what kind of triangle it is?" Instead of the teacher answering and going to the board and pointing out the names and different triangles, the teacher can simply ask, "Does someone have an answer? Or "Would someone like to offer help to Mario?" Allow students to share how they think about the math concept and any tips they have for remembering the information.

Incorporate writing activities like math journals. This is an excellent way for students to process what they've learned and what questions they still have. The journal could start with simple prompts such as, "One thing I learned today..." "One thing I still don't understand..." "One way I can get the help I need..." "The answer to this problem is..." Writing out the answer to a problem is a very important skill to develop because many state math tests require a constructed response to questions.

Challenge students to create their own math problems. This can be a fun activity if students create a problem similar to the ones you have used in class and they exchange problems with a partner. By creating the problem and checking the answer they are reinforcing their own learning.

Using technology

Technology can also be a powerful tool in math instruction for ELLs. Here are some ideas ways you can play with technology in a math lesson:

Look for educational resources that accompany your school's technology tools and programs. There may be online or software programs available for teachers. For teachers who have an electronic whiteboard in the classroom, there are many resources available on links that can be easily accessed and brought into the classroom.

Look for interactive games that offer students a chance to practice their mathematical skills. Nintendo DS has an educational game called Brain Age, which is not language specific. The game provides excellent mathematical training for numbers and tracks results, showing student improvement over time.

Find out what's available online. Vital NY (Video Teaching and Learning for NYS Educators) on Teachers' Domain offers an online library of free media resources from the best in public television. Teachers' Domain resources include video and audio segments, Flash interactives, images, documents, lesson plans for teachers, and student-oriented activities. (Free registration required.)

If students are using a graphing calculator, ensure that they learn how to use it. Based on background and prior educational experience, students might not be familiar with how to use a calculator nor some of the more sophisticated models, such as the graphing calculator. Give students a chance to practice solving problems with their calculators once you have reviewed different functions. Texas Instruments offers numerous activities and product tutorials in their educational materials.

Even if it doesn't come easily at first, there are ways to get ELLs excited about math. By keeping their language skills and needs in mind when planning mathematical instruction (and by helping your colleagues do the same), you will be taking important steps in helping students master mathematical concepts and skills - and who knows? Your students may be the next generation of economists, rocket scientists, and math teachers just waiting for the tools they need!

Note: I'd like to thank my Minnesota colleagues, Hillary Hansen at Burnsville Senior High School and Kim Olson at Hidden Valley Elementary, as well as Xiao-lin Yin-Croft from San Francisco, for providing many of the helpful math teaching tips in this article. It is inspiring to know that there are talented, creative teachers who are always finding better ways to teach and are willing to share the knowledge.
Common Core Math and ELLs: Blog Posts
These posts from our Common Core blog highlight resources that can be used in Common Core math instruction for ELLs.
EngageNY: Common Core and Math
The EngageNY website provides materials focused on curricular examples, standards for mathematical practice, and other materials for professional development.
Discovery Education: Puzzlemaker
Puzzlemaker is a free puzzle generation tool for teachers, students and parents where users can create and print customized word search, criss-cross, math puzzles, and more using their own word lists.
Catherine Snow: Word Generation
Catherine Snow's new website provides information and resources for educators who would like to learn more about Word Generation and how it is implemented. Includes links to comprehensive academic word list that students need to master to comprehend academic content.
Dave's Math Tables: English/Spanish Dictionary
List of English math vocabulary words with the Spanish equivalent.
PBS Teachers: Math Lessons
PBS Teachers offers a database of multimedia math lessons and activities that can be searched by grade or topic.
SMART Notebook Lesson Activities
Browse lesson activities and materials for classrooms using SMART notebook software. Choose the country or region, and search by curriculum standards, subject and grade level.
Colorín Colorado Webcast: English Language Learners in Middle and High School
This webcast features Dr. Deborah Short and discusses effective instructional strategies for teaching English language learner students in middle and high school, such as the SIOP model.
Translating Word Problems
This is a great site for teachers in the elementary levels, as it provides a list of keywords you can teach your ELLs to look for as they read word problems. Also included are useful ideas and tricks to better prepare students to understand written math problems.

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Math Instruction for English Language Learners

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My dad gave me one dollar bill'Cause I'm his smartest son,And I swapped it for two shiny quarters'Cause two is more than one! - Excerpt from "Smart," by Shel Silverstein As a student, I struggled with math. I didn't understand why it came so naturally to some students, but not to me.

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http://nlvm.usu.edu/en/nav/frames_asid_155_g_2_t_1.html

https://www.mathlearningcenter.org/resources/apps

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https://illuminations.nctm.org/Activity.aspx?id=3566

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Ch 6 (Van De Walle):

Addressing the needs of all means providing access and opportunity for

Students who are identified as struggling or having a disability

Students from different cultural backgrounds

Students who are English language learners

Students who are mathematically gifted

Students who are unmotivated or need to build resilience

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Evaluating – Defend their reasoning of why their work explains the subtraction problem

Analyzing- Organize their thinking in a way that makes sense to them.  – formulate which strategy works best and compare the other strategies – experiment which strategy is best to solve subtraction problems

Applying- Demonstrate their knowledge of subtraction by solving the problems with a strategy

Understanding- Discuss what a subtraction problem looks like – identify the parts of a subtraction equation (Vocabulary)

Remembering- Repeat and state the parts of the lesson – Vocabulary, I can learning target, Yes/No questions

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Decks of cards: 1-5 & 1-10
Number Lines
Ten Frames
Red/Yellow Counters
Paper Pencils
Expo Markers
Dry Erase Boards

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Bring it back to original word problem – ask students to come up and solve under doc cam & on board – explain their thinking

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High students w/ Med. Students

-Med. Students w/ Low students

-Language proficiency students grouped based on L1 language

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Strategies:

o Number line

o Draw a picture

o Ten Frame

o Fingers

o Counter Cubes

o Count-in-head/Mind movie

·Vocabulary equation

o Whole, minus, part, equals, difference

·HIGHEST number comes first always

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Students may subtract the smaller digit in a column from the larger digit, regardless of which appears in the top line. This can arise if they hear that, “for subtraction you always start with the bigger number and subtract the smaller number”.

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Content- explanation of different strategies for student choice to solve problems – TPS option for discussion times – Open ended questions for discussion

Process- Students get to choose which strategy to use when solving subtraction problem – the level of each strategy differentiates CRA model – Flexible groups for students that are ‘getting it’ vs. students that need extra practice

Product- Students that are understanding subtraction fluency 1-5 will be challenged with decks of cards 6-10. Students solve the problems based on their level of thinking. Less independent à More independent work

RWSL: Vocabulary decomposed (whole, minus, part, equals, difference) – group discussions – practice writing the equation using solution station if confused

Kinesthetic- hand gestures & go noodle activity

Tactile- manipulatives for solving problem - concrete in CRA model

Visual- solution station

Auditory- video and language cues

WIDA Language Proficiency Levels: Wait time, non-verbal responses (thumbs up & thumbs down), sentence frame (whole, minus, part, equals, difference), partners based on language proficiencies

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Video: https://youtu.be/QkPa9V2wtZs

https://www.mhschool.com/math/mathconnects/assets/asset_view.html?s=an_gk_le_subtractionsign&type=ANI

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According to achievethecore.org, the supporting standard for subtraction fluency is the ability to understand addition As putting together and adding to, and understand subtraction as taking apart and taking from.  These supporting concepts develop algebraic thinking in regard to addition. Students will transfer that knowledge when thinking in terms of subtraction.

Exit Ticket – 2 numbers written on the board, students place it into an equation and solve – formative assessment of standard – assesses the ability to form an equation, place the bigger number first, use a strategy, get the correct answer

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-Make sense of problems and persevere in solving them
-Construct viable arguments and critique the reasoning of others
-Model with mathematics

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http://www.corestandards.org/Math/Content/K/OA/A/5/

CAS K.1.2.b:

Fluently add and subtract within 5.
http://cde.state.co.us/sites/default/files/documents/comath/documents/math_pk_and_k.pdf

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Subtraction Fluency

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Cubes: Fill a box with cubes, rows of cubes, or layers of cubes. The number of unit cubes needed to fill the entire box is known as the volume of the box. Can you determine a rule for finding the volume of a box if you know its width, depth, and height?

https://illuminations.nctm.org/Activity.aspx?id=4095

Area and Perimeter: Learn how to find the area and perimeter of rectangles.

https://www.mathplayground.com/area_perimeter.html

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Common Core State Standards Sequence of teaching Measurement

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CCSS.Math.Content.K.MD.A.1
Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.

CCSS.Math.Content.K.MD.B.3
Classify objects into given categories; count the numbers of objects in each category and sort the categories by count. ¹

CCSS.Math.Content.1.MD.C.4
Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

CCSS.Math.Content.2.MD.A.1
Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

CCSS.Math.Content.2.MD.A.2
Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

CCSS.Math.Content.2.MD.A.4
Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

CCSS.Math.Content.2.MD.B.5
Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

CCSS.Math.Content.2.MD.B.6
Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.

CCSS.Math.Content.2.MD.C.8
Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?

CCSS.Math.Content.2.MD.D.9
Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

CCSS.Math.Content.2.MD.D.10
Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems ¹ using information presented in a bar graph.

CCSS.Math.Content.3.MD.A.1
Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.

CCSS.Math.Content.3.MD.A.2
Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). ¹ Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem. ²

CCSS.Math.Content.3.MD.B.4
Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units- whole numbers, halves, or quarters.

CCSS.Math.Content.3.MD.C.5.a
A square with side length 1 unit, called "a unit square," is said to have "one square unit" of area, and can be used to measure area.

CCSS.Math.Content.3.MD.C.5.b
A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.

CCSS.Math.Content.3.MD.C.6
Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

CCSS.Math.Content.3.MD.C.7.a
Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

CCSS.Math.Content.3.MD.C.7.b
Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

CCSS.Math.Content.3.MD.C.7.c
Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.

CCSS.Math.Content.3.MD.C.7.d
Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

CCSS.Math.Content.3.MD.D.8
Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

CCSS.Math.Content.4.MD.A.2
Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

CCSS.Math.Content.4.MD.C.5
Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:

CCSS.Math.Content.4.MD.C.5.a
An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a "one-degree angle," and can be used to measure angles.

CCSS.Math.Content.4.MD.C.7
Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

CCSS.Math.Content.5.MD.A.1
Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

CCSS.Math.Content.5.MD.C.3.a
A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume.

CCSS.Math.Content.5.MD.C.3.b
A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

CCSS.Math.Content.5.MD.C.5
Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

CCSS.Math.Content.5.MD.C.5.a
Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

CCSS.Math.Content.5.MD.C.5.b
Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

CCSS.Math.Content.5.MD.C.5.c
Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

CCSS.Math.Content.HSS.MD.A.1
(+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

CCSS.Math.Content.HSS.MD.A.2
(+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

CCSS.Math.Content.HSS.MD.B.5
(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

CCSS.Math.Content.HSS.MD.B.7
(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

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Measurement & Data | Common Core State Standards Initiative

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CCSS.Math.Content.K.MD.A.1Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.

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A common error is to confuse the formulas for area and perimeter. Such results are largely due to an overemphasis on formulas with little or no conceptual background.

Another common error when students use formulas comes from failure to conceptualize the meaning of height and base in two-dimensional geometric figures.

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There are four important principles of iterating units of length, whether they are nonstandard or standard (Dietiker, Gonulates, Figueras, & Smith, 2010, p. 2):

All units must have equal length—if not, you cannot accumulate units by counting.

All units must be placed on the path being measured—otherwise, a different quantity is being measured.

The units must be without gaps—if not, part of the quantity is not being measured.

The units must not overlap—otherwise, part of the quantity is measured more than once.

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Develop benchmarks or reference points for units.

Use 'chunking' when appropriate to estimate the whole length.

Use subdivisions, similar to chunking, but this is when you mentally divided in half and then in fourths or even eighths by repeated halving until a more manageable length is arrived at.

For length, area, and volume, it is sometimes easy to mark off single units visually. You might use your hands or make marks or folds to keep track as you go.

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STUDENTS with SPECIAL NEEDS

Find examples of measurements of all types in newspapers, on signs, or in other everyday situations. Present the context and measures, but without units. The task is to predict what units of measure were used. Have students discuss their choices. For students with disabilities, you may want to provide the possible units so they can sort the real-world measures into groups.

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Phases of Measurement Teaching Strategies

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Phases of Measurement Teaching Strategies

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CH 19 (Van de Walle)

Big Ideas

1. Measurement involves a comparison of an attribute of an item or situation with a unit that has the same attribute. Lengths are compared to units of length, areas to units of area, time to units of time, and so on.

2. Estimation of measures and the development of benchmarks for frequently used units of measure help students increase their familiarity with units, preventing errors and aiding in the meaningful use of measurement.

3. Measurement instruments (e.g., rulers) are devices that replace the need for actual measurement units (e.g., a physical foot measure).

4. Area and volume formulas provide a method of measuring these attributes by using only measures of length.

5. Area, perimeter, and volume are related. For example, as the shapes of regions or three-dimensional objects change while maintaining the same areas or volumes, there is an effect on the perimeters and surface areas.

There are 3 broad phases for teaching about each of the measurement attributes for objects (such as length, area, volume, capacity, mass, angle, temperature) and for events (such as time and chance).

Phase 1: Identifying the attribute

Phase 2: Learning to measure

Phase 3: Learning to calculate

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Annenberg Learner www.learner.org

This tremendous resource lists free online professional learning activities, including information about all sorts of interesting uses of mathematics in the real world, resources for free and inexpensive materials, and information about funding opportunities.

Center for Implementing Technology in Education (CITEd): Tech Matrix www.techmatrix.org

CITEd’s Tech Matrix is a useful database of technology products that supports instruction in mathematics for students with special needs. Each product evaluation includes a link to the supplier’s website.

Illuminations (NCTM) http://illuminations.nctm.org

This is an incredible site developed by NCTM to provide teaching and learning resources such as lesson ideas and digital tools that are intended to “illuminate” Principles and Standards for School Mathematics. Also at this site are multimedia investigations for students and links to video vignettes designed to promote professional reflection. In addition, Calculation Nation allows students to explore mathematics topics while playing games with one another over the Web.

Inside Mathematics http://insidemathematics.org

This site features examples of innovative teaching methods, insights into student learning, tools for mathematics instruction that teachers and specialists can use immediately, resources to support the Common Core State Standards, and video tours of the ideas and materials on the site.

International Society for Technology in Education (ISTE) www.iste.org

ISTE is the professional organization for educators interested in infusing technology into instruction. It maintains exciting resources for teachers, including website links, professional development, and publications. ISTE’s National Educational Technology Standards for students (NETS-S) can be found by clicking the NETS section from the home page. The standards address such topics as creativity and innovation; communication and collaboration; research and information fluency; critical thinking, problem solving, and decision making; digital citizenship; and technology operations and concepts.

The Math Forum http://mathforum.org

The forum has resources (Math Tools) for both teachers and students. There are suggestions for lessons, puzzles, and activities, plus links to other sites with similar information. There are forums where teachers can talk with other teachers. Two pages accept questions about mathematics from students or teachers (Ask Dr. Math) and about teaching mathematics from teachers (Teacher 2 Teacher).

National Library of Virtual Manipulatives and eNLVM http://nlvm.usu.edu

This NSF-funded site located at Utah State University contains a huge collection of applets organized by the five content strands of the Standards and also by the same four grade bands. The eNLVM section contains online units, customizable student activities, and tools to help teachers develop activities collaboratively.

National Science Digital Library http://nsdl.org

The NSDL is a portal for education and research on learning in science, technology, engineering, and mathematics (STEM) and features a collection of digital learning objects connected to the Common Core State Standards.

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CH 7 (Van De Walle):
How digital content is used in mathematics instruction will vary considerably with the topic, the grade level, and the content itself. The following are offered as considerations that you should keep in mind.

Digital content should contribute to the objectives of the lesson or unit. It should not be used as an add-on or substitute for more accessible approaches.

For individualized or small-group use, provide specific instructions for using the resource, and provide time for students to freely explore or practice.

Combine online activities with offline computer activities (e.g., collect measurement data in the classroom to enter into an online spreadsheet).

Create a management plan for using the digital content. This could include a schedule for use (e.g., during centers, during small-group work) and a way to assess the effectiveness of the resource use. Although some programs include a way to keep track of student performance, you may need to determine whether the tool is effectively meeting the objectives of the lesson or unit.

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CH 7 (Van De Walle):
Very young children can use virtual pattern blocks to create patterns for copying, continuing, transforming, and analyzing (see www.mathplayground.com/patternblocks.html). Because the supply of pattern blocks is unending, students are not restricted by the number of available materials. Copies of their designs can be printed through a screen capture so that other students can be challenged to identify the pattern.

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Build patterns, create and solve critical thinking problems, and model various math concepts with pattern blocks at MathPlayground.com!

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CH 7 (Van De Walle)

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CH 7 (Van De Walle)

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1) What did I learn that I didn't know?: I had not known what the SEARCH acronym stood for. Now realizing this, it is definitely something I would want to include in my classroom. With the ever-growing technology use in classrooms, having a simple poster with the SEARCH acronym would be a great tool to have. I am amazed by how many people, not even just kids, are unaware of fake news or sources. Seeing how vital technology has become to our lives everyday, safe searches, credibility, and internet safety should be taught as a unit in schools.

2) What was the most surprising information? As our students become consumers of online information at an elementary age, I was amazed to see the fact about real vs fake news on the infographic. Each statistic about students not being able to distinguish real vs fake news, or being skeptical is surprising. I was not taught about internet safety until middle school, so I can understand some students not realizing real vs. fake, but the numbers were quite high. It worries me seeing how many students use technology and have it with them daily. I hope since it isn't being taught as much in school, that they are discussing online safety and credibility at home.

3) How is digital literacy taught in my mentor classroom/school?: My mentor teacher uses google classroom quite a bit. The students will conduct research on certain topics and use the internet to find different news that interest them to write about. I was not there for the digital safety discussion, but I know that Julie did have now with her class. I don't know the parameters of it, but I am going to show her the SEARCH acronym to maybe use in the classroom.

As a future teacher, I think that digital citizenship needs to be taught deeper. This should be a unit, or a lesson at the beginning of the year. Before the students begin researching and using technology, they should have a 'technology agreement' or some other fun activity that gets them engaged with what online credibility and safety is.

This website below has a really interesting school wide digital citizenship program:
https://www.commonsense.org/education/digital-citizenship

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CH 18 (Van De Walle) & Ratios (Math Matters):
Proportional reasoning is difficult to define in a simple sentence or two. It is not something that you either can or cannot do. According to Lamon (2006), proportional thinkers

Understand ratios as distinct entities representing a relationship different from the quantities they compare (See earlier discussion about the composed units.)

Recognize proportional relationships as distinct from nonproportional relationships in real‐world contexts

Have a sense of covariation (That is, they understand relationships in which two quantities vary together and are able to see how the variation in one coincides with the variation in another.)

Develop a wide variety of strategies for solving proportions or comparing ratios, most of which are based on informal strategies rather than prescribed algorithms

Proportional reasoning is a unifying theme in mathematics. For each of the content strands (number, algebra, measurement, geometry, and data analysis and probability), think about content that involves proportional reasoning, and explain the connections among all of these ideas.

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Illuminations Proportional Reasoning Lesson Plan

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Illuminations Proportional Reasoning Lesson Plan

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Figure This! Math Index: Ratio and Proportions (NCTM): Several challenging ratio and proportion problems found in work and everyday life tasks.
http://figurethis.nctm.org/challenges/math_index.htm

https://www.mathplayground.com/ASB_DirtBikeProportions.html

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David Schwartz compares features of various creatures to humans. For example, in the title comparison, Schwarts deduces that if a person had a frog’s jumping ability, he could jump from home plate to first base in one hop. This short picture book contains 12 more fascinating comparisons. Schwartz also provides the factual data on which the proportions are based. Students can figure out how strong or tall they would be if they were one of the featured animals.

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Activity 18.4 Pencil‐to‐Pencil

ENGLISH LANGUAGE LEARNERS

STUDENTS with SPECIAL NEEDS

Hold up a cutout of a very large pencil (e.g., 30 inches in length). Explain to students that this is the exact size of a pencil used by a giant. Ask, “If this is her pencil, what else can you tell me about her?” For students (particularly those with disabilities) who need more structure or guidance, ask specific questions like “How tall is the giant?” “How long would her hand be?” After students have found out things about the giant, have them post their findings on posters, and have them illustrate or explain how they found the measures. ELLs are likely to be more familiar with centimeters, so have students choose what measurement system they use, or have all students use centimeters.

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Activity 18.1 Birthday Cupcakes

STUDENTS with SPECIAL NEEDS

Explain to students that they are going to be icing cupcakes and selling them at school. In a recipe for icing, the instructions say that to ice one batch of cupcakes with aqua‐colored icing, you will need 2 drops of green food coloring and 5 drops of blue food coloring. Ask students to figure out how many drops of food coloring will be needed for 1 batch of cupcakes, 2 batches, 5 batches, and so on. After students have explored and found a number of batches, ask if they can figure out how much blue for one drop of green, and how much green for one drop of blue. Ask students to think about how this information helps them in determining the number of color drops for various numbers of batches. Students—particularly students with disabilities—may need to model this situation by using green and blue color tiles or counters.

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Gifted and Talented Students

Special Education Identified Students

English Language Learners

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1. Assigning more of the same work. This is the least appropriate way to respond to mathematically gifted students and the most likely to result in students’ hiding their ability. This approach is described by Persis Herold as “all scales and no music” (quoted in Tobias, 1995, p. 168).

2. Giving free time to early finishers. Although students find this rewarding, it does not maximize their intellectual growth and can lead to hurrying to finish a task.

3. Assigning gifted students to help struggling learners. Routinely assigning gifted students to teach others what they have mastered is an error in judgment, because it puts mathematically talented students in a constant position of tutoring rather than allowing them to create deeper and more complex levels of understanding.

4. Gifted pull-out opportunities. Unfortunately, these programs are often unrelated to the regular math curriculum (Assouline & Lupkowski-Shoplik, 2011). High-ability learners can’t just get one-stop shopping in a pull-out program; they need individual attention to develop depth and more complex understanding.

5. Independent enrichment on the computer. This practice does not engage students with mathematics in a way that will enhance conceptual understanding and support their ability to justify their thinking.

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LDOnline: Offers a wealth of resources including assessment tools, teaching strategies, readings, videos, podcasts, and interesting articles on mathematical disabilities.

http://www.ldonline.org

Center for Applied Special Technology (CAST): This site contains resources and tools to support the learning of all students, especially those with disabilities, through universal design for learning (UDL).
http://www.cast.org

National Association for Gifted Children (NAGC): NAGC is dedicated to serving professionals who work on behalf of gifted students. See the “Tools for Educators” section for online articles and resources.
http://www.nagc.org

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Reflective Questions to Focus on Culturally Relevant Mathematics Instruction

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Reflective Questions to Focus on Culturally Relevant Mathematics Instruction

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Activities for Students with Moderate and Severe Disabilities

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Activities for Students with Moderate and Severe Disabilities

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What was confirmed about your beliefs/understandings around RtI?

I knew that RTI was a 3-tiered process that had a goal to help the student continue to grow academically, socially, and emotionally. It was confirmed that the process was done with the grade-level team and leadership team. I knew that there was some documentation that correlated with RTI.

What is new learning you have gained from the exploration?

I did not realize that RTI is not addressed individually but rather Tier I is based on the whole class. The entire class is receiving RTI on an evidence-based approach. I also learned that RTI is used in academic and behavioral purposes. The connection between academic and behavior is an important part of the students’ learning. RTI addresses all of the possible influencers/outcomes that comes with intervening.

What questions/comments do you have based on your personal experience with RtI in your mentor classroom and school?

How do we communicate with parents without seeming like we are forcing documentation? I know that RTI requires the documentation, but I could see some parents being against it because of this. How can we communicate to them this is a good thing for their students?

Why is there so much assessing done for monitoring the students? Also do they ever ‘pass out’ of a Tier or of RTI? What’s the ultimate end-goal of RTI to allow students to move-on?

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Real-world Percents Lesson:
https://mathgeekmama.com/add-subtract-decimals-lesson/

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Decimals - Math Matters

First, when two numbers less than one are multiplied, why is the product smaller
than either of the numbers? For example, why is 0.5 + 0.1 equal to 0.05? One way to
interpret this equation is to find one tenth of 0.5. Imagine five tenths. To take one
tenth of these five tenths, we need to divide each of the tenths into ten pieces, which
would result in hundredths. One tenth of each tenth is one hundredth; since there
were five tenths to start with, there would be five hundredths.

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When the thousand block is assigned a value of “one,” then a flat (hundred) block
has a value of one tenth, a long (ten) block has a value of one hundredth, and a
unit cube has a value of one thousandth. The decimal number represented by
these blocks is now 1.132. Notice how the blocks support the idea that 1.132 is
the sum of 1 +  0.1 + 0.03 + 0.002.

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Solving Percent Problems with Representational Drawings

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Solving Percent Problems with Representational Drawings

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Circle 3
This game challenges students to use logic as they combine decimals to add to 3. Not as easy as it sounds.
http://nlvm.usu.edu

Fraction Models
This is an engaging matching game using representations of percents, fractions, and a regional model.

https://illuminations.nctm.org/Activity.aspx?id=3519

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Student Self-Assessment Practices That Work (BLOG)

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Student Self-Assessment Practices That Work (BLOG)

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Understanding how we learn best is one of the keys to growth as human beings. This essential metacognitive skill is not innate and must be modeled and taught in school starting as early as possible.

The way it looks for different age groups will be different, obviously as younger students are not always cognitively ready to do this kind of thinking. However with the right scaffolds in place, reflection and self-assessment can be a rich way to extend the depth of independent learning and also aid in informing the teacher about valuable data around growth and understanding.

As we put more and more of the control into student hands, it's not enough to just tell them to reflect or self-assess; it is imperative that we give them tools to do these practices well and then offer many opportunities to practice and get better at it.
Here are some ways teachers can start to implement meaningful self-assessment into classes:

At the beginning of a unit, consider having students set goals for themselves based on projected outcomes and objectives. These goals will drive reflection and tracking of progress throughout each student's journey.

Scaffold the process by starting off with a google form with a list of questions that walk them through the process of thinking about their learning like:

What do you think this assignment was about? or What was your understanding of the assignment?

What did you have to do?

What steps did you take to complete the assignment?

What kinds of challenges did you face? How did you overcome these challenges?

What did you do well in this assignment? How do you know?

Did you meet the success criteria? How do you know?

Wording for these questions will likely be adjusted for the age of the students you are working with.

It's a good idea to have students reflecting throughout the process. Not just at the end. So consider allowing them to think about each day's learning as an exit ticket or something they keep in their notebooks for daily summary. In their own words they can think about what they learned in class that day and how it connects to something they learned before or how it could connect to other skills or subject areas.

Once students get a handle on these kinds of questions on a sheet or form, ask students to start sharing their learning in other formats, more narrative. Some will prefer to write, others may prefer to use multimedia options like Voxer or Screencast-o-matic.

At the end of major assignments, students should be asked to reflect on their learning and self-assess what they've learned based on exemplars and standards. As they are working through assignments, standards should be transparently connected to the learning outcomes and while they are learning specific skills, the teacher should help with the connections to the standards. It makes sense for younger students to rewrite the standards in "kid-friendly" language or if students are too young, the teacher can simplify the standards so they understand, keeping each part simple.

Self-assessments based on standards should be written like an argument paper (also aligning with the Common Core). When they say they are approaching, meeting or exceeding the standards, the student should show in their work support for this assessment. Where in their own work is there evidence of the skill at work? Showing students examples of what this looks like and providing them feedback early in the year to develop these skills is also extremely helpful. For example when I read student self-assessments and reflections, I comment on the quality of each paragraph in the narrative. If they did a good job explaining what was expected on the assignment, I write "good understanding of the task" while highlighting part of the work. If there standards aren't evident, I might say, "Good evidence of process, but need to tie evidence of learning to the standards. Make sure to review the assignment sheet where standards are listed." Sometimes I even share student examples if I notice a child isn't getting it.

Over all, self-assessment is a valuable tool as it gives students the ability to really consider their learning and equally as importantly, share that understanding with the teacher. Once we understand what students know and can do, both demonstrated in the work and their ideas about the work, we can adjust instruction and/or class pacing as needed. The more adept students become with this tool, the better tailored class time and assessments can be.
How do you teach students to self-assess? What is the greatest lesson you've learned for teaching them to do so? Please share

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Student Self-Assessment Practices That Work

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Self-assessment is a valuable tool as it gives students the ability to really consider their learning and equally as importanly, share that understanding with the teacher. Once we understand what students know and can do, both demonstrated in the work and their ideas about the work, we can adjust instruction and/or class pacing as needed.

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Multiply the Decimal Number by 100%

This is safe to do because 100% = 1, and multiplying by 1 does not change the value.

Example: Convert 0.125 to percent

Multiply 0.125 by 100%:

0.125 × 100% = 12.5%

Don't forget the % sign!

Answer 0.125 = 12.5%

The easy way to multiply by 100 is to move the decimal point 2 places to the right. so:

From Decimal

To Percent

move the decimal point 2 places to the right
(and add the "%" sign!)

More Examples:

Example: Convert 0.35 to percent

Move the decimal point two places to the right: 0.35 → 3.5 → 35%

Answer 0.35 = 35%

Example: Convert 0.985 to percent

Move the decimal point two places to the right: 0.985 → 9.85 → 98.5%

Answer 0.985 = 98.5%

You may also need to add zeros on the right to move the decimal point:

Example: Convert 1.2 to percent

Move the decimal point two places to the right: 1.2 → 12. → 120.%

Answer 1.2 = 120%

Why

These all mean the same thing:

0.5 (half) of the box is green.

50 of the 100 squares are green,

50 100 of the box is green,

50% of the box is green.

So 0.5 (a decimal number) = 50% (a percentage)

The "%" tells you that there is a hidden "/100" there.

Another example:

0.25 (a quarter) of the box is green.

25 of the 100 squares are green,

25 100 of the box is green,

25% of the box is green.

So 0.25 (a decimal number) = 25% (a percentage)

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Multiply the Decimal Number by 100% This is safe to do because 100% = 1, and multiplying by 1 does not change the value. Multiply 0.125 by 100%: 0.125 × 100% = 12.5% Don't forget the % sign! Answer 0.125 = 12.5% The easy way to multiply by 100 is to move the decimal point 2 places to the right.

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Close to a familiar fraction - benchmark fractions:

Students are given a decimal number that is close to a common fraction they know - develops familiarity with decimals and not a new concept or skill. They will have the ability to discuss and apply their knowledge of fractions, place value, and decimals in the activity.

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CH 17 (Van De Walle):

Decimal Point indicates the units position.

Percents = hundredths (3rd way to write fractions and decimals)

Addition and Subtraction with decimals aligns the numbers in like position values.

Multiplication and Division of 2 numbers will produce the same digits regardless of the alignment of the decimal points.

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Multiplying Decimals Base Ten Blocks Lesson

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Multiplying Decimals Base Ten Blocks Lesson

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Getting the Decimal Point with Blocks - National Council of Teachers of Mathematics

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Use base ten blocks to model decimal multiplication.

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Self Assessment Prompts

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Identifying Fraction Gaps

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If you're seeing this message, it means we're having trouble loading external resources on our website.

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1,100 possible mastery points

About this unit

From cooking measurements to geometry, fractions are all around us. By understanding how the numerator and denominator work together, you'll be able to break down numbers into smaller parts, compare different fractions, and get a grasp on concepts like equivalent fractions.

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Understand fractions | Arithmetic | Math | Khan Academy

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From cooking measurements to geometry, fractions are all around us. By understanding how the numerator and denominator work together, you'll be able to break down numbers into smaller parts, compare different fractions, and get a grasp on concepts like equivalent fractions.

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Cuisenaire Rod Comparison Activity

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Level 1 - It looks like it's true

"If it looks like a duck, and quacks like a duck, it's probably a duck."

It's a good phrase because it's memorable and it well-defines the concept. It also reinforces the lack of concrete data associated with this level of comprehension: "I can't prove it's true, but my understanding is that it's unlikely to not be true". This proves level 1 comprehension because... well, they have made a correct basic assumption based on the appearance of the problem.

Level 2 - I can deduce that it's true

"I have gathered some data on the objects in question, and can deduce that this is the case"
Note: Deduce, not prove.

To show Level 2 comprehension, the student must be able to provide detail to support their Duck Test. In the original example given, it would be to say "I have calculated the angles, I know they're both acute so my duck test holds". They don't have to then prove that the corollary (angles are congruent) holds to show this level of comprehension. This proves level 2 because they have used actual data to make a correct assumption (well, technically an assumption).

Level 3 - I understand the Importance of these properties

"Not only can I deduce that this is true, I can extend my deduction to related abstract concepts"

To show Level 3 comprehension (IMO, there may be disagreement here), the student must show that they understand the relevance of what they are seeing. What are the consequences of the these two angles being congruent? Can the student make the jump of comprehension to the other properties that must therefore be true (related pairs of supplementary angles, the types of geometrical shapes that would be formed from sets of congruent angles, which lines must be parallel if these two angles are congruent). Rather than just seeing it's true, how can I USE the fact that it's true?

Again, note that the student isn't required to prove any of this yet, just understand the significance of the data. I might be able to deduce from the sound on the roof that it's raining pretty hard, but will it occur to me to take an umbrella when I go outside?

This proves level 3 because they can volunteer consequences of the properties they are seeing, showing an understanding of the abstraction - they understand what it means, not just what it says.

Level 4 - I can Prove that it's true

"I know that this is true, and I can prove it"

The student shows comprehension of both the consequences of the properties and the base principles behind the properties. They can use these base principles to prove their duck test rather than just deducing from "stuff I know is true". F.Ex., I can prove that these angles are congruent using my knowledge of angles and parallel lines, rather than simply deducing it's true because I've been told that two parallel lines bisected by another line at an angle forms a set of congruent angles.

This shows level 4 comprehension because they can prove their deductions rather than relying on "facts" that they have been told (Ability to prove the theorem from axioms, definitions, and previously proven theorems, as opposed to merely knowing how to use the theorem).

Level 5 - I don't know that this is true (yet), but I can prove it

"I know that this is true (and I can prove it), but can I prove whether this is true?"

Student shows absolute comprehension of the subject, to the point that they can apply what they have been taught to prove things that they haven't been taught. Using their knowledge, they can make good assumptions about other things that might be true, and go on to prove and disprove them.

This shows level 5 comprehension because the student understands enough about the system to try and extend their knowledge beyond its current level.

"I know that this holds in this case, but can I be sure that it always holds?"

"This configuration looks interesting, can I use my knowledge of geometry to prove anything about it?".

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CH 18 (Van De Walle):

A ratio is a multiplicative comparison of two quantities or measures. A key developmental milestone is the ability of a student to begin to think of a ratio as a distinct entity, different from the two measures that made it up.

Ratios and proportions involve multiplicative rather than additive comparisons. Equal ratios result from multiplication or division, not from addition or subtraction.

Rate is a way to represent a ratio and in fact represents an infinite number of ratios.

Proportional thinking is developed through activities involving comparing and determining the equivalence of ratios and solving proportions in a wide variety of problem‐based contexts and situations without recourse to rules or formulas.

Types of Ratios:

Part to Part

9 girls to 7 boys --> 9/7 (meaning a ratio of 9 to 7, not nine-sevenths)

Part-Whole

9 girls to 16 total students --> Fraction 9/16

Ratios as Quotients

(CH18) Ratios can be thought of as quotients. For example, if you can buy four kiwis for $1.00, determining the cost for one kiwi is a rate problem that can be expressed as a quotient: 1.00 ÷ 4 = 0.25. The cost of one kiwi is 25 cents.

Ratios as Rates

MPG, roses per bouquet, passengers per busload

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Activity 17.3 Calculator Decimal Counting

ENGLISH LANGUAGE LEARNERS STUDENTS with SPECIAL NEEDS

Recall how to make the calculator “count” by pressing 1 Now have students press 0.1 … When the display shows 0.9, stop and discuss what this means and what the display will look like with the next count. Many students will predict 0.10 (thinking that 10 comes after 9). This prediction is even more interesting if, with each press, the students have been accumulating base-ten strips as models for tenths. One more press would mean one more strip, or 10 strips. Why doesn’t the calculator show 0.10? When the tenth press produces a display of 1 (calculators are not usually set to display trailing zeros to the right of the decimal), the discussion should revolve around trading 10 strips for a square. Continue to count to 4 or 5 by tenths. How many presses to get from one whole number to the next? For students with disabilities and ELLs, counting out loud along with the calculator “one tenth, two tenths,…” supports the concept (e.g., 10 tenths as being the same as 1 whole) while reinforcing appropriate mathematical language. They may need to be reminded that a place is “full” when it has 9 of any unit and the addition of another unit will push to the position that is one unit to the left (like the mileage in a car). Once students are working well with tenths, try counting by 0.01 or by 0.001. These counts illustrate dramatically how small one-hundredth and one-thousandth really are. It requires 10 counts by 0.001 to get to 0.01 and 1000 counts to reach 1.

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Ch 6 (Van De Walle):

Addressing the needs of all means providing access and opportunity for

Students who are identified as struggling or having a disability

Students from different cultural backgrounds

Students who are English language learners

Students who are mathematically gifted

Students who are unmotivated or need to build resilience

Heterogeneous classrooms, where differentiation strategies are used to provide individualized and appropriate support to students, support the learning of all students. In heterogeneous classes, expectations are often turned upside down as children once perceived as less able demonstrate understanding and work meaningfully with concepts to which they would never be exposed in a low-track class.

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CH 15 & 16 (Van De Walle): Students need to experience fractions as wholes, ratios, decimals, and division.

3 categories of models with fractions:

Area: (1/3 of a garden)

Length (3/4 of an inch)

Set or Quantity (1/2 of the class)

Partitioning and estimating fractions is critical for fractions. Teaching equivalent fractions is a benchmark before computing fractions.
CCSS Developmental Process

Grade 3—Introduction to fractions.

Grade 4—Adding and subtracting fractions with like denominators, and multiplication of fractions by whole numbers.

Grade 5– Developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions.

Grade 6—Completing understanding of division of fractions.

Reflection Journal Fraction Observation

Fraction Assessment Project

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Having students take charge of their learning and assessments allows teachers to reflect on certain students who may need more help overall. This could even be a check-in at the end of class. I have seen lots of ideas online that could be utilized in the classroom. Rather than just having data from a summative assessment, these self-assessments could be used as a formative assessment and from this information, teachers can adjust their lessons and instruction.

Is self-assessment something that should be used daily? Or like one of the articles stated, mostly used prior and during learning. I see a lot of different strategies on Pinterest and Teachers Pay Teachers, but I would like to see some more examples in a classroom.

CH 5 (Van De Valle): Building Assessment into Instruction

Student self-assessment should be used for teachers to measure their learning, but for a record to show how they perceive their strengths and weaknesses --> students should try and take responsibility for their learning

"An open-ended writing prompt such as was suggested for journals is a successful method of getting self-assessment data:

How well do you think you understand the work we have been doing on fractions during the last few days? What is still causing you difficulty with fractions?

Write two of the important things you learned in class today (or this week).

Which problem(s) on the handout/quiz did you find the most challenging? Which were the easiest?"

Reflection Journal: Self Assessment

I think the idea of implementing a ‘table’ assessment like this is a great way for students to start evaluating their progress and learning. It would need to be a classroom discussion or rule that when these stoplights are up, the students know that if they are green, they help someone who might be yellow or red. This little task would help teachers know who to intervene with as well. Rather than check the entire room, having students assess themselves and then as a teacher you can mentor those who don’t understand the topic.

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Performance-based tasks should be connected to actual problem-solving activities used in instruction. These tasks should always have students justify and explain their thinking.
GRASP: acronym used for constructing performance assessment tasks
G: real-world GOAL
R: real-world ROLE
A: real-world AUDIENCE
S: real-world SITUATION
P: real-world PRODUCT or PERFORMANCES
S: Standard

CH 1 (Van De Valle): Teaching Math in 21st Century

6 Principles of Teaching Mathematics

Equity - high expectations for ALL students

Curriculum - coherent, focused on important math, and well articulated across the grades --> math should be seen as a whole on-going teaching opportunity and not isolated into bits and pieces

Teaching - understand what students know, need to learn, and misconceptions - challenge and support them to learn it well (meaningful instructional tasks and generalizable strategies)

Learning - actively building new knowledge from experience and prior knowledge

Assessment - support the learning and importance of mathematics - should furnish useful information for student & teacher - done for students to guide and enhance learning

Technology - influences math that is taught and enhances learning

Principles should be deeply intertwined in mathematics programs.

5 Content Standards

Number and Operations

Algebra

Geometry

Measurement

Data Analysis and Probability

5 Process Standards

Problem Solving

Reasoning and Proof

Communication

Connections

Representation

GRASPS Google Doc Performance Task

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Diagnostic interviews are important for teachers when they are identifying possible gaps in students' learning. In math, a diagnostic interview reflects the standards of mathematical practice and scaffolds a students depth of understanding when given a problem. The difference between a regular graded task and a diagnostic interview is listening to the student explain their thinking. They should explain why they did what they did and how they solved the task.

Diagnostic Interview
Using Achievethecore.org

Reflection Journal: Organizing Assessment

Organizing data for assessments: Google Sheets

Assessment should be used as a “continuum” throughout the year. The on-going assessments in a classroom should:  Pre-assess, formatively assess, and summatively assess. I think the most important part of the formative assessment is analyzing the check-in data and intervening.

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After class, I have noticed how many patterns reflect in numbers. Finding patterns using prime and composite numbers, rectangles, and factors have really opened my eyes to how many patterns there are. We can infer and reflect on all different kinds of patterns. Cuisenaire rod activity (p18 Math Matters)

CH 14 (Van De Valle): Algebraic Thinking

Algebraic thinking continues from PreK - 12th Grade

3 strands of algebraic thinking:

Study of structures of the number system

Study of patterns, relations, and functions

The process of mathematical modeling, (use & meaning of symbols)

Meaning of the Equal Sign! (BALANCE)

The equal sign is a principal method of representing relationships in our number system.

Both sides of the equations are equivalent expressions

Math Matters: Number Sense

Helping students to understand the place value system

Group objects into specific-size groups: tens, hundreds, thousands... realize they can count these groups

Part-whole relationship to partition numbers into groups based on powers of ten

Multiplicative relationship among groups - 3,289 (32 hundreds)

Reflection Journal: Algebraic Thinking

The conjectures are important for students to make and realize they have refined their thinking or solving process to understand why they were right or wrong. They also are now justifying their thinking by explaining this process. The justification process comes with asking certain questions too. I asked her what types of questions she uses to encourage this higher-order thinking. Some of these were:

Is this always true?

How do you know?

What could you have done differently?

How did you prove your prediction?