MATHEMATICS GRADE 5 UNIT 1 BUILDING CULTURE AND ESTABLISHING STRUCTURE: A WEEK OF INSPIRATIONAL MATH “This week is about inspiring students through open, beautiful and creative math. We have chosen the different tasks so that students see math as a broad, interesting and visual subject that involves deep thinking. Students will learn important growth mindset messages that will help them feel confident, try harder all year, persist with open and difficult problems and embrace mistakes and challenge. All tasks are low floor and high ceiling – they are accessible to all students and they extend to high levels.” (Jo Boaler, youcubed.org) K­2nd 3rd­5th 6th­8th Week of Inspirational Math for Primary Grades: https://drive.google.com/open?id=19BpgpJnxTnmiBtvbJ_weVPKUHJvf
HAuPbMmSJD9I7lA youcubed.org (must register for lesson access: use Grades 3­4 lesson plan & media) youcubed.org (must register for lesson access: use Grades 5­9+ lesson plan & media) PLANNING FOR MATHEMATICAL EQUITY & ACCESS POINTS Backwards Mapping ●
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Identify the Learning for the UNIT of instruction Determine Success Criteria How will we ensure a balance of conceptual, procedural, and application of mathematics learning? How will I connect the math learning using progressions so there is a better understanding? Anticipate student learning, multiple representations, misconceptions/errors How will we ensure the mathematical practices are evident in the learning? Build on what they already know Environmental ●
How will we ensure students are engaged in a 21st Century Learning Environment to include collaboration, creativity, critical thinking, communication . . . Establish and practice structures As a team, what structures will be in place to ensure our students are provided a positive math culture, environment, and experience around mathematics. How do we ensure a positive math culture where students can feel comfortable and have a growth mindset? ●
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Language ● What language will students utilize to support content and language demands? ● Plan for academic conversations ● What vocabulary, sentence structures and language functions will students need to master? ​ VOCABULARY​ ​+ ​STRUCTURE ​= ​FUNCTION PLANNING FOR MATHEMATICAL EQUITY & ACCESS POINTS (continued) Instructional ●
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Set Purpose, Goals, Expectations, Objectives. How will students be engaged in behaviors that promote the mathematical practices? Prepare for think alouds and modeling that will be most effective. What questions, prompts, and cues will we use? What resources will I utilize? (framework, tasks, videos, thinking maps, tasks, text, ...) Promote perseverance Use the Gradual Release of Responsibility (GRR) and give students opportunity to enter at different points Make connections between different representations Allow all students the opportunity to enter the learning with a “hook”, a real life situation, an open­ended question, 3­ACT math task, etc Formative Assessment ●
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How will we check for understanding? How will we provide feedback? Monitor student learning and look for misconceptions/errors. Assess students in critical thinking­MP­provide tools that are necessary, they make connections, they conceptualize the math, they discover and explore, they struggle productively, engage in discourse. ​ADDRESSING THE LANGUAGE GAP VOCABULARY What words will students need to learn and apply? (Tier 2 or 3 Words) STRUCTURE How will students use those words in sentences? (Simple, Compound, Complex) PLC PLANNING What do I want my students to know and be able to do? How will we know if they have learned it? How will we respond when learning has not occurred? How will we respond when learning has already occurred? FUNCTION How will students use those sentences to DESCRIBE, RETELL, JUSTIFY, EXPLAIN Etc… (Language demands from the standards) What do I want my students to know and be able to do? UNIT OVERVIEW In the years prior to grade five, students developed an understanding of the structure of the place­value system. At the onset of grade five, instructional time should focus on extending division to two­digit divisors, integrating decimal fractions into the place­value system, developing understanding of operations with decimals to hundredths, and developing fluency with whole­number and decimal operations.
ESSENTIAL QUESTIONS MATHEMATICAL PRACTICES How does the position of a digit in a number relate to its value? What strategies can be used to multiply whole numbers? What strategies can be used to divide whole numbers? What strategies can I use to divide by a two­digit divisor? 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning COMMON MISCONCEPTIONS: COMPARING DECIMALS Some students relate comparing decimals with the idea “the longer the number, the greater the number.” With whole numbers, a five­digit number is always greater than a one­, two­, three­, or four­digit number. However, when comparing decimals, a number with one decimal place may be greater than a number with two or three decimal places. UNIT 1 LEARNING SBAC Targets Target C:​ Major Understand the place value system. Target D:​ Major Perform operations with multidigit whole numbers and with decimals to hundredths. STANDARDS FRAMEWORK NOTES 5.NBT.1 ​Recognize that in a multi­digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. Students use place value to understand that multiplying a decimal by 10 results in the decimal point appearing one place to the right (e.g., 10 × 4.2 = 42), since the result is 10 times larger than the original number; similarly, multiplying a decimal by 100 results in the decimal point appearing two places to the right, because the number is 100 times larger. Students also make the connection that dividing by 10 results in the decimal point appearing one place to the left (e.g., 4 ÷ 10 = 0.4), since the number is 10 times smaller (or of the original), and dividing a number by 100 results in the decimal point appearing two places to the left because the number is 100 times smaller (or of the original). 5.NBT.2​ Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole number exponents to denote powers of 10. Powers of 10 is a fundamental aspect of the base­ten system. Students extend their understanding of place value to explain patterns in the number of zeros of the product when multiplying a number by powers of 10, including the placement of the decimal point. The use of whole­number exponents to denote powers of 10 is new to fifth­grade students. 5.NBT.3 ​Read, write, and compare decimals to the thousandths. (a) Read and write decimals to thousandths using base­ten numerals, number names, and expanded form, e.g. 347.392 = 3 x 100 + 4 x 10 + 7 x 1 + 3 x (1/10) + 9 x (1/100) + 2 x (1/1000). (b) Compare two decimals to the thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. Base­ten blocks can be a powerful tool for seeing equivalent representations. For instance, if a “flat” is used to represent 1 (the whole or unit), then a “stick” represents , and a small “cube” represents . Students can be challenged to make sense of a number like 0.23 as being represented by both (2/10) + (3/100) and (23/100).For students who are not able to read, write, and represent multi­digit numbers, working with decimals will be challenging. Teachers can use base­ten blocks and money to provide meaning for decimals. For example, dimes can represent tenths, pennies represent hundredths, and a penny circle with a (1/10) sliver in it can represent thousandths. 5.NBT.5 ​Fluently multiply multi­digit whole numbers using the standard algorithm. In previous grades, students built a conceptual understanding of multiplication with whole numbers as they applied multiple strategies to compute and solve problems. Students can continue to use different strategies and methods learned previously—as long as the methods are efficient—but they must also understand and be able to use the standard algorithm. 5.NBT.6 ​Find whole­number quotients of whole numbers with up to four­digit dividends and two­digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Division strategies in grade five extend the methods learned in grade four to two­digit divisors. Students continue to break the dividend into base­ten units and find the quotient place by place, starting from the highest place. They illustrate and explain their calculations by using equations, rectangular arrays, and/or area models. Estimating the quotients is a difficult new aspect of dividing by a two­digit number. Even if students round appropriately, the resulting estimate may need to be adjusted up or down. Major (Priority) ­​ Areas of intensive focus where students need fluent understanding and application of the core concepts. These clusters require greater emphasis than the others based on the depth of the ideas, the time that they take to master, and/or their importance to future mathematics or the demands of college and career readiness Supporting​­ Rethinking and linking; areas where some material is being covered, but in a way that applies core understanding; designed to support and strengthen Additiona​l­​ ​Expose students to other subjects; may not connect tightly or explicitly to the major work of the grade How will we know if they have learned it? SUCCESS CRITERIA: WHAT DOES LEARNING LOOK LIKE ●
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The student represents powers of 10 by using whole­number exponents. The student reads and writes decimals to the thousandths using base­ten numerals, number names, and expanded form. The student compares two decimals to the thousandths by using >, =, and < symbols. The student multiplies multi­digit whole numbers. The student determines whole­number quotients of whole numbers with up to four­digit dividends and two­digit divisors using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. (click link and scroll to “Item and Task Specifications” for more detailed information.) ❖ Formative assessment methods to consider: immediate feedback, questioning, overheard discourse, goal­setting and attainment, multiple representations, rubric, error analysis, CFA development, self­assessment How will we respond when learning has not occurred? Professional Learning Communities will develop and implement Response to Intervention. (see progressions) How will we respond when learning has already occurred? Professional Learning Communities will develop and implement Enrichment. PROGRESSIONS Mathematics standards are not isolated concepts.​ ​COHERENCE MAP­ACHIEVE THE CORE Standards relate to one another, both within and across grades. The Coherence Map illustrates the coherent structure of the Common Core State Standards for Mathematics ●
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Build student understanding by linking together concepts within and across grades. Identify gaps in a student's knowledge by tracing a standard back through its logical prerequisites. Visualize and understand how supporting standards relate to the major work of the grade. Previous Grades In the years prior to grade five, students learned strategies for multiplication and division, developed an understanding of the structure of the place­value system, and gained fluency in adding and subtracting whole numbers within one million using the standard algorithm. Fifth Sixth Grade five involves students extending division to two­digit divisors, integrating decimal fractions into the place­value system, developing understanding of operations with decimals to hundredths, and developing fluency with whole­number and decimal operations. For students in grade five, the expected fluency is to multiply multi­digit whole numbers (with up to four digits) using the standard algorithm. The extension of the place­value system from whole numbers to decimals is a major accomplishment for a student that involves both understanding and skill with base­ten units and fractions. Skill and understanding with adding, subtracting, multiplying, and dividing multi­digit whole numbers and decimals will culminate in fluency with the standard algorithms in grade six. RESOURCES Mathematics Framework NUMBER TALKS 3­ACT Math Tasks KHAN ACADEMY McGRAW TEXT ● Chapter 1 ● Chapter 2 ● Chapter 3 Ready Common Core iReady STRATEGY BANK VOCABULARY place­value chart, period, place, place value, standard form, expanded form, decimal, decimal point, equivalent decimals, digit, value, difference, less than, prime factorization, prime numbers, exponent, base, power, squared, cubed, powers of 10, distributive property, compatible numbers, area model, estimate, multiply, add, round, product, fact family, unknown, variable, dividend, divisor, quotient, remainder, partial quotients Additional Sources: Howard County Public School System, Tucson Unified School District