MATHEMATICS GRADE 6 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 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 A critical area of instruction in grade six is to connect ratio, rate, and percentage to whole­number multiplication and division and use concepts of ratio and rate to solve problems. Students’ prior understanding of and skill with multiplication, division, and fractions contribute to their study of ratios, proportional relationships, unit rates, and percentage in grade six. ESSENTIAL QUESTIONS MATHEMATICAL PRACTICES What is the relationship between a ratio and a fraction? Why is it important to know how to solve for unit rates? How is a ratio or rate used to compare two quantities or values? How and where are ratios and rates used in the real world? How can I model and represent rates and ratios? What are the similarities and differences between fractions and ratios? 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 UNIT 1 LEARNING SBAC Targets STANDARDS Target C: Additional/Supporting Compute fluently with multi­digit numbers and find common factors and multiples. 6.NS.4​ Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. ​For example, express 36 + 8 as 4(9 + 2). Students use GCF to factor addends through distributive property. Target A: Major Understand ratio concepts and use ratio reasoning to solve problems 6.RP.1 ​Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. Students should use the language. Numerator and Denominators are whole numbers in unit rates. 6.RP.2​ Understand the concept of a unit rate ​a/b associated with a ratio ​a:b with b ≠ ​0, and use rate language in the context of a ratio relationship. FRAMEWORK NOTES 6.RP.3​ Use ratio and rate reasoning to solve real­world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. 3a​ Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. 3b​ Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? 3c​ Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. Standard calls out examples of conceptual strategies to solve problems. Students should reason and understand rates and ratios. Use tape diagrams, double number lines, and ratio tables. Draw attention to benchmark percentages as a strategy to solve percent problems Use models and tables to solve percent problems. 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 COMMON MISCONCEPTIONS: RATIOS and FRACTIONS ❖ Although ratios can be represented as fractions, the connection between ratios and fractions is subtle. Fractions express a part­to­whole comparison, but ratios can express part­to­whole or part­to­part comparisons. Care should be taken if teachers choose to represent ratios as fractions at this grade level. ❖ Proportional situations can have several ratios associated with them. For instance, in a mixture involving 1 part juice to 2 parts water, there is a ratio of 1 part juice to 3 total parts (1:3), as well as the more obvious ratio of 1:2. ❖ Students must carefully reason about why they can add ratios. For instance, in a mixture with lemon drink and fizzy water in a ratio of 2:3 , mixtures made with ratios 2:3 and 4:6 can be added to give a mixture of ratio 6:9 , equivalent to 2:3. This is because the following are true: 2 (parts lemon drink) + 4 (parts lemon drink) = 6 (parts lemon drink) 3 (parts fizzy water) + 6 (parts fizzy water) = 9 (parts fizzy water) However, one would never add fractions by adding numerators and denominators: How will we know if they have learned it? SUCCESS CRITERIA: WHAT DOES LEARNING LOOK LIKE ●
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The student uses ratio language to describe a ratio relationship. The student determines the unit rate associated with a real­world ration. The student finds missing values in tables of equivalent ratios. The student plots coordinate pairs to represent equivalent ratios. The student makes tables of equivalent ratios relating quantities with whole­number measurements. The students solves real­world problems involving unit rate. The students solves mathematical problems involving finding the whole, given a part and the percent. The student solves real­world and mathematical problems involving finding a percent of a quantity as a rate per 100. The students uses ratio reasoning to convert measurement units. The student uses ratio reasoning to manipulate and transform units appropriately when multiplying or dividing quantities. (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. Sixth Previous Grades When students enter grade six, they are fluent in addition, subtraction, and multiplication with multi­digit whole numbers and have a solid conceptual understanding of all four operations with positive rational numbers, including fractions. Seventh Students’ prior understanding of and skill with multiplication, division, and fractions contributes to their study of ratios, proportional relationships, unit rates, and percentage in grade six. ● Only non­negative numbers ● Use models ● Can be written as fractions, but are different from fractions in several ways ● As students generated equivalent ratios, they will notice how multiplication and division are related to each other In grade seven, students extend their reasoning about ratios and proportional relationships. Students use ratios in cases that involve pairs of rational number entries and compute associated rates. They identify unit rates in representations of proportional relationships and work with equations in two variables to represent and analyze proportional relationships. These concepts will extended to include scale drawings, slope, and real­world percent problem. RESOURCES Estimation 180 3 ACT MATH TASK ● 6th grade​, ​7th grade​, ​8th grade 6­8 DESMOS ● Calculator ● Student Desmos ● Teacher Desmos KHAN ACADEMY McGRAW TEXT ● Chapter 1 ● Chapter 2 Ready Common Core iReady STRATEGY BANK VOCABULARY rate, unit rate, ratio, greatest common factor, least common multiple, factors, multiply, diagram, equivalent, ratio table coordinate plane, ordered pair, x­axis, y­axis, x­coordinate, y­coordinate, origin, graph, factor, multiple, unit rates, rate, equivalent fractions, ratio, bar diagram, greatest common factor, least common multiple, range, fraction, decimal rational number, percent compare, order, decimals, percents, least common denominator (LCD), proportion, percent proportion, fractions, simplify Additional Sources: Howard County Public School System, Tucson Unified School District