MATHEMATICS GRADE 7 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_weVPKUHJv
fHAuPbMmSJD9I7lA 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 now and be able to do? UNIT OVERVIEW A critical area of instruction in grade seven is developing an understanding and application of proportional relationships, including percentages. In grade seven, students extend their reasoning about ratios and proportional relationships in several ways. 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. They also solve multistep ratio and percent problems, such as problems involving percent increase and decrease (University of Arizona [UA] Progressions Documents for the Common Core Math Standards 2011c). ESSENTIAL QUESTIONS MATHEMATICAL PRACTICES ➢
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How are fractions and ratios alike? How are they different? How do you compute a unit rate given a comparison of two quantities? How do you determine if two ratios are proportional? Given a table, graph, equation, diagram or verbal description, how do you determine if it is showing a proportional relationship? ➢ How are proportional relationships represented on a graph, a table and an equation? ➢ How does an understanding of proportionality connect to solving problems involving interest, taxes, tips and discounts? ➢ What are the key points on a proportional graph that best describe the proportional relationship? Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning UNIT 1 LEARNING SBAC Targets STANDARDS TARGET A: Major Analyze proportional relationships and use them to solve real­world 7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person 1
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walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 2 / 4
miles per hour, equivalently 2 miles per hour. FRAMEWORK NOTES Make connections to 6th grade Proportional Relationships Learning Ensure students use reasoning about specific strategies and why they work and mathematical problems. 7.RP.2 Recognize and represent proportional relationships between quantities. 2a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. 2b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 2c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. 2d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. 7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. Proportional Relationship is the foundation for Functions. Ratios involving negative numbers are important in algebra and calculus, but not a part of 7th grade. Students should use a variety of methods to solve proportional relationship problems to include tape diagrams, double number lines, using tables, using rates, and by relating proportional relationships to equivalent fractions. √ 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 PROPORTIONS ❖ Students have difficulty determining which quantities should be the numerator and denominator in a unit rate, such as $2.85 per gallon. Point out that “per” means “for each” or “apiece”. The quantity after “per” is the denominator, for example, $2.85/gallon. ❖ Students often think, because of notation, that fractions and ratios are the same thing. For example they think that because we can write the ratio of boys to girls in the class as 2/3, students think that this is fraction as opposed to the ratio. Using ratios in context is key to understanding the difference between a fraction and ratio. ❖ Students often have a difficult time understanding that fractions and ratios may represent different comparisons. It is very important when teaching that we label all ratios and fractions. ❖ Fractions always express a part­to­whole comparison, but ratios can express a part­to­whole comparison or a part­to­part comparison. For example, the ratio of boys to girls in a class is 7:10, or it could be the ratio of boys to the total class 7:17 making it a fraction. ❖ Students often misinterpret a ratio table as an INPUT/OUTPUT table. ❖ Students may try to create a rule rather than see the equivalency and multiplicative relationship among ratios. ❖ Students have difficulty understanding that every ratio can create 2 unit rates. For example, if you buy 3 oranges for $2, then you have two unit rates; 1.5 oranges per dollar or $0.66 per orange. ❖ When graphing points on a coordinate plane, student commonly go up and down first and then left or right. Students often graph the y­value before the x. Another way to think about it is student often plot the x­value as the y and visa versa. For example, given the point (3, ­2), student will go up three and then left 2 as opposed to going right 3 and then down 2. How will we know if they have learned it? SUCCESS CRITERIA: WHAT DOES LEARNING LOOK LIKE ●
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The student computes unit rates and finds the constant of proportionality of proportional relationships in various forms. The student determines whether two quantities, shown in various forms, are in a proportional relationship. The student represents proportional relationships between quantities using equations. The student interprets specific values from a proportional relationship in the context of a problem situation. The student computes with percentages in context 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 Seventh In grade six, students worked primarily with ratios involving whole­number quantities and discovered what it meant to have equivalent ratios. In grade six, students worked with many examples of proportional relationships and represented them numerically, pictorially, graphically, and with equations in simple cases. In grade six, students used ratio tables and unit rates to solve percent problems. In grade seven, students find unit rates in ratios involving fractional quantities (7.RP.1 ). In grade seven, students examine more closely the characteristics of proportional relationships. In particular, they begin to identify these facts: When proportional quantities are represented in a table, pairs of entries represent equivalent ratios. The graph of a proportional relationship lies on a straight line that passes through the point (0,0), indicating that when one quantity is 0, so is the other. Equations of proportional relationships in a ratio of a:b always take the form y= k · x , where Eighth The study of proportional relationships is a foundation for the study of functions, which is introduced in grade eight and continues through higher mathematics. In grade eight, students will understand that the k is the constant ab if the variables x and y are defined so that the ratio x:y is equivalent to a:b proportional relationships they studied in grade seven are part . (The number k is also known as the constant of proportionality [7.RP.2 ]). In grade seven, of a broader group of linear students extend their work to solve multi­step ratio and percent problems (7.RP.3 ) . functions. 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 VOCABULARY rate, unit rate, complex frac‰‱on, unit ra‰‱o, dimensional analysis, propor‰‱onal, non‐propor‰‱onal, equivalent ra‰‱os, coordinate plane, origin, x‐axis, y‐axis, ordered pair, x‐coordinate, y‐coordinate, graph, quadrants, coordinate pair, x‐coordinate, y‐coordinate, x‐axis, y‐axis,, propor‰‱on, cross product, rate of change, constant rate of change, slope, direct varia‰‱on, constant of varia‰‱on, constant of propor‰‱onality, percent, percent propor‰‱on, percent equa‰‱on A
dditional Sources: H oward C ounty Public School System, Tucson U nified School D istrict