0ce580a5bf8deaeafa17eb9c0dd9fc2d.doc Time Frame: Approximately 1-2 Weeks Connections to Previous Learning: Previously, students learned to find the area of shapes in 3rd grade, progressing to the area of a net for rectangular solids and the volume of solids as indicated in the measurement standards for K-5. It is expected that students know how to add, subtract and multiply with fractional amounts. Learning about expressions and equations in the previous units of 6th grade provides background on solving equations, the meaning of variable and making connections between expressions and equations, graphs and a real world or mathematical situations. Students already have learned the foundational ideas of data distributions, measures of center and variability, which can be displayed in visual formats.
Focus of this Unit: In this unit students will solve a variety of real world problems and communicate solutions using dot plots. They will apply their understanding of graphing on the coordinate plane to a study of the relationship between side lengths and volume. Students will determine which variables are dependent and independent in this real-world context and will create expressions/equations based on their analysis of their data and variables. Problems involving geometric shapes, like finding the volume of a right rectangular prism, will also be solved by students using their knowledge of variables and graphing techniques. For instance, students practice using numbers to describe relationships between quantities that vary together to identify and understand the relationship between the side lengths of a box and the box’s volume where the net for the box is described on a 4 quadrant grid. Students use coordinate geometry in a four-quadrant grid to determine lengths, and develop deeper understanding of the volume formula for a rectangular solid.
Connections to Subsequent Learning: The focus on the relationship between two quantities and the equation leads to work with functions and provides examples of applications for the idea of variable. In 7th grade, students continue to construct and describe the relationships between geometrical figures and solve problems with volume.
4/3/2018 12:21:28 下午 Adapted from UbD Framework Page 1 0ce580a5bf8deaeafa17eb9c0dd9fc2d.doc
Desired Outcomes Standard(s): Represent and analyze quantitative relationships between dependent and independent variables. . 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. Apply and extend previous understandings of numbers to the system of rational numbers. . 6.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. Summarize and describe distributions. . 6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. Solve real-world and mathematical problems involving area, surface area, and volume. . 6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = I w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
Supporting Standards: Solve real-world and mathematical problems involving area, surface area, and volume. . 6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. . 6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles. Use the nets to find surface areas of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
WIDA Standard: (English Language Learners) English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. English language learners benefit from: explicit instruction with regard to the context of the tasks for geometric measurement, data distributions, and dependent and independent variables. tactile and virtual tools for measuring dimensions of geometric figures and displaying data.
4/3/2018 12:21:28 下午 Adapted from UbD Framework Page 2 0ce580a5bf8deaeafa17eb9c0dd9fc2d.doc
Understandings: Students will understand that … Graphing objects in a four quadrant graph can provide ways to measure distances and identify that shapes have specific properties Expressions are a modeling tool to use when solving real word problems. This process can provide a way to describe quantitative relationships – for instance, traveling some distance (d) at a given rate of travel will take a given amount of time (t)with a constant rate. Volume of a rectangular prism can be determined by multiplying the length, width and height dimensions when the dimensions are fractional lengths.
Essential Questions: What is the relationship between the dimensions of a figure and its volume? How do ordered pairs on coordinate graphs help define relationships? How do we determine if a variable is independent or dependent in an expression or equation? What is the value of using different data representations? What models are helpful for understanding and quantifying the volume of rectangular prisms?
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.) *1. Make sense of problems and persevere in solving them. Students apply understandings developed from previous experiences with expressions and equations to represent and solve problems regarding volume and data. 2. Reason abstractly and quantitatively. Students demonstrate abstract reasoning by using formulas to create expressions and equations with numbers and variables so that they can efficiently represent and solve problems. 3. Construct viable arguments and critique the reasoning of others. Students construct and critique arguments about effective algebraic, graphic and tabular representations about geometric figures and data distributions to solve problems. *4. Model with mathematics. Students model problem situations symbolically, graphically, using a table, and contextually. Students form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations. *5. Use appropriate tools strategically. Students choose formulas appropriate to given situations. They also choose appropriate algebraic and visual models to use when representing and solving problems. 6. Attend to precision. Students label numeric values in problem-solving situations and attend to the language of contextualize problems when creating models. 7. Look for and make use of structure. Students examine the geometric structures to determine appropriate problem solving strategies. They consider the structures of data representations when creating algebraic representations to solve problems. They use the structure of problems to differentiate between dependent and independent variables. 8. Look for express regularity in repeated reasoning. Students generalize strategies based upon experiences with multiple problem solving structures, geometric problems and data distributions.
4/3/2018 12:21:28 下午 Adapted from UbD Framework Page 3 0ce580a5bf8deaeafa17eb9c0dd9fc2d.doc
Prerequisite Skills/Concepts: Advanced Skills/ Concepts: Students should already be able to: Some students may be ready to: Use variables to represent unknowns. Find relationships between two quantities and the equation as related to work Add, subtract, multiply and divide with whole numbers, decimals with functions. and fractions. Construct and describe the relationships between geometrical figures and solve problems with volume fluently.
Knowledge: Students will know… Skills: Students will be able to … Define independent and dependent variables. (6.EE.9) All standards in this unit go beyond the knowledge level. Use variables to represent two quantities in a real-world problem that change in relationship to one another. (6.EE.9) Write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable (6.EE.9) Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. (6.EE.9) Solve real-world problems by graphing points in all four quadrants of the coordinate plane (6.NS.8) Use coordinates to find distances between points with the same first coordinate or the same second coordinate. (6.NS.8) Use absolute value to find distances between points with the same first coordinate or the same second coordinate. (6.NS.8) Display numerical data in plots on a number line, dot plots, histograms and box plots. (6.SP.4) Calculate the volume of a right rectangular prism. (6.G.2) Apply the formula to solve real world mathematical problems involving volume with fractional edge lengths. (6.G.2)
4/3/2018 12:21:28 下午 Adapted from UbD Framework Page 4 0ce580a5bf8deaeafa17eb9c0dd9fc2d.doc
Academic Vocabulary:
Critical Terms: Supplemental Terms: Volume Net Independent variables Edge Dependent variables Dot plot Ordered pairs Histogram x- axis Box plot x coordinate Rectangular prism y-axis y coordinate
4/3/2018 12:21:28 下午 Adapted from UbD Framework Page 5