Interpreting and Analyzing Data

Transcription

1 Grade 5 Mathematics, Quarter 4, Unit 4.1 Interpreting and Analyzing Data Overview Number of instructional days: 5 (1 day = 45 minutes) Content to be learned Interpret a given representation (tables, bar graph, circle graphs, or line graphs) to answer questions related to the data. Analyze data of given representation to formulate or justify conclusions to make predictions or solve problems. Determine or calculate mean, median, mode, and range for data in various contexts. Select and/or describe representations and elements that best display a set of data or a situation. Use measures of central tendency (mean, median, mode) or range to analyze situations or solve problems. Essential questions How do you interpret the data you have collected? What types of information can be best expressed through a line graph? What predictions would you make based on the data given? Mathematical practices to be integrated Make sense of problems and persevere in solving them. Analyze givens, constraints, relationship and goals. Make conjectures about the solutions. Explain relationship between equations, verbal descriptions, tables, and graphs. Model with mathematics. Identify important quantities and their relationships. Draw conclusions, interpret results, and revise models if needed. What representations (bar graph, line graph, charts, etc.) would best represent the changes in temperature over a one-year period? How can determining the mean, median, mode, and range help you to analyze and interpret data? Cumberland, Lincoln, and Woonsocket Public Schools C-53

2 Grade 5 Mathematics, Quarter 4, Unit Interpreting and Analyzing Data (5 days) Written Curriculum Grade-Level Expectations M(DSP) 5 1 Interprets a given representation (tables, bar graphs, circle graphs, or line graphs) to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems. (State) (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP) 5 2.) M(DSP) 5 2 Analyzes patterns, trends, or distributions in data in a variety of contexts by determining or using measures of central tendency (mean, median, or mode) or range to analyze situations, or to solve problems. (State) M(DSP) 5 3 Identifies or describes representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M(DSP) 5 1. (State) (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP) 5 2.) Clarifying the Standards Prior Learning In grade 4, students interpreted pictographs or circle graphs to answer questions as new learning. While still interpreting line plots, tables, and bar graphs, they began to justify conclusions and continued to make predictions. Students in fourth grade began to use measures of central tendency (median or mode). They also continued to decide how to best organize and display data. Current Learning Students in grade 5 now use line graphs to interpret a given representation. This concept is taught at the developmental level. As they analyze the data, the mean is new to central tendency while other measures of central tendency are at the reinforcement level. Central tendency is now used to analyze situations and solve problems. This instruction is at the developmental stage. The Bloom s taxonomy levels are analysis, synthesis, and evaluation. Future Learning In grade 6, students will use stem-and-leaf plots to interpret a given representation and to answer questions when organizing and displaying data. They will also analyze situations and use dispersion. Additional Research Findings According to Principles and Standards for School Mathematics, As students construct graphs of ordered numerical data, teachers need to help them understand what the values along the horizontal and vertical axes represent. Using experience with a variety of graphs, teachers should make sure that students encounter and discuss issues such as why the scale on the horizontal axis needs to include values that are not in the data set and how to represent zero on a graph (p. 178). C-54 Cumberland, Lincoln, and Woonsocket Public Schools

3 Interpreting and Analyzing Data (5 days) Grade 5 Mathematics, Quarter 4, Unit The book also states, A reasonable objective for upper elementary and middle grade students is that they begin to regard a set of data as a whole that can be described as a set and compared to other data sets. As students examine a set of ordered numerical data, teachers should help them learn to pay attention to important characteristics of the data set: where data are concentrated or clumped, values for which there are no data, or data points that appear to have unusual values. Building on their informal understanding of the most and the middle, students can learn about three measures of center mode, median, and, informally, the mean. Students need to learn more than simply how to identify the mode or median in a data set. They need to build an understanding of what, for example, the median tells them about the data, and they need to see this value in the context of other characteristics of the data (p. 179). Additionally, In grade 5, once students are experienced using the mode and median as part of their data descriptions, they can begin to conceptually explore the role of the mean as a balance point for the data set, using small data sets. The idea of a mean value what it is, what information it gives about the data, and how it must be interpreted in the context of other characteristics of the data is a complex one, which will continue to develop in later grades. Data can be used for developing arguments that are based on evidence and for continued problem posing. As students discuss data gathered to address a particular question, they should begin to distinguish between what the data show and what might account for the results (p. 180). Cumberland, Lincoln, and Woonsocket Public Schools C-55

4 Grade 5 Mathematics, Quarter 4, Unit Interpreting and Analyzing Data (5 days) Notes About Resources and Materials C-56 Cumberland, Lincoln, and Woonsocket Public Schools

5 Grade 5 Mathematics, Quarter 4, Unit 4.2 Writing and Solving Algebraic Expressions Overview Number of instructional days: 10 (1 day = 45 minutes) Content to be learned Represent a mathematical situation as an expression or number sentence using a letter or symbol involving any two of the four operations. Solve linear algebraic expressions using whole numbers involving any two of the four operations. Model equivalency or use other representations to show the understanding of equality. Solve one-step linear equations (such as 2x = 12). Determine which values make an equation a true statement given possible values for x (such as 2x + 3 = 11{x : x = 2, 3, 4, 5}). Essential questions How can you represent unknown quantities in an algebraic expression? If we have two operations, how do we know how to solve the problem? What strategies can be used to solve for unknowns in algebraic equations? Will more than one value in a set make the statement true? How do you know? Mathematical practices to be integrated Make sense of problems and persevere in solving them. Identify and execute appropriate strategies to solve the problem. Evaluate progress toward the solution and make revisions if necessary. Model with mathematics. Identify important quantities in a practical situation and map their relationships using such tools as formulas. Analyze mathematical relationships to draw conclusions. How can we determine that 2n = 16? How can you determine if two algebraic expressions are equal? How do you determine if an unknown makes an algebraic equation a true or false statement? Cumberland, Lincoln, and Woonsocket Public Schools C-57

6 Grade 5 Mathematics, Quarter 4, Unit Writing and Solving Algebraic Expressions (10 days) Written Curriculum Grade-Level Expectations M(F&A) 5 3 Demonstrates conceptual understanding of algebraic expressions by using letters to represent unknown quantities to write linear algebraic expressions involving any two of the four operations; or by evaluating linear algebraic expressions using whole numbers. (State) M(F&A) 5 4 Demonstrates conceptual understanding of equality by showing equivalence between two expressions using models or different representations of the expressions (expressions consistent with the parameters of M(F&A) 5 3), by solving one-step linear equations of the form ax = c, x ± b = c, or x/a = c, where a, b, and c are whole numbers with a 0; or by determining which values of a replacement set make the equation (multi-step of the form ax ± b = c where a, b, and c are whole numbers with a 0) a true statement (e.g., 2x + 3 = 11, {x: x = 2, 3, 4, 5}). (State) Clarifying the Standards Prior Learning Fourth-graders used letters or symbols to represent unknown quantities to write simple linear algebraic expressions. They used one of the four operations. Fourth-graders evaluated simple linear algebraic expressions using whole numbers. Students in fourth grade showed equivalence between two expressions using models or different representations of the expressions. They simplified numerical expressions where left to right computations may be modified only by the use of parentheses and they solved one-step linear equations where a, b, and c are whole numbers. Current Learning Students in fifth grade use letters to represent unknown quantities to write linear algebraic expressions involving any two of the four operations. Fifth-graders evaluate linear algebraic expressions using whole numbers. Students solve one-step linear equations, including division. When fifth-graders are given a set with multiple values, they are able to evaluate which whole number makes the statement true. Future Learning Sixth-grade students will be able to write linear algebraic expressions involving any of the four operations, and consistent with order of operations. They will be able to evaluate linear algebraic expressions with more than one variable. Students in sixth grade will use the mathematical terms variable and order of operations. They will be able to evaluate an expression within an equation. Additional Research Findings Principles and Standards for School Mathematics states that by the end of fifth grade, students should be able to make generalizations by reasoning about the structure of the pattern. For example, a fifth-grade student might explain that, if you add the first n odd numbers, the sum is the same as n x n (p. 160). At this grade band, students begin to better understand the usefulness of a variable (represented by a box, letter, or symbol). As students explore patterns and note relationships, they should be encouraged to represent their thinking. Students are beginning to use the idea of a variable as they think about how to C-58 Cumberland, Lincoln, and Woonsocket Public Schools

7 Writing and Solving Algebraic Expressions (10 days) Grade 5 Mathematics, Quarter 4, Unit describe a rule from the pattern they have observed. As students become more experienced in investigating, articulating, and justifying generalizations, they can begin to use variable notation and equations to represent their thinking. Teachers will need to model how to represent thinking in the form of equations. In this way, they can help students connect the ways they are describing their findings to mathematical notation. Students should also understand the use of a variable as a placeholder in an expression or equation. For example, they should explore the role of n in the equation 80 x 15 = 40 x n and be able to find the value of n that makes the equation true (p. 161). Cumberland, Lincoln, and Woonsocket Public Schools C-59

8 Grade 5 Mathematics, Quarter 4, Unit Writing and Solving Algebraic Expressions (10 days) Notes About Resources and Materials C-60 Cumberland, Lincoln, and Woonsocket Public Schools

9 Grade 5 Mathematics, Quarter 4, Unit 4.3 Linear Relationships and Constant Rate of Change Overview Number of instructional days: 10 (1 day = 45 minutes) Content to be learned Identify, model, and describe situations with constant rates of change to demonstrate understanding of linear relationships (y = kx). Identify and extend a variety of patterns (linear and nonlinear) represented in models, tables, sequences, or in problem situations. Write a rule in words or symbols that identifies finding specific cases of a linear relationship. Essential questions How would you describe the pattern shown on the graph? Given a graph, can you tell me what is happening in the graph? How would you use a line graph to tell a story? What in the situation tells you that something is changing? What does constant rate mean? Given a car s speed in mph and the time spent traveling, how would you plot that data on a graph? Mathematical practices to be integrated Look for and make use of structure. Look for, develop, generalize, and describe a pattern orally, symbolically, graphically, and in written form. Apply and discuss properties. Look for and express regularity in repeated reasoning. Look for mathematically sound shortcuts. Use repeated applications to generalize properties. What predictions can the patterns support? Why are can be used to describe a constant rate of change? How can you describe the pattern that has a constant rate of change in a table? How would you describe this pattern using words? How can we change our words to symbols to describe the same pattern? Cumberland, Lincoln, and Woonsocket Public Schools C-61

10 Grade 5 Mathematics, Quarter 4, Unit Linear Relationships and Constant Rate of Change (10 days) Written Curriculum Grade-Level Expectations M(F&A) 5 2 Demonstrates conceptual understanding of linear relationships (y = kx) as a constant rate of change by identifying, describing, or comparing situations that represent constant rates of change (e.g., tell a story given a line graph about a trip). (Local) M(F&A) 5 1 Identifies and extends to specific cases a variety of patterns (linear and nonlinear) represented in models, tables, sequences, or in problem situations; and writes a rule in words or sc symbols for finding specific cases of a linear relationship. (State) Clarifying the Standards Prior Learning Fourth-graders are able to identify and extend to specific cases a variety of patterns both linear and nonlinear. Students are able to use models, tables, or sequences. Fourth-graders may choose to write a rule in words or symbols to find the next case. They are able to demonstrate conceptual understanding of linear relationship as a constant rate of change by identifying, describing, or comparing situations that represent constant rates of change. Current Learning Fifth-graders are able to identify and extend to specific cases a variety of patterns now including problem situations. In fifth grade, students write a rule in words or symbols for finding specific cases of a linear relationship. Student responses will be at the knowledge, comprehension, development, and reinforcement levels. Future Learning Sixth-graders will write a rule in words or symbols for finding specific cases of a nonlinear relationship. In sixth grade, students will write an expression or equation using words or symbols to express the generalization of a linear relationship. Sixth-graders will construct and interpret graphs of real occurrences and describe the slope of linear relationships in a variety of problem situations. They will be able to describe how change in the value of one variable relates to change in the value of a second variable in problem situations with constant rates of change. Additional Research Findings Principles and Standards for School Mathematics states that students should investigate numerical and geometric patterns and express them mathematically in words or symbols. They should analyze the structure of the pattern and how it grows or changes, organize this information systematically, and use their analysis to develop generalizations about the mathematical relationships in the pattern. Students should be encouraged to explain these patterns verbally and to make predictions about what will happen if the sequence is continued (p. 159). Change is an important mathematical idea that can be studied using algebraic tools. This work is a precursor to later more focused attention on what the slope of a line represents, that is, what the steepness C-62 Cumberland, Lincoln, and Woonsocket Public Schools

11 Linear Relationships and Constant Rate of Change (10 days) Grade 5 Mathematics, Quarter 4, Unit of the line shows about the rate of change. Students should have opportunities to study situations that display different patterns of change change that occurs at a constant rate, such as someone walking at a constant speed, and rates of change that increase or decrease (p. 163). Cumberland, Lincoln, and Woonsocket Public Schools C-63

12 Grade 5 Mathematics, Quarter 4, Unit Linear Relationships and Constant Rate of Change (10 days) Notes About Resources and Materials C-64 Cumberland, Lincoln, and Woonsocket Public Schools

13 Grade 5 Mathematics, Quarter 4, Unit 4.4 Statistical Probability Overview Number of instructional days: 5 (1 day = 45 minutes) Content to be learned Predict the likelihood of an event as a fraction. Test the predictions through experiments. Determine if a game is fair. Determine the experimental or theoretical probability of an event and express the results in fraction form. Mathematical practices to be integrated Make sense of problems and persevere in solving them. Identify and execute appropriate strategies to solve the problem. Evaluate progress toward the solution and make revisions if necessary. Check answers using a different method, and continually ask, Does this make sense? Reason abstractly and quantitatively. Make sense of quantities and their relationships in problem situations. Use varied representations and approaches when solving problems. Know and flexibly use different properties of operations and objects. Change perspectives, generate alternatives, and consider different options. Essential questions How do you express the probability of an event as a fraction? What predictions can you make before you start the experiments? What does it mean when the probability of an event is 3/4? Probability is a number describing the likelihood of an event with a value between which two numbers? How can you determine if a game is fair? What is the difference between experimental and theoretical probability? How can you explain the relationship between the probability of an event and its actual outcome? How do you determine the number of trials for your experiment? What do you think happens to the probability the more you experiment? What affects the likelihood of an event? Cumberland, Lincoln, and Woonsocket Public Schools C-65

14 Grade 5 Mathematics, Quarter 4, Unit Statistical Probability (5 days) Written Curriculum Grade-Level Expectations M(DSP) 5 5 For a probability event in which the sample space may or may not contain equally likely outcomes, predicts the likelihood of an event as a fraction and tests the prediction through experiments; and determines if a game is fair. (Local) M(DSP) 5 5 For a probability event in which the sample space may or may not contain equally likely outcomes, determines the experimental or theoretical probability of an event and expresses the result as a fraction. (State) Clarifying the Standards Prior Learning Fourth-grade students predicted the likelihood of an event as a part-to-whole relationship. Students determined the theoretical probability of an event and expressed the result as part to whole. Current Learning Fifth-grade students predict the likelihood of an event as a fraction. Students determine the experimental probability of an event and express the result as a fraction. The instructional levels appropriate for this current learning are the developmental and reinforcement levels. Future Learning In sixth grade, students will predict the theoretical probability of an event. Sixth-grade students will test the prediction through simulations and will design fair games. In sixth grade, students will also determine the experimental or theoretical probability of an event in a problem-solving situation. Additional Research Findings According to Principles and Standards for School Mathematics, students should begin to learn about probability as a measurement of the likelihood of events. Students should also explore probability through experiments that have only a few outcomes, such as using game spinners with certain portions shaded and considering how likely it is that the spinner will land on a particular color. They should come to understand and use 0 to represent the probability of an impossible event and 1 to represent the probability of a certain event, and they should use common fractions to represent the probability of events that are neither certain nor impossible. Through these experiences, students encounter the idea that although they cannot determine an individual outcome, such as which color the spinner will land on next, they can predict the frequency of various outcomes (p. 181). Benchmarks for Science Literacy states that by the end of fifth grade, students should understand that a small part of something may be special in some way and not give an accurate picture of the whole. How much a portion of something can help to estimate what the whole is like depends on how the portion is chosen. There is a danger of choosing only the data that show what is expected by the person doing the choosing (p. 181). C-66 Cumberland, Lincoln, and Woonsocket Public Schools

15 Statistical Probability (5 days) Grade 5 Mathematics, Quarter 4, Unit Spreading data out on a number line helps to see what the extremes are, where they pile up, and where the gaps are. A summary of data includes where the middle is and how much spread is around it. Cumberland, Lincoln, and Woonsocket Public Schools C-67

16 Grade 5 Mathematics, Quarter 4, Unit Statistical Probability (5 days) Notes About Resources and Materials C-68 Cumberland, Lincoln, and Woonsocket Public Schools

17 Grade 5 Mathematics, Quarter 4, Unit 4.5 Measuring Customary and Metric Units Including Time and Temperature Overview Number of instructional days: 5 (1 day = 45 minutes) Content to be learned Use appropriate units to measure consistently. Convert units of measure within systems when solving problems across the content strand. Essential questions What units and tools are used to measure time? What units and tools are used to measure temperature? How do you decide which unit of measure to use? What are the relationships among centimeters, millimeters, and meters? What comparisons can you make among the units of the metric system? Mathematical practices to be integrated Use appropriate tools. Attend to precision. Look for and express regularity in repeated reasoning. What comparisons can you make among the units of the customary system? What are some common benchmarks you can use to describe ounces, pints, quarts, and gallons? What is the relationship between ounces and pounds? Is there a difference between ounces used to find capacity and ounces used to find weight? Cumberland, Lincoln, and Woonsocket Public Schools C-69

18 Grade 5 Mathematics, Quarter 4, Unit 4.5 Measuring Customary and Metric Units Including Time and Temperature (5 days) Grade-Level Expectations Written Curriculum M(G&M) 5 7 Measures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across the content strands. (State) See Benchmarks in Appendix B. Clarifying the Standards Prior Learning Fourth-graders measured length to 1/4 inch, centimeters to the 1/2 centimeter, and meters to the 1/2 centimeter. Students also measured in yards. Miles and kilometers were used in scale questions. Students in fourth grade performed equivalencies of 12 inches in 1 foot, 100 centimeters in 1 meter, 3 feet in 1 yard, and 36 inches in 1 yard. Fourth-grade students measured time in five-minute intervals. Students used day and year measurements. Time equivalence included hours in a day, days in a week, days in a year, seconds in a minute, and minutes in an hour. In temperature, fourth-grade students measured to 1 degree in both Celsius and Fahrenheit. In capacity, fourth-graders measured to the whole quart. In mass, students measured to the whole kilogram and to the whole gram. Fourth-graders measured in weight to whole pounds. Current Learning In fifth grade, students measure length to 1/8 inch. Fifth-grade students know the equivalency of 10 millimeters in 1 centimeter. In time, they measure to one minute. In capacity, fifth-grade students measure to one ounce and add the gallon and pint to their measurement. Students now have equivalencies in capacity: ounce to quart, quart to gallon, and pints to quart. Weight now includes equivalencies of ounces to pounds. Fifth-graders now add angles and rotations in measurement to two degrees. Future Learning Sixth-grade students will be able to accurately measure inches to 1/16 inch, centimeter to 1/10, meters to 1/100. Sixth-graders will use the mile measurement in rate questions. They will identify equivalencies as 1,000 millimeters in 1 meter, 1000 milliliters in 1 liter, 360 degrees in a circle and 90 degrees in a right angle. Capacity will include liter units and mass will include gram measurement to 1/10 gram. Additional Research Findings According to Principles and Standards for School Mathematics, students in grades 3 5 should measure the attributes of a variety of physical objects and extend their work to measuring more complex attributes including area, volume, and angle. They will learn that length measurements in particular contexts are given specific names, such as perimeter, width, height, circumference, and distance. They begin to establish some benchmarks by which to estimate or judge the size of objects. Students in grades 3 5 should be able to recognize the need to select appropriate units for what is being measured. In these grades, more emphasis should be placed on the standard units that are used in the United States (the customary units) and around the world (the metric system). Students should become familiar with the common units of these systems and establish mental images or benchmarks for judging and comparing size. Students should gain facility in expressing measurements in equivalent forms. They should use their C-70 Cumberland, Lincoln, and Woonsocket Public Schools

19 Measuring Customary and Metric Units Grade 5 Mathematics, Quarter 4, Unit 4.5 Including Time and Temperature (5 days) knowledge of relationships between units and their understanding of multiplicative situations to make conversions, such as expressing 150 centimeters as 1.5 meters or 3 feet as 36 inches. Since students in the United States encounter two systems of measurement, they should also have convenient referents for comparing units in different systems for example, two centimeters is a little less than an inch, a quart is a little less than a liter, a kilogram is about two pounds. However, they do not need to make formal conversions between the two systems at this level (p. 172). In grades 3 5, an expanded number of tools and a range of measurement techniques should be available to students. When using conventional tools, such as rulers and tape measures, students will need instruction to use them properly. When standard measurement tools are difficult to use in a particular situation, students must learn to adapt their tools or invent techniques that will work. Students should be challenged to develop measurement techniques as needed in order to measure complex figures or objects (p. 173). Cumberland, Lincoln, and Woonsocket Public Schools C-71

20 Grade 5 Mathematics, Quarter 4, Unit 4.5 Measuring Customary and Metric Units Including Time and Temperature (5 days) Notes About Resources and Materials C-72 Cumberland, Lincoln, and Woonsocket Public Schools

21 Grade 5 Mathematics, Quarter 4, Unit 4.6 Describing Similarity Through Transformations, Congruency, and Symmetry Overview Number of instructional days: 5 (1 day = 45 minutes) Content to be learned Model, explain, and describe the proportional relationship among the sides of similar triangles and rectangles. Scale up and down, keeping the same angle measures. Solve related problem including applying scales on maps. Determine and give directions between locations on a map or coordinate grid using all four quadrants. Plot points in four quadrants in context (e.g., games, mapping, identifying the vertices of polygons as they are reflected, rotated and translated). Determine the horizontal and vertical distances between points on a coordinate grid in the first quadrant. Essential questions What does it mean when two shapes are proportional? How does scaling a polygon (without changing its angle measures) affect the dimensions of the figure? How do you find the distances between two locations on a map? How would you determine the horizontal and vertical distances between points on a coordinate grid in the first quadrant? Mathematical practices to be integrated Model with mathematics. Identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. Analyze mathematical relationships to draw conclusions. Attend to precision. Use clear definitions and state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. Specify units of measure and use label parts of graphs and charts. Strive for accuracy. How would you give directions between two locations on a map or coordinate grid? What strategy would you use to draw a similar figure on a coordinate grid? How are the quadrants alike and different? Cumberland, Lincoln, and Woonsocket Public Schools C-73

22 Grade 5 Mathematics, Quarter 4, Unit 4.6 Describing Similarity Through Transformations, Congruency, and Symmetry (5 days) Grade-Level Expectations Written Curriculum M(G&M) 5 5 Demonstrates conceptual understanding of similarity by describing the proportional effect on the linear dimensions of triangles and rectangles when scaling up or down while preserving angle measures, or by solving related problems (including applying scales on maps). Describes effects using models orsc explanations. (Local) M(G&M) 5 9 Demonstrates understanding of spatial relationships using location and position by interpreting and giving directions between locations on a map or coordinate grid (all four quadrants); plotting points in four quadrants in context (e.g., games, mapping, identifying the vertices of polygons as they are reflected, rotated, and translated); and determining horizontal and vertical distances between points on a coordinate grid in the first quadrant. (Local) Clarifying the Standards Prior Learning Students in fourth grade demonstrated conceptual understanding of similarity. Fourth-graders applied scales on maps or applied characteristics of similar figures to identify similar figures or to solve problems involving similar figures. Students in fourth grade described relationships using models or explanations. Fourth-grade students plotted points in the first quadrant in context. They found the horizontal and vertical distances between points on a coordinate grid in the first quadrant. Current Learning Students in fifth grade describe the proportional effect on the linear dimensions of triangles and rectangles when scaling up or down while preserving angle measures. Fifth-grade students use all four quadrants when interpreting and giving directions between locations on a map or coordinate grid. They plot points in four quadrants in context. Students reach the comprehension, application, development, and reinforcement levels of Bloom s Taxonomy with this content. Future Learning Sixth-grade students will describe the proportional effect on the linear dimensions of polygons or circles. Additional Research Findings Principles and Standards for School Mathematics state that an understanding of congruence and similarity will develop as students explore shapes that in some way look alike. Although students will not develop a full understanding of similarity until the middle grades, when they focus on proportionality, in grades 3 5 they can begin to think about similarity in terms of figures that are related by the transformations of magnifying or shrinking (p. 166). C-74 Cumberland, Lincoln, and Woonsocket Public Schools

23 Describing Similarity Through Transformations, Grade 5 Mathematics, Quarter 4, Unit 4.6 Congruency, and Symmetry (5 days) Students at this level also should learn how to use two numbers to name points on a coordinate grid and should realize that a pair of numbers corresponds to a particular point on a grid. Using coordinates, they can specify paths between locations and examine the symmetry, congruence, and similarity of shapes drawn on the grid. They can also explore methods for measuring the distance between locations on the grid. As students ideas about the number system expand to include negative numbers, they can work in all four quadrants of the Cartesian plane (p. 167). Cumberland, Lincoln, and Woonsocket Public Schools C-75

24 Grade 5 Mathematics, Quarter 4, Unit 4.6 Describing Similarity Through Transformations, Congruency, and Symmetry (5 days) Notes About Resources and Materials C-76 Cumberland, Lincoln, and Woonsocket Public Schools