Table of Contents Problem Solving with the Coordinate Plane

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1 GRADE 5 UNIT 6 Table of Contents Problem Solving with the Coordinate Plane Lessons Topic 1: Coordinate Systems 1-6 Lesson 1: Construct a coordinate system on a line. Lesson 2: Construct a coordinate system on a plane. Lessons 3 & 4: Name points using coordinate pairs, and use the coordinate pairs to plot points. Lesson 5 & 6: Investigate patterns in vertical and horizontal lines, and interpret points on the plane as distances from the axes. Topic 2: Patterns in the Coordinate Plane and Graphing Number Patterns from Rules 7-12 Lesson 7: Plot points, use them to draw lines in the plane, and describe patterns within the coordinate pairs. Lesson 8: Generate a number pattern from a given rule, and plot the points. Lesson 9: Generate two number patterns from given rules, plot the points, and analyze the patterns. Lesson 10: Compare the lines and patterns generated by addition rules and multiplication rules. Lesson 11: Analyze number patterns created from mixed operations. Lesson 12: Create a rule to generate a number pattern, and plot the points. Topic 3: Drawing Figures in the Coordinate Plane Lesson 13: Construct parallel line segments on a rectangular grid. Lesson 14: Construct parallel line segments, and analyze relationships of the coordinate pairs. Lesson 15: Construct perpendicular line segments on a rectangular grid. Lesson 16: Construct perpendicular line segments, and analyze relationships of the coordinate pairs. Lessons 17: Draw symmetric figures using distance and angle measure from the line of symmetry.

2 Topic 4: Problem Solving in the Coordinate Plane Lesson 18: Draw symmetric figures on the coordinate plane. Lesson 19: Plot data on line graphs and analyze trends. Lesson 20: Use coordinate systems to solve real world problems. Topic 5: Multi-Step Word Problems Lessons 21-25: Make sense of complex, multi-step problems and persevere in solving them. Share and critique peer solutions. Topic 6: The Years in Review: A Reflection on A Story of Units Lessons 26 & 27: Solidify writing and interpreting numerical expressions. Lesson 28: Solidify fluency with Grade 5 skills. Lessons 29 & 30: Solidify the vocabulary of geometry. Lesson 31: Explore the Fibonacci sequence. Lesson 32: Explore patterns in saving money. Lessons 33 & 34: Design and construct boxes to house materials for summer use.

3 Grade 5, Math Unit 6 for Parents and Students Vocabulary Familiar Terms and Symbols Angle - the joining of two different rays sharing a common vertex Angle measure - number of degrees in an angle Degree - unit used to measure angles Horizontal - parallel to the x-axis Line - two-dimensional object that has no endpoints and continues on forever in a plane Parallel lines - two lines that do not intersect; Symbol ( ) Perpendicular two lines are perpendicular if they intersect, and any of the angles formed between the lines are 90 angles; symbol ( ) Point - zero-dimensional figure that satisfies the location of an ordered pair Rule - procedure or operation(s) that affects the value of an ordered pair Vertical - parallel to the y-axis New Terms for 5 th Grade Unit 6 Axis - fixed reference line for the measurement of coordinates Coordinate - number that identifies a point on a plane, the distance from zero to the point Coordinate pair - two numbers that are used to identify a point on a plane; written (x, y) where x represents a distance from 0 on the x-axis and y represents a distance from 0 on the y-axis Coordinate plane - plane created by a horizontal number line (the x-axis) and a vertical number line (the y-axis) intersecting at a point called the origin. Each point in the coordinate plane can be specified by an ordered pair or coordinate pair of numbers. Line of Symmetry a line of symmetry divides a figure into 2 congruent parts Midpoint the half-way point on a line segment Ordered pair - two quantities written in a given fixed order, usually written as (x, y) Origin - fixed point from which coordinates are measured; the point at which the x-axis and yaxis intersect, labeled (0, 0) on the coordinate plane Quadrant - any of the four equal areas created by dividing a plane by an x-axis and y-axis

4 Lesson by Lesson Suggestions Lessons 1-6: In this topic students are introduced to the concept of a coordinate as describing the distance of a point on the line from zero. Students will also describe given points using coordinate pairs, and then use given coordinate pairs to plot points. Example 1: Plot A so its distance from the origin is 2. You need to figure out the value of each tic mark that is not labeled. You can determine that the value of each is 1. Start at zero and move 2 units to the right. Plot your point above the correct tic mark. Example 2: Plot L so its distance from the origin is 20. First you need to figure out the value of each tic mark that is not labeled. From 35 to 50 there is a difference of 15. Divide the 15 by 3 (3 sections between 35 and 50) and you get 5. So each tic mark changes by 5. Once you find the value of each tic mark you can then place the letter L on the line. Example 3: What is the coordinate of point S? First find the value of each tic mark. Since there are 6 spaces between 4 and 5, each tic mark would represent 1/6. When moving from the origin, the coordinate for point S is 4 1/6. Plotting a Coordinate Pair: How would you plot the point (2,5) on the coordinate grid? Start at the origin and move 2 units over on the x-axis. Then move 5 units up on the y-axis. (2, 5) (x, y)

5 Identifying a point on a Coordinate Grid: Find the coordinate pair for point B. Start at the origin and move along the x-axis. You will move 6 spaces on the x-axis to get to 3 units. (Each space equals ½ unit.) Then move up 2 spaces on the y-axis to get to the 1 unit. (Each space equals ½ unit.) Point B is at (3,1) Patterns in Coordinate Pairs Horizontal lines Vertical lines Look at line p, what do you notice about the 3 points and their coordinates? They have different x coordinates but the y coordinates are all 8. ***Any time the y- coordinates are the same in a set of coordinate pairs, the line created will always be horizontal. Look at line n, what do you notice about the 3 points and their coordinates? They have different y coordinates but the x coordinates are all 4. ***Any time the x -coordinates are the same in a set of coordinate pairs, the line created will always be vertical.

6 Lessons 7-12: Graphing in the Coordinate Plane Students will plot points and use them to draw lines in the plane by investigating patterns relating the x- and y- coordinates of the points on the line and reason about the patterns in the ordered pairs. They will also use given rules to generate coordinate pairs, plot the points, and investigate relationships. Patterns in the Coordinate Plane Step 1: Complete the chart. Use the numbers in the x column as the x coordinate and the numbers in the 7 column as the y coordinate. When writing a coordinate pair always put them inside parenthesis and separate the two numbers with a comma. Step 2: Plot the points on the coordinate plane below and then use a straight edge to draw a line connecting these points. Step 3: Write a rule showing the relationship between the x- and y- coordinates of points on the line. *This is related to finding a pattern and similar to an input output table. Each y- coordinate is 2 times greater than its corresponding x-coordinate. Step 4: Name two other points on this line. (2 ½, 5) (1 ¼, 2 ½)

7 Generate Two Number Patterns from Given Rule Plot the Points / Analyze the Patterns Step 1: Complete the tables for the given rules. Step 2: Plot the points and then construct lines l and m on the coordinate plane. Compare and contrast these lines. Lines l and m are parallel. I noticed that the y values on Line m are 3 units greater than the y values on Line l. Example: Line l has a point at (5, 6). The point above it on Line m has the coordinates of (5, 9). The y-coordinate of 9 on Line m is 3 units greater than y-coordinate of 6 on Line l. Another point on Line l is (9, 10). Three units above it is a point with coordinates (9, 13) on Line m. The y- coordinate of 13 on line m is 3 units greater that the y- coordinate of 10 on Line Based on the patterns you see, predict what line p, whose rule is 7 more than x, would look like. Draw your prediction on the plane above. I think that Line p will be parallel to Lines l and m. These are the x- and y- coordinates I will plot and then construct line p. (o, 7) (4, 11) (7, 14) (10, 17) Once I constructed Line p, I noticed that the y values on Line p are 3 units greater than the y values on Line m and 6 units greater than Line l.

8 Lessons 13-17: Drawing Figures in the Coordinate Plane Students will draw figures in the coordinate plane by plotting points to create parallel, perpendicular, and intersecting lines. Construct parallel line segments in a coordinate plane a. Identify the locations of E and F Locations E: (1, 3 ½ ) F: (3, 1 ½ ) a. b. Draw line EF (EF). b. c. Determine coordinate pair for L and M, such that EF LM and then draw LM. Coordinate Pairs: L: (2 ½, 3 ½ ) M: (4 ½, 1 ½ ) d. Explain the pattern you used when determining coordinate pairs for L and M. I shifted x-coordinates three ½ units to the right, but I kept the y- coordinates the same. I did not shift up or down. *NOTE: In creating LM EF, the student could have shifted 1, 2, 4, etc. units to the left or right.

9 Example 1: AB and ST are parallel. Compare the coordinates of points S and T to the coordinates of points A and B. a. Why is each x-coordinate in points in A and B 2 less than the x-coordinates in points S and T? The x-coordinates for points S and T shifted 2 units to the left. b. Why is each y-coordinate in points in A and B 1 more than the y-coordinates in points S and T? The y-coordinates for points S and T shifted 1 unit up. Constructing Perpendicular Segments Things to remember: A triangle that has one 90 angle is called a right triangle. The sum of the three angles of a triangle is equal to 180. Therefore the sum of the other two angles in a right triangle is equal to 90, since = 180. These two angles each measure less than 90, so they are called acute angles. Step 1: Draw a right triangle that has ST as its longest side. Step 2: The right triangle has a height of 2 units and a base of 3 units. Dashed lines show the height and base. T and S are acute angles whose sum is 90. Angle R is a right angle whose measure is 90. Step 3: Triangle RST is used to draw a segment perpendicular to ST by visualizing sliding triangle RST and rotating it so it appears standing up. It now has a base of 2 units and a height of 3 units. - Sketch another triangle the same as RST. Use dashed lines to sketch RT and RS and a solid line to sketch the longest side, ST. A straight angle has a measure of 180. T and S add up to 90, so the angle formed by the two solid segments must have a measure of = 180. Since the two longest sides of these triangles form a right angle, we can say that we have constructed perpendicular segments.

10 Draw Symmetric Figures from the Line of Symmetry Step 1: Draw a line of symmetry. This line will be used to draw symmetrical points, line segments, and/or figures. Step 2: Draw a point, A, above the line. Step 3: Draw a second point, B, on the same side of the line as A. Step 4: Draw AB. Step 5: Measure the distance from point A to the line of symmetry. Measure the same distance on the opposite side of the line of symmetry. Make certain that the edge of the ruler is perpendicular to the line of symmetry. Draw a point and name it C. Since point C was drawn using the ruler that was placed perpendicular to the line of symmetry and it is the same distance from the line of symmetry as point A, we can say that point C is symmetric to point A or point A is symmetric to point C. Step 6: Repeat step 5 with point B. Step 7: Draw CD.

11 Lessons 18-20: Problem Solving in the Coordinate Plane Students will use the coordinate plane to show locations, movement, and distance on maps. Line graphs are also used to explore patterns in the coordinate plane and make predictions based on those patterns. Draw Symmetric Figures on the Coordinate Plane Step 1: Record the ordered pair for each point on the grid. Step 2: Construct a line of symmetry, l, whose rule is x is always 5. Then plot points symmetric to the Points A to D. Since A and B are 4 units from the line of symmetry, then the points symmetric to A and B would be 4 units to the right of the line of symmetry. (F and G) Points C and D are 2 units from the line of symmetry so the points symmetric to C and D would be 2 units to the right of the line of symmetry. (I and H) Step 3: Connect the points to create symmetrical figures across the vertical line of symmetry.

12 Use Coordinate Plane to Solve Word Problems Example 1: The line graph below tracks the rain accumulation, measured every half hour, during a rainstorm that began at 2:00 p.m. and ended at 7:00 p.m. Use the information in the graph to answer the questions that follow. 1. How many inches of rain fell during this five-hour period? 2 ¼ inches fell during the five-hour period. 2. During which half-hour period did inch rain fall? Explain how you know. From 2:30 p.m. to 3:00 p.m. a ½ inch of rain fall. As the line moves up, each grid line increases by a ¼ inch. It takes 2 one-fourths to equal ½ inch. 3. During which half-hour period did rain fall most rapidly? Explain how you know. Rain fall most rapidly from 4:45 p.m. to 5:15 p.m. because the line is very steep. 4. Why do you think the line is horizontal between 3:30 p.m. and 4:30 p.m.? The line is horizontal between 3:30 p.m. to 4:30 p.m. since no rain fall. 5. For every inch of rain that fell here, a nearby community in the mountains received a foot and a half of snow. How many inches of snow fell in the mountain community from 5:15 PM and 7:00 PM? From 5:15 PM to 7:00 PM a total of ½ inch of rain fell. A foot is the same as 12 inches and a half of foot is 6 inches. So a foot and a half of snow is equivalent to 18 inches. The community got ½ of the 18 inches which is 9 inches or ¾ of a foot.

13 IXL skills covered in this unit: Recommended Resources R.1Coordinate graphs review - whole numbers only R.2Coordinate graphs with decimals and negative numbers R.3Graph points on a coordinate plane R.4Coordinate graphs as maps R.5Relative coordinates: follow directions R.6Quadrants S.1Read a table S.2Interpret line graphs S.3Create line graphs T.3Patterns involving addition and multiplication T.4Numeric patterns: word problems Videos Use a model to identify parts of a coordinate plane Plot points on a coordinate plane Read coordinates of a point on the coordinate plane Or use quick code: LZ1700 Or use quick code: LZ1701 Or use quick code: LZ1702