1. 7 th and 8 th Grade Mathematics Curriculum Supports Eric Shippee College of William and Mary Alfreda Jernigan Norfolk Public Schools

2. Perimeter, Area, Volume, and Surface Area

3. NCTM Focal Points In grade 6 students should: solve problems involving area and volume to extend grade 5 work and provide a context for equations

4. NCTM Focal Points In grade 7 students should: develop an understanding of and using formulas to determine surface areas and volumes of three- dimensional shapes connect their work on proportionality with their work on area and volume by investigating similar objects.

5. NCTM Focal Points In grade 7 students should: develop an understanding of and using formulas to determine surface areas and volumes of three- dimensional shapes Students connect their work on proportionality with their work on area and volume by investigating similar objects. They understand that if a scale factor describes how corresponding lengths in two similar objects are related, then the square of the scale factor describes how corresponding areas are related, and the cube of the scale factor describes how corresponding volumes are related.

6. The Perimeter is 24 Inches. What’s the Area? Working in groups of two or three, cut off a strip of adding machine paper that is at least 24 inches long. Glue it together so that it forms a paper collar that is about 24.1 inches around. Count out 38 one-inch cubes. Using as many of these cubes as needed, construct rectangular arrays that fill the paper collar you just made. Record the dimensions of each array and record its area and perimeter. Note any patterns you find.

7. The Perimeter is 24 Inches. What’s the Area? LengthWidth PerimeterArea 1 2 6 5 10 11 3 4 24 6 7 8 9 20 27 32 35 36

8. The Area is 24 Inches. What’s the Perimeter? Working in groups of two or three, count out 24 one-inch cubes. Form these 24 cubes into a rectangular array, using all 24 cubes. Record the dimensions of each array and record its area and perimeter. Note any patterns you find.

9. The Area is 24 Square Inches. What’s the Perimeter? LengthWidth PerimeterArea 1 212 24 3 228 28 50 24 4 620 24

10. Let’s try surface area.

11. Scale FactorLengthWidthHeight Surface Area in one-inch squares Starting prism Doubling the width Tripling the width Doubling the height Tripling the height 232 Counted 32 one-inch squares 2 62 2 12 + 12 + 4 + 4 + 12 + 12 = 56 one-inch squares 92 2(lw+lh+wh) = 2(2*9+2*2+9*2) = 80 one-inch squares 234 6 + 6 + 8 + 8 + 12 + 12 = 52 one-inch squares 2(lw+lh+wh) = 2(2*3+2*6+3*6) = 72 one-inch squares 236

12. Conclusion When we doubled or tripled one measurement, it increases the total surface area, but we did not see any constant change. There is no direct relationship to changing one measured attribute to its changing surface area.

13. Now that we have calculated surface area, let’s try volume.

14. Scale FactorLengthWidthHeightVolume in one-inch cubes Starting prism Doubling the height Tripling the height Doubling the width Tripling the width 232Counted 12 one-inch cubes 2 34 2 12 cubes + 12 cubes = 24 cubes 2 layers = 2 * 12 = 24 cubes 36 12 + 12 + 12 = 36 cubes 3 layers = 3 * 12 = 36 cubes 262 12 cubes + 12 cubes = 24 cubes 2 vert. layers = 2 * 12 = 24 cubes 12 + 12 + 12 = 36 cubes 3 vert. layers = 3 * 12 = 36 cubes 292

15. Conclusion When we doubled or tripled one measurement it doubled, or tripled the volume. So if we know the volume of the first prism we can multiply its volume by whatever scale factor we are increasing one measurement by to find out how much the volume of a new prism will be. There is a direct relationship to changing one measured attribute to its change in Volume.

16. What about other shapes?

17. Volume = (8*6*5) divided by 3 = 80 if we double any of the measurements we double the volume Volume (16*6*5) divided by 3 = 160 Volume (8*12*5) divided by 3 = 160 Volume (8*6*10) divided by 3 = 160

18. Surface Area = 4(1/2bh) + lw SA = 2( ½ * 6 * 5.83 ) + 2 ( ½ * 8 * 6.40 ) + 8 * 4 = 118.18 If we double the length SA = 2( ½ *6* 5.83 ) + 2 ( ½ * 16* 9.43 ) + 16 * 6 = 281.86 If we double the width SA = 2( ½ * 12 * 7.8 ) + 2 ( ½ * 8 * 6.40 ) + 8 * 12 = 240.8

19. Height = 2 in Diameter = 1 in Volume = 1.57 cu in Surface Area = 7.85 sq in If you double the height Height = 4 in Diameter = 1 in Volume = 3.14 cu in Surface Area = 14.13 sq in Volume = r 2 h Surface Area = 2( r 2 ) + 2 r h

20. Height = 2 in Diameter = 2 in Volume = 6.28 cu in Surface Area = 18.84 sq in If you double the diameter Volume = r 2 h Surface Area = 2( r 2 ) + 2 r h

21. Area of Circles

24. Shifting Gears

25. SOL 7.4 The student will solve single-step and multistep practical problems, using proportional reasoning. Write proportions that represent equivalent relationships between two sets. Solve a proportion to find a missing term. Apply proportions to convert units of measurement between the U.S. Customary System and the metric system. Calculators may be used. Apply proportions to solve practical problems, including scale drawings. Scale factors shall have denominators no greater than 12 and decimals no less than tenths. Calculators may be used. Using 10% as a benchmark, mentally compute 5%,10%, 15%, or 20% in a practical situation such as tips, tax and discounts. Solve problems involving tips, tax, and discounts.

26. Proportional Reasoning Determine whether two quantities are in a proportional relationship, e.g, by testing for an equivalent ratios in a table or graphing in a coordinate plane (does the graph go the through the origin) Identify the constant proportionality (unit rate) in table, graphs, equations, diagrams, and verbal description of proportional relationships

27. Grab A Handful Reach into the bag or box of color tiles and grab one handful of color tiles! Describe your handful of color tiles using ratios. Determine how many of each color you would have if you grabbed another handful (exact). What about four handfuls? What if you had …..?

28. Grab A Handful How did you organize your thinking? ▫Organize your thinking in table format? ▫What do you notice? ▫How could the table help you determine the number of each color of 8 handfuls?

29. Fraction Riddle Create a quadrilateral using the following clues: ▫½ green ▫ ⅓ red ▫Remainder a color of your choice!

30. Fraction Riddle Create a quadrilateral using the following clues: 40% yellow 2 / 5 green 1 / 10 red 10% blue What if…?

31. Fraction Riddle Create your own riddle Use a minimum of 16 tiles Use at least three colors Describe your riddle using percents and fractions What do we know about student thinking?

32. Fraction Strips How can we use fraction strips to make sense of percents? Identify the fraction pieces Identify percents What represents 100%? What about…….?

33. Transitioning to Models Represent 100% Show me…..10%, 20%, 5%, 1%, 50% Using your models or fraction strips find 25% of 78

34. Method Drawings and Percents Find 20% of 25. Find 10% of 120 12 is what percent of 36? 45 is what percent of 180?

35. Method Drawings and Percents 21 is 75% of what number? 12 is 66. 6…% of what number?

36. Where is the context? Create a scenario for each of the following Find 5% of 25. Find 10% of 120 12 is what percent of 36? 45 is what percent of 180? 21 is 75% of what number? 12 is 25% of what number?

37. Percent of Change and Model Drawings The price of gas rose from $1.60 per gallon in January to $2.00 per gallon in April. What was the percent increase?

38. $1.60 100% Model Drawing

39. $1.60 100% 50% Model Drawing

40. $1.60 100% 50% Model Drawing.80

41. $1.60 100% 50% Model Drawing.40 25%

42. $1.60 100% Model Drawing $2.00 25%.40

43. On your own! Justin wants to buy a wristwatch that normally sells for $60.00. If the wristwatch goes on sale for $45.00, what is the percent decrease in the price of the wristwatch?

44. Making Model Drawings a Reality What supports are needed? What resources are needed? Where do you find the questions?

45. Contact Information The power point will be available on our website http://tidewaterteam.wm.edu. http://tidewaterteam.wm.edu Eric Shippee: ewship@wm.eduewship@wm.edu Alfreda Jernigan ajerniga@nps.k12.va.usajerniga@nps.k12.va.us Thank you and have a great day!