This week in math we are continuing to explore how to find percents. We have already used tape diagrams to visually represent finding percents, and now we have used double line diagrams as well. We are moving into using equations now that we can think about what percentages represent.
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This week in math we will be learning more about about ratios, proportions, and finally - percents. We will use tape diagrams to show simple relationships, or equations, between more and less amounts.
This week students are beginning Unit 4 on Proportional Ratios and Percents. We will be using fractions - especially multiplying and dividing with those fractions.
Today in class students worked on the area of circles on a grid by counting the full grid squares and estimating the partial ones.
Answers to HW: 1. The first graph shows the relationship between the diameter and area of a circle, because it is not a proportional relationship. The second graph shows the relationship between the diameter and the radius, because it is proportional and the constant of proportionality is 12. The third graph shows the relationship between the diameter and the circumference, because is it proportional and the constant of proportionality is π. 2. a. The square inside the circle has an area of 2 square units because it is made of 4 triangles each with area 1/2 square unit, and 4/2 = 2. The square outside the circle has an area of 4 square units, because 2 squared = 4. b. The square inside the circle has an area of 18 square units because 12 + 12/2 = 18 (the square inside the circle contains 12 full grid squares and 12 half grid squares). The square outside the circle has an area of 36 square units because 6 squared = 36. 3. About 4,500 in. squared If the diameter is 3 times greater, the area must be 3 squared, or 9 times greater. 4. About 1.82 meters because 100 ÷ 55 ≈ 1.82 5. About 47 cm because 15 ⋅ π ≈ 47 6. 75 cm Today in class students worked to rearrange numbers and spaces to make calculations more simple. We estimated the area of a house by breaking the house floor plan into "chunks" and finding the area of the rectangles. We then added these together to find an estimate of the area of the house. Answers to HW:
1. 20 cm squared, since the shape can be divided (vertically) into rectangles of area 2, 6, and 12 square centimeters. 2. a. You would divide the state into triangles and squares to estimate the area. b. For rectangles, parallelograms, and triangles, you need both base and height 3. 37 ⋅ 20 ⋅ π or about 2,325 in. 4. 6400 ⋅ 2 ⋅ π is about 40,000 km 5. Recipe 1: 4/12 or 1/3 tablespoons lemonade mix per cup of sparkling water Recipe 2: 4/6 or 2/3 tablespoons lemonade mix per cup of sparkling water Recipe 3: 3/5 or 0.6 tablespoons of lemonade mix per cup of sparkling water Recipe 4: 1/2 ÷ 3/4 or 2/3 tablespoon of lemonade mix per cup of sparkling water Today in class students learned more about the relationship between the circumference and the diameter of circles.
Answers to HW: 1. 405 ⋅ π or about 1,272 inches (106 feet) 2. About 169 times. There are 36 inches in a yard so 10,800 inches in 300 yards and 10,800 ÷ 64 ≈ 169. 3. Circle A: radius 2.5, diameter 5, circumference 15.7 Circle B: radius 7.6, diameter 15.2, circumference 47.7 4. 4 m. If the diameter of Circle B is 112 times larger than Circle A, its circumference must be as well. We can rewrite to calculate: (83)(32) = 4. 5. a. 10 cm b. 5 cm c. CA, AF, AD, AG, AE Today in class students reinforced their definitions of the structures of circles and that circumference/ diameter is equal to pi (3.14) the constant of proportionality.
Answers to HW: 1a. You should label the Ferris wheel with an 80 m diameter. b. 80 (3.14) = about 251 m. The gondola the riders sit in looks a little further out than the diameter, but this is close. 2a. Radius; diameter: 10 in, circumference: about 31 in b. Diameter; radius: 1.9 cm, circumference: about 12 cm c. Diameter; radius: 7 ft, circumference: about 44 ft d. Circumference; diameter: about 24 ft, radius: about 12 ft e. Radius; diameter: 14 in, circumference: about 44 in f. Diameter; radius: 5 cm, circumference: about 31 cm g. Circumference; diameter: about 60 mm, radius: about 30 mm h. Radius; diameter: 120 mm, circumference: about 380 mm 3. The two sides of the triangle each contribute 12 units and the semi-circle has a perimeter of 6 ⋅ π or about 18.84 units. So 12 + 12 + 18.84 = 42.84 units 4. Circle B is bigger. There are 12 inches in 1 foot. The circumference of Circle A is about 95.8 cm because 1 ft ⋅ 12 in ⋅ 2.54 cm ⋅ π ≈ 95.8. The circumference of Circle B is 100 cm because there are 100 cm in 1 m. 5. 72/ 3.14 = about 23 inches Today in class students learned that the diameter and the circumference of circles are proportionate. The constant of proportionality is 3.14.
Answers to HW: 1. Diego's flying disc measurements are way off; the circumference should be a little more than 3 times the diameter. 2. Using 3.14 for the constant of proportionality...and rounding your answers.... object diameter circumferencehula hoop 35 in. 110 in. circle pond 177 ft 556 ft magnify glass 5.2 cm 16 cm car tire 22.8 in. 71.6 in. 3.a. AC, AD, AB, AE, or AG; 7.5 cm b. CD; 15 cm 4. a. plot the points on the graph b. No, the relationship is not proportional because the line does not go through the origin. 5.a. Lin - 3 cups, Noah - 4 cups b. Lin - 10 cups, Noah - 7 1/2 cups Today in class students checked their understanding of what makes a circle a circle, and defined key terms about the features of circles: diameter, radius, center, and circumference.
Answers to HW: 1. You can use a compass, or trace around a cylinder at home. 2. a. line segments that go through the center of the circle are AE and DG b. line segments that start from the center and go to the edge of the circle are AH, DH, EH, and GH 3. The diameters should have the same length regardless of where you measure as long as you go through the center every time, round your answer the same way every time, and have a way to draw/trace a perfect circle.4. 2.5 cups of sugar; 10 (1/4) = 2.5 5. 4 is the constant of proportionality Today in class students began Unit 3 Measuring Circles. We got a new workbook today, and determined that the perimeter of a square is proportional to the diagonal line across the square. The area of the square is NOT proportional to the diagonal of the square. It was good practice to graph measured data and determine proportionality.
Answers to HW: 1. 6.3 cm; the perimeter is 9 (2.8) = 25.2 cm, and 25.2 cm / 4 sides = 6.3 cm 2. B and C 3. Nope. Diego needs to realize that measurements are not always perfect, and the line could still be straight and go through the origin, so it's possibly proportional. 4. a. about 380 gallons b. after about 35 minutes c. about 15 5. a. Tyler ran 2/100 of a mile further b. Tyler had a pace of 8.33333 or 8 1/3 minutes per mile; Elena had a pace of 8.5 minutes per mile (see it in the equation?) c. Tyler ran faster because it took him less time to run 1 mile than Elena. |
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