ME7-18, ME7-19, ME7-20, ME7-21

Transcription

1 1 Measurement Grade 7 Part 1 Introduction In this unit, students will determine the relationships between units of length, area, volume, and capacity. They will develop and use formulas for the areas of parallelograms, triangles, trapezoids, circles, and composite shapes and formulas for the perimeters of circles. Students will also develop an understanding of ratios and rates and solve problems involving ratios and rates. Questions for Extra Practice Many lessons include questions for extra practice. You can write these questions on the board or photocopy them onto transparencies and use an overhead projector to display them. Summary BLMs Area formulas used in the unit are summarized on BLMs, for easy reference. This chart lists the summary BLMs available and the lesson(s) they relate to. Summary BLM Area of Parallelogram and Triangle (p XXX) Area of Trapezoid (p XXX) Circles Lesson(s) ME7-4, ME7-5, ME7-6 ME7-9 ME7-18, ME7-19, ME7-20, ME7-21 Curriculum Differences Lessons ME7-4 to ME7-6 address WNCP core curriculum expectation 7SS2 (area of triangles and parallelograms). The material covered in these lessons is on the Grade 6 curriculum in Ontario. Teachers in Ontario may choose to condense these lessons drastically: review the formulas for the area of a parallelogram and the area of a triangle, as well as methods for finding the height of a parallelogram or a triangle, and then use the worksheets for diagnostic purposes. Students in Ontario need to demonstrate their understanding of the concepts in ME7-4 to ME7-6 because they will use them when working with trapezoids and composite shapes in lessons ME7-8 to ME7-10. Lessons ME7-8 to ME7-10 address curriculum expectations which do not appear on the WNCP curriculum and so are optional for those who follow it. Lesson(s) Material Ontario WNCP ME7-1 Changing units of length, mass, capacity 7m35 diagnostic, used throughout the unit ME7-2 Dividing decimals 7m18, 7m19 7N2 ME7-3 Changing units of area 7m36 7N2 ME7-4, ME7-5, ME7-6 Area of parallelogram Area of triangle Diagnostic, used in ME7-8, ME7-9, ME7-10, ME7-20, ME7-21 7SS2 ME7-7 Using formulas 7m1, 7m3, 7m4, 7m5, 7m6, 7m7 7SS2, [C, CN, PS, R, V] ME7-8, ME7-9, ME7-10 Area of trapezoids Area of composite shapes 7m37, 7m38, 7m39 optional

2 2 ME7-1 Changing Units Workbook pages Goals Students will convert measurements of mass, length, and capacity. Prior Knowledge Required Is familiar with decimals to ten thousandths Can multiply or divide decimals by 10, 100, and Is familiar with units of measurement for mass, capacity, and length Knows the relationships between the units in the vocabulary list Vocabulary metre, kilometre, millimetre, centimetre litre, millilitre kilogram, gram, milligram Curriculum Expectations Ontario: 5m38; 6m34; 7m5, 7m35 WNCP: 5SS2, [CN] (diagnostic only) Make sure your students know how to multiply and divide decimals by 10, 100, and by shifting the decimal point. You can use Workbook Question 1 as a diagnostic test. If necessary, review NS7-45 to NS7-48. Review relationships. Review the relationships between units of length (metre, kilometre, millimetre), capacity (litre and millilitre) and mass (gram, kilogram, milligram). Discuss the meaning of the prefixes kilo and milli (both mean 1 000, but kilo is used for larger units and milli is used for smaller units). Do we multiply or divide? Draw the following diagram on the board: km ASK: Which unit is the largest? Which unit is the smallest? When we have a measurement in metres, say metres, how many kilometres is that? (2 km) Do we need more or fewer of the new units? (fewer) How many times fewer? (1 000 times fewer) To get the measurement in kilometres from a measurement in metres, should we multiply by or divide by 1 000? (divide) Why? (The new unit is larger than the old unit, so we need fewer of the new units.) Add an arrow from m to km on the diagram and label the division by m mm Discuss the conversion from kilometres to metres, then from metres to millimetres and vice versa, adding more arrows to the diagram. The finished diagram should look like this: Create similar diagrams for the units of mass and capacity. km m mm 1 000

3 3 Choose pairs of units and ask students if they need to multiply or divide by to convert a measurement from one to the other. EXAMPLES: from m to km, from kg to g, from mm to m. Convert between units by multiplying or dividing by See Questions 2 and 3. Extra practice: Change the units: a) 245 m = km b) 2.67 L = ml c) 0.76 km = m d) 345 mg = g e) 36.9 g = kg f) m = mm g) ml = L h) mm = m i) g = mg j) 0.09 kg = g Review the relationships between m, cm, and mm. Draw the diagram at right on the board and have students add arrows and multiplication or division prompts, explaining why they should multiply or divide and by how much. If your students are familiar with decimetres (dm), add them to the empty step. m cm mm Ask students to tell whether they need to multiply or divide, and by how much, to convert measurements from: a) m to cm b) cm to mm c) cm to m d) mm to cm BONUS e) mm to km f) km to cm SAMPLE ANSWER: a) We need more cm than m (and there are 2 steps from one to the other in the diagram), so we have to multiply by 100 to convert a measurement in m to a measurement in cm. Now have students perform some conversions between these units, as in Question 5. Extra practice: Change the units: a) 240 m = cm b) 2.61 m = cm c) 0.78 cm = m d) 38.5 cm = m BONUS e) 36.9 km = cm f) cm = km g) mm = km Word problems that require conversion of units. Students can solve the following problems and other similar problems that you create. PE Connecting 1. Order the fern leaves from longest to shortest: oak fern 18 cm ostrich fern 1.5 m bracken fern 90 cm royal fern 1.30 m 2. The CN Tower in Toronto, Ontario, is about 533 m high. A school ruler is 31 cm long. About how many rulers long is the CN Tower? 3. A bull elephant weighs pounds. If 1 pound = 454 g, what is the mass of the elephant in kilograms? 4. A small carton holds 250 ml of milk. A vat contains 500 L of milk. How many small cartons can be filled from that vat? 5. What is heavier: a pair of male mountain gorillas weighing 150 kg each or ants weighing 3 mg each? How many ants will weigh the same as a pair of male gorillas?

4 4 BONUS There are about ants in the world. There are only about 720 mountain gorillas. Pretend that all ants weigh 3 mg and all gorillas weigh 120 kg. What weighs more: all the gorillas in the world or all the ants? ANSWERS: 1. ostrich fern, royal fern, bracken fern, oak fern 2. Since 10 rulers are about 3 m long, = 177 2/3, or about , so the CN Tower is about rulers long. A more exact answer is given by converting the height of the tower from metres to centimetres and dividing by the length of a ruler: rulers kg cartons ants weigh 30 kg, so a pair of gorillas is heavier. It will take ants to match the weight of a pair of male gorillas. BONUS The ants weigh 300 billion kg = kg, whereas the gorillas weigh only kg. Extensions 1. Write 1.25 m on the board and underline the 1. What quantity does the underlined digit represent? (1 m) Repeat with the other digits. (2 dm, 5 cm) What is the measurement of the underlined digit? a) $2.57 b) 2.57 m c) 2.57 dm d) $2.57 e) 2.57 m f) 2.57 dm ANSWERS: a) dimes b) dm c) cm d) pennies e) cm f) mm 2. Which is a greater length: mm or 3.8 km? ANSWER: 3.8 km 3. John has a strip of paper 1 dm long. He folds the strip of paper so that it has a crease in its centre. What measurements can John make in centimetres using the strip? ANSWER: measurements that are any multiple of 5 cm 4. Write a measurement in decimetres that is between these measurements: 1 3 a) 320 mm and 437 mm b) 507 mm and 622 mm c) 1 m and 1 m 2 4 d) 3 cm and 4 cm e) 47 mm and 48 mm f) mm and mm g) 2.8 m and 2.9 m SAMPLE ANSWERS: a) 4 dm b) 6 dm c) 16 or 17 dm d) 0.35 dm e) dm f) 53 dm g) 28.7 dm 5. Which insect travels faster: an insect moving 42 mm per second on an insect moving 24 m per hour? ANSWER: 24 m/hr = mm / s = mm/s = about 6.67 mm/s, so the first insect is faster. Another way to see this is: The first insect is moving at 42 mm/s = m/s. There are seconds in an hour, so in an hour this insects moves m, meaning its speed is m/hr = m/hr, which is more than 24 m/hr. Again, the first insect is faster.

5 5 ME7-2 Changing Units to Divide Decimals Workbook page 168 Goals Students will use their knowledge of units of length to divide decimals by decimals. Prior Knowledge Required Is familiar with decimals to ten thousandths Can multiply or divide decimals by 10, 100, and Is familiar with units of measurement of length Knows the relationships between the units in the vocabulary list Vocabulary metre, kilometre, millimetre, centimetre Curriculum Expectations Ontario: 7m2, 7m5, 7m18, 7m19 WNCP: 7N2, [CN, R] Review the relationships between units of length: m, cm, mm, and km. Multiplying or dividing the dividend and the divisor by the same number does not change the answer. Draw a line 80 cm long and a line 20 cm long on the board. Write the measurements beside the lines. ASK: How many times will the smaller line fit into the larger line? How do you know? How can we check? Invite volunteers to check using the strategies they suggest. Have materials such as paper, scissors, string, and a measuring tape on hand for the students to use. If no one suggests it, create a benchmark 20 cm long and check how many times it fits, marking where it ends each time on the larger line. Invite a volunteer to write the division statement showing how many times the smaller line will fit into the larger line. (80 cm 20 cm = 4) Point out that the number 4 does not have a measurement because it is not a measure of length, it is a measure of how many times the smaller line fits into the larger line. Ask another volunteer to rewrite the measurements of the line segments in metres. ASK: Does rewriting the measurements change the fact that the small line segments fits into the large line segment 4 times? (no) How does the division statement change? (.8 m.2 m = 4) Ask students to think of the following numbers as measurements in metres and convert them to centimetres (multiply by 100) to find the quotient, as in Questions 1 and 2. PE Connecting a) b) c) d) ANSWERS: a) = = 41 b) 9 c) 12 d) 8 Now ask students to think of the numbers in as measurements in centimetres and convert them to millimetres (multiply by 10). Will this method produce the same quotient? (yes) How do you know? (because multiplying and dividing both numbers by the same number doesn t change the quotient) Have students practise finding quotients using this method (see Question 3). Repeat with measurements in metres and convert them to millimetres (multiply by 1 000). EXAMPLES:

6 6 Now have students solve division problems by changing both numbers to whole numbers. Students will need to decide whether to multiply by 10, 100, or EXAMPLES: ASK: Do you think it is more useful to convert from bigger units to smaller units or vice versa? Why? (from bigger to smaller is more convenient smaller units means more whole numbers, which are easier to work with) You can have students solve the following problem using metres and centimetres to see this: How many times will a string that is 60 cm long fit into a string that is 120 cm long? ANSWER: 120 cm 60 cm = 2 and 1.2 m 0.6 m = 2, but it s easier to solve the first division problem in your head. (To do the second one your head, you would multiply by 10.) The need to multiply the divisor and the dividend by the same number. Present the following problem: To 7.29 by 0.9, Juno converts both terms to whole numbers as follows: = ( ) (0.9 10) = = 81 Is she correct? Ask students to pretend that Juno s numbers are measurements in metres. About how many lines 0.9 m long will fit into a line 7.29 m long? How do you know without dividing? (0.9 is slightly less than 1 and 7.29 is slightly more than 7, so the answer should be around 8) What was Juno s mistake? PA 7m2, [R] Have students explain why they need to multiply both the dividend and the divisor by the same number. SAMPLE ANSWERS: you have to convert to the same unit; by multiplying by the same number you use equivalent ratios; multiplying a dividend by a number is equivalent to multiplying the quotient by the same number, and multiplying the divisor by a number is equivalent to dividing the quotient by the same number, so to keep the quotient the same you need to multiply and divide by the same number.

7 7 ME7-3 Changing Units of Area Workbook page 169 Goals Students will convert measurements of area. Prior Knowledge Required Is familiar with decimals to ten thousandths Can multiply or divide decimals by 10, 100, 1 000, and Is familiar with units of measurement for mass, capacity, and length Can convert measurements in metres to centimetres and vice versa Can find the area of rectangles and squares Vocabulary metre, centimetre metre squared (m 2 ) centimetre squared (cm 2 ) Materials BLM Square Metre (p 1) BLM Area and Perimeter Review (p 2) Curriculum Expectations Ontario: 6m39; 7m2, 7m3, 7m36 WNCP: 7N2, [R, C] Make sure your students know how to multiply and divide decimals by 100 and by shifting the decimal point. You can use Question 1 as a diagnostic test. If necessary, review NS7-45 to NS7-48. Students who need to review the area of rectangles can complete BLM Area and Perimeter Review. Review the names of the units of area: metres squared (m 2 ) and centimetres squared (cm 2 ). 1 m 2 = cm 2. Draw a very large square on the board and mark its sides as 1 m. ASK: What is the area of this square? (1 m 2 ) Write the area below the square. Divide the square into a grid, and ASK: How many squares are in the large square? (100) How do you know? ASK: How many centimetres are in a metre? Change the markings on the sides to show 1 m = 100 cm. Ask whether the smaller squares are 1 cm 1 cm. (no) What is the side length of each smaller square? (10 cm) What do you have to do to get squares 1 cm 1 cm? (divide the smaller squares into another grid) Show the division on one of the squares. (As an alternative, you can photocopy BLM Square Metre on a transparency and project it on the board; cover the labels to start and uncover them as you go.) How many small squares are in the medium square? (10 10 = 100) How do you know? What is the area of the medium square? (100 cm 2 ) How many small squares will fit into the largest square? (10 000) What is the area of the largest square in cm 2? ( cm 2 ) How do you know? (There are 100 medium

8 8 squares in the large square, and 100 small squares can fit in each medium square, so in total there will be cm 2 = cm 2 ). Write the equation for the area of the largest square on the board. Find the area a different way and compare the two methods. ASK: How many small squares fit along the side of the large square? (100) Why? (because 1 m = 100 cm) How do you find the area of a square? What would that give for the large square? (100 cm 100 cm = cm 2 ) What is the area of the large square in metres? Ask students to write the equality for the area: 1 m 2 = 1 m 1 m = 100 cm 100 cm = cm 2. Now compare these equalities: cm 2 = cm 2 and 100 cm 100 cm = cm 2. How are they the same? How are they different? In the first equality, the first number does not have units it is the number of times the medium square is taken and the second number is the area of the medium square. In the second equality, both numbers are length measurements, so they are measured in centimetres and produce centimetres squared when multiplied. PE Reflecting on other ways to solve a problem Converting between m 2 and cm 2. Draw a rectangle on the board and mark the sides as 120 cm and 200 cm. Ask students to find the area of the rectangle in cm 2. ( cm 2 ) Then ask them to convert the lengths to metres and to find the area in m 2. (1.2 2 = 2.4 m 2 ) Ask students to compare the answers. Which measurement has a larger numeral? Which measurement has a larger unit? Do you need more or fewer cm 2 than m 2? To get the measurement in cm 2, what do you have to do to the measurement in m 2? (multiply by ) Why do you multiply? (Because the new unit is smaller, we need more units, so we have to multiply by ) Repeat with a rectangle 60 cm 70 cm, and again with a rectangle 25 cm 40 cm. Write several measurements on the board and ask students to convert the units between cm 2 and m 2. Practise converting measurements of area. At first, ask students to explain why they need to multiply or to divide by , as in Question 3, then have students just do the conversion, as in Question 4. PA 7m2, [C] Extra practice: Change the units: a) 23 m 2 = cm 2 b) 2.61 m 2 = cm 2 c) cm 2 = m 2 d) 38.5 cm 2 = m 2 e) m 2 = cm 2 f) 0.54 m 2 = cm 2 g) cm 2 = m 2 h) 0.5 cm 2 = m 2 i) m 2 = cm 2 j) m 2 = cm 2 k) 456 cm 2 = m 2 l) 4.72 cm 2 = m 2 Extensions 1. a) Write each measurement as a fraction (in lowest terms) of a metre. i) 20 cm ii) 50 cm iii) 75 cm iv) 70 cm v) 40 cm vi) 25 cm vii) 180 cm viii) 300 cm b) Write each measurement as a fraction of a metre to decide which areas represent more than 1 m 2. EXAMPLE: 20 cm 300 cm = 1 5 m 3 m = 3 5 m2 < 1 m 2 i) 40 cm 200 cm ii) 60 cm 300 cm iii) 80 cm 150 cm

9 9 2. What has larger area? Which has larger perimeter? a) a rectangle 30 cm 1.5 m or a rectangle 920 mm 45 cm b) a square 1 m 1 m or a rectangle 70 cm 130 cm c) a square 1 m 1 m or a rectangle 90 cm 110 cm d) a square 1 m 1 m or a rectangle 99 cm 101 cm ANSWER: a) = 4500 cm 2 and = 4140 cm 2, so the first rectangle is larger b) the rectangle has area =.91 m 2, so the square is larger c) the rectangle has area =.99 m 2, so the square is larger d) the rectangle has area =.9999 m 2, so the square is larger In a), the first rectangle has a larger perimeter. All three rectangles in b), c), and d) have the same perimeter as the 1 m 1 m square, so the square has the largest area of all rectangles with the same perimeters. PE Connecting

10 10 ME7-4 Area of Parallelograms Workbook pages Goals Students will develop the formula for the area of parallelograms. Prior Knowledge Required Is familiar with decimals to ten thousandths Can multiply or divide decimals by multiples of 10 Can convert units of length and area Knows the formula for the area of a rectangle Can make a simple geometric sketch Can draw a perpendicular to a line through a point Can draw parallel lines Vocabulary metre, kilometre, millimetre, centimetre base height area Materials a large, flexible rectangle made from cardboard strips and paper fasteners (see picture below) protractors or set squares paper parallelograms for every student BLM Parallelograms (p 4) BLM Area of Parallelograms (p 5) BLM Distance Between Parallel Lines (p 3) Curriculum Expectations Ontario: 6m37; 7m3, 7m5, 7m7 (diagnostic only) WNCP: 7SS2, [C, CN, ME] Parallelograms are not defined by length and width. Draw several parallelograms, like the ones shown at right, on the board. ASK: Can you find the length and width of these shapes? Discuss with the class how they might first identify the length and the width. Point out that the length of a side cannot be considered the length of the shape, since the shape is actually longer than the side itself. Is the longest diagonal the length of the shape? Should length be perpendicular to width? These questions, and the resulting discussion, should

11 11 make it clear that parallelograms don t have an easily identifiable length and width, as do rectangles. As a result, we use different concepts to define a parallelogram. The area of a parallelogram is not defined by side lengths. Have on hand a large, flexible rectangle made from cardboard strips and paper fasteners. ASK: Do you think two parallelograms with the same side lengths have the same area? How could we find out? Take various answers, then hold up the rectangle and begin to deform it into a shape that has area close to zero. Point out that you aren t changing the length of the sides, you re just moving them and creating different parallelograms. ASK: Have we answered the question? How? (Students should see, qualitatively if not quantitatively, that parallelograms with the same side lengths can have different areas.) Emphasize that the length of adjacent sides does not define the area of a parallelogram the way it does a rectangle. Review the necessary geometry concepts. Review drawing the perpendicular to a line through a point (G7-4). Students can use protractors or set squares to draw the perpendiculars. For practice and as a lead-in to the concept of height, ask students to draw a pair of slant (neither horizontal, nor vertical) parallel lines (review drawing parallel lines from G7-6 if necessary) and then draw several perpendiculars to both lines. Then ask students to measure the distance between the lines along the perpendiculars. Ask students to explain why the distance does not depend on where the perpendicular is drawn. (If necessary, review this concept using Activity 2 of G7-6 or BLM Distance Between Parallel Lines.) Introduce base and height. Take any parallelogram that you have drawn on the board and mark the base and height. Point out that you can choose any side of a parallelogram to be the base (and the line opposite is also considered a base). Once a pair of bases is chosen, the distance between the bases (measured along the line perpendicular to the bases) is fixed. Point out that base is short for the length of the base and height refers to both the distance between the bases and the perpendicular it is measured along. ASK: Does the height of a parallelogram depend on where it is measured? (no) Why or why not? (because the lines are parallel and the height is the distance between the parallel lines, so it is the same everywhere) PE Connecting Turning parallelograms into rectangles that have the same area. Give your students at least two paper parallelograms. (You can create your own or use parallelograms 1 and 2 from BLM Parallelograms.) Ask students to measure the sides, to choose a pair of bases, and to find the heights. To find the heights, students can either fold the parallelograms to create a perpendicular folding each base back onto itself will create a crease perpendicular to both bases or use protractors or set squares. ASK: Are there parallelograms where you could not find the height for the base you chose? Why did that happen? Could you find a height for the other base? (In Parallelogram 1, if the shorter side is chosen as the base, the height falls outside the parallelogram. However, choosing the longer side as a base allows finding the height easily.) ASK: How could you cut a parallelogram into two pieces and rearrange it to create a rectangle? Ask students to think about this, to discuss with a partner, and to try out their ideas. Give everyone a chance to find the answer, and, if necessary, invite one or more students to explain it to the class. ASK: Are the sides of the rectangles the same length as the sides of the parallelogram? How many sides preserve

12 12 their length after being rearranged? (2 the bases) What about the height of the parallelogram where do you see that in the rectangle? (the height becomes the width of the rectangle) Students will discover that if they know the height and the base of a parallelogram, they can create a rectangle by shifting part of the parallelogram parallel to the base, as in Question 1. ASK: Did the area of the parallelogram change when you cut and rearranged the pieces? How can we calculate the area of the parallelogram using what we ve learned? (The width of the rectangle is the same as the height of the initial parallelogram, the length of the rectangle is the same as the base of the parallelogram, and the area did not change. The area of a rectangle is length width, so the area of the parallelogram is base height.) Finding the area of parallelograms. Have students find the area of these parallelograms: a) base 3 cm, height 4 cm b) base 3 m, height 4.5 m c) base 3.5 km, height 2.3 km d) base 3.4 cm, height 4.5 cm e) base 4 cm, height 2.5 cm f) base m, height 0.3 m SAMPLE ANSWER: c) 3.5 km 2.3 km = 8.05 km 2 Finding areas when measurements are given in different units. Remind students that we must have the same units for base and height when multiplying. Have students find the areas of parallelograms with measurements given in different units two ways: first by converting to the smaller unit, then by converting to the larger unit. Ask students to check whether the areas are the same both ways by converting units of area. Use the questions below and make up more if necessary. PE Reflecting on the reasonableness of the answer 1. Find the area of these parallelograms: a) base 30 cm, height 0.4 m b) base 3 m, height 45 cm c) base 1.5 m, height 23 cm d) base 7 cm, height 0.04 m e) base 43.5 cm, height 1.1 m f) base.035 m, height 2.4 cm g) base 8.17 m, height 410 cm h) base 32 cm, height 0.5 m i) base 3.5 m, height cm SAMPLE SOLUTION: a) 30 cm 40 cm = cm 2 and 0.3 m 0.4 m = 0.12 m 2 Converting from cm 2 to m 2, cm 2 = 0.12 m 2 so these answers are the same. ANSWERS (in m 2 ): b) 1.35 m 2 c) m 2 d) m 2 e) m 2 f) m 2 g) m 2 h) 0.16 m 2 i) m 2 2. Find the area of the following parallelograms in cm 2 and in m 2. Which unit is more convenient in each case? a) base 1 m, height 1 cm b) base 1 km, height 1 m c) base 1 km, height 1 cm ANSWERS: a) 100 cm 2 = 0.01 m 2, cm 2 is more convenient b) 1000 m 2 = cm 2, m 2 is more convenient c) cm 2 = 10 m 2, m 2 is more convenient Students should use the formula for the area of a parallelogram in their answers to Question 8. PA [C], 7m7 SAMPLE ANSWER for a): Area of parallelogram = base height. The parallelograms have the same height. The base of one is three times larger than the base of the other, so the area of the larger parallelogram is 3 smaller base height, so the area of the larger parallelogram is three times the area of the smaller parallelogram.

13 13 Finding areas of parallelograms in different ways. Ask students to draw a parallelogram and to find its area in two ways, using different sides as bases. Should the two methods give the same answer? Why? (Yes, because the area is the same no matter how you measure it.) Are the two answers the same? If not, what could make the answers different? (imprecise measurements, a mistake in drawing the heights) PE Reflecting on the reasonableness of the answer Parallelograms with height falling outside the parallelogram. Give students copies of Parallelograms 3 and 4 from BLM Parallelograms. Explain that you want to find the area of these parallelograms using the shorter (thick) pair of sides as bases. Is it possible to find the height by folding? Why not? (the height falls outside the parallelogram) Ask students to trace the parallelograms in their notebooks and to extend the bases to create two longer parallel lines. Remind students that the height is the distance between parallel lines measured along a perpendicular line. Ask students to find the distance between the lines and the area of the parallelograms. Cutting and rearranging parallelograms with height outside the parallelogram to find the area. ASK: Will cutting and rearranging the parallelogram change the area? (no) Ask students to try to create a different parallelogram by cutting and rearranging the pieces of their paper parallelograms without turning the short sides into lines inside the parallelogram. Encourage multiple solutions and discuss ways to find the area as a class. In each case, ask students to measure the heights and the bases of the new parallelograms and to compare them with the height and the base of the old parallelograms that they traced in their notebooks. Students can use BLM Area of Parallelograms to consolidate their findings. Extension Two rectangles with the same area and perimeter are congruent (are exact copies of each other, have the same size and shape). Are two parallelograms with the same area and perimeter congruent? See BLM Investigating Area of Parallelograms (p 6). PE Revisiting conjectures that were true in one context

14 14 ME7-5 Area of Triangles Workbook pages Goals Students will develop and apply the formula for the area of triangles using area of rectangles. Prior Knowledge Required Is familiar with decimals to tenths Can multiply decimals or convert centimetres to millimetres Can find the area of a rectangle using a formula Can find the area of a right triangle Understands that area is additive Can identify congruent triangles Is proficient in measuring with a ruler Can identify and draw right angles using a protractor or set square Vocabulary base height area Materials paper triangles (acute scalene) for every student protractors or set squares grid paper Curriculum Expectations Ontario: 6m36, 6m37, 6m38; 7m2, 7m3 (diagnostic only) WNCP: 7SS2, [C, R, V] Review how to find the areas of a rectangle and a right triangle. Emphasize the connection between the two: a right triangle is half of a rectangle with the same length and width as the sides of the triangle adjacent to the right angle. area of triangle Splitting triangles into two right triangles. Draw several acute and obtuse triangles on a grid so that the longest side is horizontal. = area of rectangle divided by 2 This way: Not this way: Ask students to split the triangles into two right triangles. Then invite students to draw a rectangle around each triangle and to use the rectangle to find the area of

15 15 both the original triangle and the two right triangles. ASK: What fraction of the area of the big rectangle is the area of the big triangle? (half) Finding the area of obtuse triangles on a grid using subtraction of areas. Find the area of the triangle at right as a class, following the progression of steps in Workbook Question 4. PA Workbook Question 3, [R, V], 7m2 B A E D C Introduce base and height. Point out that when we split a triangle into two right triangles, we drop a perpendicular from one of the vertices to the opposite side. The perpendicular is called the height of the triangle and the side we draw the perpendicular to is called the base. Ask students to draw an acute scalene triangle and to draw a perpendicular to each of its sides. Then show the picture at right and have students explain how they could find the area of the triangle. What fraction of the rectangle is the triangle? (half) Ask students to identify the base and the height in the picture, then write the area of the rectangle in terms of this base and height. What is the area of the triangle in terms of base and height? (base height 2) Draw a right triangle and choose one of the legs (legs are sides adjacent to the right angle) as a base. Ask students to find the height. Since the triangle is a right triangle, the height is the second leg. Does the formula base height 2 produce the same answer as length width 2? (Yes: in the case of a right triangle we can talk about length and width as the length and width of the rectangle that was cut in half to make the triangle and they are the same as base and height if we regard one of the legs as the base.) Turning triangles into rectangles of the same area. See the Activity below, which can be completed with the Investigation in the Workbook (page 173). Part C of the Investigation can be used for assessment. PA 7m3, [C, R, V] Introduce the case of an obtuse triangle with one of the shorter sides chosen as the base. Ask students to suggest how to draw the height to that base. If the solution does not arise, explain that when we draw perpendiculars, they are perpendicular to the whole line containing the base, not just the line segment itself. We can draw a perpendicular to the line that contains the base by first extending the base. Draw several triangles on a grid in a variety of orientations and mark different sides as bases. Ask students to copy the triangles and to draw heights for the chosen bases using protractors or set squares. Then draw and label several triangles with heights already drawn for the students and have them identify the base in each case, as in Workbook Question 10. Activity Give your students at least two copies of an acute scalene triangle and ask students to try to cut and rearrange the each triangle to create a rectangle with the same area. Students can use tape to hold the pieces of the rectangles together. Students can discuss possible solutions in pairs. Debrief as a class. If 3 cm 4 cm 2 cm

16 16 the solutions shown in the Investigation do not arise, show them. Ask students to compare the dimensions of the resulting rectangles (they will be different) and to check that the areas of the rectangles are the same by cutting one of the rectangles in half and rearranging the pieces. Extension This extension goes with the last question on page 174 in the Workbook, where students are likely to get three different answers because the measurements are approximations. When you find the area of a triangle by drawing heights and measuring the lengths of the sides and the heights, you perform two imprecise operations: drawing a perpendicular and measuring with a ruler. Suppose the triangle on the worksheet was much larger (for example, drawn on the board) and the lengths were measured to the nearest centimetre, with area calculated to the nearest centimetre squared (cm 2 ). Are the three methods of calculating the area (using different sides as bases) more likely to give the same answer in this case than in the original question? Why or why not? ANSWER: Yes. The measurements would need to be less precise, so they would produce answers that are closer to each other.

17 17 ME7-6 Area of Triangles and Parallelograms Workbook page 175 Goals Students will develop and apply the formula for the area of triangles using parallelograms. Prior Knowledge Required Is familiar with decimals to tenths Can multiply decimals Can find the area of a parallelogram using a formula Understands that area is additive Can identify and draw congruent triangles on a grid Can convert centimetres to millimetres Vocabulary base height area Materials various paper triangles or BLM Triangles (p 7) (see Activity) grid paper Curriculum Expectations Ontario: 6m37; 7m2 (diagnostic only) WNCP: 7SS2, [R, C] Students can use cut-outs of triangles drawn on grid paper to help create the parallelograms in Workbook Questions 1 and 2. Before tackling the Bonus question, students can draw several triangles on grid paper so that one of the sides is horizontal and draw the median to the horizontal side. They can find the areas of both parts of the divided triangle, and compare them. Students will quickly see that the areas are the same. ASK: Will this always be the case for any triangle, not just triangles drawn on a grid? (yes) Ask students to justify their answer using the formula for the area of triangles. PE Justifying the solution (SAMPLE ANSWER: Let s use the horizontal sides as the base for both triangles. The bases are equal, because they are both halves of the base of the initial triangle. The triangles have a common vertex opposite the bases, so the height is the same for both triangles. Therefore area = base height 2 is the same for both triangles.) Activity Divide students into groups of three, and give each student two copies of an isosceles triangle and two copies of a scalene triangle as follows: Student 1 gets right triangles, Student 2 gets acute triangles, and Student 3 gets obtuse triangles. (Samples shapes are provided on BLM Triangles.)

18 18 Each student should try to create as many shapes as possible from two copies of the same triangle by joining the triangles along a pair of sides of the same length. Ask students to trace the shapes they make and to identify them if possible. In their groups, students should discuss the shapes they produced. How many shapes does each triangle make? Are all the shapes quadrilaterals? When is the resulting shape a triangle? What types of quadrilaterals can be created? What special features do they have? ANSWERS: It is easier to check the answers if you have the cut-outs before you. Each isosceles triangle produces 3 shapes and each scalene triangle produces 6 shapes. Student 1: The isosceles right triangles produce a right isosceles triangle, a square, or a parallelogram. The scalene right triangles can create two different isosceles triangles, two different parallelograms, a kite, or a rectangle. Student 2: The isosceles acute triangles produce a rhombus, a parallelogram, or a kite. The scalene acute triangles produce three different kites and three different parallelograms. Student 3: The isosceles obtuse triangles produce a rhombus, a parallelogram, or a shape with two pairs of equal adjacent sides and an indentation. The scalene obtuse triangles produce a kite, two shapes with two pairs of equal adjacent sides and an indentation, and three different parallelograms. All shapes for all three students have either a line of symmetry or rotational symmetry. Ask students to think about the area of the shapes produced from the same pair of triangles. ASK: What can you say about the area of the shapes you ve obtained? (the areas are the same) For each pair of triangles, which quadrilateral can you use to find the area? If there is more than one, which one is easiest to use? Have students find the areas of the parallelograms and triangles. Extensions 1. Do the triangles T 1, T 2, T 3, and T 4 all have the same area? Explain. ANSWER: Yes, because they all have the same base and the same height. 2. Find the areas of the triangles below, then fill in the table. a) b) c) d) T 1 T 2 T 3 T 4

19 19 Triangle Number of dots on the perimeter of the triangle Half the number of dots on the perimeter of the triangle Number of dots inside the triangle a) What do you notice? Hint: How can you get the last column from the two previous columns? Mathematicians have proven that the area of any polygon on dot paper can be found using the number of dots inside and on the perimeter of the polygon. Pick s theorem: Area = # of dots inside the shape + # of dots on the perimeter the triangle 2 1. Students can check Pick s theorem with simple shapes (parallelograms, rectangles, squares) on dot paper. Area

20 20 ME7-7 Problem-Solving Using a Formula Workbook pages Goals Students will develop strategies for solving problems and solve problems using formulas. Prior Knowledge Required Is familiar with decimals to tenths Can multiply decimals Can find the area of a parallelogram, rectangle, and triangle using a formula Understands that area is additive Can make a sketch Can convert units of length and area Vocabulary base height area Curriculum Expectations Ontario: 6m38; 7m1, 7m3, 7m4, 7m5, 7m6, 7m7 WNCP: 7SS2, [C, CN, PS, R, V] Discuss with students subjects and situations in which they might need to use a formula. Examples: in physics, chemistry and other sciences; to find area and volume. Have students complete the worksheets page by page. You can use the questions below as extra practice for students who work more quickly than others (to keep the class working through the questions at approximately the same pace) or for students who are struggling with a particular concept or step. PA Workbook Questions 2, 3 [C, V], 7m6 Question 6 have students explain the choice of the unit [R], 7m4 Question 12 [C, R], 7m3 Question 15 [C, CN], 7m5, 7m7 Extra practice for page 176: For each problem below, answer these questions: What do you need to find? Which formula will you use? Which of the sketches given is the most helpful for solving the problem? Explain your choice.

21 21 a) A parallelogram has base 1 m and height four times shorter than that. What is the area of a parallelogram? A B C 1 m 4 1 m 4 1 m 1 m 1 m 1 m 4 b) In triangle ABC, C = 90, AC = 4 cm, and BC = 2 cm. D is the midpoint of AC. What is the area of ABD? i) ii) iii) B B B 2 cm C 4 cm D A 2 cm C 4 cm D A 2 cm C D 4 cm A Extra practice for page 177: For each problem below: Write out the measurements you are given. Identify what you need to find. Write the formula you will use. Make a sketch for the problem and mark the information you know on the sketch. (Note: Students do not need to solve the problems.) a) A parallelogram has base 6 m and height 7 m. What is the area of the parallelogram? b) A desk is 1 m deep and 120 cm wide. What is the area of the desk? c) A flag is a rectangle 1 m long and 80 cm wide. How much cloth is needed to make the flag? d) A chevron is made of two parallelograms joined along a common base 10 cm long. Each parallelogram has height 5 cm. What is the total area of the chevron? BONUS Build a kite as follows: Take two thin strips of wood, one 40 cm long and the other 25 cm long. Nail the wooden strips together perpendicularly at the centre of the shorter one to make a cross. Join the adjacent vertices of the cross with more strips of wood. Then nail a piece of silk to the wooden strips along the perimeter of your structure. How much silk is needed to build the kite? Extra practice for page 178: Len decides to rearrange the parallelogram in Question 5 into a rectangle: a) Write the formula Len uses to find the area. b) Finish the sketch using what you are given in the problem. c) Which value is not given directly in the problem?

22 22 d) Fill in the blanks: length of rectangle = width of rectangle: e) Find the area of the parallelogram as the area of the new rectangle. Is the answer the same as in Question 5? ANSWERS: a) length width c) length of rectangle d) length = 15 cm, width = 10 cm e) 150 cm 2 Extra practice for page 179: Solve the problems given as extra practice for page 177. ANSWERS: a) 42 m 2 b) 1.2 m 2 c) 0.8 m 2 d)100 cm 2 BONUS 0.05 m 2 = 500 cm 2 Extra practice for page 180: Make a sketch for each problem. Mark the information you know on the sketch. Mark x for the piece of information you do not know. Write and solve the equation for x. a) The area of a rectangular sheet of foil is 25 m 2. The width of the sheet is 75 cm. What is the length of the sheet? b) A parallelogram is 25 cm high and its area is 1.5 m 2. What is the length of the base of the parallelogram? c) An isosceles triangle has base 5 cm and area 30 cm. What is the height of the triangle? ANSWERS: a) m b) 6 m c) 12 cm Extra practice for page 181: Decide whether what you need to find is given by a formula or is part of a related formula. Write the formula, then use it to solve the problem. a) A can of paint can cover 10 m 2 of area. The paint is used to draw a solid centre line on the road. The line is 10 cm wide. How long is the line? b) A traffic island is a parallelogram with a distance of 3 m between sides that are each 5 m long. What is the area of the traffic island? c) A parking spot is a parallelogram with longer side 5.5 m and area 16.5 m 2. What is the width of the parking spot? BONUS The Canada Post logo is made up of a parallelogram and a triangle that have the same height and share a base. The base is 16 mm long, and the total area of the logo is 168 mm 2. What is the area of the triangle? What is the height of the triangle and the parallelogram? ANSWERS: a) The length is part of the formula length width = area. Let x be the length in metres. The width is 0.1 m. Then the formula gives x 0.1 = 10, so x = 100 m. b) Area is given by the formula base height = area; area = 15 m 2. c) The width of the parking spot is the height of a parallelogram, part of the formula base height = area; 5.5x = 16.5, so width = 3 m.

23 23 BONUS The triangle is half the parallelogram, so the area of the triangle is 1/3 of the area of the logo, or = 56 mm 2. The height is part of two formulas here: area of parallelogram and area of triangle. If x is the height of the triangle, the equation for the area of the triangle will be: x 16 2 = 56. Rewriting the equation gives 8x = 56, so x = 7 mm. Extensions Extensions 1 and 2 can be used to assess Ontario Expectation 7m33 research and report on real-life applications of area measurements. Students can search for information in both print and online resources. Remind students to use more than one source and to keep a record of their sources. 1. Research project: Farming Alpacas A. Background questions: What is an alpaca? Why do people farm alpacas? Are there alpaca farmers in Canada? B. An alpaca farmer has 8 males, 8 females, and 8 babies. About how much area does the farmer need for this many alpacas? (Convert the answers you find to square metres if necessary. Note: 1 square foot 0.9 m 2, 1 hectare = m 2, 1 acre m 2.) Why do you think there are recommended minimums for the area the animals need to live on? C. Fencing comes in 1 m sections. Using the grids on graph paper, with 1 cm representing 10 m, show as many different rectangular pens as you can that will hold all the alpacas. Fencing costs $30 per metre and grass seed costs $15 for 100 m 2. Complete a chart with these headings to determine how much each rectangular pen will cost: Length Width Perimeter (m) Area (m 2 ) Cost of fencing and grass seed ($) Which size pen do you recommend the farmer use and why? D. The farmer will keep the animals in 3 different pens: one for adult males and two for females and babies (4 females and 4 babies in each pen). The farmer needs to add fencing to separate the large pen into 3 smaller pens. Look at your best rectangular pen from part C. Find 4 different ways of dividing the pen into 3 equal parts. Which method uses the least total amount of fencing? Try at least two other rectangular pens from your chart to make sure you have made the best possible choice. Think: If one pen uses less exterior fencing, does it necessarily use less total fencing? Why or why not? 2. Research project: Building a Swimming Pool A. You want to build a rectangular swimming pool with the largest possible area. You have 300 square tiles, each 30 cm 30 cm, that you want to place around the pool as a border. What should

24 24 the perimeter of the pool be? Remember: tiles need to be placed at the corners as well, so the perimeter can t be cm. Find several possible rectangles for your pool, remembering that the length and width must each be a multiple of 30 cm, and record their perimeters and areas in a chart with these headings: Length Width Perimeter (cm) Area (cm 2 ) Remember to think about what your pool will look like in real life. Rectangles such as 30 cm 4410 cm are not practical a person can t fit into a pool that is only 30 cm wide! Which of your pools will have the largest area? B. How does your pool compare to an Olympic-size pool in area? In shape? Do most existing pools maximize the area given the perimeter? Why or why not? 3. Ancient Mayans used units of length called kaans and belkaans. A field is a rhombus with the area of 500 squared kaans. The side of the rhombus is 1 belkaan, and the height of the rhombus is 1.25 belkaans. How many kaans are in each belkaan? SOLUTION: The area of a rhombus is = 1.25 squared belkaans = 500 squared kaans. This means 1 squared belkaan is = 400 squared kaans. If 1 belkaan = x kaans, then 1 squared belkaan = x 2 squared kaans. So x 2 = 400, and x = 400 = 20. This means 1 belkaan = 20 kaans.

25 25 ME7-8 Area of Trapezoids Workbook pages Goals Students will develop and apply the formula for the area of trapezoids using area of rectangles. Prior Knowledge Required Is familiar with decimals to tenths Can multiply decimals Can find the area of a parallelogram, rectangle, and triangle using a formula Understands that area is additive Can make a sketch Can convert units of length and area Vocabulary base height area Curriculum Expectations Ontario: 7m1, 7m2, 7m3, 7m5, 7m37, 7m38, 7m39 WNCP: optional [CN, R] Review trapezoids. Trapezoids are quadrilaterals with exactly two parallel sides, called bases. Like the height of a parallelogram, the height of a trapezoid is the distance between the bases. Draw several trapezoids on the board, label the vertices, and have students identify the bases. Then invite volunteers to draw the heights. Review the fact that the height is the distance between two parallel lines (measured along a perpendicular), therefore its value does not depend on its placement. Review the term right trapezoid a trapezoid with two right angles. Finding areas of trapezoids by subdividing them into rectangles and triangles. Draw several right trapezoids on grid paper, as in Workbook Question 1, and ask students to copy them on grid paper. Ask students to split each right trapezoid into a rectangle and a right triangle. Review how to find the area of rectangles and right triangles. Ask students to find the areas of their trapezoids by finding the areas of the rectangles and triangles and adding them. Repeat with several trapezoids that are not right trapezoids and must therefore be divided a different way (see Workbook Question 2). Draw several right trapezoids not on a grid, and mark the bases and the side perpendicular to them. Invite volunteers to once again divide the trapezoids into triangles and rectangles. Ask students to label all four sides of the rectangles with their lengths, and then figure out the short sides of the right triangles