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Unit 1 – Square Roots and Surface Area Grade 9 Mathematics Unit 1 – Square Roots and Surface Area Section 1.1 – Square Roots and Perfect Squares Review: Changing Decimals to Fractions To change a decimal to a fraction, we move the decimal to the right, and put the number over a power of 10. For example, let’s look at 0.6. To get rid of the decimal we would move it one place to the right. Because we move the decimal one place to the right, we add one zero on the bottom. Therefore, Try the following: a) 0.08 b) 0.25 c) 0.379 Recall, a perfect square is the result of a whole number multiplied by itself. For example, 16 is a perfect square since L. Clemens Page |1 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics To determine if a decimal is a perfect square, we can change it into a fraction, or remember this rule: Rule: If the decimal has an even number of decimal places and consists of a perfect square, the decimal number is a perfect square. For example, a) 0.49 has an even number of decimal places (two numbers after the decimal) fcp://@fc.nlesd.ca,%2318870007/Mailbox/004b262b-011feef7and 49 is a perfect square, therefore 0.49 is also a perfect square. b) 0.00016 is NOT a perfect square. Even though 16 is a perfect square it has an odd number of decimal places (three numbers after the decimal). Some fractions can also be perfect squares. In order for a fraction to be a perfect square, both the numerator and denominator have to be perfect squares. Let’s consider the following situations. a) Is a perfect square? We can also be show this by drawing a diagram using squares. L. Clemens Page |2 Unit 1 – Square Roots and Surface Area b) Is 16 c) Is Grade 9 Mathematics a perfect square? a perfect square? You can also check to see if a number is a perfect square in your calculator. If the square root terminates (ends) it is a perfect square. Decimal Value of Square Root Type of Decimal Is it a perfect square? 1.69 3.5 70.5 5.76 0.25 2.5 L. Clemens Page |3 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics Try the following: . 1. Use a diagram to determine the value of 2. Which of the following are perfect squares? Explain. 3. Find each square root: a) L. Clemens b) c) d) Page |4 Unit 1 – Square Roots and Surface Area 4. Which of the following are perfect squares? a) 1.96 5. Grade 9 Mathematics b) 0.9 c) 0.036 Calculate the number whose square root is: a) 0.6 L. Clemens b) 0.04 c) 1.4 d) Page |5 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics Section 1.2 – Square Roots and Perfect Squares Recall: To estimate a perfect square, we locate the perfect square just above and just below the number you are looking for. For example, =5 a) =4 b) L. Clemens Page |6 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics We can use the same way we estimate non-perfect square whole numbers, to estimate the square root of non-perfect fractions and decimals. Consider . Between which two decimal perfect squares does 0.38 fall? 0.6 0.7 Example 1: Estimate each square root and show workings. a) b) c) b) L. Clemens Page |7 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics To estimate a fraction, we find the closest perfect squares to the numerator and the denominator and find the square root. Consider . The closest perfect square to 8 is 9. The closest perfect square to 15 is 16. Therefore, we can estimate as which is then equal to . Try the following: a) b) We can also estimate fractions by changing the fraction to a decimal first, and then estimating. Consider . is the same as 0.3. Since 0.30 falls between the perfect squares 0.25 and 0.36, and so we estimate in between these numbers. 0.5 L. Clemens falls between 0.6 Page |8 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics Using a method of your choice, estimate each of the following: Find each square root: a) b) c) Recall: Pythagorean Theorem A right triangle is a triangle with one 90o angle. The Pythagorean Theorem states that for any right triangle, the area of the square on the hypotenuse is equal to the sum of the area of the squares on the other two sides (legs). The side directly across from the 90o angle is called the _______________________________. This is always the longest side in a right triangle. The two remaining sides are called ______________________________. c2 a2 b2 L. Clemens Page |9 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics Example 1: a) Find the missing value: a) c b) L. Clemens P a g e | 10 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics Section 1.3 – Surface Area of Objects Made from Right Rectangular Prisms Investigation Assume each face of a linking cube has 1 unit2. What is the surface area of 1 cube? Now link 2 cubes together and determine the surface area. Continue to do so, linking each additional cube to the end creating a “train”. Fill in the chart below. Number of Cubes Surface Area (square units) 1 2 3 4 5 What happens to the surface area each time you add a cube? Why do you think it changes this way? L. Clemens P a g e | 11 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics Here is an object made from 4 cubes. What is the surface area? How did you determine the surface area? We could easily find the surface area by finding the total surface area of each block, adding them together and then subtracting the _______________________! Find the surface area of the object below: L. Clemens P a g e | 12 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics Surface Area of Square/Rectangular Prisms To find the surface area of any prism, we need to individually find the area of all surfaces and add them together. In the case of a rectangular prism, all the faces are rectangles so we use the same formula for all six rectangles. Formula for the Area of a Rectangle: _________________________________ Example 1: 8 cm Front/Back: 10 cm 6 cm Top/Bottom: Side/Side: Total Surface Area: L. Clemens P a g e | 13 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics Try these: a) 5 cm 6 cm 8 cm b) 12 cm 12 cm 12 cm A _____________________________________ object is composed of two or more objects. L. Clemens P a g e | 14 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics To find the surface area of a composite object, imagine dipping the object in paint. What happens when you separate the blocks below once they have been painted? – You will have an area on each block that is not painted because it was touching the other block. This unpainted part is called the overlap. To find the surface area, we calculate the area of all the faces covered in paint. The overlap is not painted, so it is not part of the surface area; thus we find the total surface area of each block, add them together and subtract the overlap from the total at the end. Example 1: 1. First calculate the surface area of the larger prism. Front/Back: 10 m 6m Top/Bottom 2m 4m 5m 2m Side/Side: Total Surface Area: L. Clemens P a g e | 15 Unit 1 – Square Roots and Surface Area 2. Grade 9 Mathematics 10 m Second, calculate the surface area of the smaller prism. Front/Back: 6m 2m 4m Top/Bottom 5m 2m Side/Side: Total Surface Area: 10 m 3. 6m Now calculate the overlap. 2m 2m 4m 2m 5m 4m Total Surface Area of the Composite Figure: ________________________ + ________________________ - ______________ = ________________ SArea of Large Object + SArea of Small Object – 2 Overlap L. Clemens P a g e | 16 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics Try These: 1) 2) 6 4 1 4 2 4 4 L. Clemens 2 2 2 2 P a g e | 17 2 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics Section 1.4 – Surface Area of Other Composite Objects To find the surface area of a triangular prism, we need to use two formulas. Area of a Rectangle =____________________ Area of a Triangle = ___________________ Example 1: 2 Triangular Faces: 3 Rectangular Faces: First: Second: Third: Total Surface Area: L. Clemens P a g e | 18 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics Try These: A) B) L. Clemens P a g e | 19 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics Finding the Area of a Composite Object When finding the surface area of a composite figure involving triangular prisms, we use the same process we did before. Step 1: Calculate the surface area of the larger prism. Front/Back : Top/Bottom: Side/Side: Total Surface Area: L. Clemens P a g e | 20 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics Step 2: Calculate the surface area of the smaller prism. 2 Triangular Faces: 3 Rectangular Faces: First: Second: Third: Total Surface Area: Step 3: Calculate the overlap. What is the shape of the overlap? Be careful with these composite objects the overlap could be a triangle OR rectangle. Area of Triangle: Total Surface Area: L. Clemens P a g e | 21 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics Try These: A) B) L. Clemens P a g e | 22 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics Recall: Surface Area Formula for a Cylinder Example 1: Two cakes are arranged as shown below. These cakes are to be covered in frosting. What is the area of the frosting? L. Clemens P a g e | 23 Unit 1 – Square Roots and Surface Area Grade 9 Mathematics Example 2: L. Clemens P a g e | 24