Unit 3 (Chapter 1): Number Relationships The 5 Goals of Chapter 1 I will be able to: model perfect squares and square roots use a variety of strategies to recognize perfect squares use a variety of strategies to estimate and calculate square roots explain and apply the Pythagorean theorem solve problems by using a diagram Your Task With a partner work through the following: Write answers on a piece of paper 1. One partner take 30 squares 2. Together, build the top four levels of the pyramid you see here 3. On a piece of paper each person needs to write down the chart below and fill it in Pyramid level Total # of squares in # of squares on each side level = 1 1 = 2 4 = 3 9 = 4 16 = 4. You have just practiced making square roots (the relationship between the number of cubes in a square and the number of cubes that make up each side). In your own words, write the definition of a square root.
pg. 2 Individually, try to solve the following: The Side Length of the Half Mat Remember, to find the side length of a square, we must take the square root of it's area ( 8100 = ) Half-mat The area is 8100 cm squared. I know 9 x 9 = 81, so each side must be 90 cm. The Side Lengths and area of the full mat The width of two sides is the same as the half mat, so I know it must be 90 cm. The length of the other two sides is twice the width, so I know it must be 180 cm. Whole Mat The area of the half mat is 8100 cm 2 1. Find the area of each mat. (Example, the area of the half mat is 8100 cm 2 ) 2. Find the area of the whole floor 3. Find the side lengths of each mat 4. Find the side length of the whole floor Area is L x W, so the area would be 180 x 90. The area of the full mat is 16 200 cm squared. Dimensions of the Room If the width of each mat is 90 cm and it takes 3 mats to cover the width of the room, 3 x 90 = 270 cm. Since the room is a square, I know the length will also be 270 cm. Area is L x W, so 270 x 270 = 72 900 cm squared.
pg. 2 A triangular number is a number that can be arranged in a triangle. Each row is 1 greater than the row above it. With your learning partner, you have 5 minutes to calculate the following: A square number is a number that can be arranged in a square. side length of the half mat side lengths of the full mats the area of the full mats the dimensions of the entire room the area of the entire room
With your learning partner, complete the following: What is the smallest square number? Explain your reasoning What is the smallest triangular number? Explain your reasoning. Use graph paper and create squares for 1, 6, 10 and 15. Can you see a way to divide it into 2 triangular numbers? Recognizing Perfect Squares Prime your brain: What is a prime number? Can you come up with the first 10 prime numbers?
24 What are 2 factors of 24? When can you say that a number is divisible by another number? 12 2 Which of these 2 factors is a prime number? 4 3 Which of these 2 factors is a prime number? What does it mean for a number to be a prime number? 2 2 Is there any whole number multiplied by itself that results in a product of 24? What do you think it means if a number is considered to be a 'Perfect Square'?
A perfect square is the product of a whole number multiplied by itself, which is a square number. Is 400 a perfect square? 2 x 2 = 4 7 x 7 = 49 In your notebook, show how you could figure this out. 6 x 6 = 36
With a partner, create tree diagrams to find out which of the following numbers are perfect squares Square Roots of Perfect Squares Prime Your Brain 64 120 Write today's learning outcome in your notebook: I can use a variety of strategies to determine the square root of a perfect square. 100 1000 10 000 900 Think of 3 numbers that are perfect squares. Check with 2 other people to see what they chose.
Let's look at some strategies to solve this problem. The mat Vanessa needs to Find the side length of a mat that has an area of 144 m 2 (Green part) A = 144m 2 Things to think about... 1. To find the side length of a square, we take the square root of it's area 2. As a class, calculate the side lengths of the green mat Think about perfect squares and square numbers. Vanessa's mat is a square. Is it a perfect square? New term: Square Root A square root is one of two equal factors of a number. For example: the square root of 81 is 9 because 9 x 9, or 9 2 = 81 The square root symbol is If the floor (Purple) is 15 m by 20 m, how far will it be from the edge of the floor to the edge of the mat. Remember, it must be perfectly centered. 3. With a partner, see if you can determine the distance between the sides of the floor mat and the walls of the gym.
121 chairs need to be arranged for the audience of a play. Is it possible to arrange the chairs so that there are an equal number in each row and column? Prove it Is 121 a perfect square? How do you know? Remember: A perfect square is a whole number product of 2 equal, whole number factors! Are these perfect squares?
Judo mats are squares with a minimum area of 36 m 2 and a maximum area of 64 m 2. The side length of each mat is a whole number in metres. Sketch each possible mat on grid paper. What are the possible side lengths of the mats? Today's Assignment Questions on Pages 14 & 15 #3, 9, 11, 12, 13, 15, 18
Estimating Square Roots Prime your Brain... Copy today's learning outcome into your notes. Aim: I can estimate the square root of numbers that are not perfect squares. Alyssa, her brother and her dad drilled a hole in the ice on the lake to measure its thickness. The ice was 30 cm thick. Their total mass was 125 kg. They used this formula to check if the ice is safe. Required thickness (cm)=0.38 load in kilograms Can the ice support them safely?
The multiplication symbol is often omitted from formulas when the meaning is clear. For example, 0.38 means the same as 0.38 x The symbol " = " means "approximately equal to". For example, 2 = 1.414 Alyssa, her brother and her dad drilled a hole in the ice in the lake to measure its thickness. The ice was 30 cm thick. Their total mass is 125 kg. They used this formula to check. Required thickness (cm)=0.38 Can the ice support them safely? load in kilograms Key Information! Alyssa and her dad's combined weight is the load on the ice. To find the required thickness of the ice that will support that weight, we need to estimate the square root of the 125 kg load and multiply that number by 0.38
Things to think about! Draw a 10 by 10 square, an 11 by 11 square and a 12 by 12 square. Calculate the area of each square. 1. How can you calculate the area of a square? 2. If you know the area of a square, can you calculate its side length? 3. Between what two perfect squares does 125 lie? 4. Is 125 closer to 121 or 144? 5. What must be substituted into the thickness formula? 6. When you determine the square root of 125 using a calculator, will you get a whole number? 7. What is the square root of 125? How can we determine the square root of 125 using a calculator? Different calculators use different key sequences to calculate square root. Most use the number, then the 125 Determine 125 to two decimal places using a calculator. You can also use a calculator to check the square root of a number that is not a perfect square. The number you get when finding the square of a square root may be an approximation.
Trying it out! The square root of 125 is 11.18 (to 2 decimal places). *Remember, we can find this by keying in 125 and then pushing However, when you square your answer, 11.18, does it equal exactly 125? How can you check this? Your calculator also has an x 2 function. Try entering 11.18, and then push the x 2. If your calculator does not have this function, you can multiply 11.18 by 11.18. Estimating Square Roots Now, let's look back at the original problem to see if the ice can support them safely. Alyssa, her brother and her dad drilled a hole in the ice in the lake to measure its thickness. The ice was 30 cm thick. Their total mass is 125 kg. They used this formula to check. Required thickness (cm)=0.38 load in kilograms Can the ice support them safely?
Talk the problem through. The 11 by 11 square has an area of 121 square units. The 12 by 12 square has an area of 144 square units. Since the area of 125 square units is between 121 and 144, we know that the side length is between 11 and 12. A square floor has an area of 85 m 2. About how long are its sides? Is 85 a square number? How can we find the square root of 85? 125 = 11.18033989 or about 11.18 required thickness = 0.38 x 11.18 = 4.25 cm Since the ice is 30 cm thick, the ice is more than thick enough to support the weight of Alyssa and her father.
A truck has a mass of 5000 kg. What thickness of ice is needed to support the truck? Use the formula: Required thickness (cm) = 0.38 5000 Assignment: 5, 8, 9, 10, 13 and 14
Pythagorean Theory Pythagorean Theory
Pythagorean Theory Pythagorean Theory
Pythagorean Theory Pythagorean Theory
Pythagorean Theory Pythagorean Theory
Pythagorean Theory
The Pythagorean Theorem Aim: I can model, explain and apply the Pythagorean Theorem Critical thinking question: What are the 3 kinds of angles? Write them down, then turn to your learning partner and see what he/she thinks Pythagoras was a mathematician who lived 2500 years ago. He is known for his Pythagorean theorem, which is used to solve problems involving the lengths of right triangles. Before we look at his theorem, let's do a quick review on angles and triangles!
What is an acute angle? Types of Triangles An angle whose measure is less than 90 o What are acute triangles? Acute triangles have 3 acute angles. What is an obtuse angle? An angle whose measure is greater than 90 o but less than 180 o What are obtuse triangles? Obtuse triangles have 1 obtuse angle and 2 acute angles. What is a right angle? An angle whose measure is 90 o and is formed by perpendicular lines What are right triangles? Right triangles have 1 right angle and 2 acute angles
In a right angle, the two shortest sides are called the legs. The longest side, opposite the right angle, is called the hypotenuse. Basically, the Pythagorean Theorem states that if one leg is squared (let's call it 'a') and the second leg is squared, (let's call it 'b'), when you add them together, the answer will be the length of the hypotenuse (let's call it 'c') squared. It will look like this in an algebraic equation: a 2 + b 2 = c 2 http://www.learnalberta.ca/content/mejhm/index.html?l=0&id1=ab.math.jr.shap&id2 =AB.MATH.JR.SHAP.PYTH&lesson=html/video_interactives/pythagoras/pythagorasInteractive.html leg hypotenuse e.g. If a = 4m; b = 3m, what will be the length of the hypotenuse, or c? a 2 + b 2 = c 2 4 2 + 3 2 = c 2 16 + 9 = 25 25 = 5 So the length of side c is 5m leg
What strategy did you use to calculate this distance? A cowhand rode a horse along the diagonal path, instead of around the fence of the ranch. What distance did he save by riding the diagonal path? a = 9km and b = 12km The formula is a 2 + b 2 = c 2 So, 9 2 + 12 2 = c 2 start b = 12km c =? km a = 9km 81 + 144 = 225 Because 225 is the squared number, we need to find the square root of it. 225 = 15 Path With your learning partner, calculate the distance the cowhand saved. Remember to prove your work.
The distance he travelled was 15km. However, the question asked, how many kilometres did he save by taking the diagonal path. With your learning partner, calculate the distance he would have gone had he travelled along sides 'a' and 'b'. Today's Assignment start b = 12km c =? km Path a = 9km What strategy will you use? Pages 30 and 31 #5, 7, 8, 9, and 13
Solve Problems Using Diagrams Aim: I can use diagrams to solve problems about squares and square roots. 5m 3m? A 5m ladder is leaning against a wall. The bottom of the ladder is 3m from the wall. How high is the top of the ladder from the ground? With a partner, figure out how high the ladder is from the ground.
We can use the Pythagorean Theorem to determine the height. We know the base, and we know the hypotenuse. We can use 3 2 + b 2 = 5 2 9 + b 2 = 25 25-9 = 16 b 2 = 16 16 = 4 Therefore, b= 4 and the height of the wall from the ground is 4m. Joseph is building a model of the front of a famous Haida longhouse. He wants the model to have the above measurements. How can Joseph calculate the two lengths at the top of the model? With a partner, draw a model to show how you would find 'c' at the top of the model.
Page 35 Questions # 1, 2, and 5 You may work with a partner. Use diagrams to show your work.