1.0 Fractions Review Name: Date: Goal: to review some key fractions skills in preparation for the upcoming unit. Main Ideas: b)!!

Transcription

1 1.0 Fractions Review Name: Date: Goal: to review some key fractions skills in preparation for the upcoming unit Toolkit: Working with integers Operations with fractions Main Ideas: Reducing Fractions To reduce fractions: Find a whole number that divides the numerator (top) and the denominator (bottom). Reduce!! Ex1) Ex1) Put each fraction in lowest terms (a.k.a. reduce the fractions). a)!!" b)!!!" c)!"!!" d)!""!""" Adding & Subtracting Fractions 1) To add or subtract fractions, you need a. 2) Then add or subtract the numerators 3) Leave the denominators the same 4) Reduce if possible Ex2) Ex2) Add or Subtract. Reduce if necessary. a)! +!!! b)!!!! c)! +!!!!!" d)!!!!!!

2 Multiplying Fractions Ex3) To multiply fractions, 1) Multiply numerators 2) Multiply denominators 3) Reduce if possible Ex3) Multiply. Reduce if necessary. a)!!!! b) 136!!" c) 45!!.!" Dividing Fractions To divide fractions, 1) Change to a multiplication by FLIPPING the second fraction 2) Multiply numerators and denominators 3) Reduce if possible Ex4) Ex4) Divide. Reduce if necessary.! a)!!! b) 3!! c)!!!!! Cross Multiplication 1) To solve with Cross Multiplication, you need 1 fraction = 1 fraction. 2) Then multiply ACROSS (see example) 3) Solve for x (if you have a number times x, divide both sides by that number). Ex5) Ex5) Cross Multiply to solve for x (decimal answers possible!)! a) =!! b) =! c)!!!.!"!"!".!"! =!!" Reflection: What type of fraction question do you think you could explain to someone else? What type of fraction question would you like to have some explain to you?

3 1.1 Imperial Measures of Length Name: Date: Goal: to estimate imperial measures of length and to convert between imperial units Toolkit: Anything you know about imperial measures Multiplying fractions Perimeter: distance around Main Ideas: Common Imperial Measures of Length Warm-up: Use your formula sheet. How many inches are in a foot? How many yards are in a mile? What are the short forms for: inch foot yard mile A referent is something (usually handy, or common) used to help estimate or describe a measurement. Inch: Foot: Yard: Mile: Imperial and fractions: How long is this pencil (in inches)? Use the picture, not your own ruler. Ex1) Ex1) Using your formula sheet (and/or prior knowledge), convert 6 yards to a) feet b) inches Now, we will find the answers using Unit Analysis: 1) Write 6 yards x (the x means multiplication) 2) Find the conversion from yards to feet, and write it as a fraction: Two ways: 3) Because you start with yards, you want yards on the bottom of the conversion fraction so that the yards unit cancels, leaving you with feet. a) b)

4 Ex2) Ex2) Use Unit Analysis to convert 4 pounds to ounces: Ex3) Ex3) Using your formula sheet, convert 63 in. to: a) feet & inches b) yards, feet, inches When converting to two units, set up unit analysis, write the answer as a fraction, and then convert it to a mixed fraction. When converting to three units, do the same as converting to two units, then take the whole number and repeat the process. Note: we often write feet and inches in short-hand: Ex 4) Ex4) You take a ribbon, wrap it around the bottom of an oddly shaped sculpture, and find that the perimeter of the base is 142 in. What is the perimeter in feet and inches? Ex5) Money! (HW qs #14, 15, 21) Ex5) In Ex4, we needed 142 inches of ribbon. We decide to buy a fancier ribbon (of the same length). Ribbons-R-Us sells the fancy ribbon by the foot, at $1.12 per foot. How much will the ribbon cost, before taxes? Scale Diagrams (HW qs #16, 17, 20) Ex6) Hint: Maps often provide a scale that looks like 1 : We write scales as distance on drawing : actual distance If we go from a map to real life, the distance gets bigger, so If we go from real life to a map, the distance gets smaller, so Ex6) On a map with a scale of 1: , the distance between Seward and Anchorage in Alaska is 1! inches. What is the distance between these towns, to! the nearest mile? Reflection: In example 4, we used a ribbon to find out the perimeter of something with a strange shape. Why is this a useful trick? Write steps explaining to someone how to use a string to measure the distance around an object.

5 1.3 Converting Imperial and SI Units Name: Date: Goal: to practice converting between imperial and SI units of measurement Toolkit: Formula sheet Unit analysis = or or Main Ideas: Definitions Imperial units units of measurement such as mile, yard, foot, inch, ounce, pound commonly used in the United States and some industries in Canada SI units Système International a system of measurement using powers of 10 (10, 100, 1000, 0.1, 0.01 ) used in Canada and much of the world. It is also called the metric system. base Tera Giga Mega kilo hecto deca unit deci centi milli micro Ex1) Ex1) A bowling lane is approximately 19 m long. What is this measurement to the nearest foot? Use unit analysis and your formula sheet: Ex2) What if you can t find the exact conversion on the formula sheet? Ex2) Convert 720cm to yards. Use unit analysis, but this time it will be a two-step conversion: Ex3) Ex3) You use a website coupon for a free 24 oz bag of dog food for your new puppy. How many kilograms of dog food did you buy?

6 Ex4) Ex4) Convert 452 ounces to the nearest tenth of a kilogram: Ex5) Using conversions to compare measurements Ex5) Two students compare their heights: Marie is 160cm and Louise is 5'2". Who is taller, and by how much (to the nearest inch)? For all comparisons: Key In order to compare, we need Marie: Louise: Ex6) Ex 6) Convert 3 feet 8 inches to the nearest tenth of a centimetre: Reflection: What things do you measure using the imperial system? What things do you measure using the metric system?

7 1.5 Volumes of Prisms, Pyramids and Cones Name: Date: Goal: to solve problems involving the volumes of pyramids and cones AND to connect them to the volumes of prisms and cylinders. Toolkit: Pythagoras: Main Ideas: Area of a triangle: + other formulas on your formula sheet Volume: what happens to units? Using the calculator carefully! Ex: π button (not 3.14) Ex:!"#!! à typeà 615 / (4π ) Definitions Rectangular Prism: a 3-dimensional object made up of six rectangular face Triangular Prism: a 3-dimensional object made up of 2 triangular faces and 3 rectangular faces. Cylinder: a 3-dimensional object that looks like a soup can. Volume formula? Volume formula? Volume formula? Right pyramid: a 3-dimensional object that has triangular faces and a base that is a polygon (polygon = many sides) Regular tetrahedron: a tetrahedron is a triangular pyramid that has 4 congruent equilateral triangular faces. Right cone : a right circular cone is a 3-D object with a circular base and a curved surface. Apex the vertex ( ) farthest from the base of an object

8 Formulas Volume of a Prism = (Area of base)(height) (see formula sheet General Prism) A Pyramid has a volume that is one- third V Rectangular Prism = Volume of a Pyramid = (see formula sheet General Pyramid) the volume of the corresponding prism! V Rectangular Pyramid = V Cylinder = V Cone = Ex1) Find the volume of a rectangular prism with length 2.6cm, width 4.1cm, and height of 6.3cm (to the nearest tenth). Step 1: Sketch and label a diagram Step 2: Write the formula you need: Step 3: List the values you know Step 4: Substitute into the formula (use brackets!) Step 5: Evaluate and remember units! Ex 2) Find the volume of a square pyramid that has a base dimension of 7.2m and a height of 9.9m (to the nearest tenth). Ex 3) Find the volume of a cylinder with diameter 62cm and height 46cm (to the nearest tenth).

9 What if you don t know the height? Ex4) Determine the volume of this cone to the nearest cubic inch. Working Backwards Ex5) A cone has a height of 9.2m and a volume of cubic metres. Determine the radius of the base of the cone to the nearest tenth of a metre. Same steps!(almost J ) Step 1: Sketch and label a diagram Step 2: Write the formula you need: Step 3: List the values you know Step 4: Substitute into the formula (use brackets!) Step 5: SOLVE for what s missing and remember units! Reflection: If you could not remember the formula for the volume of a cone or pyramid, what do you know that could help you find the volume?

10 1.4 Surface Area of Pyramids and Cones Name: Date: Goal: to solve problems involving the surfaces areas of pyramids and cones. Toolkit: Pythagoras Using formulas carefully Working backwards/solving Units for area: Using the calculator carefully! Ex: π button Main Ideas: Definitions Surface Area the sum of the areas of all the surface faces of the shape Lateral Area the sum of the area of all the sides of a 3D object except its top and bottom bases. Refer to the 3D pictures on your formula sheet and identify the lateral area of each shape. Regular tetrahedron a tetrahedron is a triangular pyramid. A regular tetrahedron has 4 congruent equilateral triangular faces, as shown: Ex 1) Investigate the surface area formula for a rectangular prism (on your formula sheet). Then, find the surface area of a rectangular prism with length 7cm, width 3cm, and height 4cm. Ex 2) Investigate the surface area formula for a cylinder (on your formula sheet). Then, find the surface area of a cylinder with a radius of 2m and a height of 5m (to the nearest tenth).

11 Finding the SA of a pyramid when you know SLANT HEIGHTS (easier) Ex3) Find the surface area of a regular tetrahedron with edge length 7.0m and slant height 6.1m. 6.1m 6.1m 7.0m Finding SA of a pyramid when you DON T know SLANT HEIGHTS (harder) Ex4) Investigate the surface area formula of a square based pyramid (on your formula sheet). A pyramid has a square base with 12-foot sides and a height of 8ft. Calculate the surface area of the pyramid to the nearest square foot. **It is always very important to realize if you need height (h) for your formula, or slant height (s). Then, see what has been given in the question. If the one you need has NOT been given, you will have to use PYTHAGORAS to find it! Warning! Not on formula sheet! If you have a pyramid with a regular polygon base (pentagon, hexagon, etc), find the area of the base by splitting it into a series of triangles, and then find the lateral area by counting up the lateral triangles. Ex 5) Find the surface area of the pyramid. 9cm 5cm 4cm

12 (see p for a development of the formula for SA of a cone) Surface Area of a Cone = (from formula sheet) Surface area of a Cone Ex6) Find the surface area of a cone with base diameter 6.0 inches and a height of 4.0 inches (to the nearest square inch). Working backwards Ex7) A farmer unloads rice onto the ground and the rice forms the shape of a cone. The lateral surface area that the farmer must cover with a tarp is 200ft 2. The diameter of the cone is 12 ft. How tall is the cone? (To the nearest foot.) Reflection: What are some assumptions we made in the example with the farmer s pile of rice?

13 1.6 Surface Area and Volume of a Sphere Name: Date: Goal: to solve problems involving the surface area and volume of a sphere Toolkit: Main Ideas: Using formulas and your formula sheet Working backwards Definitions Hemisphere Formulas pgs.46,48 have explanations: Surface Area Sphere = 4πr! Volume of a Sphere =!! πr! Ex1) Calculate the surface area and volume of this sphere:

14 Ex2) The surface area of a golf ball is approximately 57.2 cm 2. What is the volume of the ball (to the nearest tenth?) Ex3) Determine the surface area and volume of a hemisphere with a radius of 5yards. Reflection: Think back over the past 3 topics. Which do you find easier to work with surface area or volume? Why?

15 1.7 Volume and Surface Area of Composite Objects Name: Date: Goal: to solve problems involving the volume and surface area of composite objects Toolkit: Main Ideas: Helpful hints: What formula(s) do you need? What values do you know? How will you combine the information of the different objects (add, subtract, plan!) Remember units Read carefully Composite Object: 2 or more objects Ex1) Determine (to the nearest tenth) the volume of this cylinder with a cone removed: Ex 2) Find the volume to the nearest tenth:

16 Ex3) Determine (to the nearest tenth) the surface area of this cylinder with a cone removed: Ex 4) Find the surface area to the nearest tenth: Reflection: How would the problem be different if the cone were ON TOP of the cylinder, instead of removed from the cylinder?