9 Academic Review

Number Sense Order of Operations Unit Price Proportion Fractions Percent Ratios

B E D M A S Order of Operation: RACKETS XPONENTS IVISION LEFT TO RIGHT ULTIPLICATION A DDITION S LEFT TO RIGHT UBTRACTION

Example: Solve 6×3÷2−(4−9) =6×3÷2−(−5) =6×3÷2+5 =18÷2+5 =9+5 =14

Unit Price MONEY ÷ ITEM Jennifer Ahmad $0.08/g + $0.02/g = $0.10/g Ahmad bought nuts from a local market and paid $40 for 500 g. Jennifer went to the grocery store to purchase her apples and paid $0.02/g more. If Jennifer bought 350 g of apples, how much did she pay? Jennifer Ahmad $0.08/g + $0.02/g = $0.10/g $40 500g =$0.08/g $0.10/g ×350g = $350 ∴ Jennifer paid $350 for her apples.

Fractions Change all mixed fractions to improper Multiplication/Division → NO COMMON DENOMINATOR To multiply invert (flip) the fraction after the division sign and multiply i.e. ÷ 𝟐 𝟑 is the same as× 𝟑 𝟐 Follow BEDMAS To convert to a decimal: NUMERATOR ÷ DENOMINATOR If you’re stuck use the fraction button on your calculator.

Percent Calculating Percentage 21 25 ×100 =0.84×100 =84% Robbie got a run 21 out of his last 25 times at bats, calculate his batting average. 21 25 ×100 =0.84×100 =84% ∴ Robbie’s batting average is 84%.

Percent Percent OF 24÷100=0.24 0.24×250=60 24% of Grade 9 students have decided to take drama next year. If there are 250 grade 9 students, how many have signed up for drama? 24÷100=0.24 0.24×250=60 ∴ 60 grade 9 students signed up to take drama next year.

Percent Percent IS 30% off →she is paying 70% Becky paid $60 before tax for a Champion Sweater. If the sweater was on sale for 30% off what was the original price? 30% off →she is paying 70% $60 𝐢𝐬 70% of the original price 70÷100=0.70 $60÷0.70=$85.71 ∴ the original price of the sweater was $85.71.

Ratios Jordan Kim Mike TOTAL 3 5 4 12 𝑥 𝑦 𝑧 $245,862 Jordan, Kim and Mike share a business and split their profits in the ratio of 3:5:4 respectively. If they earned $245,862 last year, how much did each person make? Jordan Kim Mike TOTAL 3 5 4 12 𝑥 𝑦 𝑧 $245,862 3 𝑥 = 12 245,862 5 𝑦 = 12 245,862 4 𝑧 = 12 245,862 𝑥=$61,465.50 𝑦=$102,442.50 𝑧=$81,954.00

Algebra Simplify Expressions Follow BEDMAS Expand using the distributive property Collect Like terms Adding/Subtracting – Do NOT change exponents Multiplying – ADD exponents Dividing – SUBTRACT exponents Solve equations Isolate the variable by using inverse operations. Move “letters” to one side, “numbers” to the other and divide.

Example: Simplify (4𝑥)(3 𝑥 2 )−2𝑥(5 𝑥 2 −9𝑥) =12 𝑥 3 −10 𝑥 3 +18 𝑥 2 =2 𝑥 3 +18 𝑥 2

Example: Solve 15 𝑥−3 =3𝑥−9 15 𝑥−3 =3𝑥−9 15𝑥−45=3𝑥−9 15𝑥−3𝑥=−9+45 12𝑥=36 12 12 𝑥=3

Example: Application Determine a simplified expression for the perimeter of the rectangle shown below. 𝑥+1 3𝑥−5 3𝑥−5 𝑥+1 𝑃= 8𝑥 −8 8𝑥−8=112 8𝑥=112+8 Solve for 𝑥 when the perimeter is 112 cm. 8𝑥=120 8 8 𝑥=15 cm

Linear Relations SLOPE = RATE (OF CHANGE) 𝑦−intercept = INITIAL VALUE Table of Values Graph Equation Description

Linear Relations – Table of Values 𝒙 𝒚 16 97 18 106 22 124 30 160 32 169 Rate: ∆𝑦 ∆𝑥 Initial Value: 4.5×16=72 = 106−97 18−16 97−72=25 = 9 2 Partial Variation =4.5 Equation: 𝑦= 9 2 𝑥+25 Note: This is linear as there is a constant rate of change i.e. (160-124)/(30-22)=4.5.

Linear Relations – Graph Rate: ∆𝑦 ∆𝑥 Initial Value: = 75−0 3−0 Direct Variation = 75 3 =25 mi/h Equation: 𝑑=25𝑡 Note: This is linear as there is a constant rate of change i.e. it’s a straight line.

Linear Relations – Equation 5𝑥−3𝑦+45=0 Rate: 5 3 −3𝑦=−5𝑥−45 −3 −3 −3 Initial Value: 15 𝑦= 5 3 𝑥+15 Partial Variation

Linear Relations – Equation from 2 points Determine the equation of the line that passes through the points 𝐴 2,−5 and 𝐵 −4,7 . (𝑥,𝑦) 𝑚= 7−(−5) −4−2 𝑦=−2𝑥+𝑏 −5=−2(2)+𝑏 = 12 −6 −5=−4+𝑏 −5+4=𝑏 ∴𝑦=−2𝑥−1 =−2 𝑏=−1

Linear Relations – Equation Example 2 Determine the equation of the line that is perpendicular to the line 5𝑥−2𝑦+10=0 and has the same 𝑥−intercept as the line 3𝑥+5𝑦−30=0. 𝑦=− 2 5 𝑥+𝑏 slope 𝑥−intercept 5𝑥−2𝑦+10=0 3𝑥+5 0 −30=0 0=− 2 5 (10)+𝑏 3𝑥−30=0 −2𝑦=−5𝑥−10 0=−4+𝑏 3𝑥=30 𝑦= 5 2 𝑥+5 𝑥=10 𝑏=4 ∴𝑦=− 2 5 𝑥+4 𝑚=− 2 5 point:(10,0)

Scatter Plots & Motion Scatter Plot Know how to read and interpret points. Draw a line of best fit. Create the equation of the line of best fit. Distance/Speed/Time Graphs Remember to state 4 details for every section: Direction Change in distance Change in time Speed (Distance ÷ Time)

Example: Scatter Plot 𝑚= 1300−100 55−0 𝑚= 1200 50 𝑚=24 𝑏 ∴𝑙=24ℎ+100

Example: Distance/Speed/Time Distance Away From Home Away from home 150 m 100 mins. Distance Away From Home (m) 1.5 m/min. Stopped 0 m 100 mins. 0 m/min. Towards home 150 m 50 mins. 3 m/min.

Geometry 2D Shapes: Perimeter and Area Don’t use the formula page for the perimeter just add the outside (Recall: 𝐶=𝜋𝑑). For the area of a composite shape, use the formula page to find the area of each shape and then add and/or subtract. 3D Shapes: Surface Area and Volume Remember for Surface Area just find the Area of each side and add them together (lateral surface – curved surface). For the volume of a composite shape, use the formula page to find the volume of each shape then add and/or subtract.

Angles Straight Angles add to 1800. Right Angles add to 900 . Opposite Angles are equal (X-Pattern) Parallel lines (Z-Pattern, F-Pattern, C-Pattern) The sum of the angles in a triangle equals 1800. The sum of the angles in a quadrilateral equals 3600. The sum of the interior angles in a polygon is 𝑛−2 × 180 𝑜 . The sum of the exterior angles in a polygon is 360 𝑜 .

Example: Interior Angles in a Polygon The sum of the interior angles of a polygon is 46800. How many sides does the polygon have? 𝑛−2 × 180 0 = 4680 0 𝑛−2 = 4680 0 ÷ 180 0 𝑛−2=26 𝑛=26+2 𝑛=28 ∴ the polygon has 28 sides.

Example: Exterior Angles of a Polygon Determine the value of on this regular nonagon. 𝑥=360 0 ÷9 𝑥 𝑥= 40 0