Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
John Wallis wikipedia , lookup
Positional notation wikipedia , lookup
Approximations of π wikipedia , lookup
Location arithmetic wikipedia , lookup
History of trigonometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Mathematics and architecture wikipedia , lookup
Mixed Numbers / Quick Review Tyla arranged 7 trapezoids. Her arrangement shows 7 haivesofahexagon:Z H It also shows 3 whole hexagonspiusi 3j and 3.- represent the same amount. They are equivalent. f = 3 -} An improper fraction shows an amount greater than 1 whole. is an improper fraction. C A mixed number has a whole number part and a fraction part. isa mixed number. Try These a a a a a a a a a a • a • a •. • . • • a a a . a a a a a a a a 1. Write an improper fraction and a mixed number for each picture. b) c) 58 AAAA ØOOQOOO a a a • • a a ___ ___ ___ Converting between Mixed Numbers and Improper Fractions These plates have ) These plates have 1+ sandwiches. -- -- sandwiches. - ----1and4 reprsentthesameannount. --- is a mixed number. * is an improper fraction. 2 X 8 = 16 16 + 7 = 23 > To write 2-- as an improper fraction, multiply the whole number by the denominator and add the numerator. = 13 ÷ 2 as a mixed number, divide the numerator by the denominator. > To write 7 23 SO,T - 6 Ri = 13 1 So,6j- 2- = T \ Try These . . . 1. Write each mixed number as an improper fraction. 7 a) 3= 3 b) 4= 6 C) = 1 7y d) 19 12O 2. Write each improper fraction as a mixed number. 8 a)= 60 39 b)--= 48 -= c)9 16 d)—-= J Comparing Mixed Numbers and Improper Fractions Quick Review You can compare and order mixed numbers and improper fractions. I 3 39 > Order 1,ä-,and I 1 0 114 from least to greatest. I Use numberlinesof I I I I I 1 equal length. 8 I I I I I A V I I I I I I ‘ I 0 The order frofl1 least to 933 greatest is --,-j-, lij-. ‘T’ I i + I I I A 1 3 2 I I I 2 I 2 Compare 3and-. 3 15 . . Write 3 as an improper fraction: -iWrite - 15 45 4 12 I as an equivalent fraction with denominator 12: 17 17 45 45 Compareand-j-j-- > -j 17 3 So,3> . . . . , . . , . . . . . 1 1. Use these number lines to order II II I II I . . I . . . and . . . . . . . . . . . . . . . 4 from least to greatest. I 1 0 I,llllllllIlI 0 1 2 I I I 1 2. Write>,<,or= 7 a)1 62 7 b) 4 13 4 c)- 5 3 . . . . . . _____ Exploring Ratios Quick Review A ratio is a comparison of 2 quantities with the same unit. Here are 3 squares and 5 circles. LI LI L100 00 0 Here are some ways to compare the shapes. - > - Part-to-PartRatios squares to circles is 3 to 5 or 3 : 5. • circles to squares is 5 to 3 or 5 :3. - Thenumbers3and5are the terms of the ratio. - - > Part-to-Whole Ratios squares to shapes is 3 to 8 or 3 :8 or circles to shapes is 5 to 8 or 5 :8 or Try These . . . . , , S 1. Write each ratio in as many ways as you can. a) balls to bats — b) batstoballs_ c) balls to all toys d) bats to all toys 64 You can write a part-to-whole ratio as a fraction. --. . . — • S • S S S • • S I • S S I S S S S _____ _____ _____ _____ _____ _____ _____ _____ Equivalent Ratios 4 Quick Review -____ > The ratio 3 :2 means that for every 3 apples there are 2 pears. The ratio 6 :4 means that for every 6 apples there are 4 pears. 3 :2 and 6:4 are equal. 3 :2 and 6 :4 are equivalent ratios. ii ) You can use a table and patterns to find equivalent ratios. The numbers in the Apples column are multiples of 3. The numbers in the Pears column are multiples of 2. The ratios of apples to pears are: 3:2,6:4,9:6,12:8,15:10, Apples Pears Ratio 3 2 3:2 6 4 6:4 9 6 9:6 12 8 12:8 15 10 15:10 j Try These WI t I F 4•W*P I I W I I I IllS , I S I IIII IS ISa 1. Write 2 equivalent ratios for each ratio. 66 a) 5 :3 b)7:4 c)3:9 d)4:11 e)2:6 f) 8:5 I Exploring Percents / _% Quick Review This hundredths grid has 100 small squares. of the grid. Each square represents Twenty-seven squares are shaded. You can describe the shaded part of the grid. )‘ 27 out of 100 squares are shaded. . ) 0.27 of the grid is shaded. 27% of the grid is shaded. Percent means “per hundred” or “out of 100.” This is a percent symbol. You read 27% as 27 percent. Try These . . . . . . a * t a a a • b a a • • a • • • • • I • • • • I I I • 1. Write a fraction with hundredths, a decimal, and a percent to describe the• shaded part of each grid. a) f b)_ C) d)fWWW 2. Write a fraction with hundredths, a decimal, and a percent to describe the unshaded part of each grid in question 1. a) 68 b) c) d) I Relating Fractions, Decimals, and Percents 1’ Quick Review ‘F 27 100 Fractions, decimals, and percents are 3ways to describe parts of a whole. / 0.274 N. [-1j p27% of this shape is shaded. ,/P.XlOWN = - I Iw1i = 30% --- =0.30 30% of the shape is shaded. of the squares are shaded. ,X25 I ----25°/ 25% of the squares are shaded. Try These. . . •t a a a 1. Write each fraction as a percent and as a decimal. 7 9 a)-y b) d) e)-j 4 2. What percent is shaded? a) .: b) 00000 00000 70 c) a a •m • S• S Exploring Triangles Quick Review ——___________ We can name triangles by the number of equal sides. symmetry. Try These a a)A a a a a a A scalene triangle has no equal sides, no equal angles, and no lines of symmetry. An isosceles triangle has 2 equal sides. It has 2 equal angles. It has 1 line of symmetry. An equilateral triangle has 3 equal sides. It has three 60° angles. It has 3 lines of a a a a • a a a a a a a • a j a a a • a a a a a a a a a . a a a 1 a Name each triangle as equilateral, isosceles, or scalene. d)N 72 b)/\ C) > ey j b Naming and Sorting Triangles by Angles An obtuse triangle has one angle greater than 90°. A right triangle has one 90° angle. An acute triangle has all angles less than 90° B D AA E NF We can sort triangies in a Venn diagram. Right Triangles Isosceles Triangles Try These . 4 4 4 4 4 4 4 4 I, • 4 4 4 4 4 4 4 4 4 4 • 4 4 1. Name each triangle as an acute,a right, or an obtuse triangle. a) C) b)N 2. Which triangle in question 1 is isosceles? How do you know? 74 4 j 44•• _______ Drawing Triangles / 7- Quick Review You can use a ruler and a protractor to construct a triangle. Construct triangle ABC with these measures • AB=3cm • • LA80° AC=2.5cm chthetriang(erst Draw side AB. Make it 3 cm long. Measure an 80°angle at A. - Draw side AC. Makeit2.5cm long. JoinCtoBto rn:keside BC. 4. ,L. H Try These . . e . . . . . . . . . S I S 1. Use a ruler and protractor. Construct triangle EFG. Side EF is 7 cm long. Angle F is 900. Side FG is 5.3 cm long. 2. What is the measure of: a) angleE? 3. How long is side EG? 76 b) angleG? • • S I S I • S I S • I • S • • • p a • S • S I UNIT 6 lila 1 Investigating Polygons Quick Review line segments. A polygon is a closed shape with sides that are straight at the Exactly 2 sides meet at each vertex. The sides intersect only vertices. These shapes are non-polygons. This shape is a polygon. > R A regular polygon has all sides and all angles equal. It also has line symmetry. 0 h.-- An irregular polygon does not have all sides equal and all angles equal. d A convex polygon has alt angles less than 1800. A concave polygon has at least one angle greater than 180°. HN TryThese 1. Circle each polygon. 78 1 UNIT 6 Congruence in Regular Polygons Quick Review Here are 2 ways to show 2 squares are congruent. Place one square on top of the other. If they match exactly, they are congruent. ) Compare the side and angle measures. If all sides are equal and all angles are equal, the squares are congruent. H 2cm 2cm 2cm 2 cm B 2cm 2cm C 2cm G TryThese , 1. Triangles LMN and QPQ are congruent. Write the measure of each angle and the length of each side in OPQ. 2cm . •1 . * 0 3 cm P Q 2. Which of these polygons are congruent? Explain how you know. z2 80 c7 ; Perimeters of Polygons ;5; Quick Review We can find the perimeter of any polygon by adding the side lengths. For this pentagon: 2.0 cm 4.0 cm + + 2.0 + 2.5 + 2.0 1.5 = 4.0 Perimeter 1.5 The perimeter is 12 cm. cm 2.0 cm We can use a formula to find the perimeter of some polygons. ParaI1eIograh, Square - 2c__ 2cm 3cm P=2X(€+s) P=2X(3+2) P=sX4 P=2X4 =2X5 =10 The perimeters of the polygons are 8 cm and 10 cm. =8 Try These * I 4 $ 4 1. Find the perimeter of each polygon. a) b) 2.5 cm 82 5 2.5 cm 5; UNIT 6 100 Area of a Rectangle A * - (ISSOII rj Quick Review Here is one way to find the area of a rectangle. 8cm ) Multiply the length by the width. 8 X 4=32 So, the area of the rectangle is 32 cm . 2 cm Tà findthe thea of a rectangle, muftiply the length by the width. Rule. Formula: Area = A = length X width exw Try These Find the area of each rectangle. Complete the chart. Figure A 7cm A 6 4mB B C 2m 7cm 3m D D 5 cm E E 20cm lim 1km F F 10km 84 1 Area