Dimensions Math® 6–8 Curriculum

Dimensions Math® 6–8 Curriculum

The Singapore math method for middle school.

Singapore Math Inc. imensions Math® 6–8 brings the Singapore math approach into middle school. The program emphasizes problem solving and empowers students to think mathematically, both inside and outside the classroom. Pre-algebra, algebra, geometry, data analysis, probability, and some advanced math topics are included in this rigorous series. Dimensions Math 6–8 is a logical next step for students who completed Dimensions Math PK–5, or for students ready to gain a solid foundation for higher level math.

Singapore Math Inc. TIMSS TIMSS singaporemath.com standardized andstate assessments. student performance internationally andathomeon Our Singapore Math international mathtesting. in Singapore, whichconsistently ranks atthetop in teaching approach basedonresearch of mathmastery The Singapore mathapproach isahighly effective Math? Singapore Why 629 520 Number Average MathCognitive DomainScores Average MathContent DomainScores 633 528 Knowing 623 525 Algebra ® curricula aimto raise U.S. Grade 8 Grade 8 619 515 Applying 500 Geometry 617

Reasoning 616 514 522 617 Data and Chance Introduces concepts inatangible way andprogresses CPA (Concrete Pictorial Abstract) Approach features include: using equationsto solve algebraic word problems. Key using barmodelsto understand algebraic concepts to of mathematicalthinking.Students transition from Singapore mathapproach instills adeepunderstanding The intentional progression of concepts inthe Approach The TIMSS 2015 International Results in Mathematics in Results International 2015 TIMSS unknowns inagiven situation. concepts. Allows students to determine theknowns and Bar Modeling to increasing levels of abstraction. 54% 10% Benchmark United States United Singapore International Benchmarks inMathematics Advanced Percentage of Students Reaching : Helpsstudents visualizearange of math 37% 81% Benchmark High Grade 8 Intermediate 70% 94% Benchmark : 99% 91% Benchmark Low

TIMSS Textbooks

Textbooks provide a systematic, well-rounded 3 Try It!: Gives students an opportunity to answer approach to math that facilitates the internalization a similar question to check how well they have of concepts and instills curiosity. Lessons grasped the concept. engage students with different levels of problem solving and real world application of math topics. 4 Class Activity: Introduces new concepts through Textbooks A and B for each grade correspond to cooperative learning methods. the two halves of the school year. 5 Basic Practice: Provides simple questions that 1 Chapter Opener: Introduces a topic through a real involve the direct application of concepts. world example and identifies learning objectives. 6 Further Practice: Provides more challenging 2 Example: Helps students understand and master questions that involve the direct application of a concept through a worked example. concepts.

7 Math@Work: Provides questions that involve the 1 Percent application of integrated concepts to practical

Try It! 11 The following two histograms show the distributions of situations.lengths of ropes (in centimeters) from 2 boxes – A and B.

Histogram A Histogram B

25 25 20 1 20 4 15 15

Objective:Frequency 10To explore the properties of equations.Frequency 10 5 5

Materials: Balance0 scale labeled with an equal0 sign in the middle. 50 60 70 80 90 100 50 60 70 80 90 100 Two differentLength (cm) colored connecting cubesLength such (cm) as yellow and green to represent the variables and constants.

Small stickers with x to label the yellow (variable) cubes and = to For each of the histograms, answer the following questions. label the middle of the scale. REMARK (a) Is the shape of this histogram approximately symmetric, skewed left, or skewed right? The class with the highest Questions frequency is referred to as (b) Which interval describes the center of the lengths of 1. Put 3 yellow cubes together and label them with an x, as shown. the modal class. ropes in this box? In a grouped frequency (c) Which is the modal class? How many ropes are there table or a histogram, we are x This shows that x = 3. in this class? unable to find the mode as a single data value as the data (d) What does this histogram tell you about the lengths of Place 3 yellow cubes together on the left side and 3 green cubes on theis grouped right in classes. So ropes in this box? instead, we find the modal side of the balance scale, as shown. class.

x (b) D

8 cm = A C EXERCISE 13.2 Let’s Learn to… Imagine you have to compare two 2. Add 2 green cubes to the left side of the balance scale, so we have x + 2 on the 21 cm fractions with different denominators, You can leftround side your of the answers scale. to one decimal place when necessary. epress fratios a eimals as perets say, 3 and 7 . Which is larger? At a 11 18 a ie ersa B 19 cm E glance, it is not easy to tell because the BASIC PRACTICE fi the peretage of a atit a sole fractions are not written as equivalent 5 problems iolig perets fractions with the same denominator. 1. The dot plot below showsx the weights of (a) How many students are represented in To solve this, you can convert the some sixth grade students. the dot plot? fractions or decimals to percents, which Weights of 6th Grade Students (b) State the unit of weight used in the dot Example 4 A garden, in the shape of a parallelogram, arehas fractionsan area with a denominator of 100. = plot. 2 of 54.6 m2. The perpendicular distance between the two The news media bombards us with (c) State the greatest and the lowest weights longer sides of the garden is 6.5 m. What is the length of of the students. percentages to back up their claims, so 3. Next, add 2 green cubes to the right side of the scale. Fill in the  with the each longer side? (d) Which is the most commonly occuring do the banking and financial sectors. appropriate number. weight? What other instances of percentages can 30 40 45 50 55 Solution x + 2 = 3 +  . you find in your daily life? (e) Around which values do the data cluster? Weight (kg) (f) What does this dot plot tell you about the 6.5 m 4. Now, the balance scale shows the equation x + 2 = 5.weights of these students? Area = 54.6 m² Take 2 green cubes off the left pan. (a) What happens to the scale? 220 (b) Take 2 green cubes off the right pan now. What happens to the scale? DMCC G6A C07 new.indd 189 ? 6/21/16 4:37 PM 2. The (c)frequency Fill in thetable  withbelow the showsappropriate the number. (b) What is the size of the class 5 < x , 15? The length of the side which we are finding is the base of the number of points xthat + 2 –students 2 = 3 + 2in – a sixth. (c) Are the amounts $44.99 and $45 parallelogram corresponding to the given perpendicular DMCC_G6B_Chp13 new.inddgrade 220 class scored in a ring toss game. grouped in the same interval? If not, 2/8/17 11:03 AM height. which interval is each in? Points Scored in Ring Toss Game Area of garden = length of side × perpendicular distance (d) What is the modal class of this data? 2 54.6 m = length of side × 6.5 m Area of parallelogram Points Number of Students (e) What does Chapterthe 9histogram Equations and tell inequalities you 33 = base × height about the amounts these people have 54.6m2 1 to 10 2 Length of side = spent in the supermarket? 6.5m 11 to 20 5 = 8.4 m DMCC_G6B_Chp09 new.indd 33 2/2/17 6:42 PM 6 The length of each longer side is 8.4 m. FURTHER PRACTICE 6 31 to 40 8 41 to 50 4 4. For each data set below, will a dot plot or a Try It! 4 A piece of cardboard is in the shape histogram be a better display for the data? (a) How many students played the game? of a parallelogram. If the length of Draw the suitable diagram to display the 3 (b) What is the span of each class interval? one side is 80 cm and the area of data. Then describe the distribution of the (c) What should the interval be? the cardboard is 3,600 cm2, find the data. (d) Which class has the highest number of perpendicular distance between Area = 3,600 cm² (a) The ages (in years) of children in a 80 cm students? the given pair of opposite sides. playground are (e) What percent of the students scored at least 31 points? 1 2 3 5 4 1 3 8 4 2 3 3. Mila asked people leaving the supermarket how much they spent and recorded their answers. The histogram below shows the (b) The daily high temperatures (in °C) in amount spent by people in Mila’s survey. Florida over two weeks in May are 118 Amount Spent in the Supermarket 33 30 31 32 33 31 34 32 34 29 30 32 33 30 DMCC6B_Chp11_new.indd 118 2/8/17 11:27 AM 8 6 (c) The heights (in cm) of plants in a 4 garden are

Frequency 2 Singapore Math Inc. 89 105 114 92 99 0 5 15 25 35 45 55 65 102 87 95 88 80 Amount Spent ($) 129 90 81 85 108 (a) Copy and complete the following (d) The weekly wages (in dollars) of grouped frequency table for the data. workers are Amount Spent, $x Frequency 138 126 130 135 125 133 5 < x , 15 1 124 133 123 128 130 140 135 136 125 124

55 < x , 65 2

Chapter 13 displaying andChapter comparing 13 statistics data 221

DMCC_G6B_Chp13 new.indd 221 2/8/17 11:03 AM Workbooks

8 Brainworks: Provides higher-order thinking questions Workbooks offer the necessary practice to hone that involve an open-ended approach to problem concepts covered in the textbooks. Exercises help solving. students polish their analytical skills and develop a stronger foundation. Workbooks A and B for 9 In A Nutshell: Consolidates important rules and each grade correspond to the two halves of the concepts for quick and easy review. school year.

10 Extend Your Learning Curve: Extends and applies 1 Basic Practice: Simple questions that drill concepts to problems that are investigative in nature comprehension of concepts. and engages students in independent research. 2 Further Practice: More complex questions that

MATH@ WORK 11. The distributions of the ages of 50 boys and 7 50 girls in a music school are displayed in involve direct application. 9. The masses (in grams) of 12 eggs each in the following histograms. Carton A and Carton B are displayed in the Ages of boys in the music school dot plots below. Masses of Eggs in Carton A 20 3 Challenging Practice: Hard questions that require 15 synthesis.

56 57 58 59 60 61 62 63 Frequenc y 10 Mass (g) 5

Masses of Eggs in Carton B 0 6810 12 14 Age (years) 4 Enrichment: QuestionsIntroduction that demand To Algebra higher order

3 Ages of girls in the music school 56 57 58 59 60 61 62 63 thinking, analysis, and reasoning. Mass (g) 20 (a) Describe briefly the distribution of the masses of eggs in each carton. 15 (b) Compare the two distributions. What Basic Practice

conclusions can you draw? Frequenc y 10 1 1. Simplify the following. 10. The following list shows the shot put 5 (a) (2w)2 (b) 3p × 4p distances (in metres) of 32 boys. (c) 3q2 × 5q (d) 2r × (4r)2 2 3 0 6810 12 14 (e) 12x ÷ 4 (f) 24y ÷ 2y BRAIN5.3 6.2 WORKS6.8 3.2 5.5 6.1 7.7 5.9 Age (years) (g) 21w2 ÷ 7w2 (h) 18z2 ÷ (3z)2 8 4.2 5.9 6.4 7.6 6.5 3.9 4.8 5.0 15. The figure shows a square and a shaded 6.6 5.6 5.1 6.9 5.3 7.9 6.4 4.1 rectangle.(a) What What percent is the of thearea boys of thein the shaded music 2. Simplify the following. 13. The7.2 picture3.5 below6.0 shows4.8 6.7 the 5.9national4.7 flag6.8 rectangle?school are between 6 and 10 years old? (a) 2x × 3y (b) 18y ÷ 3x of the Republic of the Congo. The flag is a How about the girls? (c) 6x ÷ 2y × 3w (d) 8y × 3y ÷ 2x rectangle(a) Group consisting the data ofusing a yellow a frequency diagonal table (b) Describe2 cm briefly each5 cm distribution of (e) p × 5q – 2 × 3r (f) 3x + 8y ÷ 2z 2 2 band, andwith with uniform a green class upper intervals triangle starting and ages. (g) (3p) + 5q × 2r (h) (5b) – 3c × 2d Further Practice red lowerfrom triangle. 3 < x , The4. area of the yellow (c) Compare the distributions of the ages 2 cmof the boys and girls of the music school. 2 band(b) inDraw the flaga withhistogram dimensions to display as shown the 3. When x = 3 and y = 5, evaluate the following expressions. 1 11. (a) Find the sum of below isgrouped 3 ft2. data. (a) 4x – 5y (b) 8y + 2x (i) 2 8x + 215y and 6x – 10y, (ii) 7a3 – 3b, –43 a + 9b, and –9a – 10b, (c) Which8 interval best describes the BRAIN WORKS (c) 3y + (2x) (d) 2y – (2x) 1 x 41x 3 center2 ft of the distances?y (e) (iii) 2(4p – 5q) and 3(–4q + 3p), (iv)(f) of (8x – 12y) and of (4x + 10y). 2 y y42 2 (d) Boys who get shot put distances less 2 2 12. Ashna was doing a survey to find the ages (b) Subtractxy + xy + than 5.0 m are sent for a training 5 cm (g) (h) of people attending a baseball game. She (i)xy – 4s + 9t from 3s – t, (ii) (8xyr – – )53w from 7w + 12r, program. How many boys should 2 1 drew a grouped frequency table for the (iii) – (3x + 9y) from (8x + 14y). attend the training program? ages with class intervals of 10 – 20, 20 – 30, 4. When x = –2,3 y = –5, and z =2 3, evaluate the following expressions. 2 1 ft (c) Subtract 7m – 8n from the sum of 7n – 8m and 20m – 9n.2z 2 30 – 40, 40 – 50, and 50 – 60. Why might (a) 2.5x – 3y + 4z (b) 3x + these class intervals be incorrect? y 2 Write in your journal 12. (c)Simplify3xy –each yz of the following. (d) 2y × (z – xy) 16. The diagram below shows a seven-piece 2x 3 (e)(a) x(32m+ –y 27)+ +z2 2(4m – 5n) – 3(1 – 2n) (b)(f) (3a + 5b – 7) + (4a – 6b + 5) (zy + )2 A student claims that points with the same x- and y-coordinatessquare puzzle. must Given that the area of the  1 2 3  3 7 1 2 lie in Quadrant I or Quadrant III. Is he correct? Explainsquare your puzzle answer. ABCD displaying is 1 mand, whatcomparing statisticsis the data 223 (c) (43p – 73 q – 39) – ( p + 5 + 3q) (d)  –3xy + 3 1– 3  –  xy – +  y 1 Chapter 13 Chapter 13 (g) x + y + z (h) –3 x2 – y 3+ z4  2 3 4 2 ft 9 2 area of each shape a, b, c, d, e, f, and g ? Enrichment(e) 5(x + 4y – 1) + 4(–4x + 6y – 2) (f) –5(3p – 2q – 8) – 4(–10 + 3p – q) l y (a) Forml an equation in y and solve it to 3 9 5. Find the  1 value1 of   5 9  8  5 3 5 2  6  DMCC_G6B_Chp13 new.indd 223 e A D 2/8/17 11:03 AM (g) 3 ab + – 2 + 4 ab + – 1 (h)  st – –  – 12st + – 3 find the value of y. The Coordinate Plane 26. 3 26p 4 8 116 5 2 4 8 3 5 h y (a) when p = 16 and q = , (b) Whats is the ratio2nd Quadrantof the width of1st Quadrantthe c q 2 t We can show the positions of g flag to its length? 2 2 22 Nu points in two-dimensional space 5 13. Simplify(b) p(R –each r ) whenof the pfollowing. = , R = 25, and r = 24, (c) Express the yellow area as a percentage d 7 a on a coordinate plane with 2nd Quadrant 4 1st Quadrant (a) 4[–2t a + 4 – 2(a + 3)] AB(b) 6w – CD5 + 3[(4 – 3w) – 2(Ew – 8)] In of the area of the flag. (c) kx when k = 5, x = 7, and t = 2, ordered pairs such as (3, 2). The (c) 4 – 7c – z2[(c + 4) + 2(2c – 5)] 6x (d)5x 2s + 9 – 3(s4 x– 5) – 2[3(32x – s) + 2(4 – 3s)] x 3 (d) (kx ++2 2y) when k = 3.5, x = 4, y = –5, and z = 3, first number is the x-coordinate P(3, 2) (e) 3[5 – 3w – 5(2w + 1)] (f) –y + 3x + 2[3x – y + 2( y – 2x)] a2 e k 1 14. In the diagramand the below, second parallelogramsnumberOrigin is the In(g)(e) the4(3 figure,p + when7q) ABCDE– k5[4 = p3 –and is(q +xa =4portionp ) ,+ 5q ]of a road from(h) the–21 exitm + 8An of– 3[2(an mexpressway – 2n) – 3(3 tom –a 2buildingn)] E. Q(– 4, 1) ()x 3 4 ABCD, EBCFy-coordinate., and GBCH shareThe asigns common of 1 f AB = 6x km, BC = 5x km, CD = 4x km, and DE = 2x km. A car drives at the speed limits, 111 1 1 1 base, BC, andthe their coordinates opposite sidesare aredifferent, all on x i.e.,(f) 100 +km/hr, + 90 when km/hr, a = 60, bkm/hr, = – ,and and 50c = km/hr. in each section from A to E respectively. Let O 14. (a) (i) Simplify the expression 3a + 9 – 5a – 6. a line paralleldepending to the base. on the quadrant in −5 −4 −3 −2 −1 1 2 3 4 5 abc 21 5 9 3rd Quadrant 4th Quadrant −1 b T minutes(ii) Hence, be the findtime the taken value by ofthe the car expression to reach E when from a A =. 2.5. which the point is located. S(4, –1) (a) Express T in terms of x. G H A D E F −2 (b) (i) Simplify the expression 2(4b – 7c) – 3(2c – 3b). R(–3, –2) (b) When x = 0.45, find the value of T. 1 15 B −3 C (ii) Hence, find the value of the expression when b = –6 and c = . 2 3rd Quadrant 4th Quadrant x x −4 27. The(c) (i)sides Simplifyof ABC the are expression AB = (3x + (64)y cm,– 9) BC – =(8 (4y x– –6). 5) cm, and CA = (x + 13) cm. 3 2 −5 (a) Express the perimeter of ABC in terms of x. Give the answer in factored form. (ii) Hence, find the value of the expression when x = 5 and y = –3. (b) A square PQRS has the same perimeter as ABC. Express the length of PQ in terms of x. 3 1 3 (c)(d) When(i) Simplify x = 7, find the expression p – q + (2p – q). 5 4 10 (i) the perimeter of ABC, Challenging (ii) Hence, Practice find the value of the expression when p = 15 and q = –10. 10 (ii) the area of PQRS. PairsB and LinesC Equations, Table of Values and 4 (e) (i) Simplify the expression 40 – z – 3[2(4 + 3z) – 3(3z – 1)]. Graphs (ii) Hence, find the value of the expression when z = 4. • The(a) If two middleDo points the pointparallelograms have of the this same line x -coordinate,haveis (6, 9),the half-way same they between P(3, 3) and Q(9, 15). Without 24. The following table shows Kenneth’s results in 4 tests. lie on a vertical line. 28. plottingheight? the points Explain on youra graph answer. paper, how would you • find We the can coordinates graph equationsof the point on the • If two points have the same y-coordinate, they coordinate plane by making a table of x Test Number Score Maximum Possible Score R(b), which Are is theone-third areas way offrom the P to Qthree? What are the coordinates of the point S, 15. Express each of the following in its simplest form. whichlie on ais horizontal one-third line. way from Q to P? and y values. 2x + 1 x – 3 x 4y – 3 y – 5 parallelograms equal? Explain why. (a) 1 + 6.5 (b) 10 – • We can use the absolute values to find the • An equation whose graph is a straight 3 4 3 2 distance between points on a horizontal or line is called a linear equation. The 4z + 22 1 – 5z 12 3(202 – 3)w 6(4w – 3) Q(9, 15) (c) + x (d) + vertical line. graph of the equation shows all the 4 3 5 19 252 5 122 possible values for x and y that satisfy 3(4p + 5) 2(3p + 1) q + 5 2q + 7 (e) 4 – 28 (f) 40 + – 1 S the equation. 5 3 112 5 Independent and Dependent Variables • We can use equations, tables, and 2(2pq – ) 3(qp + 4 ) 1  mm + 2 mn – 3 mn +  (a)(g) In which –test was Kenneth’s + performancex the best?(h) Explain121  your – answer. –  DMCC6B_Chp11_new.indd 122 (6, 9) graphs to show the relationship between 2/8/17 11:27 AM 3 2 4  3 6 2  • An independent variable is one whose values (b) For each test, grade ‘A’ is given if the score is more than or equal to 70% of the maximum possible quantities in real-life situations. you can freely choose and have control over. It (a) Thescore. figure Find, shows as a percentage, 1 square tile the of number x by x ofunits, times 5 rectangularKenneth was tiles given of xgrade by 1 ‘A’.unit, and 6 square tiles R y (c) ofSuppose 1 by 1 thatunit. 67.5% Arrange of thethe tilesstudents to form in Kenneth’sa rectangle class and statewere itsgiven dimensions. grade ‘A’ at least once in the does not depend on other variables. 2 A dependent variable is one whose values 7 (b) Hence,4 tests. orFind otherwise, the number express of studentsx + 5x +who 6 in were the formnot given (x + agrade)(x + b‘A’), where in any a ofand the b aretests integers. if there are 2 depend on the independent variable.P(3, 3) (c) Express40 students x + in 8 xthe + 15class. in the form (x + p)(x + q), where p and q are integers. 23 6 independent variable 5 3 3 Hint: Consider the change in x and y coordinates, then use proportions to work out 29.25. The(a) volumesA fruit crate of two contains glasses a of mix water of are80 apples(7ax – and3bx +oranges. 6ay – 4 Ifby )21.25% cm and of (11 thebx fruits + 5ax are – 6rotten,by – 21 finday) cmthe 04_G7A_DMWK_Ch04 new.indd 23 3 6/25/12 6:27 PM the required( x , ycoordinates. ) 4 respectively.number ofLet rotten V cm fruits. be the total volume of water in the two glasses. (a) Express V in terms of a, b, x, and y in1 factored form. 3 (b) Suppose that 30% of the apples and of the oranges are rotten. Find the number of (b) If both x and y are doubled, determine5 whether V will be doubled. dependent variable 2 (i) rotten apples, When drawing graphs to represent a relationship (ii) rotten oranges. between two variables, the independent variable is 1 (c) Hence, express the number of apples as a percentage of graphed on the horizontal axis and the dependent x (i) the number of fruits, O variable is graphed on the vertical axis. 1 2 Chapter3 104 Coordinates5 6 and graphs 107 (ii) the number of oranges. singaporemath.comIn general, the x-coordinate is the independent variable and the y-coordinate is the dependent 26. Eligible clients of a bank are offered 2 repayment schemes for a one-year loan. DMCC6B_Chp10 new.inddvariable. 107 2/8/17 12:08 PM Scheme A: Pay $50 and 105% of the loan at the end of the one-year period Scheme B: Pay 103% of the sum of $200 and the loan at the end of the one-year period 106 (a) (i) Which is a better scheme for Mr. Martin to use if he is eligible for the loan and wants to borrow 26 $10,000? (ii) How much will he save if he selects the better scheme?

DMCC6B_Chp10 new.indd 106 2/8/17 12:08 PM (b) Mr. Carter, another eligible client, also borrowed from the bank. Find his loan amount if his payment 04_G7A_DMWK_Ch04 new.inddby either 26 of the schemes is the same. 6/25/12 6:27 PM

27. (a) If X is 25% less than Y, by how many percent is Y more than X? (b) If X is 25% more than Y, by how many percent is Y less than X? (c) If X is decreased by 10% and then increased by 10%, find the percentage change in X. (d) If Y is increased by 10% and then decreased by 10%, find the percentage change in Y.

43

07_G7A_DMWK_Ch07 new.indd 43 6/25/12 6:52 PM Teacher’s Workbook Guides, Notes Solutions and Solutions

Teacher’s Guides (Grade 6) provide teaching Workbook Solutions (Grades 6–8) contain fully worked suggestions and important information for educators to solutions for problems in workbooks. help students achieve math mastery.

11. Method 1 2.2C Division of a Fraction by a Fraction 7 Teaching Notes and Solutions (Grades 7–8) contain lb 8 Basics

detailed notes and fully worked solutions for all 14. (a) 15 3

questions and problems in textbooks. From the model, 7 7 units lb 8 7 1 1 unit lb ÷ 7 = lb 5 5 5 8 8 3 3 3 Method 2 1 From the model, there are 3 groups Lesson 6 7 7 1 1 5 lb ÷ 7 = lb × = lb 8 8 7 8 of 3 . 1 15 5 Therefore, 3 ÷ 3 = 3. 1 lb of rice is in each bag. 8 Objective: (b) 2 r It r 4 It is impossible to write every solution for an inequality because inequalities have an infinite number of solutions. Since a number line extends infinitely 7 7 1 7 • Graph inequalities using a number line. in each direction, we can graph all the solutions of an inequality on a number 12. yd ÷ 9 = yd × = yd line. 9 9 9 81 The number line below shows the solution to x Ͼ –3. The open circle 7 indicates that the solution does not include –3. Each piece is yd long. REMARKS 81 1. Introduction The arrow indicates that there are an infinite number of solutions in a given direction. Even the numbers that are ï ï ïï ï 0 12 not shown on the number line 1 1 Draw a number line similar to the one on Every number to the right of –3 is a solution to x Ͼ –3. here, such as 5 and 100, are –3 is not part of the solution because –3 = –3. part of the solution. Challenge textbook page 46. Ask students: 4 4 The number line below shows the solution to x р 5. The closed circle 1 indicates that the solution includes 5. 1 . • Where would the solutions to x > −3 be on the From the model, there are 2 groups of 4 REMARKS 13. 5 2 1 line? (To the right of −3.) The symbol “р” means “less than or equal to,” and the Therefore, ÷ = 2. ï 0 1 32 4567 symbol “у” means “greater 4 4 • Would −3 be part of the solution? (No, because than or equal to.” Every number to the left of 5 is a solution to x р 5. all solutions must be greater than −3.) 5 is part of the solution because 5 = 5. (c) 3 4 Repeat with x < 3. 18 Graph the solution for x Ͼ –1.

Introduce the symbol ≥ (greater than or equal t Draw a number line that includes –1 and several integers washing to the right and left of –1. Then, draw an open circle above to). Ask students where the solutions to x ≥ −3 –1 and an arrow going to the right. machine 4 would be on the line, and how the solution is 5 ïï ï 10 2 34 different from x > −3 (i.e., it includes −3). 1 1 1 Try It! 18 Graph the solution for each of following inequalities. (a) x Ͼ – 4 4 4 4 Read and discuss the top of page 46 and the (b) x у 8 1 From the model, there are 3 groups of . REMARKS. 4 3 1 Student Textbook page 46 ? Therefore, ÷ = 3. DMCC_G6B_Chp09 new.indd 46 1/24/17 12:35 PM 4 4 Give students graph paper and rulers. Have them draw number lines to graph the solutions (d) 1 2. Development to x > −3, x < −3, x ≥ −3, and x ≤ −3. Make sure 2 students understand that we use an open circle Have students study Examples 18–20 and do when we have > or <, and a closed circle when Try It! 18–20 on their own, and then compare we have ≥ or ≤. their solutions with partners or in a group. ? 4 4 1 2 Notes: Note: ÷ 2 = × = 2 • Remind students that on a horizontal number • Make sure students are using the open and 5 5 2 5 4 2 2 1 2 2 line, the numbers become greater going to the closed circles correctly, that the arrows ÷ 3 = × = . 5 5 3 15 From the model, there is 1 group of 4 right and become less going to the left. are going in the proper direction, and that 2 1 2 Therefore, ÷ = 1 Each child received of her savings. 4 2 • The solution includes all the values indicated they are graphing the solutions neatly and 15 by the arrow, not just whole number values. accurately. 1 Thus, the solution to x < 5 includes 4.3, −2, etc. Try It! 18 Answers Chapter 2 fractions 17 • When graphing the solutions to inequalities, (a) make sure students draw the number lines directly on top of the horizontal lines on −5 −4 −3 −2 −1 0 1 2 the graph paper, and use the vertical lines (b) to draw the hash marks for the integers. 5 6 7 8 9 10 11 12 Emphasize neatness and accuracy.

60 ©2017 Singapore Math Inc. Dimensions Math® Teacher’s Guide 6B

Singapore Math Inc. Scope & Sequence

6A 7A 8A

1 Whole Numbers 1 Factors and Multiples 1 Exponents and Scientific Notation

2 Fractions 2 Real Numbers 2 Linear Equations in Two Variables

3 Decimals 3 Introduction to Algebra 3 Expansion and Factorization of Algebraic Expressions 4 Negative Numbers 4 Algebraic Manipulation 4 Quadratic Factorization and 5 Ratios 5 Simple Equations in One Variable Equations 6 Rate 56 Ratio, Rate, and Speed 5 Simple Algebraic Fractions 7 Percent 7 Percentage 6 Congruence and Similarity 6B 8 Angles, Triangles, and 7 Parallel Line and Angles in Quadrilaterals 8 Algebraic Expressions Triangles and Polygons 7B 9 Equations and Inequalities 8B 9 Number Patterns 10 Coordinates and Graphs 8 Graphs of Linear and 10 Coordinates and Linear Graphs Quadratic Functions 11 Area of Plane Figures 11 Inequalities 9 Graphs in Practical Situations 12 Volume and Surface Area of Solids 12 Perimeters and Areas of 10 Pythagorean Theorem 13 Displaying and Comparing Data Plane Figures 11 Coordinate Geometry 13 Volumes of Surface Areas 12 Mensuration of Pyramids, of Solids Cylinders, Cones and Spheres 14 Proportions 13 Data Analysis 15 Data Handling 14 More About Quadric Equations 16 Probability of Simple Events

17 Probability of Combined Events

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Dimensions Math 6–8 brings the highly effective Singapore math method to middle school. This rigorous series includes pre-algebra, algebra, geometry, data analysis, probability, and some advanced math topics. Set your students up for success in higher math with a strong Singapore math foundation!

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