Dimensions Math®

6–8 Curriculum

Singapore Math Inc.

The Singapore math method for middle school.

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

TIMSS

SingaporeUnited States

633

Knowing Applying Reasoning

619 616

528 515 514

Average Math Cognitive Domain Scores

54%Advanced

BenchmarkHigh

BenchmarkIntermediateBenchmark

LowBenchmark

81% 94% 99%

10% 37% 70% 91%

Percentage of Students Reaching International Benchmarks in Mathematics

Grade 8

Grade 8

629

AlgebraNumber Geometry Data and Chance

623 617 617

520 525 500 522

Average Math Content Domain Scores

Grade 8

singaporemath.com

TIMSS 2015 International Results in Mathematics

TheApproach

WhySingapore Math?

The intentional progression of concepts in the Singapore math approach instills a deep understanding of mathematical thinking. Students transition from using bar models to understand algebraic concepts to using equations to solve algebraic word problems. Key features include:

CPA (Concrete Pictorial Abstract) Approach: Introduces concepts in a tangible way and progresses to increasing levels of abstraction.

Bar Modeling: Helps students visualize a range of math concepts. Allows students to determine the knowns and unknowns in a given situation.

The Singapore math approach is a highly effective teaching approach based on research of math mastery in Singapore, which consistently ranks at the top in international math testing. Our Singapore Math® curricula aim to raise U.S. student performance internationally and at home on standardized and state assessments.

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TextbooksTextbooks provide a systematic, well-rounded approach to math that facilitates the internalization of concepts and instills curiosity. Lessons engage students with different levels of problem solving and real world application of math topics. Textbooks A and B for each grade correspond to the two halves of the school year.

Chapter Opener: Introduces a topic through a real world example and identifies learning objectives.

Example: Helps students understand and master a concept through a worked example.

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Singapore Math Inc.

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2

(b)

19 cm

8 cm

21 cm

D

E

C

B

A

A garden, in the shape of a parallelogram, has an area of 54.6 m2. The perpendicular distance between the two longer sides of the garden is 6.5 m. What is the length of each longer side?

Solution

6.5 mArea = 54.6 m²

?

The length of the side which we are finding is the base of the parallelogram corresponding to the given perpendicular height.

Area of garden = length of side × perpendicular distance 54.6 m2 = length of side × 6.5 m Area of parallelogram

= base × height

Length of side = 54.6 m6.5 m

2

= 8.4 m

The length of each longer side is 8.4 m.

A piece of cardboard is in the shape of a parallelogram. If the length of one side is 80 cm and the area of the cardboard is 3,600 cm2, find the

perpendicular distance between the given pair of opposite sides.

Area = 3,600 cm² 80 cm

Example 4

Try It! 4

118

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6

5

Objective: To explore the properties of equations.

Materials: Balance scale labeled with an equal sign in the middle. Two different colored connecting cubes such as yellow and green to

represent the variables and constants. Small stickers with x to label the yellow (variable) cubes and = to

label the middle of the scale.

Questions1. Put 3 yellow cubes together and label them with an x, as shown.

x This shows that x = 3.

Place 3 yellow cubes together on the left side and 3 green cubes on the right side of the balance scale, as shown.

=

x

2. Add 2 green cubes to the left side of the balance scale, so we have x + 2 on the left side of the scale.

=

x

3. Next, add 2 green cubes to the right side of the scale. Fill in the with the appropriate number.

x + 2 = 3 +  .

4. Now, the balance scale shows the equation x + 2 = 5. Take 2 green cubes off the left pan. (a) What happens to the scale? (b) Take 2 green cubes off the right pan now. What happens to the scale? (c) Fill in the with the appropriate number.

x + 2 – 2 = 3 + 2 –  .

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Chapter 9 Equations and inequalities 33

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Chapter 13 statistics 221Chapter 13 displaying and comparing data 221

2. The frequency table below shows the number of points that students in a sixth grade class scored in a ring toss game.

Points Scored in Ring Toss Game

Points Number of Students

1 to 10 2

11 to 20 5

6

31 to 40 8

41 to 50 4

(a) How many students played the game? (b) What is the span of each class interval? (c) What should the interval be? (d) Which class has the highest number of

students? (e) What percent of the students scored at

least 31 points?

3. Mila asked people leaving the supermarket how much they spent and recorded their answers. The histogram below shows the amount spent by people in Mila’s survey.

Amount Spent in the Supermarket

Amount Spent ($)

Freq

uenc

y

0 5 15 25 35 45 55 65

24

68

(a) Copy and complete the following grouped frequency table for the data.

Amount Spent, $x Frequency

5 < x , 15 1

55 < x , 65 2

(b) What is the size of the class 5 < x , 15? (c) Are the amounts $44.99 and $45

grouped in the same interval? If not, which interval is each in?

(d) What is the modal class of this data? (e) What does the histogram tell you

about the amounts these people have spent in the supermarket?

FURTHER PRACTICE

4. For each data set below, will a dot plot or a histogram be a better display for the data? Draw the suitable diagram to display the data. Then describe the distribution of the data.

(a) The ages (in years) of children in a playground are

1 2 3 5 4 1

3 8 4 2 3 (b) The daily high temperatures (in °C) in

Florida over two weeks in May are

33 30 31 32 33 31 34

32 34 29 30 32 33 30

(c) The heights (in cm) of plants in a garden are

89 105 114 92 99

102 87 95 88 80

129 90 81 85 108

(d) The weekly wages (in dollars) of workers are

138 126 130 135 125 133

124 133 123 128 130

140 135 136 125 124

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

Try It!: Gives students an opportunity to answer a similar question to check how well they have grasped the concept.

Class Activity: Introduces new concepts through cooperative learning methods.

Basic Practice: Provides simple questions that involve the direct application of concepts.

Further Practice: Provides more challenging questions that involve the direct application of concepts.

Math@Work: Provides questions that involve the application of integrated concepts to practical situations.

Let’s Learn to…1 express fractions and decimals as percents

and vice versa

2 find the percentage of a quantity and solve problems involving percents

Percent7

Imagine you have to compare two fractions with different denominators,

say, 311

and 718

. Which is larger? At a

glance, it is not easy to tell because the fractions are not written as equivalent fractions with the same denominator. To solve this, you can convert the fractions or decimals to percents, which are fractions with a denominator of 100.

The news media bombards us with percentages to back up their claims, so do the banking and financial sectors. What other instances of percentages can you find in your daily life?

DMCC G6A C07 new.indd 189 6/21/16 4:37 PM220

You can round your answers to one decimal place when necessary.

220

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

Histogram A

Length (cm)

Freq

uenc

y

0 50 60 70 80 90 100

510152025

Histogram B

Length (cm)

Freq

uenc

y

0 50 60 70 80 90 100

510152025

For each of the histograms, answer the following questions. (a) Is the shape of this histogram approximately

symmetric, skewed left, or skewed right? (b) Which interval describes the center of the lengths of

ropes in this box? (c) Which is the modal class? How many ropes are there

in this class? (d) What does this histogram tell you about the lengths of

ropes in this box?

Try It! 11

EXERCISE 13.2

BASIC PRACTICE

1. The dot plot below shows the weights of some sixth grade students.

Weight (kg)

40 45 50 5530

Weights of 6th Grade Students

(a) How many students are represented in the dot plot?

(b) State the unit of weight used in the dot plot.

(c) State the greatest and the lowest weights of the students.

(d) Which is the most commonly occuring weight?

(e) Around which values do the data cluster? (f) What does this dot plot tell you about the

weights of these students?

The class with the highest frequency is referred to as the modal class.

In a grouped frequency table or a histogram, we are unable to find the mode as a single data value as the data is grouped in classes. So instead, we find the modal class.

REMARK

DMCC_G6B_Chp13 new.indd 220 2/8/17 11:03 AM

WorkbooksWorkbooks offer the necessary practice to hone concepts covered in the textbooks. Exercises help students polish their analytical skills and develop a stronger foundation. Workbooks A and B for each grade correspond to the two halves of the school year.

Basic Practice: Simple questions that drill comprehension of concepts.

Further Practice: More complex questions that involve direct application.

Challenging Practice: Hard questions that require synthesis.

Enrichment: Questions that demand higher order thinking, analysis, and reasoning.

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singaporemath.com

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Chapter 13 statistics 223Chapter 13 displaying and comparing data 223

MATH@ WORK

9. The masses (in grams) of 12 eggs each in Carton A and Carton B are displayed in the dot plots below.

Mass (g)

Masses of Eggs in Carton A

56 58 60 6261 635957

Mass (g)56 58 60 6261 635957

Masses of Eggs in Carton B

(a) Describe briefly the distribution of the masses of eggs in each carton.

(b) Compare the two distributions. What conclusions can you draw?

10. The following list shows the shot put distances (in metres) of 32 boys.

5.3 6.2 6.8 3.2 5.5 6.1 7.7 5.94.2 5.9 6.4 7.6 6.5 3.9 4.8 5.06.6 5.6 5.1 6.9 5.3 7.9 6.4 4.17.2 3.5 6.0 4.8 6.7 5.9 4.7 6.8

(a) Group the data using a frequency table with uniform class intervals starting from 3 < x , 4.

(b) Draw a histogram to display the grouped data.

(c) Which interval best describes the center of the distances?

(d) Boys who get shot put distances less than 5.0 m are sent for a training program. How many boys should attend the training program?

11. The distributions of the ages of 50 boys and 50 girls in a music school are displayed in the following histograms.

0

10

15

5

6 8 10 12 14

20

Ages of boys in the music school

Age (years)

Freq

uenc

y

0

10

15

5

6 8 10 12 14

20

Ages of girls in the music school

Age (years)

Freq

uenc

y

(a) What percent of the boys in the music school are between 6 and 10 years old? How about the girls?

(b) Describe briefly each distribution of ages.

(c) Compare the distributions of the ages of the boys and girls of the music school.

BRAIN WORKS

12. Ashna was doing a survey to find the ages of people attending a baseball game. She drew a grouped frequency table for the ages with class intervals of 10 – 20, 20 – 30, 30 – 40, 40 – 50, and 50 – 60. Why might these class intervals be incorrect?

DMCC_G6B_Chp13 new.indd 223 2/8/17 11:03 AM

8 BRAIN WORKS

13. The picture below shows the national flag of the Republic of the Congo. The flag is a

rectangle consisting of a yellow diagonal band, and with a green upper triangle and red lower triangle. The area of the yellow band in the flag with dimensions as shown

below is 3 18

ft2.

y

y

2 12

ft

2 12

ft

2 12

ft

(a) Form an equation in y and solve it to find the value of y.

(b) What is the ratio of the width of the flag to its length?

(c) Express the yellow area as a percentage of the area of the flag.

14. In the diagram below, parallelograms ABCD, EBCF, and GBCH share a common base, BC, and their opposite sides are all on a line parallel to the base.

G H A D E

B C

F

(a) Do the parallelograms have the same height? Explain your answer.

(b) Are the areas of the three parallelograms equal? Explain why.

15. The figure shows a square and a shaded rectangle. What is the area of the shaded rectangle?

5 cm2 cm

2 cm

5 cm

16. The diagram below shows a seven-piece square puzzle. Given that the area of the square puzzle ABCD is 1 m2, what is the area of each shape a, b, c, d, e, f, and g ?

D

C

A

B

c

d

b

e

f

a

g

122

DMCC6B_Chp11_new.indd 122 2/8/17 11:27 AM

1

15

3 Introduction To Algebra

Basic Practice

1. Simplify the following.(a) (2w)w)w 2 (b) 3p 3p 3 × 4p 4p 4(c) 3q2 × 5q (d) 2r × (4r)r)r 2

(e) 12x2 ÷ 4 (f) 24y24y24 3 ÷ 2y÷ 2y÷ 2(g) 21w2 ÷ 7w2 (h) 18z2 ÷ (3z)z)z 2

2. Simplify the following.(a) 2x × 3y 3y 3 (b) 18y18y18 ÷ 3x(c) 6x ÷ 2y÷ 2y÷ 2 × 3w (d) 8y8y8 × 3y 3y 3 ÷ 2x(e) p × 5q – 2 × 3r (f) 3x + 8y+ 8y+ 8 ÷ 2z(g) (3p(3p(3 )p)p 2 + 5q × 2r (h) (5b)b)b 2 – 3c × 2d

3. When x = 3 and y = 5, evaluate the following expressions.(a) 4x4x4 – 5x – 5x y (b) 8y8y8 + 2x(c) 3y3y3 2 + (2x)x)x 2 (d) 2y2y2 3 – (2x)x)x 3

(e) xy

(f) xy4

2

(g) x yx y + –

(h) x yx y

+ ( – )

2 2

3

4. When x = –2, x = –2, x y = –5, and z = 3, evaluate the following expressions.

(a) 2.5x – 3y– 3y– 3 + 4z (b) 3x + x + x zy

2

(c) 3xy – yz (d) 2y 2y 2 × (z (z ( 2 – xy)xy)xy

(e) x2 + y2 + z2 (f) xz y2

( + )

3

2

(g) x3 + y3 + z3 (h) –3x3 – y3 + 19

z3

5. Find the value of

(a) pq23

when p = 16 and q = 12

,

(b) p(R(R( 2 – r2) when p = 227

, R = 25, and r = 24,

(c) kxt when t when t k = 5, k = 5, k x = 7, and x = 7, and x t = 2,t = 2,t(d) (kx + 2kx + 2kx y + 2y + 2 )y)y z when k = 3.5, k = 3.5, k x = 4, x = 4, x y = –5, and z = 3,

(e) kx( )3

when k = 3 and k = 3 and k x = x = x 14

,

(f) + + a b c1 1 1 when a = 1

21, b = – 1

5, and c = 1

9.

2

23

Further Practice

11. (a) Find the sum of (i) 8x + 15y and 6x – 10y, (ii) 7a – 3b, –4a + 9b, and –9a – 10b,

(iii) 2(4p – 5q) and 3(–4q + 3p), (iv) 14

of (8x – 12y) and 32

of (4x + 10y).

(b) Subtract(i) 4s + 9t from 3s – t, (ii) 8r – 5w from 7w + 12r,

(iii) – 23

(3x + 9y) from 12

(8x + 14y).

(c) Subtract 7m – 8n from the sum of 7n – 8m and 20m – 9n.

12. Simplify each of the following.(a) (3m – 7) + 2(4m – 5n) – 3(1 – 2n) (b) (3a + 5b – 7) + (4a – 6b + 5)

(c) (4p – 7q – 9) – ( p + 5 + 3q) (d)

x y– + – 12

23

34

–

x y – + 32

73

14

(e) 5(x + 4y – 1) + 4(–4x + 6y – 2) (f) –5(3p – 2q – 8) – 4(–10 + 3p – q)

(g) 3

a b + – 216

14

+ 4

a b + – 158

916

(h) 85

s t – – 52

34

58

– 23

s t12 + – 365

13. Simplify each of the following.(a) 4[–2a + 4 – 2(a + 3)] (b) 6w – 5 + 3[(4 – 3w) – 2(w – 8)](c) 4 – 7c – 2[(c + 4) + 2(2c – 5)] (d) 2s + 9 – 3(s – 5) – 2[3(3 – s) + 2(4 – 3s)](e) 3[5 – 3w – 5(2w + 1)] (f) –y + 3x + 2[3x – y + 2( y – 2x)](g) 4(3p + 7q) – 5[4p – (q + 4p) + 5q] (h) –21m + 8n – 3[2(m – 2n) – 3(3m – 2n)]

14. (a) (i) Simplify the expression 3a + 9 – 5a – 6. (ii) Hence, find the value of the expression when a = 2.5. (b) (i) Simplify the expression 2(4b – 7c) – 3(2c – 3b).

(ii) Hence, find the value of the expression when b = –6 and c = 12

.

(c) (i) Simplify the expression x3

(6y – 9) – x2

(8y – 6). (ii) Hence, find the value of the expression when x = 5 and y = –3.

(d) (i) Simplify the expression 35

p – 14

q + 310

(2p – q).

(ii) Hence, find the value of the expression when p = 15 and q = –10. (e) (i) Simplify the expression 40 – z – 3[2(4 + 3z) – 3(3z – 1)]. (ii) Hence, find the value of the expression when z = 4.

15. Express each of the following in its simplest form.

(a) x2 + 13

+ x – 34 (b) y4 – 3

3 – y – 5

2

(c) z4 + 24

+ z1 – 55

(d) w3(2 – 3 )2

+ w6(4 – 3)5

(e) p3(4 + 5)5

– p2(3 + 1)3

(f) q + 52

+ q2 + 75

– 1

(g) p q2(2 – )3

– q p3( + 4 )2

+ 14

(h) 12

– – m m m n m n + 23

– 36

+ 2

04_G7A_DMWK_Ch04 new.indd 23 6/25/12 6:27 PM

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26

Enrichment

26.

A B C D E6x 5x 4x 2x

In the figure, ABCDE is a portion of a road from the exit A of an expressway to a building E. AB = 6x km, BC = 5x km, CD = 4x km, and DE = 2x km. A car drives at the speed limits, i.e., 100 km/hr, 90 km/hr, 60 km/hr, and 50 km/hr in each section from A to E respectively. Let T minutes be the time taken by the car to reach E from A.

(a) Express T in terms of x. (b) When x = 0.45, find the value of T.

27. The sides of ABC are AB = (3x + 4) cm, BC = (4x – 5) cm, and CA = (x + 13) cm. (a) Express the perimeter of ABC in terms of x. Give the answer in factored form. (b) A square PQRS has the same perimeter as ABC. Express the length of PQ in terms of x. (c) When x = 7, find (i) the perimeter of ABC, (ii) the area of PQRS.

28.

1 1

1x

x

x

(a) The figure shows 1 square tile of x by x units, 5 rectangular tiles of x by 1 unit, and 6 square tiles of 1 by 1 unit. Arrange the tiles to form a rectangle and state its dimensions.

(b) Hence, or otherwise, express x2 + 5x + 6 in the form (x + a)(x + b), where a and b are integers. (c) Express x2 + 8x + 15 in the form (x + p)(x + q), where p and q are integers.

29. The volumes of two glasses of water are (7ax – 3bx + 6ay – 4by) cm3 and (11bx + 5ax – 6by – 21ay) cm3 respectively. Let V cm3 be the total volume of water in the two glasses.

(a) Express V in terms of a, b, x, and y in factored form. (b) If both x and y are doubled, determine whether V will be doubled.

04_G7A_DMWK_Ch04 new.indd 26 6/25/12 6:27 PM

4

43

Challenging Practice

24. The following table shows Kenneth’s results in 4 tests.

Test Number Score Maximum Possible Score1 6.5 102 12 203 19 254 28 40

(a) In which test was Kenneth’s performance the best? Explain your answer.(b) For each test, grade ‘A’ is given if the score is more than or equal to 70% of the maximum possible

score. Find, as a percentage, the number of times Kenneth was given grade ‘A’.(c) Suppose that 67.5% of the students in Kenneth’s class were given grade ‘A’ at least once in the

4 tests. Find the number of students who were not given grade ‘A’ in any of the tests if there are 40 students in the class.

25. (a) A fruit crate contains a mix of 80 apples and oranges. If 21.25% of the fruits are rotten, find the number of rotten fruits.

(b) Suppose that 30% of the apples and 15

of the oranges are rotten. Find the number of(i) rotten apples, (ii) rotten oranges.

(c) Hence, express the number of apples as a percentage of(i) the number of fruits, (ii) the number of oranges.

26. Eligible clients of a bank are offered 2 repayment schemes for a one-year loan.Scheme A: Pay $50 and 105% of the loan at the end of the one-year periodScheme B: Pay 103% of the sum of $200 and the loan at the end of the one-year period

(a) (i) Which is a better scheme for Mr. Martin to use if he is eligible for the loan and wants to borrow $10,000?

(ii) How much will he save if he selects the better scheme?(b) Mr. Carter, another eligible client, also borrowed from the bank. Find his loan amount if his payment

by either of the schemes is the same.

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.

07_G7A_DMWK_Ch07 new.indd 43 6/25/12 6:52 PM

9The Coordinate Plane

We can show the positions of points in two-dimensional space on a coordinate plane with ordered pairs such as (3, 2). The first number is the x-coordinate and the second number is the y-coordinate. The signs of the coordinates are different, depending on the quadrant in which the point is located.

1st Quadrant

4th Quadrant

2nd Quadrant

3rd Quadrant

x

y

−3

− 4

−5

−2

−1

1

2

3

4

5

1O 2 3 4 5−1−2−3− 4−5

P(3, 2)

S(4, –1)

R(–3, –2)

Q(– 4, 1)

Pairs and Lines

• Iftwopointshavethesamex-coordinate, they lie on a vertical line.

• Iftwopointshavethesamey-coordinate, they lie on a horizontal line.

• We can use the absolute values to find thedistance between points on a horizontal or vertical line.

Independent and Dependent Variables

• An independent variable is onewhose valuesyou can freely choose and have control over. It does not depend on other variables.

A dependent variable is one whose values depend on the independent variable.

independent variable

( x , y )

dependent variable

When drawing graphs to represent a relationship between two variables, the independent variable is graphedonthehorizontalaxisandthedependentvariableisgraphedontheverticalaxis.In general, the x-coordinate is the independent variable and the y-coordinate is the dependent variable.

In a Nutshell

Equations, Table of Values and Graphs

• We can graph equations on thecoordinate plane by making a table of x and y values.

• An equationwhose graph is a straightline is called a linear equation. The graph of the equation shows all the possible values for x and y that satisfy the equation.

• We can use equations, tables, andgraphs to show the relationship between quantities in real-life situations.

x

1

3

4

5

6

7

1 2 4 5 6O

y

2

3

106

DMCC6B_Chp10 new.indd 106 2/8/17 12:08 PM

10The middle point of this line is (6, 9), half-way between P(3, 3) and Q(9, 15). Without plotting the points on a graph paper, how would you find the coordinates of the point R, which is one-third way from P to Q? What are the coordinates of the point S, which is one-third way from Q to P?

P(3, 3)

(6, 9)

Q(9, 15)

R

S

Hint: Consider the change in x and y coordinates, then use proportions to work out the required coordinates.

A student claims that points with the same x- and y-coordinates must lieinQuadrantIorQuadrantIII.Ishecorrect?Explainyouranswer.

1st Quadrant

4th Quadrant

Origin

2nd Quadrant

3rd Quadrant

x

y

Write in your journal

Chapter 10 Coordinates and graphs 107

DMCC6B_Chp10 new.indd 107 2/8/17 12:08 PM

Brainworks: Provides higher-order thinking questions that involve an open-ended approach to problem solving.

In A Nutshell: Consolidates important rules and concepts for quick and easy review.

Extend Your Learning Curve: Extends and applies concepts to problems that are investigative in nature and engages students in independent research.

Singapore Math Inc.

Workbook Solutions

Teacher’s Guides, Notes and SolutionsTeacher’s Guides (Grade 6) provide teaching suggestions and important information for educators to help students achieve math mastery.

Teaching Notes and Solutions (Grades 7–8) contain detailed notes and fully worked solutions for all questions and problems in textbooks.

Workbook Solutions (Grades 6–8) contain fully worked solutions for problems in workbooks.

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

The arrow indicates that there are an infinite number of solutions in a given direction. Even the numbers that are not shown on the number line here, such as 5 and 100, are part of the solution.

REMARKS

The symbol “�” means “less than or equal to,” and the symbol “�” means “greater than or equal to.”

REMARKS

B Graphing Inequalities Using a Number LineIt is impossible to write every solution for an inequality because inequalities have an infinite number of solutions. Since a number line extends infinitely in each direction, we can graph all the solutions of an inequality on a number line.

The number line below shows the solution to x � –3. The open circle indicates that the solution does not include –3.

1 20

Every number to the right of –3 is a solution to x � –3. –3 is not part of the solution because –3 = –3.

The number line below shows the solution to x � 5. The closed circle indicates that the solution includes 5.

1 32 4 5 6 70

Every number to the left of 5 is a solution to x � 5. 5 is part of the solution because 5 = 5.

Graph the solution for x � –1.

Solution Draw a number line that includes –1 and several integers to the right and left of –1. Then, draw an open circle above –1 and an arrow going to the right.

210 3 4

Graph the solution for each of following inequalities.

(a) x � – 4 (b) x � 8

Example 18

Try It! 18

46

DMCC_G6B_Chp09 new.indd 46 1/24/17 12:35 PMStudent Textbook page 46

Lesson 6

Objective:

• Graph inequalities using a number line.

1. Introduction

Draw a number line similar to the one on textbook page 46. Ask students:• Where would the solutions to x > −3 be on the

line? (To the right of −3.)• Would −3 be part of the solution? (No, because

all solutions must be greater than −3.)Repeat with x < 3.

Introduce the symbol ≥ (greater than or equal to). Ask students where the solutions to x ≥ −3 would be on the line, and how the solution is different from x > −3 (i.e., it includes −3).

Read and discuss the top of page 46 and the REMARKS.

Give students graph paper and rulers. Have them draw number lines to graph the solutions to x > −3, x < −3, x ≥ −3, and x ≤ −3. Make sure students understand that we use an open circle when we have > or <, and a closed circle when we have ≥ or ≤.

Notes:• Remind students that on a horizontal number

line, the numbers become greater going to the right and become less going to the left.

• The solution includes all the values indicated by the arrow, not just whole number values.

Thus, the solution to x < 5 includes 4.3, −12, etc.

• When graphing the solutions to inequalities, make sure students draw the number lines directly on top of the horizontal lines on the graph paper, and use the vertical lines to draw the hash marks for the integers. Emphasize neatness and accuracy.

2. Development

Have students study Examples 18–20 and do Try It! 18–20 on their own, and then compare their solutions with partners or in a group.

Note:• Make sure students are using the open and

closed circles correctly, that the arrows are going in the proper direction, and that they are graphing the solutions neatly and accurately.

AnswersTry It! 18

(a)

(b)

−5 0−1−2−3−4 1 2

5 109876 11 12

17Chapter 2 fractions

?

?

Challenge

13.

Each child received 215 of her savings.

45 ÷ 2 = 4

5 × 12 = 2

5 25 ÷ 3 = 2

5 × 13 = 2

15

washingmachine

15

45

11. Method 1

From the model,

7 units 78 lb

1 unit 78 lb ÷ 7 = 1

8 lbMethod 2

12. 79 yd ÷ 9 = 7

9 yd × 19 = 7

81 yd

Each piece is 781 yd long.

78 lb

78 lb ÷ 7 = 7

8 lb × 17 = 1

8 lb1

1

18 lb of rice is in each bag.

14. (a)

(b)

(c)

(d)

2.2C Division of a Fraction by a Fraction

Basics

From the model, there are 3 groups of 5

3 . Therefore, 15

3 ÷ 53 = 3.

From the model, there are 2 groups of 14 .

Therefore, 24 ÷ 1

4 = 2.

From the model, there are 3 groups of 14 .

Therefore, 34 ÷ 1

4 = 3.

From the model, there is 1 group of 24 .

Therefore, 24 ÷ 1

2 = 1

53

53

53

14

14

24

24

34

12

14

14

14

153

Ratio, Rate, and Speed

Percentage

Angles, Triangles, and

Quadrilaterals

7A

Factors and Multiples

Real Numbers

Simple Equations in One Variable

Introduction to Algebra

Algebraic Manipulation

1

2

3

4

5

56

7

8

Proportions

Data Handling

Parallel Line and Angles in

Triangles and Polygons

7B

8A

Number Patterns

Probability of Simple Events

Probability of Combined Events

Exponents and Scientific Notation

Coordinates and Linear Graphs

Linear Equations in Two Variables

Perimeters and Areas of

Plane Figures

Simple Algebraic Fractions

Expansion and Factorization of

Algebraic Expressions

Volumes of Surface Areas

of Solids

Congruence and Similarity

Inequalities

Quadratic Factorization and

Equations

Scope &Sequence

6A

6B

Algebraic Expressions

Equations and Inequalities

Volume and Surface Area of Solids

Coordinates and Graphs

Displaying and Comparing Data

Percent

Whole Numbers

Fractions

Ratios

Decimals

Rate

Negative Numbers

Area of Plane Figures

More About Quadric Equations

Graphs of Linear and

Quadratic Functions

Graphs in Practical Situations

Mensuration of Pyramids,

Cylinders, Cones and Spheres

Pythagorean Theorem

Coordinate Geometry

Data Analysis

8B

singaporemath.com

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Think,Learn,Do,Succeed.

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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!