LESSON 1 Number Line: Comparing and Ordering Integersblog.wsd.net/jeelzey/files/2014/08/Math-7-Lessons-1-10.pdf · Number Line: Comparing and Ordering Integers (page 6) ... Which
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
The plot in a word problem tells us what equation to write to solve the problem.
• A formula is an equation written with letters, also called variables.
• Each kind of word problem has a formula.
• Stories about combining have an addition pattern:
some � more � totals � m � t
• Stories about separating have a subtraction pattern:
starting amount � some went away � what is lefts � a � l
• Stories about comparing also have subtraction patterns:
greater � lesser � differenceg � l � d
later � earlier � differencel � e � d
• To solve a word problem:
1. Look for keywords that will help you find the plot of the story:combining, separating, or comparing.Use the key words chart on page 35 in the Student Reference Guide.
2. Write an equation for the problem using the formula and numbers from the story.Use a variable for the missing number
3. Find the missing number.Use the missing numbers chart on page 4 in the Student Reference Guide.
4. Check to see that your answer makes sense.
Teacher Notes:• Introduce Hint #12, “Word
Problem Cues,” Hint #13, “Finding Missing Numbers,” and Hint #14, “Abbreviations and Symbols.”
• Refer students to “Equivalence Table for Units” on page 1, “Time” on page 2, “Missing Numbers” on page 4, and “Word Problem Keywords” on page 35 in the Student Reference Guide.
• Post reference chart, “Word Problem Keywords.”
Addition and SubtractionWord Problems (page 19)
Practice Set (page 22)
a. What are the three kinds of word problems described in this lesson?
c s c
b. In example 2 on † pg. 21, we solved a word problem to find how much money Alberto spent on
milk and bread. Using the same information, write a word problem that asks how much money Alberto gave to the clerk.
At the store, Alberto bought milk and bread that cost $ . The clerk gave Alberto $
in change. How much money did give to ?
† When you see , refer to your Saxon Math Course 3 textbook.
c. Write a story problem for this equation.$20.00 � a � $8.45
Abby went to the store with $ . She bought a . The clerk gave her $ in
change. money did Abby spend?
For problems d –f, identify the plot, write an equation, and solve the problem.
d. From 1990 to 2000 the population of Garland increased from 180,635 to 215,768. How many more people lived in Garland in 2000 than in 1990?
plot: equation: � � d
answer: people
e. Binh went to the theater with $20.00 and left the theater with $10.50. How much money did Binh spend at the theater? Explain why your answer is reasonable.
plot: equation: � a �
answer: ; The answer is reasonable because half of $20 is $ . The money left,
$10.50, is a little m than half, so the money spent should be a little l than half.
f. In the three 8th-grade classrooms at Washington school, there are 29 students, 28 students, and 31 students. What is the total number of students in the three classrooms?
plot: equation: � � � t
answer: students
g. Circle the equation that shows how to find how much change a customer should receive from $10.00 for a $6.29 purchase.
• Fractions have two parts: a numerator (top number) and a denominator (bottom number).
numerator number of parts describeddenominator number of equal parts
• To find a fraction of a group:1. Divide by the denominator.2. Multiply by the numerator.
Example: Two fifths of the 30 questions on the test were multiple-choice. How many questions were multiple-choice?
1. Divide by the denominator: 30 � 5 � 62. Multiply by the numerator: 6 � 2 � 12
12 of the questions were multiple choice.
• One way to compare fractions is to compare each fraction to 1 _ 2 .
• Take half of the denominator and compare it to the numerator.If the numerator is greater than half the denominator, the fraction is greater than 1 _ 2 .If the numerator is less than half the denominator, the fraction is less than 1 _ 2 . If the numerator is equal to half the denominator, the fraction is equal to 1 _ 2 .
Example: Arrange these fractions from least to greatest.
3 __ 6 , 3 __
5 , 3 __
8
Compare each fraction to 1 _ 2 .
3 _ 6 : half of 6 is 3. The numerator is equal to 3, so 3 _ 6 is equal to 1 _ 2 .
3 _ 5 : half of 5 is 2 1 _ 2 . The numerator is more than 2 1 _ 2 , so 3 _ 5 is more than 1 _ 2 .
3 _ 8 : half of 8 is 4. The numerator is less than 4, so 3 _ 8 is less than 1 _ 2 .
3 __ 8 < 3 __
6 < 3 __
5
Teacher Notes:• Introduce Hint #15, “Naming
Fractions/Identifying Fractional Parts,” Hint #16, “Fraction of a Group,” and Hint #18, “Comparing Fractions.”
• Students who are not ready for the abstract nature of fractions will benefit from fraction manipulatives. Fraction tower manipulatives are available in the Adaptations Manipulative Kit. Hint #17, “Fraction Manipulatives,” describes how to make your own fraction tower manipulatives.
• Refer students to “Fraction Terms” on page 12 in the Student Reference Guide.
b. Three fifths of the 30 questions on the test were multiple-choice. How many multiple-choice questions were there? Explain why your answer is reasonable.
Divide by the denominator. Multiply by the numerator.
questions; 3 _ 5 is than 1 _ 2 , and is greater than 1 _ 2 of 30.
c. Greta drove 288 miles and used 8 gallons of fuel. Greta’s car traveled an average of how many miles
per gallon of fuel?
d. Arrange these fractions from least to greatest. , , Compare each to 1 __ 2 .
5 ___ 10
, 5 __ 6 , 5 ___
12
, , least greatest
Written Practice (page 33)
1. Divide by the denominator.Multiply by the numerator.
1 _ 4 of a mile
1 mi � yd
2. least to greatestCompare to 1 __ 2 .
, ,
3. 2 _ 3 of 600
Divide by the denominator.Multiply by the numerator.
• Rates use the word per to mean “in one.”65 miles per hour (65 mph) means “65 miles in one hour.”32 feet per second (32 ft/sec) means “32 feet in one second.”
• To solve rate problems, use the loop method from Lesson 6.
• The average (or mean) describes what number is in the “center” of a group of numbers.
1. Add the numbers.2. Divide by the number of items.
• The average must be between the smallest and largest numbers.Example: Find the average of 5, 1, 3, 5, 4, 8, and 2.
1. Add the numbers. 5 � 1 � 3 � 5 � 4 � 8 � 2 � 282. Divide by the number of items. 28 � 7 � 4
The average is 4. Four is between the smallest number (1) and the largest number (8).
• The median is the middle number when the numbers are put in order.Example: Find the median of 5, 1, 3, 5, 4, 8, and 2.
1. Write the numbers in order. 1, 2, 3, 4, 5, 5, 82. Count the numbers. There are 7 numbers.
Counting from the first number or the last number, 4 is the middle number.
The median is 4. In this group of numbers, the average and the median are the same. This is not always true.
• The mode is the number that occurs most often.Example: Find the mode of 5, 1, 3, 5, 4, 8, and 2.The mode is 5. Five is the only number that occurs more than once in these numbers.
• The range is the difference between the largest and smallest numbers in a group.Example: Find the range of 5, 1, 3, 5, 4, 8, and 2.The range is 7.The largest number is 8 and the smallest number is 1.8 � 1 � 7.
• A line plot is a way to show a group of numbers. Each number is shown by an “X” above anumber line.
Example: Display this group of numbers on a line plot. {5, 1, 3, 5, 4, 8, 2}
Teacher Notes:• Introduce Hint #21,
“Average.”
• Refer students to “Average” on page 7 and “Statistics” on page 23 in the Student Reference Guide.
• Review Hint #19, “Converting Measures and Rate.”
Rates and Average Measures of Central Tendency (page 41)
b. How far can Freddy drive in 8 hours at an average speed of 50 miles per hour?
mi hr
c. If a commuter train averages 62 miles per hour between stops that are 18 miles apart, about how
many minutes does it take the train to travel the distance between the two stops? 62 miles per hour is about 60 miles per hour and 60 miles per hour is 1 mile per minute.
mi min
d. If the average number of students in three classrooms is 26, and one of the classrooms has 23 students, then which of the following must be true? (Circle one.) The average must be between the smallest
and largest numbers.
A At least one classroom has fewer than 23 students.
B At least one classroom has more than 23 students and less than 26 students.
C At least one classroom has exactly 26 students.
D At least one classroom has more than 26 students.
e. What is the mean of 84, 92, 92, and 96?
f. The heights of five basketball players are 184 cm, 190 cm, 196 cm, 198 cm, and 202 cm. What is the
average height of the five players?
The price per pound of apples sold at different sold at different grocery stores is reorted below. Use this information to answer problems g–i.
$0.99 $1.99 $1.49 $1.99
$1.49 $0.99 $2.49 $1.49
g. Display the data in a line plot.
___ 1 ? __
8
1 __ 1 ? __
8
Practice Set (page 44)
a. Alba ran 21 miles in three hours. What was her average speed in miles per hour?
per Multiply the loop.Divide by the outside number.
Divide by the denominator. Multiply by the numerator.
3. Convert pounds to ounces.Then add 7 ounces.
1 lb. � oz.
lb. oz.
4. Multiply the loop.Divide by the outside number.
km hr
7. average
8121619
� 20
8. plot:
equation: � � t
Written Practice (page 45)
1 ___
7 __
?
___ ?
__ 1
5. mi hr
6. See p. 1 in the Student Reference Guide.
___ 1
? __
5
) _____
t �
Practice Set (continued) (page 45)
h. Compute the mean, median, mode, and range of the data.
i. Rudy computed the average price and predicted that he would usually have to pay $1.62 per pound of apples. Why is Rudy’s prediction incorrect?The mode is the most common price for apples. The mode is $ .
A factor is a counting number that divides evenly into another number.
• Think of factors in pairs that multiply to make the number:6 has two factor pairs: 1 � 6 � 6 and 2 � 3 � 6The factor pairs can be shown with rectangles:
The factors of 6 are 1, 2, 3, and 6.
• A prime number is a counting number greater than 1 that has exactly two factors (one factor pair).
7 has one factor pair: 1 � 7 � 7
The factors of 7 are 1 and 7. Therefore, 7 is a prime number.
• The factors of a prime number are always 1 and the number itself.
Some prime numbers: 2, 3, 5, 7, 11, 13, …
• A counting number that has more than two factors is a composite number.
Any composite number can be written as the product of factors that are prime numbers. This is called prime factorization.
• We can use a factor tree to write the prime factorization of a number:
1. List two factors of the given number under “branches” of the tree. (If you have trouble, start with 2, 3, or 5.)
2. Check if either factor is prime. If the factor is prime, circle it. If it is not, continue to draw branches and factor until each number is prime.3. Write the prime factors in order.
You may have to write some numbers more than once.
Teacher Notes:• Introduce Hint #24, “Factors
of Whole Numbers,” Hint #25, “Tests for Divisibility,” Hint #26, “Prime Factorization Using the Factor Tree,” and Hint #27, “Prime Factorization Using Division by Primes.”
• Refer students to “Factors” and “Tests for Divisibility” on page 5 and “Prime Numbers” on page 9 in the Student Reference Guide.
• Post reference chart, “Primes and Composites.”
1
63
2
1
7
Prime Numbers (page 54)
420
4210
6 752
52
420 � 2 � 2 � 3 � 5 � 7
a. Is 9 a prime or composite number? Use 9 color tiles to make a square.
b. Write the first 10 prime numbers. 2, , , , 11, 13, , , 23, See p. 9 in the Student Reference Guide.
c. Complete this factor tree for 36.
d. Find the prime factors of 60 using division by primes.
60 � � � �
Write the prime factorization of each number in e–g.
e. 25 � �
• Another way to do prime factorization is division by primes:
1. Write the number in a division box.2. Divide by the smallest prime number that is a factor. (Try 2, 3, or 5.)3. Divide that answer by the smallest prime number that is a factor.4. Repeat until the quotient is 1.5. The divisors are the prime factors of the number.
• Tests for divisibility will help you find factors of numbers or tell if a number is prime:
• To compare fractions that do not have common denominators, cross-multiply:
Example: Compare: 2 __ 3 3 __
5
2 __ 3
3 __ 5
• An improper (top heavy) fraction is a fraction that is greater than 1 or equal to 1.
• A mixed number is a whole number and a fraction.
• To write an improper fraction as a mixed number or integer: 1. Divide the denominator into the numerator.2. Write the quotient as the whole number.3. Write the remainder as the numerator of the fraction.4. Keep the same denominator.
Example: Express 3 ___ 10
as a mixed number.
The quotient is 3, so that is the whole number. The remainder is 1, so that is the numerator. The denominator stays the same: 3.
Practice Set (page 65)
Each number in a–c is a member of one or more of the following sets of numbers. Write all the lettersthat apply. A Whole numbers B Integers C Rational numbers
a. 5 b. 2 c. � 2 __ 5
Use prime factorization to reduce each fraction in d–f.
Complete each equivalent fraction in g–i.Multiply the loop.Divide by the outside number.
g. 3 __ 5 � ___
20
h. 3 __ 4 � ___
20
i. 1 __ 4 � ____
100
j. Compare: Cross-multiply. 3 __
5 3 __
4
k. Draw and label points on the number line for these numbers: �1, 3 __ 4
, 0, 3 __ 2
, �1 __ 2
l. Convert the improper (top-heavy) fraction to a mixed number. Shade the circles to show that the numbers are equal.
9 __ 4 �
m. Equivalent fractions can be formed by multiplying by a fraction form of 1 (same numerator and denominator). What property of multiplication states that any number multiplied by 1 is equal to the original number?
Property of Multiplication
n. Write a subtraction equation using whole numbers and a difference that is an integer.