Algebraic Expressions and Equations Unit 4 begins with a review of exponential notation and order of operations. These topics are integral to algebra, which is introduced in this unit.
To develop algebraic thinking, students work with numerical expressions and number sentences. They use expressions with variables to write simple equations and to generalize patterns. Students apply properties of numbers and operations and the order of operations to identify and generate equivalent expressions.
In several lessons in this unit, students extend their work with number sentences to inequalities, comparing two quantities that are not equal. They learn how to write and graph inequalities.
Students apply these algebraic concepts to real-world situations. For example, amusement park rides sometimes have a minimum height requirement of 4 feet. In real life, an individual can be only so tall, so there is also a probable maximum height. In this example, a rider must be 4 feet tall, and we could consider the maximum height for a person to be 9 feet. Students first define a variable to work with: let h represent the height of a rider in feet. Then they write and graph inequalities that model this situation: 4 ≤ h and h ≤ 9.
The unit concludes with lessons on absolute value and the uses of absolute value. Students explore the absolute value of a number as its distance from 0 on a number line. Absolute value emphasizes magnitude (distance) rather than whether the number is negative or positive. This emphasis is important in real-life contexts. For example, students want to compare estimates for the number of jelly beans in a jar. Some estimates will be too high and some too low. To evaluate whose estimate is closest, students compare the distances the estimates are from the actual total, disregarding whether a guess is above or below the actual total. This situation could be represented using absolute value.
One focus of the absolute-value lessons is to help students make distinctions between using integers to represent situations and using absolute value. For example, a balance of -$50 in a bank account indicates that the owner of the bank account owes the bank $50. Students use an integer to represent the bank balance. If they want to answer the question of how much money is owed, one approach is to consider the distance the balance is from 0. They can use |-50|, or the absolute value of -50, to represent the amount of money that is owed, or the total amount of the debt.
Please keep this Family Letter for reference as your child works through Unit 4.
VocabularyImportant terms in Unit 4:
algebraic expression An expression that contains one or more unknowns or variables. For example, if Maria is 2 inches taller than Oliver, and if the variable a represents Maria’s unknown height, then the algebraic expression a - 2 represents Oliver’s height.
Distributive Property of Multiplication over Addition A property that relates multiplication of a sum of numbers by distributing the factor over the terms in the sum.
In symbols: For any numbers a, b, and c, a ∗ (b + c) = (a ∗ b) + (a ∗ c).
Distributive Property of Multiplication over Subtraction A property that relates multiplication of the difference of numbers by distributing the factor over the terms in the difference.
In symbols: For any numbers a, b, and c, a ∗ (b - c) = (a ∗ b) - (a ∗ c).bottom of p. 152.
equation A number sentence with an =. For example, 15 = 10 + 5 is an equation.
exponential notation A way to show repeated multiplication by the same factor. For example, 23 is exponential notation for 2 ∗ 2 ∗ 2. The 3 is the exponent. It tells how many times the number 2, called the base, is used as a factor. You read this number as two to the third power.
inequality A number sentence with a relation symbol other than =, such as >, <, ≥, ≤, ≠, or ≈.
order of operations A set of rules that tells the order in which operations in an expression are performed. The accepted order is:
1. Follow Steps 2–4 for expressions inside parentheses or grouping symbols. Work from the innermost to the outermost grouping symbols.
2. Calculate all expressions with exponents.
3. Multiply and divide in order from left to right.
Calculate inside the parentheses.Simplify the exponential expression.Divide.Add.
variable A letter or symbol that can be replaced by a value. A variable can represent one particular number or many different numbers.
23
Base
Exponent
Do-Anytime Activities Work with your child on some of the concepts taught in Unit 4:
1. Ask your child how to play the game Name That Number. Play a few rounds of the game several times during this unit. You can play the game online if your child’s class has access to the Everyday Math online games. If not, you can use a regular deck of 54 playing cards as follows:
• Let all number cards represent their face values.
• Let the four aces be 1s.
• Change the four queens to 0s.
• Remove the four jacks, four kings, and two jokers. Label each of these ten cards with one of the numbers from 11 through 20.
2. Think of a number between 25 and 100 and ask your child to guess it by asking questions that involve inequalities: less than, greater than, less than or equal to, or greater than or equal to. Encourage them to use operations when guessing. For example, your child could ask questions such as “Is 10 times your number greater than 500? Is 25 less than your number greater than 10?” Your child should keep asking questions until he or she figures out the number. The goal is to find the number using as few questions as possible. Keep track of the number of questions it takes to figure out a number, and then switch roles with your child.
3. Ask your child to give the magnitude and direction for situations that have numbers below 0. For example, a temperature reading of -10˚F means that the temperature is below 0 and its magnitude is 10.
Building Skills through GamesIn Unit 4, your child will work on a variety of skills by playing the following games:
R atio Memory Match Students match ratio picture cards to ratio notation cards.
Name That Number Students use order of operations in number sentences to make a target number.
Fraction Top-It (Division) Students divide whole numbers and fractions by fractions. Then they compare fractions.
Doggone Decimal Students estimate products of whole numbers and decimals.
Solution Search Students find solutions to inequalities.
Absolute Value Sprint Students explore absolute value of numbers on a number line.
Percent Spin Students calculate percents.
As You Help Your Child with HomeworkAs your child brings assignments home, you might want to go over the instructions together, clarifying as necessary. The answers listed below will guide you through the unit’s Home Links.
