Download Unit 3 Expressions and Equations: Equivalent

Unit 3
Expressions and Equations:
Equivalent Expressions
Introduction
In this unit, students will review the order of operations and will apply properties of operations to
make calculations easier. Students will evaluate expressions at given values of the variable and
will translate situations into mathematical expressions. Students will add, subtract, expand, and
factor linear expressions with integer coefficients. Students will recognize equivalent expressions
using properties of operations, using substitution at rational values, and using pictures.
Fraction notation. We often use slashes for fractions (such as 1/2 or −1/2) to save time and
space. You do not need to display the fractions to students this way. But, if you do, introduce
the notation first.
Furthermore, in this unit, negative fractions are always written with the negative sign in front of
the whole fraction, not in front of the numerator or denominator.
1
2
Example: - , not
-1
1
or
2
-2
Materials. We recommend that students always work in grid paper notebooks. Paper with
1/4-inch grids works well in most lessons. In this unit, grid paper will be very helpful when
multiplying multi-digit numbers and using standard algorithms. If students do not use grid
paper notebooks in general, you will need to have lots of grid paper on hand throughout
Lesson EE7-5. If students who have difficulties in visual organization will be working without
grid paper, they should be taught to draw a grid before starting to work on a problem.
Students will often be using concepts from the previous unit. Provide students with
BLM Adding, Subtracting, and Multiplying Review at the beginning of this unit, so that they
have it handy whenever they are doing independent work in this unit.
Terminology. The word “expression” is used throughout this unit and can refer to either a
numeric expression or an algebraic expression.
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-1
EE7-1
Order of Operations
Pages 68–70
Standards: preparation for 7.EE.1
Goals:
Students will evaluate expressions involving more than one operation by using the correct order
of operations.
Prior Knowledge Required:
Can add and subtract fractions
Can multiply a fraction by a whole number
Vocabulary: brackets, denominator, equation, numerator, operation
Introduce brackets. Remind students that addition, subtraction, multiplication, and division are
called operations. Write on the board:
(8 − 5) + 2
8 − (5 + 2)
SAY: Do the operation in brackets first. SAY: (8 − 5) + 2 equals 3 + 2, which is 5; 8 − (5 + 2)
equals 8 − 7, which is 1.
Exercises: Evaluate.
a) (8 + 4) − 3
b) 8 + (4 − 3)
c) (8 − 2) × 3
e) 12 ÷ (2 × 3)
f) (12 ÷ 2) × 3
g) (5 + 3) × 4
Answers: a) 9, b) 9, c) 18, d) 2, e) 2, f) 18, g) 32, h) 17
d) 8 − (2 × 3)
h) 5 + (3 × 4)
Point out that the operation that you do first often changes the answer, but not always.
Adding and subtracting from left to right. Tell students that mathematicians have come up
with shortcuts so that they do not have to write brackets all the time. SAY: When there are no
brackets, do addition and subtraction from left to right. For example, 8 − 5 + 2 means 3 + 2. If
you mean 8 − 7, you have to add brackets: 8 − (5 + 2).
Exercises: Add or subtract from left to right.
a) 6 + 3 − 2
b) 6 − 3 + 2
c) 14 − 5 + 6
d) 9 + 4 − 4
Bonus: Do you need to add brackets to get the answer?
e) 9 − 4 + 2 = 3
f) 7 + 5 − 3 = 9
g) 2 + 6 − 5 = 3
h) 8 − 3 + 2 = 3
i) 7 − 3 + 2 = 6
j) 2 + 8 + 3 = 13
k) 9 − 4 − 2 = 7
l) 9 − 4 − 3 = 2
Answers: a) 7, b) 5, c) 15, d) 9, Bonus: e) yes, f) no, g) no, h) yes, i) no, j) no, k) yes, l) no
D-2
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
Remind students that adding and subtracting fractions with the same denominator is easy—
just add and subtract the numerators and keep the denominator the same. Demonstrate the
first part of the exercise below, by first adding 3/5 + 7/5 = 10/5 and then subtracting 1/5 to get
9/5. Point out that it doesn’t make it easier to change 10/5 to 2 at this intermediate stage—it’s
easier to keep it as fifths when subtracting fifths from fifths. Then have students complete the
rest.
Exercises: Add or subtract from left to right.
a)
3 7 1
+ 5 5 5
b)
7 3 4
- +
5 5 5
c)
3 5 6
+ 8 8 8
d)
8
5
4
+
15 15 15
Answers: a) 9/5, b) 8/5, c) 2/8 or 1/4, d) 7/15
Ask volunteers if any answers could be reduced. (only c), 2/8 = 1/4) Write on the board:
1 3 2
+ 2 4 3
SAY: When the denominators are different, you can use equivalent fractions to make them
equal. Start with the first two fractions, one half plus three quarters. Write on the board:
2 3 2 5 2
+ - = 4 4 3 4 3
15 8
=
12 12
7
=
12
Exercises: Add or subtract from left to right unless brackets tell you otherwise. Write the
answer in lowest terms.
a)
2 3 1
+ 3 4 2
b)
5 1 1
- +
8 2 4
c)
2 3
1
+
5 10 2
d)
4 æç 1 1ö÷
-ç + ÷
5 çè 4 6 ø÷
Answers: a) 11/12, b) 3/8, c) 3/5, d) 23/60
Multiplying and dividing from left to right. SAY: When there are no brackets, multiply and
divide from left to right.
Exercises: Multiply or divide from left to right.
a) 6 ÷ 3 × 2
b) 4 × 4 ÷ 2
c) 2 × 6 ÷ 3
d) 12 ÷ 3 × 4
Bonus: Do you need to add brackets to get the answer?
e) 3 × 6 ÷ 2 = 9
f) 12 ÷ 6 × 2 = 1
g) 5 × 8 ÷ 2 = 20
h) 8 ÷ 2 × 2 = 2
i) 2 × 3 × 4 = 24
j) 16 ÷ 4 ÷ 2 = 2
Answers: a) 4, b) 8, c) 4, d) 16, Bonus: e) no, f) yes, g) no, h) yes, i) no, j) no
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-3
Multiply or divide before adding or subtracting. SAY: When there are no brackets, multiply
or divide before adding or subtracting.
Exercises: Evaluate.
a) 4 + 5 × 2
b) 3 × 4 + 5
c) 12 − 2 × 5
d) 3 × 8 − 5
e) 3 + 6 ÷ 3
f) 8 ÷ 2 + 2
g) 15 ÷ 5 − 2
h) 14 − 6 ÷ 2
Bonus:
i) 34 + 50 × 2
j) 13 × 100 − 10
k) 96 − 6 ÷ 3
Answers: a) 14, b) 17, c) 2, d) 19, e) 5, f) 6, g) 1, h) 11, Bonus: i) 134, j) 1,290, k) 94
Review multiplying a fraction by a whole number. Write on the board:
3´
3×2
2 2 2 2 6
= + + =
5 5 5 5 5
SAY: When multiplying a whole number and a fraction, multiply the whole number by the
numerator and keep the denominator the same.
Exercises: Do the operations one at a time in the standard order.
1
5
a) 3 ´ ´
4
5
æ1
è5
4ö
5ø
b) 3 ´ççç + ÷÷÷
c)
2 1
- ´3
3 5
æ 2 1ö
d) ççç - ÷÷÷´ 3
è3
5ø
Answers: a) 3/5 × 4/5 = 7/5, b) 3 × 1 = 3 or 3 × 5/5 = 15/5 = 3, c) 2/3 − 3/5 = 1/15,
d) 7/15 × 3 = 21/15 or 7/5
Summarizing the order of operations without brackets. SAY: The order of operations gives
priority to some operations over others by assigning which ones are done first. Remember that
multiplication and division have the same priority because they are in the same fact family. You
can get 12 ÷ 4 = 3 from 4 × 3 = 12. So adding and subtracting also have the same priority. But
multiplication and division are not in the same fact family as addition and subtraction, so they
don’t have the same priority. Mathematicians decided to put multiplication and division ahead of
addition and subtraction.
Exercises: Do the operations in the standard order.
a) 9 − 3 × (6 − 4)
b) 5 × 3 − 5 − 3
d) 6 × 4 − (6 + 4)
e) 8 ÷ 2 × 3 − 4
Answers: a) 3, b) 7, c) 4, d) 14, e) 8, f) 4
c) 2 + 3 × 4 ÷ 6
f) 2 × (5 + 3) ÷ 4
Encourage students who are struggling to start by circling the operation they would do first.
More than one set of brackets (that are not nested). SAY: If there is more than one set of
brackets, all the brackets have the same priority, so you can do those operations all in one step
from left to right. Write on the board:
8 − (2 + 1) × (5 − 3) = 8 − 3 × 2
SAY: Now this is a simpler expression to evaluate.
D-4
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
Exercises:
1. Do all the operations in brackets. Then write the simpler expression.
a) (5 + 3) × (6 − 1)
b) (5 + 7) ÷ (5 − 1)
c) 5 + (4 − 1) × 2 − (6 − 3)
Bonus:
d) (3 + 4) × (6 − 1) + (7 + 5) ÷ (6 − 4) − (8 − 5)
e) (3 + 2) + (4 ÷ 2) × (6 + 3) − (9 + 3) ÷ (4 + 2)
Answers: a) 8 × 5, b) 12 ÷ 4, c) 5 + 3 × 2 − 3, Bonus: d) 7 × 5 + 12 ÷ 2 − 3,
e) 5 + 2 × 9 − 12 ÷ 6
2. Evaluate the expression.
a) (3 + 5) ÷ (5 − 3)
b) (10 + 2) ÷ (6 ÷ 3)
c) (38 − 8) − (10 + 8)
Bonus: (2 + 3) × (7 − 1) + (12 + 8) ÷ (7 − 3) − (6 − 4)
Answers: a) 4, b) 6, c) 12, Bonus: 33
Evaluating expressions where brackets include more than one operation.
SAY: Sometimes brackets include more than one operation. Do those operations in the
standard order, too. Write on the board:
(3 + 4 × 5) × 2
SAY: You have to start by evaluating the expression in brackets first. ASK: Which operation do
you do first, 3 + 4 or 4 × 5? (4 × 5) Write on the board:
(3 + 4 × 5) × 2 = (3 + 20) × 2
Have a volunteer finish evaluating the expression. (23 × 2 = 46)
Exercises: Evaluate the expression.
a) 8 − (4 + 6 ÷ 2)
b) 4 × (9 − 4 × 2)
c) 42 ÷ (10 − 1 × 3)
e) 54 ÷ (2 × 4 + 1)
f) (5 + 4 × 2) × 3
g) 2 × (17 − 5 × 2)
Bonus: (9 + 6 × 5 ÷ 2) ÷ (7 × 3 − 4 × 4 + 1)
Answers: a) 1, b) 4, c) 6, d) 2, e) 6, f) 39, g) 14, h) 3, Bonus: 4
d) (5 × 4 − 6) ÷ 7
h) 15 − (6 × 4 ÷ 2)
Evaluating expressions that have brackets inside brackets. Write on the board:
9 − (6 ÷ (2 + 1)) =
evaluate this first
SAY: The part inside the brackets needs to be evaluated first, but it has brackets, too, so you
have to start with the inside brackets. Write on the board:
6 ÷ (2 + 1)
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-5
SAY: The first step to evaluating this expression is evaluating the part inside brackets.
ASK: What is the operation that needs to be done first? (2 + 1) Write on the board:
9 − (6 ÷ (2 + 1)) = 9 − (6 ÷ 3)
Ask a volunteer to finish evaluating the expression. (9 − 2 = 7)
Exercises: Do the operations in the standard order.
a) 24 ÷ (12 − (5 × 2))
b) 20 − ((2 + 4) × 3)
c) 72 ÷ 8 − (5 − (2 × 3) ÷ 2)
Bonus:
d) (7 + 3) × (15 ÷ (7 − 2) + 11)
e) 20 ÷ (9 − (2 + 4 − (6 ÷ 3)))
f) (9 + 2) × ((20 + 10) ÷ (2 × 3) − (5 − 3))
Solutions:
a) 24 ÷ (12 − 10) = 24 ÷ 2 = 12,
b) 20 − (6 × 3) = 20 − 18 = 2,
c) 9 − (5 − 6 ÷ 2) = 9 − (5 − 3) = 9 − 2 = 7,
Bonus: d) 10 × (15 ÷ 5 + 11) = 10 × (3 + 11) = 10 × 14 = 140,
e) 20 ÷ (9 − (2 + 4 − 2)) = 20 ÷ (9 − (6 − 2)) = 20 ÷ (9 − 4) = 20 ÷ 5 = 4,
f) 11 × (30 ÷ 6 − 2) = 11 × (5 − 2) = 11 × 3 = 33
Extensions
(MP.1) 1. Use the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9, once each, to make the equations true.
÷(
×
)=1
−(
+
)=2
+
)÷
=4
(
Sample answers:
6 ÷ (3 × 2) = 1
8 − (5 + 1) = 2
(7 + 9) ÷ 4 = 4
or
8 ÷ (2 × 4) = 1
9 − (1 + 6) = 2
(5 + 7) ÷ 3 = 4
(MP.1) 2. Using exactly four 4s each time, make expressions equal to each number from 0
through 10. You may use brackets and any of the four operations.
Example: (4 × 4) ÷ (4 + 4) = 16 ÷ 8 = 2
How many different expressions can you think of? Hint: You may use the 2-digit number 44, but
it counts as two 4s.
Sample answers:
0 = (4 − 4) × (4 + 4)
6 = (4 + 4) ÷ 4 + 4
1=4÷4×4÷4
7 = 44 ÷ 4 − 4
2 = 4 × 4 ÷ (4 + 4)
8 = 4 × 4 − (4 + 4)
3 = (4 + 4 + 4) ÷ 4
9=4÷4+4+4
4 = (4 − 4) × 4 + 4
10 = (44 – 4) ÷ 4
5 = (4 × 4 + 4) ÷ 4
D-6
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
3. Make up an expression that has answer 10 that uses all four operations and at least one set
of brackets. Have a partner check that it has answer 10.
Sample answer: (7 + 3) × (2 − 8 ÷ 8)
4. (MP.7) a) Evaluate both expressions in each row.
i) 1 × 1
(1 + 1) × (1 − 1)
ii) 2 × 2
(2 + 1) × (2 − 1)
iii) 3 × 3
(3 + 1) × (3 − 1)
iv) 4 × 4
(4 + 1) × (4 − 1)
v) 5 × 5
(5 + 1) × (5 − 1)
b) What do you notice about your answers to part a)?
c) Predict 201 × 199.
Answers:
a) i) 1, 0; ii) 4, 3; iii) 9, 8; iv) 16, 15; v) 25, 24;
b) the second column is always 1 less than the first;
c) 201 × 199 = (200 + 1) × (200 − 1), so 201 × 199 is 1 less than 200 × 200 = 40,000,
so 201 × 199 = 39,999
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-7
EE7-2
The Associative and Commutative Properties
Pages 71–73
Standards: 7.EE.1
Goals:
Students will determine which operations have the associative and commutative properties and
will use the properties of operations to do mental calculations.
Prior Knowledge Required:
Knows the correct order of operations
Can multiply by 10 and 100
Can recognize pairs that add or multiply to 10 or 100
Vocabulary: associative, associative property, brackets, commutative, commutative property,
operation
Which operations are commutative? Draw on the board:
3+4=4+3
Tell students that you can count the shaded circles first or the unshaded circles first, and you
will get the same answer both ways. So addition has the commutative property. We can say that
addition is commutative. ASK: Is subtraction commutative? (no) PROMPT: Does 4 − 3 = 3 − 4?
(no) SAY: You can’t switch the numbers and get the same answer, so subtraction is not
commutative. Draw on the board:
2×3=3×2
SAY: You can look at this array as 2 rows with 3 in each row or 3 columns with 2 in each
column. ASK: Is multiplication commutative? (yes) How do you know? (because the same
picture can show 2 × 3 or 3 × 2) ASK: Is division commutative? (no) PROMPTS: Is 6 ÷ 2 the
same as 2 ÷ 6? (no) Does dividing 6 pizzas between 2 people give the same amount as dividing
2 pizzas between 6 people? (no) Write on the board:
+
−
×
÷
Ask students whether each operation satisfies the commutative property. Point to each symbol
in turn and have students signal thumbs up for yes (+ and ×) and thumbs down for no (− and ÷).
D-8
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
Introduce the associative property of addition and multiplication.
Exercises: Calculate both expressions and write = or ≠.
a) (3 + 4) + 5
3 + (4 + 5)
b) (2 × 3) × 4
2 × (3 × 4)
c) (8 − 3) − 1
8 − (3 − 1)
d) (8 ÷ 4) ÷ 2
8 ÷ (4 ÷ 2)
Answers: a) 12 = 12, b) 24 = 24, c) 4 ≠ 6, d) 1 ≠ 4
Tell students that when the order they do the operations in doesn’t change the answer, the
operation satisfies the associative property. We can say that addition is associative. Write on
the board the four operation symbols (+, −, ×, ÷), and ask students to signal whether each
operation satisfies the associative property. (+, yes; −, no; ×, yes; ÷, no)
Contrasting the associative and commutative properties. Point out that the commutative
property tells you that the order the numbers are in doesn’t matter. The associative property
tells you that the order the operations are done in doesn’t matter. Write on the board:
A. the commutative property
B. the associative property
In the exercises below, students can signal their answers by pointing to the correct word.
Exercises: What property is the equation an example of, commutative (A) or associative (B)?
a) 3 × 7 = 7 × 3
b) (3 + 4) + 5 = 3 + (4 + 5)
c) 2 + 43 = 43 + 2
d) 2 × (5 × 11) = (2 × 5) × 11
Answers: a) A, b) B, c) A, d) B
(MP.7) Using the commutative property by thinking of an expression as a number. Write
on the board:
5 × (3 − 1) = (3 − 1) × 5
ASK: What is the same about both expressions in the equation? (they are both multiplying 5 and
3 − 1) SAY: Even though 3 − 1 is an expression, it is still just a number. The expression tells you
how to get the number, but you can also think of it as the result. The two numbers being
multiplied are 5 and 3 − 1 or 5 and 2, but they are being multiplied in different orders.
ASK: What property does this equation show? (the commutative property)
Exercises: Use the commutative property to finish the equation.
a) (2 + 3) × 4 = ____ × (2 + 3)
b) 3 × (8 − 1) = (8 − 1) × _______
c) (7 − 2) × 6 = 6 × (______)
d) 2 × (3 + 8 − 4) = (__________) × 2
Answers: a) 4, b) 3, c) 7 − 2, d) 3 + 8 − 4
Write on the board:
(3 + 4) × (6 ÷ 2) =
Ask a volunteer to finish the equation using the commutative property. ((6 ÷ 2) × (3 + 4))
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-9
Exercises:
1. Use the commutative property of multiplication to complete the equation.
a) (9 − 7) × (3 + 4) = ______________
b) (8 − 5) × (8 ÷ 4) = ______________
Bonus: (10 + 3 ÷ 3) × (5 × 2 − 3) = ______________
Answers: a) (3 + 4) × (9 − 7), b) (8 ÷ 4) × (8 − 5), Bonus: (5 × 2 − 3) × (10 + 3 ÷ 3)
2. Use the commutative property of addition to complete the equation.
a) (8 ÷ 2) + (15 ÷ 5) = ______________
b) (3 × 4) + (8 ÷ 2) = ______________
Bonus: (9 − 2 × 3) + (16 − 10 ÷ 5) = ______________
Answers: a) (15 ÷ 5) + (8 ÷ 2), b) (8 ÷ 2) + (3 × 4), Bonus: (16 − 10 ÷ 5) + (9 − 2 × 3)
(MP.7) Using the commutative property on part of an expression. SAY: You can use the
commutative property on part of an expression, too. Write on the board:
(2 + 5) × 3 = (
)×3
ASK: Using the commutative property, what is 2 + 5 equal to? (5 + 2) Finish the equation on the
board:
(2 + 5) × 3 = (5 + 2) × 3
Exercises: Use the commutative property to complete the equation.
a) (3 + 5) ÷ 2 = __________
b) 40 − 5 × 7 = __________
c) 9 − (2 + 3) = __________
Answers: a) (5 + 3) ÷ 2, b) 40 − 7 × 5, c) 9 − (3 + 2)
(MP.3) Using properties to check if equations are correct. Write on the board:
(5 − 2) × 4 = (2 − 5) × 4 by the commutative property.
Tell students that you saw someone make this claim. ASK: Is this correct? (no) What mistake
did the person make? (they thought subtraction was commutative) Write on the board:
(2 + 3) × 7 = 7 × (2 + 3)
9 ÷ (2 + 1) = 9 ÷ (1 + 2)
7−5=5−7
(8 − 3) × 5 = (3 − 8) × 5
(+3) + (−2) = (−2) + (+3)
4×5−3=5×4−3
4×5−3=4×3−5
2×3+4=2×4+3
SAY: These equations were all justified using the commutative property, but I think some of
them are wrong. Ask students to decide whether each equation is using the commutative
D-10
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
property correctly. Students can signal their answer by showing thumbs up or thumbs down as
you point to each equation. (yes, yes, no, no, yes, yes, no, no)
Now write on the board:
(2 × 3) × 4 = 2 × (3 × 4)
(18 ÷ 6) ÷ 3 = 18 ÷ (6 ÷ 3)
(16 ÷ 4) ÷ 2 = 16 ÷ (4 ÷ 2)
(5 − 2) − 1 = 5 − (2 − 1)
(8 + 2) + 4 = 8 + (2 + 4)
SAY: These were all justified using the associative property, but I think some of them might be
wrong. Have students evaluate both sides to check if these equations are correct. Have
students signal whether each equation is correct (yes, no, no, no, yes). Ask volunteers to
explain why the incorrect ones are not correct. (division and subtraction are not associative)
Now write on the board:
23 + (19 + 47) = 23 + (47 + 19)
= (23 + 47) + 19
Tell students that you want to calculate the first sum, so you used the commutative property to
get the second expression and then the associative property to get the third expression. Write
on the board:
23 + (19 + 47)
23 + (47 + 19)
(23 + 47) + 19
Have volunteers evaluate each expression using the order shown by the brackets.
(23 + 66 = 89, 23 + 66 = 89, 70 + 19 = 89) Once they are finished, have all students vote on the
expression that was easiest to evaluate. Point out that the third one can be done mentally,
because pairs that add to 10 are easy to recognize and multiples of 10 are easy to add to.
Tell students that before they can pick the easier expression to evaluate and say that they
evaluated both expressions, they have to be able to prove that both expressions have the same
value.
(MP.3) Exercises: What property was used to justify the equation?
a) 18 + (25 + 32) = 18 + (32 + 25) ____________________________
= (18 + 32) + 25 ____________________________
b) (25 × 37) × 4 = (37 × 25) × 4 ____________________________
= 37 × (25 × 4) ____________________________
c) 5 × (13 × 4) = 5 × (4 × 13) ____________________________
= (5 × 4) × 13 ___________________________
Answers: a) commutative, associative; b) commutative, associative;
c) commutative, associative
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-11
Tell students that multiples of 10 and 100 are easy to add and to multiply, so some of these
expressions will be easier to calculate mentally than others.
Exercises:
(MP.1) 1. Pick the easier expression to evaluate; then evaluate it mentally.
a) 18 + (25 + 32)
b) (25 × 37) × 4
c) (5 × 4) × 13
(18 + 32) + 25
37 × (25 × 4)
5 × (13 × 4)
Answers: a) (18 + 32) + 25 = 50 + 25 = 75, b) 37 × (25 × 4) = 37 × 100 = 3,700,
c) (5 × 4) × 13 = 20 × 13 = 260
(MP.3) 2. Evaluate the sum mentally by changing it to an easier problem. Justify how you know
each change to the expression doesn’t change the value.
a) (17 + 19) + 23
b) 19 + (34 + 1)
Bonus: 797 + (419 + 3)
Answers:
a) (17 + 19) + 23 = (19 + 17) + 23 (commutative) = 19 + (17 + 23) (associative) = 19 + 40 = 59,
b) 19 + (34 + 1) = 19 + (1 + 34) (commutative) = (19 + 1) + 34 (associative) = 20 + 34 = 54,
Bonus: 797 + (419 + 3) = 797 + (3 + 419) (commutative) = (797 + 3) + 419 (associative)
= 800 + 419 = 1,219
(MP.3) 3. Evaluate the product mentally by changing it to an easier problem. Justify how you
know each change to the expression doesn’t change the value.
a) 11 × (3 × 10)
b) (19 × 5) × 2
c) (4 × 53) × 25
Answers:
a) 11 × (3 × 10) = (11 × 3) × 10 (associative) = 33 × 10 = 330,
b) (19 × 5) × 2 = 19 × (5 × 2) (associative) = 19 × 10 = 190,
c) (4 × 53) × 25 = (53 × 4) × 25 (commutative) = 53 × (4 × 25) (associative) = 53 × 100 = 5,300
Extensions
(MP.1) 1. Evaluate 1 × 2 × 3 × 4 × 5 × 20 × 25 × 30 × 50 × 100.
Answer: 9,000,000,000
Solution: By making pairs as shown: 1 × 2 × 3 × 4 × 5 × 20 × 25 × 30 × 50 × 100,
the product is: 100 × 100 × 90 × 100 × 100 = 9,000,000,000.
(MP.1) 2. When multiplied out, how many digits does 99,999,999,999 × 99,999,999 have?
Solution: The answer will be close to, but slightly less than,
100,000,000,000 × 100,000,000 = 10,000,000,000,000,000,000, which has 20 digits, so the
answer will have 19 digits.
3. Sarah adds 61 + 25 by writing
60 + 1 + 20 + 5 = 60 + 20 + 1 + 5
Which property is she using? Where is she using it?
Answer: She is using the commutative property to write 1 + 20 as 20 + 1.
D-12
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
(MP.3) 4. Which value for w makes the equation true? Justify your answers.
a) 2 × 5 = w × 2
b) w × 6 = 6 × 3
c) 2 × (3 × w) = (2 × 3) × 7
d) 3 × (w × 6) = 3 × 24
e) 10 × 3 = (2 × w) × 3
f) (3 × w) × 4 = 3 × 20
g) (2 × 3) × w = 2 × 12
Bonus: 14 × w = 2 × 35
Selected solutions: a) Using the commutative property, 2 × 5 = 5 × 2, so w = 5; d) w × 6 = 24,
so w = 4; g) By the associative property, (2 × 3) × w = 2 × (3 × w), so 3 × w = 12 and w = 4.
Answers: b) 3, c) 7, e) 5, f) 5, Bonus: 5
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-13
EE7-3
The Distributive Property
Pages 74–76
Standards: preparation for 7.EE.1
Goals:
Students will use the distributive property to calculate expressions mentally.
Prior Knowledge Required:
Can recognize applications of the commutative and associative properties
Can use the correct order of operations to evaluate expressions
Can multiply integers and fractions by whole numbers
Vocabulary: associative, commutative, denominator, distributive property, numerator
Review making equations by counting dots in two ways. Draw on the board:
6+8=7+7
Tell students that there are many ways to find out how many dots there are: 6 + 8 and 7 + 7 are
only two of them. ASK: Where do the 6 and 8 come from? (the shaded dots and the unshaded
dots) Where do the 7 and 7 come from? (the number in each row) SAY: Because we have two
different ways to write the same number, we know that 6 + 8 and 7 + 7 are equal.
Introduce the distributive property of addition and multiplication. SAY: Now I’m going to
make it harder. Write on the board:
2 × 3 + 2 × 4 = 2 × (3 + 4)
Point to each expression in turn and ASK: How does this count the dots? (the first expression
counts the 2 rows of 3 shaded dots and the 2 rows of 4 unshaded dots; the second expression
counts the rows separately—there are 3 shaded dots and 4 unshaded dots in each of the 2
rows)
Exercise: Using only the numbers 2, 3, and 4, make an equation from the picture.
Answer: 4 × 2 + 4 × 3 = 4 × (2 + 3)
D-14
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
Bonus: Draw a picture to show that (3 × 2) + (3 × 4) = 3 × (2 + 4).
Answer:
Tell students that the answers on both sides of the equal sign will be the same no matter what
numbers you substitute for 2, 3, and 4 in the equation above. SAY: This property is called the
distributive property, and you can say that multiplication “distributes” over addition.
Exercises: Finish the equation using the distributive property.
a) (4 × 5) + (4 × 7) = __________
b) (3 × 8) + (3 × 7) = __________
c) 5 × (2 + 4) = __________
d) 7 × (10 + 2) = __________
Answers: a) 4 × (5 + 7), b) 3 × (8 + 7), c) 5 × 2 + 5 × 4, d) 7 × 10 + 7 × 2
Contrasting the distributive property with the associative and commutative properties.
Point out that the commutative and associative properties relate to multiplication or addition
separately. The distributive property shows you one way that multiplication and addition are
related to each other.
Multiplication distributes over subtraction. Draw on the board:
5–2
4 × (5 − 2)
4×5−4×2
SAY: Adding four “5 − 2”s is the same as adding four 5s and subtracting four 2s.
Exercises: Evaluate both expressions. Make sure your answers are equal.
a) 2 × (7 − 4) and 2 × 7 − 2 × 4
b) 4 × (8 − 3) and (4 × 8) − (4 × 3)
Answers: a) 2 × 3 = 6, 14 − 8 = 6; b) 4 × 5 = 20, 32 − 12 = 20
Using the commutative property to write different forms of the distributive property. Write
on the board:
7 × (2 + 3) = 7 × 2 + 7 × 3
(2 + 3) × 7 = 2 × 7 + 3 × 7
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-15
Point out that the two left sides are the same because of the commutative property of
multiplication, and the parts of the right side are the same, also because of the commutative
property of multiplication. Then write on the board:
2 × (7 − 4) = 2 × 7 − 2 × 4
(7 − 4) × 2 =
SAY: The two left sides are equal by the commutative property of multiplication. What is 2 × 7
equal to? (7 × 2) Write that on the board:
(7 − 4) × 2 = 7 × 2 −
ASK: What is 2 × 4 equal to? (4 × 2) Finish the equation:
(7 − 4) × 2 = 7 × 2 − 4 × 2
Tell students that it doesn’t matter which factor is repeated, the first or the second; multiplication
distributes over both the first and second factors.
Exercises: Finish the equation using the distributive property.
a) (2 + 3) × 7 = __________
b) (4 + 6) × 9 = __________
c) (3 × 10) + (4 × 10) = __________
d) (4 × 8) + (5 × 8) = __________
e) (9 − 2) × 5 = __________
f) 5 × 3 − 1 × 3 = __________
Bonus:
g) 7 × 7 + 7 × 7 = __________
h) 5 × 4 − 5 × 4 = __________
Answers: a) 2 × 7 + 3 × 7, b) 4 × 9 + 6 × 9, c) (3 + 4) × 10, d) (4 + 5) × 8, e) 9 × 5 − 2 × 5,
f) (5 − 1) × 3, Bonus: g) (7 + 7) × 7 or 7 × (7 + 7), h) (5 − 5) × 4 or 5 × (4 − 4)
Choosing the easier expression to evaluate. Now write on the board:
4 × (52 − 49) and 4 × 52 − 4 × 49
ASK: Do these have the same answer? (yes) How do you know? (multiplication distributes over
subtraction) Which one would be easier to calculate mentally? (the first one) Why? (because
52 − 49 is easy to calculate, and so is 4 × 3, but 4 × 52 and 4 × 49 require more work)
(MP.1) Exercises: Pick the easier expression to evaluate and evaluate it mentally.
a) 3 × 587 − 3 × 287 or 3 × (587 − 287)
b) 3 × 587 + 3 × 413 or 3 × (587 + 413)
c) 17 × 100 − 17 × 1 or 17 × (100 − 1)
Answers: a) 3 × (587 − 287) = 3 × 300 = 900; b) 3 × (587 + 413) = 3 × 1,000 = 3,000;
c) 17 × 100 − 17 × 1 = 1,700 − 17 = 1,683
D-16
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
(MP.3) Identifying properties. Write the names of the different properties on the board:
1. the commutative property
2. the associative property
3. the distributive property
For the exercises below, have students signal 1 if the equation shows the commutative
property, 2 if it shows the associative property, or 3 if it shows the distributive property.
Exercises: Which property is the equation an example of: commutative (1), associative (2), or
distributive (3)?
a) 8 × 3 = 3 × 8
b) 2 × (7 − 4) = 2 × 7 − 2 × 4
c) 2 + 81 = 81 + 2
d) (5 × 2) × 4 = 5 × (2 × 4)
e) 9 + (3 + 2) = (9 + 3) + 2
f) (5 + 2) × 9 = 5 × 9 + 2 × 9
Answers: a) 1, b) 3, c) 1, d) 2, e) 2, f) 3
Does addition distribute over multiplication?
Exercises: Evaluate both expressions to check whether addition distributes over multiplication.
a) 2 + (3 × 4) and (2 + 3) × (2 + 4)
b) 3 + (4 × 5) and (3 + 4) × (3 + 5)
Bonus: Make up your own example to check whether subtraction distributes over multiplication.
Answers: a) 14 and 30, b) 23 and 56; addition does not distribute over multiplication. Bonus:
Subtraction does not distribute over multiplication.
A new notation for multiplication. Tell students that sometimes multiplication is written
without a times sign, just with brackets. Write on the board some examples:
3(4) = 3 × 4
3(2 + 5) = 3(2) + 3(5)
Exercises: Multiply.
a) 2(5)
b) 3(2)
c) 7(9)
Answers: a) 10; b) 6; c) 63; d) 1,000; e) 64
d) 10(100)
e) 8(8)
Review multiplying negative numbers by a whole number. SAY: You can multiply 3 times
−4 by just taking the negative of 3 times 4. Write on the board:
3(4) = 12, so 3(−4) = −12.
4 + 4 + 4 = 12,
so − 4 − 4 − 4 = −12.
Exercises: Multiply.
a) 2(−7)
b) 3(−3)
c) 4(−5)
Answers: a) −14, b) −9, c) −20, d) −56
d) 7(−8)
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-17
Multiplying a negative fraction by a whole number. Write on the board:
æ 2ö
2 2 2
6
3 ´çç- ÷÷÷ = - - - = çè 5 ø
5 5 5
5
3×2
SAY: When multiplying a whole number and a fraction, you can multiply the whole number by
the numerator and keep the denominator the same. When the fraction is negative, the answer is
negative.
Exercises: Multiply.
æ 4ö
è 5ø
a) 3 ççç- ÷÷÷
æ 3ö
è 7ø
b) 2 ççç- ÷÷÷
æ 2ö
è 7ø
c) 5 ççç- ÷÷÷
æ 7 ÷ö
÷
è 10 ÷ø
d) 3 ççç-
Answers: a) −12/5, b) −6/7, c) −10/7, d) −21/10
Extensions
(MP.3) 1. What property of division is being used?
730 ÷ 10 = 700 ÷ 10 + 30 ÷ 10 = 70 + 3 = 73
Answer: Division distributes over the second factor (i.e., when the second number stays the
same).
(MP.1, MP.7) 2. Evaluate 7,354 ÷ 11 − 7,134 ÷ 11 without using a calculator; then check on a
calculator. Hint: Use the distributive property.
Answer: 220 ÷ 11 = 20
(MP.7) 3. Fill in the blank.
a) 5 + 10 + 15 + 20 + 25 + 30 = 5 × ___
b) 24 + 46 + 68 = 2 × _____
c) 1,224 + 2,436 + 3,648 = 12 × _____
Solutions:
a) 5 × (1 + 2 + 3 + 4 + 5 + 6) = 5 × 21, so the answer is 21;
b) 2 × (12 + 23 + 34) = 2 × 69, so the answer is 69;
c) 12 × (102 + 203 + 304) = 12 × 609, so the answer is 609
(MP.1) 4. Evaluate.
a) 23 × 8 + 23 × (−6)
b) 23 × 82 + 23 × (−62)
Answers: a) 46, b) 460, c) −69
c) 23 × 17 + 23 × (−20)
5. Write three expressions equivalent to (9 + 6) × 2.
Sample answers: (6 + 9) × 2, 2 × (9 + 6), 9 × 2 + 6 × 2, 9 + 6 + 9 + 6
(MP.1) 6. a) Finish the equation using the distributive property: 2 × (7 + (−3)) = ________
b) The given expression in part a) is equal to 2 × (7 − 3). Is the expression you made in
part a) equal to 2 × 7 − 2 × 3? Explain how you know.
Answers: a) 2 × 7 + 2 × (−3), b) 2 × (−3) = −(2 × 3) because 2 × (−3) = − 3 − 3 = −(2 × 3),
so 2 × 7 + 2 × (−3) = 2 × 7 + (−(2 × 3)) = 2 × 7 − 2 × 3.
D-18
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
(MP.1) 7. a) Use the distributive property to fill in the blanks. Then write < or > in the box.
i) 8 × 12 = 8 × 10 + ____ × _____
ii) 135 × 135 = 132 × 135 + _____ × _____
10 × 10 = 8 × 10 + ____ × _____
132 × 138 = 132 × 135 + _____ × _____
So 8 × 12
So 135 × 135
10 × 10.
132 × 138.
b) Without calculating either product, decide which is larger, 17 × 23 or 20 × 20.
Answers: a) i) 8 × 12 = 8 × 10 + 8 × 2 and 10 × 10 = 8 × 10 + 2 × 10. Since 8 × 2 < 2 × 10, we
know that 8 × 12 < 10 × 10. ii) 3 × 135 > 132 × 3, so 135 × 135 > 132 × 138. b) 20 × 20 should
be larger, which can be seen when you compare both to 17 × 20. (17 × 23 is 17 × 3 more than
17 × 20, whereas 20 × 20 is 20 × 3 more than 17 × 20.)
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-19
EE7-4
More Properties of Operations
Pages 77–78
Standards: preparation for 7.EE.1
Goals:
Students will explore properties of operations and recognize when two numerical expressions
are equivalent.
Prior Knowledge Required:
Can use the correct order of operations to evaluate expressions
Vocabulary: dividend, divisor, equivalent expressions, evaluate
Multiplying the dividend and divisor by the same number gives the same answer.
Exercises: Evaluate all expressions. Make sure your answers are equal.
a) 20 ÷ 5
b) 30 ÷ 3
(20 × 2) ÷ (5 × 2)
(30 × 2) ÷ (3 × 2)
(20 × 3) ÷ (5 × 3)
(30 × 3) ÷ (3 × 3)
Answers: a) 4, 4, 4; b) 10, 10, 10
Dividing the dividend and divisor by the same number gives the same answer.
Exercises: Does dividing each number in a division question by the same number give the
same answer? Investigate by evaluating all expressions.
a) 60 ÷ 12
b) 90 ÷ 30
(60 ÷ 2) ÷ (12 ÷ 2)
(90 ÷ 2) ÷ (30 ÷ 2)
(60 ÷ 3) ÷ (12 ÷ 3)
(90 ÷ 3) ÷ (30 ÷ 3)
Answers: a) 5, 5, 5; b) 3, 3, 3; yes, they all have the same answer
Using the property to make easier questions. Write on the board:
85 ÷ 5
= _____ ÷ 10 = ______
Tell students you think this would be easier to do if it was dividing by 10, so you want to change
it. ASK: What do I have to multiply the 5 by to get 10? (2) What do I have to do to the 85 to keep
the answer the same if I multiply the 5 by 2? (multiply by 2) Have a volunteer fill in the blanks.
(170 and 17)
(MP.1) Exercises: Use dividing by 10 to make an easier problem. Then divide.
a) 120 ÷ 5 = ____ ÷ 10 = _____
b) 75 ÷ 5 = _____ ÷ 10 = _____
c) 135 ÷ 5 = ____ ÷ 10 = _____
d) 745 ÷ 5 = _____ ÷ 10 = _____
Answers: a) 240 ÷ 10 = 24, b) 150 ÷ 10 = 15, c) 270 ÷ 10 = 27, d) 1,490 ÷ 10 = 149
D-20
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
If students struggle with Question 3 on AP Book 7.1 p. 77, provide this hint:
(smaller number × smaller number) < (greater number × greater number)
Multiplying and dividing with 0. Remind students that they can always write two division
equations from a multiplication equation. Write on the board:
3 × 5 = 15
so
15 ÷ 3 = 5 and 15 ÷ 5 = 3
0×2=0
Have a volunteer write the two division equations. (0 ÷ 2 = 0 and 0 ÷ 0 = 2)
Repeat for 0 × 3 = 0. (0 ÷ 3 = 0 and 0 ÷ 0 = 3) ASK: Does anyone see anything wrong here?
PROMPT: Where do you see a question with two different answers? (0 ÷ 0 has both 2 and 3 as
an answer) Tell students that you cannot divide 0 ÷ 0 because any numerical expression can
only have one value. Now write on the board:
____ × 0 = 3
ASK: Can any number go in the blank? (no) Have volunteers use this multiplication equation to
write two division statements that also don’t have a number that will go in the blank:
3 ÷ 0 = ____ and 3 ÷ _____ = 0
SAY: If either of these has an answer, then something times 0 would be 3, but something times
0 is always 0, so no number goes in the blank.
Exercises: Write the answer or write X if there is no answer.
a) 0 × ____ = 9
b) 0 × 9 = _____
c) 9 ÷ _____ = 0
d) 0 × 1 = ____
e) 0 × ____ = 1
f) 1 ÷ 0 = ______
g) 0 ÷ 1 = _____
h) 8 ÷ 8 = _____
i) 0 ÷ 0 = _____
Answers: a) X, b) 0, c) X, d) 0, e) X, f) X, g) 0, h) 1, i) X
Subtracting 1 more makes the answer 1 less. Write on the board:
7 − 2 = 5 and 7 − 3 = 4
SAY: If I subtract 2 and 3 from the same number, subtracting 3 will always result in 1 less than
subtracting 2. Write on the board:
7 − (2 + 1) = 7 − 2 − 1
SAY: Compare both to 7 – 2. Taking away 1 more results in 1 less.
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-21
(MP.1, MP.7) Exercises: Evaluate both expressions. Are your answers the same? If not,
find the mistake.
a) 9 − (3 + 1) and 9 − 3 − 1
b) 10 − (7 + 1) and 10 − 7 − 1
c) 10 − (2 + 1) and 10 − 2 − 1
d) 9 − (6 + 1) and 9 − 6 − 1
Answers: a) 9 − 4 = 5 and 6 − 1 = 5, b) 10 − 8 = 2 and 3 − 1 = 2, c) 10 − 3 = 7 and 8 − 1 = 7,
d) 9 − 7 = 2 and 3 − 1 = 2
Subtracting more results in less by the same amount. Write on the board:
143 − 17 and 143 − 10
ASK: Which expression is greater? (143 − 10) How do you know? (because you are not
subtracting as much) How much greater is 143 − 10 than 143 − 17? (7 greater) SAY: If you
subtract 7 more, the answer is 7 less. Write on the board:
143 − (10 + 7) = 143 − 10 − 7
(MP.1, MP.7) Exercises: Evaluate both expressions. Are your answers the same? If not, find
the mistake.
a) 12 − (3 + 2) and 12 − 3 − 2
b) 18 − (5 + 5) and 18 − 5 − 5
c) 13 − (1 + 2 + 3) and 13 − 1 − 2 − 3
d) 20 − (4 + 6 + 5) and 20 − 4 − 6 − 5
Bonus: 100 − (1 + 2 + 3 + 4 + 5 + 6) and 100 − 1 − 2 − 3 − 4 − 5 − 6
Answers: a) 12 − 5 = 7 and 9 − 2 = 7, b) 18 − 10 = 8 and 13 − 5 = 8, c) 13 − 6 = 7 and
12 − 2 − 3 = 10 − 3 = 7, d) 20 − 15 = 5 and 16 − 6 − 5 = 10 − 5 = 5, Bonus: 79 and 79
Subtracting less results in more by the same amount. Write on the board:
18 − (10 − 2) = 18 – 8 and 18 − 10
ASK: Which expression is greater? (18 − (10 − 2)) How do you know? (because I am
subtracting less) SAY: If I subtract 8 instead of 10, I get an answer that is greater by 2. We
can write this as
18 − (10 − 2) = 18 − 10 + 2
Tell students that when two expressions have the same value, the expressions are said to be
equivalent.
Exercises: Evaluate both expressions. Are the expressions equivalent? If not, find the mistake.
a) 12 − (3 − 2) and 12 − 3 + 2
b) 10 − (5 − 3) and 10 − 5 + 3
Bonus: 15 − (10 − 7 − 1) and 15 − 10 + 7 + 1
Answers: a) 12 − 1 = 11 and 9 + 2 = 11, b) 10 − 2 = 8 and 5 + 3 = 8,
Bonus: 15 − (3 − 1) = 15 − 2 = 13 and 5 + 7 + 1 = 12 + 1 = 13
D-22
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
(MP.7) Writing subtraction expressions without brackets. Write on the board:
10 − (2 + 3 − 4) =
Tell students that you want to rewrite the expression without brackets, but using all the same
numbers. SAY: I want to start by comparing this to 10 − 2. Because I added 3 to what I am
subtracting, that will make the answer 3 less than 10 − 2, but because I am taking away 4 from
what is being subtracted, I am adding 4 to the answer. Write on the board:
10 − 2 − 3 + 4
SAY: When I add to the part being subtracted, I take that amount from the answer. When I
subtract from the part being subtracted, I add that amount to the answer. Write on the board:
10 − (1 + 3 − 2) =
10 − (7 − 4 + 1) =
10 − (8 − 7 + 6 − 5 + 4 − 3) =
For each expression, have volunteers write an expression without brackets that uses all the
same numbers. (10 − 1 − 3 + 2, 10 − 7 + 4 − 1, 10 − 8 + 7 − 6 + 5 − 4 + 3) Then have other
volunteers evaluate both expressions to make sure they are equal. (8 and 8, 6 and 6, 7 and 7)
(MP.7) Exercises: Write an equivalent expression without brackets that uses all the same
numbers. Check by evaluating both expressions.
a) 18 − (10 + 4 − 7)
b) 15 − (9 − 2 + 3)
Bonus: 20 − (10 − 8 − 1 + 5 + 6 − 3 + 2)
Answers: a) 18 − 10 − 4 + 7 = 11, b) 15 − 9 + 2 − 3 = 5, Bonus: 20 − 10 + 8 + 1 − 5 − 6 + 3 − 2 = 9
Writing division expressions without brackets. Write on the board:
48 ÷ (8 ÷ 2) and 48 ÷ 8
ASK: Which expression do you expect to be bigger? (48 ÷ (8 ÷ 2)) Why? (because you are
dividing by a smaller number) How much bigger do you expect it to be? (twice as big) Have
volunteers evaluate both expressions (48 ÷ 4 = 12 and 48 ÷ 8 = 6)
Exercises: Evaluate both expressions. Are the expressions equivalent?
a) 40 ÷ (10 ÷ 2) and 40 ÷ 10 × 2
b) 30 ÷ (6 ÷ 2) and 30 ÷ 6 × 2
c) 60 ÷ (10 × 2) and 60 ÷ 10 ÷ 2
d) 72 ÷ (2 × 3) and 72 ÷ 2 ÷ 3
Bonus: 60 ÷ (20 ÷ 2 ÷ 2) and 60 ÷ 20 × 2 × 2
Answers: a) 40 ÷ 5 = 8 and 4 × 2 = 8, b) 30 ÷ 3 = 10 and 5 × 2 = 10, c) 60 ÷ 20 = 3 and 6 ÷ 2 = 3,
d) 72 ÷ 6 = 12 and 36 ÷ 3 = 12, Bonus: 60 ÷ (10 ÷ 2) = 60 ÷ 5 = 12 and 3 × 2 × 2 = 12
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-23
SAY: Dividing by half the number gets an answer twice as big. Dividing by one third of a number
gets an answer three times bigger. And dividing by a number three times bigger gets an answer
one third as big. Write on the board:
60 ÷ (15 ÷ 3) = 60 ÷ 15 × 3
60 ÷ (2 × 3) = 60 ÷ 2 ÷ 3
Have volunteers verify these equations. (60 ÷ 5 = 12 and 4 × 3 = 12, 60 ÷ 6 = 10 and
30 ÷ 3 = 10)
(MP.7) Exercises: Use all the same numbers to write an equivalent expression without
brackets. Check by evaluating both sides.
a) 36 ÷ (6 ÷ 2)
b) 36 ÷ (6 × 2)
c) 36 ÷ (6 ÷ 2 × 3)
d) 36 ÷ (6 × 2 ÷ 3)
e) 80 ÷ (4 × 4 ÷ 2 × 5)
Answers: a) 36 ÷ 6 × 2 = 6 × 2 = 12 and 36 ÷ 3 = 12, b) 36 ÷ 6 ÷ 2 = 6 ÷ 2 = 3 and 36 ÷ 12 = 3,
c) 36 ÷ 6 × 2 ÷ 3 = 4 and 36 ÷ 9 = 4, d) 36 ÷ 6 ÷ 2 × 3 = 9 and 36 ÷ 4 = 9, Bonus:
80 ÷ 4 ÷ 4 × 2 ÷ 5 = 2 and 80 ÷ 40 = 2
Extensions
(MP.1) 1. Remind students that dividing both numbers in a division by the same number doesn’t
change the answer. Teach students that this follows from the fact that multiplying both numbers
by a fraction doesn’t change the answer. This is because, for example, dividing by 2 gives the
same answer as multiplying by 1/2.
Fill in the blanks.
30 ÷ 6 = 5
30 ÷ 6 = 5
1
1
( 30 ´ ) ÷ ( 6 ´ ) = ___ ÷ ___ = ____
2
2
1
1
( 30 ´ ) ÷ ( 6 ´ ) = ___ ÷ ___ = ____
3
3
(30 ÷ 2) ÷ (6 ÷ 2) = ___ ÷ ___ = ____
(30 ÷ 3) ÷ (6 ÷ 3) = ___ ÷ ___ = ____
Answer: all answers are 5
(MP.7) 2. Use all the same numbers to write an equivalent expression without brackets:
600 ÷ (60 ÷ (5 × 3)). Check by evaluating both expressions.
Answer: 600 ÷ 60 × 5 × 3 = 10 × 5 × 3 = 150 and 600 ÷ (60 ÷ 15) = 600 ÷ 4 = 150.
3. (MP.7) a) Does 60 ÷ (4 × 5) equal 60 ÷ (5 × 4)? How do you know?
b) Does 60 ÷ 4 ÷ 5 = 60 ÷ 5 ÷ 4? How do you know?
c) Rewrite without brackets: (30 ÷ 2) ÷ (10 ÷ 2). Show that the expression is equal to 30 ÷ 10.
Answers:
a) Yes, because 4 × 5 = 5 × 4, so both expressions are dividing 60 by the same number;
b) Yes, because 60 ÷ 4 ÷ 5 = 60 ÷ (4 × 5) and 60 ÷ 5 ÷ 4 = 60 ÷ (5 × 4);
c) 30 ÷ 2 ÷ 10 × 2 = 30 ÷ 10 ÷ 2 × 2, and since dividing by 2 and multiplying by 2 doesn’t change
the answer, the expression equals 30 ÷ 10
D-24
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
EE7-5
The Standard Method for Multiplication
Pages 79–82
Standards: preparation for 7.EE.1
Goals:
Students will multiply large numbers by breaking them into smaller numbers. Students will use
the standard algorithm to multiply two multi-digit numbers.
Prior Knowledge Required:
Can apply the distributive property of multiplication
Knows the commutative property of multiplication
Vocabulary: product, standard algorithm, sum
The standard method for multiplication with no regrouping. Tell students that the standard
method is a shortcut way to multiply the ones first and then the tens. Write on the board:
Step 1: Multiply the ones.
Step 2: Multiply the tens.
43
×
43
2
6=3×2
×
2
8 6 = 40 × 2 + 3 × 2
ASK: What property is being used here? (the distributive property)
Exercises: Multiply the ones and tens separately.
a) 41 × 2
b) 23 × 3
c) 31 × 4
Answers: a) 82, b) 69, c) 124
Multiplying the ones when regrouping is required. SAY: Sometimes you have to regroup.
Write on the board:
47 × 5
Step 1: Multiply the ones.
7 × 5 = 35
Step 2: Regroup 30 ones as 3 tens.
3
47
× 5
5 7 × 5 = 35 = 3 tens + 5 ones
SAY: Write the 3 over the tens column because we are regrouping 30 ones as 3 tens. The 5
goes in the ones column of the answer.
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-25
Exercises: Do Steps 1 and 2. Don’t multiply the tens yet.
a) 32 × 6
b) 42 × 8
c) 53 × 7
Answers: a) 1 ten and 2 ones, b) 1 ten and 6 ones, c) 2 tens and 1 one
Multiplying the tens when regrouping the ones is required. Write on the board:
47 × 5
SAY: We’ve already multiplied the ones. Now we have to multiply the tens. There are 4 tens in
47 and we are multiplying by 5. Write on the board:
5 × 4 tens = 20 tens
SAY: We already have 3 tens from the 35 ones, so altogether we have 23 tens. Write on the
board (using a different color for the “23” in bold):
3
47
×
5
235
5 × 4 tens + 3 tens = 23 tens
Exercises:
1. Finish the multiplication.
a) 5
b) 6
27
19
× 8
× 7
6
3
Answers: a) 216, b) 133, c) 170, d) 261
2. Multiply.
a) 75 × 3
b) 85 × 4
Answers: a) 225, b) 340, c) 201, d) 480
c) 2
34
× 5
0
d) 2
87
× 3
1
c) 67 × 3
d) 96 × 5
SAY: You can multiply any number of digits by a 1-digit number in the same way: Multiply the
ones first, then the tens, then the hundreds, and so on. Write on the board:
multiply the ones
2,301
2,301
×
×
2
2
D-26
tens
2
02
hundreds
thousands
2,301
2,301
×
2
602
×
2
4,602
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
Exercises: Multiply.
a) 2,034 × 2
b) 3,103 × 3
Bonus: 721,201 × 4
Answers: a) 4,068; b) 9,309; Bonus: 2,884,804
Tell students that sometimes they need to regroup. Demonstrate the first example in each set
below. You may need to help some students by providing a box on top of the tens column or
hundreds column so that they know where to put the regrouped tens or hundreds.
Exercises:
1. Regroup the ones as tens or the tens as hundreds.
a) 129 × 3
b) 314 × 3
c) 927 × 2
d) 281 × 4
e) 372 × 3
f) 741 × 2
Answers: a) 387; b) 942; c)1,854; d) 1,124; e) 1,116; f) 1,482
2. Regroup where you need to. Sometimes you will need to regroup more than once.
a) 5,152 × 3
b) 7,238 × 4
c) 38,124 × 2
d) 347 × 4
e) 283 × 5
f) 53,474 × 3
Answers: a) 15,456; b) 28,952; c) 76,248; d) 1,388; e) 1,415; f) 160,422
Students who need extra practice can copy parts of Question 10 on AP Book 7.1 p. 81 onto
grid paper, solve them again using regrouping, and check that they got the same answer both
times.
Multiplying by 10. Write on the board:
734 × 10 = 7,340
SAY: 700 × 10 is 7,000, 30 × 10 is 300, and 4 × 10 is 40. So to multiply by 10, you make every
digit worth 10 times as much, and write 0 as the ones digit.
Exercises: Multiply.
a) 3 × 10
b) 7 × 10
c) 10 × 92
d) 10 × 41
e) 187 × 10
f) 4,306 × 10
Answers: a) 30; b) 70; c) 920; d) 410; e) 1,870; f) 43,060
Multiplying by 1-digit multiples of 10. Write on the board:
34 × 20 = 34 × (2 × 10)
= (34 × 2) × 10
= 68 × 10
= 680
ASK: Which property of multiplication is being used here? (the associative property)
SAY: Because 20 splits as 2 × 10, you can first multiply 34 × 2 and then multiply the result
by 10.
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-27
Write on the board:
56 × 30 = 56 × 3 × 10
Have a volunteer evaluate 56 × 3 (168); then ask another volunteer to find 56 × 30. (1,680)
Exercises: Multiply.
a) 23 × 30
b) 412 × 20
c) 816 × 40
d) 125 × 80
Bonus:
e) 324 × 200,000
f) 320 × 6,000
Answers: a) 690; b) 8,240; c) 32,640; d) 10,000; Bonus: e) 64,800,000; f) 1,920,000
Multiplying 2-digit numbers by 2-digit numbers. Write on the board:
28
× 36
Tell students that they can do this question by splitting the problem into two easier problems.
Write on the board:
28 × 30
28 × 6
ASK: Are these problems easier than 28 × 36? (yes) If you know the answer to both of these
problems, how can you get the answer to 28 × 36? (add the answers) Write on the board:
28 × 36 = 28 × 30 + 28 × 6
ASK: What property does this show? (the distributive property) Tell students that to use the
standard algorithm for multiplication, they need to multiply 28 × 6 first and then multiply 28 × 30.
Write on the board (you can project grid paper onto the board to make this easier):
2
+
4
2
8
×
3
6
1
6
8
28 × 6
8
4
0
28 × 30
Point out that the 4 is showing the regrouping for 8 × 6 = 48 and the 2 is showing the regrouping
for 8 × 30 = 240. Then SAY: Then you add the results. Have a volunteer add the results to finish
the multiplication. (1,008)
D-28
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
Exercises: Practice multiplying the first number by the ones digit of the second number.
a) 23 × 17
b) 46 × 19
c) 25 × 32
d) 42 × 26
Answers: a) 161, b) 414, c) 50, d) 252
SAY: The next step is to multiply the first number by the tens in the second number. So in
23 × 17, you would multiply 23 × 10. Write your answer underneath your answer to 23 × 7, so
that the place values are lined up.
Exercises: Practice multiplying the first number by the tens in the second number.
a) 23 × 17
b) 46 × 19
c) 25 × 32
d) 42 × 26
Answers: a) 230, b) 460, c) 750, d) 840
SAY: Now you need to complete the multiplication by adding the two answers.
Exercises:
1. Complete the multiplication.
a) 23 × 17
b) 46 × 19
c) 25 × 32
d) 42 × 26
Answers: a) 391; b) 874; c) 800; d) 1,092
2. Multiply.
a) 85 × 32
b) 76 × 39
c) 34 × 82
d) 58 × 76
Answers: 2,720; b) 2,964; c) 2,788; d) 4,408
Tell students that they can multiply multi-digit numbers in the same way, multiplying one place
value at a time. Write on the board:
7,324
× 547
7,324
×
7
7,324
×
40
7,324
× 500
= 324 × 7
= 324 × 40
= 324 × 500
Have volunteers do each multiplication (51,268; 292,960; 3,662,000) and another volunteer do
the adding (4,006,228).
Exercises: Multiply.
a) 512 × 316
b) 723 × 841
Answers: a) 161,792; b) 608,043; c) 349,950,256
c) 52,138 × 6,712
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-29
Extensions
1. Have students calculate the total number of days in a non-leap year in two ways:
a) By adding the number of days in each month (a sequence of 12 numbers):
31 + 28 + 31 + … + 30 + 31 = 365
(MP.4) b) By changing the numbers in the sequence to make a multiplication and addition
statement:
31 + 28 + 31 + 30 + 31 + 30 + 31 + 31 + 30 + 31 + 30 + 31
= 30 + 30 + 30 + 30 + 30 + 30 + 30 + 30 + 30 + 30 + 30 + 30 + 1 − 2 + 1 + 1 + 1 + 1 + 1 + 1
= (30 × 12) + 5 = 360 + 5
= 365
Guide students in part b) by asking them to first find a round number close to 28 and 31 that
they can change all 12 numbers to. Then have them think about how to adjust for the changes
they made.
(MP.1) 2. Fill in the missing numbers.
2
2
a)
3
4
´
b)
9
1
6
5
´
6
4
c)
5
3
´
1
d)
´
6
1
4
9
1
3
0
0
4
4
Answers:
2
a)
4
´
9
D-30
3
4
6
b)
5
´
1
6
6
3
5
c)
1
9
4
´
5
6
1
4
4
1
2
0
0
1
3
4
4
4
3
´
2
2
d)
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
EE7-6
Variables and Expressions
Pages 83–84
Standards: 7.EE.1
Goals:
Students will substitute values for the variable in algebraic expressions involving one operation
and translate simple word problems into algebraic expressions.
Prior Knowledge Required:
Can add, subtract, multiply, and divide whole numbers
Can multiply fractions and integers by whole numbers
Vocabulary: algebraic expression, equation, evaluate, numeric expression, value, variable
A variable represents a changing number.
Exercises: There are 4 people in one family. Write an addition expression for the number of
chairs needed to seat everybody for the given number of guests.
a) 1 guest
b) 2 guests
c) 3 guests
d) 11 guests
e) n guests
Answers: a) 4 + 1, b) 4 + 2, c) 4 + 3, d) 4 + 11, e) 4 + n
ASK: What is changing? (the number of guests) SAY: You can use a letter to represent a
number that changes. The letter is called a variable because it represents a number that varies,
or changes. ASK: If there are n guests, how many chairs are needed? (4 + n) Tell students that
an expression that has a variable is called an algebraic expression and an expression that
includes only numbers is called a numeric expression. In the example above, the variable
represents the number of guests.
(MP.2, MP.4) Exercises: Write a subtraction expression for the change (in dollars) from $20 if
the price of a CD is …
a) $8
b) $13
c) $18
d) $n
Answers: a) 20 − 8, b) 20 − 13, c) 20 − 18, d) 20 − n
Evaluating expressions with addition or subtraction. Tell students that if they know that the
number of chairs needed is 4 + n, where n is the number of guests, then they can figure out how
many chairs are needed for any number of guests. Using a different color for the variable, write
on the board:
4+n
SAY: If I want to evaluate the expression when n = 7, for example, I can just replace the
variable with 7. Using the same color for 7 as you did for n, write on the board:
4 + 7 = 11
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-31
SAY: Writing 11 is called “evaluating the expression,” because we are saying the value of the
expression. Write the words “evaluate” and “value” on the board with underlining as shown, to
emphasize the connection.
Exercises:
1. Replace n with 3 in the expression. Then evaluate the expression.
a) n + 2
b) n − 1
c) 5 − n
d) 7 + n
Answers: a) 5, b) 2, c) 2, d) 10
2. Evaluate the expression when n = 2.
a) 5 + n
b) 5 − n
c) n + 6
d) − 3 + n
e) − 5 − n
Bonus: Evaluate the same expressions at n = −2 instead of n = 2.
Answers: a) 7, b) 3, c) 8, d) −1, e) −7, Bonus: a) 3, b) 7, c) 4, d) −5, e) −3
Write on the board:
n+6
Tell students that the expression represents the number of chairs in the nth row of a theater.
ASK: What does 7 + 6 represent? (the number of chairs in the 7th row) How many chairs are in
the 7th row? (13) How many chairs are in the 12th row? (18) How many are in the 30th row? (36)
(MP.4) Exercises:
a) There are n + 5 chairs in the nth row of a theater. How many chairs are in the 20th row?
b) The nth person in a line-up waits for n + 7 minutes. How many minutes does the 14th person
in line have to wait?
c) A taxi ride for n minutes costs $(n + 3). How much does a 24-minute taxi ride cost?
Answers: a) 25, b) 21 minutes, c) $27
Bonus: The Celsius and Kelvin scales are two temperature scales. When the temperature is
n kelvin, the temperature is (n − 273)°C. What is the temperature in degrees Celsius when the
temperature is …
a) 300 kelvin
b) 250 kelvin
c) 15 kelvin
d) 3,000 kelvin
Answers: a) 27°C; b) −23°C; c) −258°C; d) 2,727°C
Writing expressions with multiplication only. Tell students that it costs $3 to rent a pair of
skates for 1 hour. Write on the board:
a) 2 hours
b) 3 hours
c) 10 hours
d) h hours
Have volunteers write an expression on the board for how much it costs (in dollars) to rent the
skates for each part. (a) 3 × 2, b) 3 × 3, c) 3 × 10, d) 3 × h)
SAY: In this case, h is being used to represent the number of hours.
D-32
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
(MP.4) Exercises: Which quantity changes?
a) Poppies are on sale for 5¢ each.
b) An Internet café charges $2 for each hour.
c) A grocery store charges 5¢ for each plastic bag.
Answers: a) poppies, b) hours, c) plastic bags
(MP.6) Tell students that the quantity that is changing is what the variable will represent in the
expression. When using algebraic expressions to represent situations, it is important to write
what the variable represents and what the expression represents. Refer students to the exercise
above: “Poppies are on sale for 5¢ each.” ASK: What expression could we write to represent the
situation? (n × 5) What does n represent? (the number of poppies) What does n × 5 represent?
(the cost of the poppies) Demonstrate on the board how to write it:
The cost for n poppies is n × 5¢.
Exercises: Write the expression, what it represents, and what the variable represents.
a) An internet café charges $2 for each hour.
b) A grocery store charges 5¢ for each plastic bag.
c) Each soccer team has 11 players.
Answers: a) The cost for using the Internet for h hours is $2 × h. b) The cost for n bags is
n × 5¢. c) The number of players on n teams is 11 × n.
Writing multiplication without the multiplication sign. Tell students that when letters are
used in an expression, the multiplication sign (×) is often omitted to avoid confusion with the
letter “x” and to make the notation shorter. In this case, instead of writing 2 × n, you can simply
write 2n.
Exercises: Write the expression without multiplication signs.
a) 3 × n
b) 7 × r
c) 8 × T
Answers: a) 3n, b) 7r, c) 8T, d) 3mn or 3nm
d) 3 × m × n
Write on the board:
3 × n = n × 3, so 3n = n3.
SAY: Since multiplication is commutative, 3n and n3 mean the same thing. When a
multiplication expression involves numbers and variables, the convention is to write the number
first all the time. So you write 3n, not n3.
Exercises: Write the expression without multiplication signs.
a) n × 9
b) n × 7 × m
c) 5 × p
Answers: a) 9n, b) 7nm or 7mn, c) 5p, d) 2qr or 2rq
d) q × r × 2
Evaluating expressions with multiplication. Write on the board:
Renting skis for n hours costs $8n.
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-33
ASK: How much does renting skis for 3 hours cost? ($24) Tell students that you want to write
that as an expression, but replacing n with 3 in 8n looks like 83 when we mean 8 × 3. Write on
the board:
n = 3, so 8n = 8(3), not 83
SAY: We include brackets when replacing a variable directly with a number could cause
confusion. This happens whenever a variable is being multiplied by a number.
Exercises:
1. Evaluate the expression at n = 3.
a) 7n
b) 10n
c) 12n
d) 500n
Answers: a) 7(3) = 21; b) 10(3) = 30; c) 12(3) = 36; d) 500(3) = 1,500
2. Evaluate the expression at n = −2.
a) 3n
b) 10n
c) 8n
d) 8,000n
Answers: a) 3(−2) = −6; b) 10(−2) = −20; c) 8(−2) = −16; d) 8,000(−2) = −16,000
(MP.4) 3. a) Renting a canoe for n hours costs $7n. How much does it cost to rent the canoe for
6 hours?
b) A boat travels 7h km in h hours. How far does it travel in 4 hours?
c) The nth figure uses 3n squares. How many squares are in the 12th figure?
Answers: a) $42, b) 28 km, c) 36 squares
Variable side lengths. Tell students that sometimes in math problems, the lengths of the sides
of shapes are given in terms of variables. Write on the board:
x+3
3
ASK: What are the lengths of the sides of the rectangle when x is 4? (3 and 7)
Exercises: a) Use grid paper to draw the rectangle above when
i) x = 5
ii) x = 2
iii) x = 0
b) When is the rectangle a square?
Answers: a) i) 3 by 8, ii) 3 by 5, iii) 3 by 3; b) when x = 0
Draw on the board:
x+4
x
ASK: Can you draw this rectangle for x = 0? (no) Why not? (a side length cannot be 0)
D-34
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
Exercises: a) Use grid paper to draw the rectangle when …
i) x = 1
ii) x = 2
iii) x = 3
b) When is the rectangle a square?
Answers: a) i) 1 by 5, ii) 2 by 6, iii) 3 by 7; b) never, because x can never be equal to 4 more than x
Draw on the board:
x
x
ASK: When is this rectangle a square? (for any side length x)
Exercises: Use grid paper to draw the rectangle when
a) x = 1
b) x = 2
c) x = 3
Answers: a) 1 by 1, b) 2 by 2, c) 3 by 3
Extensions
1. Jim has n dimes. He has the same amount of money in nickels. Write an expression for how
many nickels Jim has.
Answer: 2n
(MP.1) 2. Draw the rectangle below for x = 1 cm, 2 cm, and 3 cm using centimeter grid paper.
Measure the diagonal. Write an expression for the diagonal.
3x
4x
Answers: 5 cm, 10 cm, 15 cm; the diagonal is 5x
3. Tasha types at least 30 words per minute. She is paid $(n − 30) per hour, where n is the
number of words per minute. How much does Tasha make in 7 hours if she types 50 words
per minute?
Answer: $140
(MP.2) 4. a) Sketch a circle divided into the following fractions.
i) thirds
ii) fourths
iii) fifths
b) Evaluate the expression 360x for
i) x =
1
3
ii) x =
1
4
iii) x =
1
5
c) Use your answers to b) and a protractor to check the accuracy of your sketches in a).
Answers: b) i) 120, ii) 90, iii) 72, c) Since there are 360° in a full turn, students just have to
check how close the angles they sketched are to 120°, 90°, and 72°.
360°
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-35
EE7-7
Expressions with Two Operations
Pages 85–87
Standards: 7.EE.1
Goals:
Students will substitute values for the variable in algebraic expressions involving two operations
and translate simple word problems into algebraic expressions.
Prior Knowledge Required:
Knows the order of operations
Can substitute values for the variable in algebraic expressions involving one operation
Can translate simple word problems into algebraic expressions involving one operation
Vocabulary: denominator, expression, integer, negative, numerator, positive, unknown,
variable
Replacing a variable with a number and evaluating. Using a different color for the variable,
write on the board:
5n + 3
Tell students you want to evaluate the expression when n = 8, so you need to replace n with 8.
Demonstrate doing so by using the same color for 8 as you used for the variable:
5(8) + 3
Ask a volunteer to evaluate the expression. (40 + 3 = 43)
Exercises: Replace n with 5, and then evaluate.
a) 3n
b) 10n
c) 10n + 1
d) 10(n − 2)
e) 10n + 4
f) 8n − 7
Answers: a) 15, b) 50, c) 51, d) 30, e) 54, f) 33
Remind students how to multiply a negative integer by a whole number. Write on the board:
3(5) = 15, so 3(−5) = −15.
Exercises:
1. Evaluate the expression.
a) 4(−5)
b) 2(−8) + 6
Answers: a) −20, b) −10, c) −10, d) −11
D-36
c) 5(4 − 6)
d) 3(−2) − 5
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
2. Replace n with −2. Then evaluate the expression.
a) 4n
b) 13n
c) 42n
d) 83(n + 2)
Answers: a) −8, b) −26, c) −84, d) 0, e) −21, f) −11
e) 7(n − 1)
f) 7n + 3
SAY: Fractions are easy to multiply by a whole number. Multiply the numerator by the whole
number and keep the denominator the same. Demonstrate with parts a) and b) of the exercises
below, and have students do the rest.
Exercises: Evaluate the expression at n =
a) 4n
b) 5n
c) 2n
3
.
5
d) 3n
e) 7n
Answers: a) 12/5 or 2 2/5, b) 15/5 or 3, c) 6/5 or 1 1/5, d) 9/5 or 1 4/5, e) 21/5 or 4 1/5
SAY: Fractions with the same denominator are easy to add and subtract. Just add or subtract
the numerators. Demonstrate with part a) of the exercises below and have students do the rest.
Exercises: Evaluate the expression at n =
a) n + 2
b) n +
4
5
c) 2n + 1
3
.
5
d) 3 n -
1
5
e)
2
- 2n
5
Answers: a) 2 3/5 or 13/5, b) 7/5 or 1 2/5, c) 11/5 or 2 1/5, d) 8/5 or 1 3/5, e) −4/5
SAY: You can add or subtract fractions with different denominators by making equivalent fractions
that have the same denominator. Demonstrate with part a) of the exercises below and have
students do the rest. Remind students that when the question has mixed numbers or improper
fractions, the answer should, too. If not, they have a choice of how to express the answer.
Exercises: Evaluate the expression at n =
a) 3 n -
1
2
b) -n -
2
3
3
.
5
c) 3 − 4n
1
2
d) 2 - 7 n
Answers: a) 13/10 or 1 3/10, b) −19/15 or −1 4/15, c) 3/5, d) −1 7/10
Review adding and subtracting negative numbers. Write on the board:
+(+) = +
+(−) = −
−(+) = −
−(−) = +
SAY: Subtracting a negative number is like adding a positive number, and adding a negative
number is like subtracting a positive number. Write on the board:
5 − (−1) = 5 + 1 = 6
5 + (−2) =
−3 + (−4) =
−3 − (−4) =
Have volunteers write the answers. (5 − 2 = 3 or +3, −3 − 4 = −7, −3 + 4 = 1 or +1)
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-37
Exercises: Evaluate the expressions.
a) −3 − (−5)
b) 9 + (−5)
c) 7 + (−8)
Answers: a) 2 or +2, b) 4 or +4, c) −1, d) −1, e) −7
d) −4 − (−3)
e) −5 − (+2)
Write on the board:
7 − 2(−3) = 7 − (
)
ASK: What number is 2 times −3? (−6) Write that in the brackets. ASK: What is 7 − (−6)?
(7 + 6 = 13) Finish the calculation on the board:
7 − 2(−3) = 7 − (−6) = 7 + 6 = 13
Exercises:
1. Finish the equation. Then evaluate the expression.
a) 8 − 2(−4) = 8 − (___)
b) 5 + 3(−4) = 5 + (___)
c) −9 − 2(−5) = − 9 − (___)
Answers: a) 8 − (−8) = 16 or +16, b) 5 + (−12) = −7, c) − 9 − (−10) = 1 or +1
2
3
2. Evaluate the expression at n = - .
a) 8 − 3n
b) 7n − 5
c)
8
- 2n
5
d)
3
+ 4n
4
Answers: a) 10, b) −29/3 or −9 2/3, c) 44/15 or 2 14/15, d) −23/12 or −1 11/12
Introduce flat fees. Tell students that a flat fee is a fixed charge that does not depend on how
long you rent an item. Because the fixed charge doesn’t change, or vary, you don’t need a
variable.
(MP.4) Exercises:
1. What quantity must be represented by a variable?
a) A skate rental company charges a $2 flat fee and $3 for each hour.
b) A boat rental company charges a $10 flat fee and $5 per hour.
c) A taxi company charges a $5 flat fee and $2 for each kilometer.
Bonus: A bus company charges 10¢ per kilometer and $5 per passenger.
Answers: a) hours, b) hours, c) kilometers, Bonus: kilometers and passengers
2. Write an expression for the cost of renting a boat at a flat fee of $9 and an hourly rate of $5 for…
a) 1 hour
b) 3 hours
c) 4 hours
d) 11 hours e) h hours
f) m hours
Answers: a) 9 + 1 × 5, b) 9 + 3 × 5, c) 9 + 4 × 5, d) 9 + 11 × 5, e) 9 + h × 5, f) 9 + m × 5
D-38
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
Using mass to model expressions. Draw on the board:
weighs 1 kg.
weighs x kg.
Tell students that a circle has mass 1 kg but you don’t know the mass of the triangle. Let’s call
its mass x kg, an unknown, because we don’t know what it is.
Exercises: Write the total mass, in kilograms.
a)
b)
c)
d)
f)
e)
Answers: a) 5 kg, b) 4x kg, c) 6 kg, d) 5x kg, e) x kg + 3 kg, f) 3 kg + x kg
Applying the commutative property of addition to algebraic expressions. Point students to
the answers for parts e) and f). ASK: Will the masses always be the same for parts e) and f)?
(yes) How do you know? (because the mass of a triangle and 3 circles is the same as the mass
of 3 circles and a triangle) Write on the board:
5+w
w+5
w=1
w=2
w=3
w=4
Ask volunteers to dictate what to put in each box. (see completed table below)
5+w
w+5
w=1
6
6
w=2
7
7
w=3
8
8
w=4
9
9
Point out that the two expressions are always equal. ASK: Why is that? (because addition is
commutative)
Equivalent expressions. Remind students that two numerical expressions—expressions with
numbers only—are equivalent when they have the same value. Tell students that two algebraic
expressions—expressions that have at least one variable—are equivalent if they have the same
value, no matter what you use for the variable. The expressions 5 + w and w + 5 are equivalent.
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-39
Exercises: Are these expressions equivalent? Check when the variable is 0, 1, 2, and one
other value of your choice.
a) 7 + x and x + 7
b) n × 5 and 5n
c) 6w + 5 and 5 + 6w
Answers: a) yes, b) yes, c) yes
Applying the associative property to algebraic expressions. Have volunteers draw a picture
to show the masses of 3(5x) and (3 × 5)x. ASK: Are these expressions equivalent? (yes) How
do you know? (both pictures show 15 triangles, so they both have the same mass, no matter
what the mass of a triangle is)
Exercises: Evaluate the expressions 3(5x) and (3 × 5)x when
a) x = 0
b) x = 1
c) x = 2
Bonus: x = −4
Answers: a) 0 and 0, b) 15 and 15, c) 30 and 30, Bonus: −60 and −60
ASK: In the last exercise, which property of multiplication were you checking every time you
showed that two expressions were equal for the same value of x? (the associative property)
Extensions
1. A rectangle has area xy. Find the area if x = 5 and y = 7.
Answer: 35
2. A triangle has area 1/2 bh. Find the area if b = 8 and h = 3.
Answer: 12
3. a) Write an expression for the area of the rectangle:
2x + 1
3
b) Use your expression to find the area when
i) x = 1
ii) x = 2
iii) x = 3
iv) x = 49
Answers: a) 3(2x + 1); b) i) 9, ii) 15, iii) 21, iv) 297
4. Evaluate (x − 80)(80 − x) when x = 90.
Answer: −100
D-40
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
EE7-8
Adding and Subtracting Expressions
Pages 88–89
Standards: 7.EE.1
Goals:
Students will add and subtract expressions of the form ax where a is an integer.
Prior Knowledge Required:
Can add and subtract integers
Can evaluate expressions at given values
Knows the distributive property
Vocabulary: coefficient, distributive property, equivalent expressions, integer, variable
Multiplication by a whole number is repeated addition. Remind students that multiplication
by a whole number is short for repeated addition. Write on the board:
2+2+2+2=4×2
3+3+3+3+3=
4+4+4=
5+5+5+5+5+5+5=
x+x+x+x+x=
Have volunteers write each addition as a multiplication. (5 × 3, 3 × 4, 7 × 5, 5x or 5 × x)
Exercises:
1. Write the sum as a product of a number and a variable.
a) x + x + x + x + x
b) n + n + n
c) m + m + m + m + m + m
Answers: a) 5x, b) 3n, c) 6m
2. Write the expression as repeated addition.
a) 5n
b) 4x
c) 7y
Answers: a) n + n + n + n + n, b) x + x + x + x, c) y + y + y + y + y + y + y
Adding expressions of the form ax. Write on the board:
3x + 4x =
x+x+x
+
x+x+x+x
ASK: How many x’s are being added altogether? (7) How do we write that as a multiplication?
(7x) Write on the board:
3x + 4x = 7x
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-41
Exercises: Add by writing out how many x’s there are altogether.
a) x + 5x
b) 2x + 2x
c) 3x + 5x
d) 6x + x
Answers: a) 6x, b) 4x, c) 8x, d) 7x
Write on the board:
3x + 4x = (3 + 4)x = 7x
SAY: To find the total number of x’s, add the numbers in front of the x’s. These numbers are
called coefficients. We’ll learn more about coefficients later in this unit.
(MP.8) Exercises: Add the x’s by adding the coefficients.
a) 2x + 8x
b) 4x + 7x
c) 3x + 3x
Bonus: 2x + 2x + 3x + 5x
Answers: a) 10x, b) 11x, c) 6x, d) 9x, Bonus: 12x
d) 5x + 4x
Tell students that when there is no number in front of the x, there is just one x, so they can think
of x as 1x.
Exercises: Add the x’s by adding the coefficients.
a) x + 3x
b) 7x + x
c) 5x + x + x
d) x + 2x + 3x
(MP.4) Bonus: Pizza costs $5 per student and drinks cost $2 per student. Write two
expressions for the total cost of x students buying pizza and drinks.
Answers: a) 4x, b) 8x, c) 7x, d) 6x, Bonus: 5x + 2x or 7x
Using mass to verify the sum of expressions. Write on the board:
3x + 4x = 7x
Remind students that they can use mass to model expressions. For example, if each triangle
weighs x kg, then 3 triangles weigh 3x kg, 4 triangles weigh 4x kg, and there are 7 triangles
altogether. So altogether they weigh 7x kg. So that’s another way of seeing that 3x + 4x = 7x.
Using substitution to verify the sum of expressions. SAY: Another way to verify that 3x + 4x
is 7x is to check that they have the same value for every x. SAY: That means I can try putting
any number in, 2 for example, and both expressions should be equal. Write on the board:
3(2) + 4(2) or 7(2)
ASK: What is 3(2)? (6) What is 4(2)? (8) What is 7(2)? (14) Does 6 + 8 equal 14? (yes)
SAY: OK, so both sides are equal for x = 2, but what about other values of x?
D-42
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
(MP.1) Exercises: Evaluate both expressions. Are your answers the same?
3x + 4x
7x
a) x = 1
b) x = 3
c) x = 5
d) x = 10
Answers: a) 3 + 4 = 7 and 7 × 1 = 7, yes; b) 9 + 12 = 21 and 7 × 3 = 21, yes;
c) 15 + 20 = 35 and 7 × 5 = 35, yes; d) 30 + 40 = 70 and 7 × 10 = 70, yes
(MP.6) ASK: What property of multiplication does each row show? (that it distributes over
addition) To guide students, you could write the names of the properties on the board:
associative
commutative
distributive
SAY: The distributive property is telling us that no matter what number I pick, adding 3 of them
to 4 of them gives 7 of them. So numbers are just like any other objects. Just like 3 apples + 4
apples is 7 apples, it’s also the case that 3 twos + 4 twos is 7 twos. Since this is true for any
number, we can say that 3 x’s + 4 x’s = 7 x’s.
SAY: When two expressions are equivalent, they are equal even for negative numbers and
fractions.
(MP.1) Exercises: Evaluate both expressions for more values of x. Are your answers the same?
3x + 4x
7x
a) x = 0
b) x = −1
c) x = −3
d) x = 2/3
e) x = −1/5
Answers: a) 0 + 0 = 0 and 7 × 0 = 0, yes; b) −3 − 4 = −7 and 7 × (−1) = −7, yes;
c) −9 − 12 = −21 and 7 × (−3) = −21, yes; d) 6/3 + 8/3 = 14/3 and 7 × 2/3 = 14/3, yes;
e) −3/5 − 4/5 = −7/5 and 7 × (−1/5) = −7/5, yes
ASK: Which expression was easier to evaluate, 3x + 4x or 7x? Point out how much easier it is to
put all the x’s together before evaluating it. SAY: We get the same answer both ways, so, if we
need to evaluate an expression, we might as well put all the x’s together first.
Subtracting expressions of the form ax by taking away x’s. Write on the board:
7x − 3x
x+x+x+x+x+x+x
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-43
SAY: The picture shows 7 x’s being added. I want to subtract 3 of them. Show doing so on the
board:
7x − 3x
x+x+x+x+x+x+x
ASK: How many x’s are left? (4) Point out that when you start with 7 of anything, and take away
3 of them, you end up with 4 of them. Write the subtraction on the board:
7x − 3x
x+x+x+x+x+x+x
7x − 3x = 4x
SAY: You’re really just subtracting the coefficients.
Exercises: Subtract.
a) 3x − 2x
b) 5x − 2x
c) 6x − 3x
d) 9x − 5x
e) 83x − 81x
Bonus: 500x − 491x
Answers: a) 1x or x, b) 3x, c) 3x, d) 4x, e) 2x, Bonus: 9x
Remind students that if there is no number in front of the x, then there is only one x, so they can
assume the coefficient is 1.
(MP.8) Exercises:
1. Subtract.
a) 7x – x
b) 8x − x
c) 9x − x
d) 87x − x
Bonus: 500x − x
Answers: a) 6x, b) 7x, c) 8x, d) 86x, Bonus: 499x
2. Write the expression so that x appears only once.
a) 8x − 3x + 2x
b) 9x − 2x + 3x
c) 7x − 2x + 4x
d) 3x + 4x − 2x
Bonus: 9x − 8x + 7x − 6x + 5x − 4x + 3x − 2x + x
Sample solution: b) 9x − 2x + 3x = 10x because 9 − 2 + 3 = 10.
Answers: a) 7x, c) 9x, d) 5x, Bonus: 5x (Students might notice that each pair of consecutive
terms (9x − 8x, 7x − 6x, 5x − 4x, and 3x − 2x) makes x.)
Adding and subtracting combined. Remind students that when adding and subtracting longer
sequences of numbers, it is easier to look at the numbers being added separately from the
numbers being subtracted. Write on the board:
7−4+3+2−1
ASK: What is the total of the numbers being added? (12) What is the total of the numbers being
subtracted? (5) Write on the board:
= 12 − 5 = 7
D-44
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
SAY: You can do the same thing with variables. Write on the board:
7x − 4x + 3x + 2x − x
Remind students that when there is no number in front of the variable, the term is really 1x.
ASK: Is the 1x being added or subtracted here? (subtracted) Have a volunteer circle all the x’s
that are being added.
7x − 4x + 3x + 2x − x
ASK: How many x’s are being added in total? (12) How many x’s are being subtracted? (5)
Write on the board:
= 12x − 5x = 7x
Repeat for 8x – 3x −5x. This time, all eight x’s are being subtracted, so there are no x’s left.
Write on the board:
0x = 0
Exercises: Add and subtract to simplify the expression.
a) 9x − 5x − 6x + 2x − 3x
b) 2x − 5x − 3x + x
c) − 4x + 2x + 7x − x
d) 8x − x − 3x − 4x
Bonus: x − 2x + 3x − 4x + 5x − 6x + 7x − 8x
Answers: a) −3x, b) −5x, c) 4x or +4x, d) 0x = 0, Bonus: −4x
Review adding and subtracting sequences of integers. Remind students that they can add
and subtract integers by using the notation of gains and losses and then adding all the gains
and all the losses. Write on the board:
(−3) + (−2) − (+4) − (−5) = − 3 − 2 − 4 + 5
=5−9
= −4
Adding and subtracting expressions of the form ax, where a is an integer. SAY: You can
do the same thing with variables. Write on the board:
7x − (+3x) − (−4x) + (+2x) + (−3x)
Start rewriting the expression without brackets and have a volunteer finish:
7x − 3x + 4x + 2x − 3x
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-45
Have a volunteer circle all the x’s that are being added. ASK: How many x’s are being added
altogether? (13) How many are being subtracted? (6) Write on the board:
= 13x − 6x = 7x
Exercises: Add and subtract.
a) −2x + (−8x) + (+4x) − (+3x) − (−5x)
b) 5x − (−x) + (+x) − (−3x) + (−2x)
c) 8x − (−5x) − (+3x) + (+2x) + (−7x)
Answers: a) −4x, b) 8x, c) 5x
Extension
(MP.1) Simplify.
a) x − 2x + 3x − 4x + 5x − 6x + 7x − 8x + … + 99x − 100x
b) x − 2x + 2x − 3x + 3x − 4x + 4x − 5x + … + 99x − 100x
1
1
1
1
x
2
4
8
16
1
1
1
1
d) x - x - x - x - x
3
6
9
18
1
1
1
1
e) x - x - x - x - x
2
4
6
12
c) x - x - x - x -
Selected solution: a) (x − 2x) + (3x − 4x) + (5x − 6x) + (7x − 8x) + … + (99x − 100x) =
− x − x − x − x − … − x = −50x
Answers: b) −99x, c) 1/16 x, d) 6/18 x or 1/3 x, e) 0
D-46
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
EE7-9
Like Terms
Pages 90–91
Standards: 7.EE.1
Goals:
Students will add and subtract expressions of the form ax + b, where a and b are integers.
Prior Knowledge Required:
Can add and subtract expressions of the form ax, where a is an integer
Knows that subtracting more results in less, and that subtracting less results in more
Vocabulary: like terms, simplify, unknown
Simplifying expressions. Write on the board:
x+x+x=
3+4+x=
For each expression, ASK: What shorter expression is this equal to? (3x and 7 + x) Tell
students that writing a shorter, easier expression that involves fewer operations is called
simplifying the expression, so what you just did was to simplify the expression. Write on the
board:
8+r−5
SAY: It might be easier to simplify the expression if you circle all the terms that are just
numbers. Demonstrate doing so:
8+r−5
SAY: 8 − 5 tells you the amount that isn’t part of the variable term. It is always the same, so we
can just calculate that amount. Write on the board:
8 − 5 = 3, so 8 + r − 5 = 3 + r or r + 3
Exercises: Simplify the expression.
a) 2 + 7 + w
b) 5 + r − 2
c) 9 − w + 5
Answers: a) 9 + w, b) 3 + r, c) 14 − w, d) p − 1
d) p + 4 − 5
SAY: When there is more than one term with the same variable, you can combine them all
together the same way you combined all the number terms. Write on the board:
2+x+x+x
ASK: What shorter expression is this equal to? (2 + 3x)
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-47
Exercises: Simplify.
a) 6 + x + 3x
b) 5x + 3 − 2x
c) 8x − 12x + 4
Answers: a) 6 + 4x, b) 3x + 3, c) −4x + 4, d) 5 + x
d) 5 − 3x + 4x
Tell students that all the terms with the same variable are called like terms, and all the number
terms without a variable are also like terms. Write on the board:
8x + 5 − 3x + 2 + x − 9
SAY: When the expression is longer, it is easier to write all the variable terms together and all
the other terms together. Write on the board:
= 8x − 3x + x + 5 + 2 − 9
SAY: This is called combining like terms. Have a volunteer simplify the x’s and another
volunteer evaluate the numbers:
=
6x
−2
(MP.7) Exercises: Combine like terms to simplify.
a) 2x + 3 − 5x + 2 − x − 4
b) −7x + 3 + 4 + 2x − 8x − 5
Bonus: 3x + 2y − 8 + 4x − 5y + 3
Answers: a) −4x + 1, b) −13x + 2, Bonus: 7x − 3y − 5
Finding perimeters of shapes with unknown sides. Draw on the board:
5
x
x
5
Remind students that the perimeter of a shape is the distance around the outside of the shape.
So the perimeter of this shape is
x+5+x+5
Have a volunteer simplify the expression. (2x + 10)
Exercises: Find the perimeter. Simplify the expression.
a)
b)
3x
c)
3x
x
x
x
x
7
5
3x
Answers: a) 2x + 5, b) 8x, c) 4x + 14, d) 2x + 8
D-48
d) x
7
4
x
4
x
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
Combining like terms uses the commutative property. Write on the board:
9 + 7x + 5 + 4x = 9 + 5 + 7x + 4x
(MP.3) ASK: What property of addition is being used? (the commutative property) Show this on
the board as follows:
9 + 7x + 5 + 4x = 9 + 5 + 7x + 4x
SAY: What is really happening when we combine like terms is we are moving all the number
terms together and all the x terms together. We can change the order of the numbers we are
adding because addition is commutative.
(MP.1) SAY: If you find this hard to see, you could replace the variable with a number and see
whether the equation makes sense. Let’s replace the variable with x = 1 because it’s easy to
multiply by. Write on the board:
9+7+5+4=9+5+7+4
SAY: It doesn’t matter if you add 7 + 5 or 5 + 7. You get the same answer both ways.
Subtracting expressions. Write on the board:
7 − (2 + 1) = 7 − 2 − 1
SAY: Compare both expressions to 7 – 2. Subtracting 1 more results in 1 less. Now write on the
board:
7 − (3 + 1) =
Have a volunteer write an equivalent expression without brackets. (7 − 3 − 1) Repeat for
7 − (4 + 1) (= 7 − 4 − 1) and 7 − (x + 1) (= 7 − x − 1).
(MP.7) Exercises: Write an equivalent expression without brackets.
a) 12 − (1 + 4)
b) 12 − (2 + 4)
c) 12 − (3 + 4)
d) 12 − (x + 4)
Answers: a) 12 − 1 − 4, b) 12 − 2 − 4, c) 12 − 3 − 4, d) 12 − x − 4
Write on the board:
3x − (4 + 5x)
Tell students that you are subtracting everything in the brackets, so what this expression really
means is:
3x − 4 − 5x
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-49
SAY: You subtract the 4 and you subtract the 5x. Have a volunteer simplify the expression:
−2x − 4
(MP.7) Exercises: Write an equivalent expression without brackets. Then simplify your
expression.
a) 3x − (5 + 6x)
b) 5x + 4 − (2x + 9)
c) 3x + 2 − (4x + 1)
Bonus: (4x + 2y + z + 3) − (x + 5y + 2z + 1)
Answers: a) 3x − 5 − 6x = − 3x − 5, b) 5x + 4 − 2x − 9 = 3x − 5, c) 3x + 2 − 4x − 1 = − x + 1,
Bonus: 4x + 2y + z + 3 − x − 5y − 2z − 1 = 3x − 3y − z + 2
Write on the board:
3x − (5 − 2x)
SAY: Here you are taking away from the amount that you are subtracting, so you have to add
that amount back. Write on the board:
= 3x − 5 + 2x
Have a volunteer simplify the expression:
= 5x − 5
(MP.7) Exercises: Write an equivalent expression without brackets. Then simplify the expression.
a) 8x − (9 − 3x)
b) 6x − (3x − 5)
c) 5x − (6x − 3)
d) 4 − (7 − 2x)
e) (4x + 3) − (3x − 2)
f) 2x + 7 − (5x + 2)
g) (2x + 4) − (x + 7)
h) (9 − 5x) − (7x − 4)
i) (3x − 5) − (6 − 3x)
Bonus: (2x − 3y + 5) − (x + 4y − 6)
Answers: a) 8x − 9 + 3x = 11x − 9, b) 6x − 3x + 5 = 3x + 5, c) 5x − 6x + 3 = − x + 3,
d) 4 − 7 + 2x = − 3 + 2x, e) 4x + 3 − 3x + 2 = x + 5, f) 2x + 7 − 5x − 2 = − 3x + 5,
g) 2x + 4 − x − 7 = x − 3, h) 9 − 5x − 7x + 4 = − 12x + 13, i) 3x − 5 − 6 + 3x = 6x − 11,
Bonus: 2x − 3y + 5 − x − 4y + 6 = x − 7y + 11
Encourage students to check their answers for one or two questions by plugging x = 1 into both
the given expression and their expression without brackets. Are the two answers the same? If
not, students should look for a mistake.
Extensions
(MP.1) 1. Two integers are consecutive if there are no integers between them: 3 and 4 are
consecutive integers, but 3 and 5 are not. Write and simplify an expression for the sum of three
consecutive integers when …
a) the variable represents the smallest integer
b) the variable represents the middle integer
c) the variable represents the largest integer
Answers: a) x + x + 1 + x + 2 = 3x + 3, b) x − 1 + x + x + 1 = 3x, c) x − 2 + x − 1 + x = 3x − 3
D-50
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
(MP.1) 2. Two even numbers are consecutive if there is no even number between them: 96 and
98 are consecutive even numbers, but 96 and 100 are not. The largest of five consecutive even
numbers is a + 3. What is the smallest of the five numbers?
Solution: If a + 3 is the largest, then a + 1 is the next largest, then a − 1, then a − 3, then a − 5.
So the smallest is a − 5.
3. a) Evaluate 4 – 3x and 3x – 4 at …
i) x = 0
ii) x = 1
iii) x = 5
b) What do you notice about your answers to part a)?
c) Add the expressions: (4 – 3x) + (3x – 4). Simplify.
Answers:
a) i) 4 and −4, ii) 1 and −1, iii) −11 and 11;
b) they are always opposite;
c) 4 − 3x + 3x – 4 = 0x + 0 = 0
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-51
EE7-10
Coefficients and Constant Terms
Pages 92–93
Standards: 7.EE.1
Goals:
Students will identify variables, coefficients, and constant terms in real-world situations.
Prior Knowledge Required:
Can add expressions of the form ax + b
Can evaluate expressions of the form ax + b
Can translate simple word problems into expressions of the form ax + b
Vocabulary: coefficient, constant term, equation, expression, variable
Differences and similarities between equations and expressions. Explain to students that
equations and expressions both contain numbers, symbols, and variables. Equations have
equal signs that separate two expressions. Write on the board:
Expression
Equation
Write some expressions and equations on the board:
3 + 4x + 5x + 6
3 + 4x = 5x + 6
5 − 2x = 7
5 − 2x + 7
Have students point to the correct word for each. (expression, equation, equation, expression)
Coefficients. Tell students that in an expression, the coefficient is the number of times that the
variable is added. Write on the board:
5x + 3 = x + x + x + x + x + 3
7x − 2 = x + x + x + x + x + x + x − 2
ASK: What is the coefficient of 5x + 3? (5) Where do you see that in the expression? (right
before the x) What is the coefficient of 7x − 2? (7) Where do you see that in the expression?
(right before the x) SAY: The number that you multiply by the variable is called the coefficient. It
can be written either before or after the variable, but it is usually written before the variable.
Write on the board:
x−2
ASK: How many times is the variable added? (once) Write on the board:
x − 2 = 1x − 2, so the coefficient is 1.
SAY: When there is no number in front of the variable, the number 1 is understood.
D-52
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
Exercises: Write the coefficient.
a) 3x + 2
b) x − 5
c) 2w + 3
Answers: a) 3, b) 1, c) 2, d) 5, e) 1
d) 5r − 7
e) m + 8
Negative coefficients. Write on the board:
5 − 2x
SAY: The variable here is being subtracted. In that case, the coefficient is written as a negative
number. So the coefficient is −2. Write on the board:
3 − 5x
7−x
− 3w + 9
− r + 10
Ask volunteers to tell you the coefficient of each expression. (−5, −1, −3, −1)
Exercises: Write the coefficient.
a) 5 − 3x
b) 3x − 5
c) 8x − 2
Answers: a) −3, b) 3, c) 8, d) −8, e) −1
d) − 8x + 4
e) − 5 − x
Expressions with two variables. SAY: When there are two variables, each variable has a
coefficient. Write on the board:
7x − 3y − 5
ASK: What is the coefficient of x? (7) What is the coefficient of y? (−3)
Exercises: Write the coefficient of x.
a) 3x + 4y + 5
b) 5w − x + 7
Answers: a) 3, b) −1, c) 1
c) x − 2y − 3
Write on the board, using different colors for the two variables:
3x + 4y + 6
x=2
y = −1
Demonstrate substituting in the numbers for the variables, using the same colors for the
numbers and the variables they replace:
= 3(2) + 4(−1) + 6
ASK: What is 3 × 2? (6) What is 4 × (−1)? (−4) Continue evaluating the expression:
=6−4+6
=2+6
=8
Exercises: Evaluate the expression for x = 3 and y = −1.
a) 2x + 5y
b) 3x − y + 4
c) − x + 4y − 2
Answers: a) 1, b) 14, c) −9
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-53
Combining like terms when there is more than one variable. Write on the board:
2x + 7y − 8 + x − 2y + 1
Tell students that they can think of each term, such as 2x or 7y, as standing for a number, even
when they don’t know which number it replaces. So these terms can be added and subtracted in
any order and still result in the same answer. SAY: Let’s put all the x’s together, all the y’s
together, and all the number terms together. Write on the board:
= 2x + x + 7y − 2y − 8 + 1
Have a volunteer continue the simplifying:
= 3x + 5y − 7
Point out that no matter what number x stands for, 2 of them plus 1 of them is 3 of them, and no
matter what number y stands for, 7 of them minus 2 of them is 5 of them.
(MP.7) Exercises: Simplify.
a) 2x + 5y − 4 − 2x + y − 1
b) 3x − 2y − 4x + y + 7
c) 3x − 2y − 8 − 5x − 7y + 4
d) − 8x + 3y + 1 − x + y − 4
Answers: a) 6y − 5, b) − x − y + 7, c) − 2x − 9y − 4, d) − 9x + 4y − 3
Introduce the term “constant term.” Tell students that the term without a variable is called the
constant term because it never changes. Write on the board:
3x + 4
3(1) + 4 = 3 + 4
3(2) + 4 = 6 + 4
3(3) + 4 = 9 + 4
SAY: No matter what the variable is, the 4 never changes. So, in this case, 4 is the constant
term. Write on the board:
2x + 5
5x + 2
3x + 1
x+3
Have students signal the constant term for the expressions by holding up the correct number of
fingers. (the constant terms are 5, 2, 1, and 3)
Write on the board:
3x = 3x + 0
D-54
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
SAY: When there is no term without a variable, you can always assume you are just adding 0,
so 0 is the constant term. Write on the board:
3x − 4 = 3x + (−4)
SAY: When the part of the expression without the variable is negative, the constant term is
negative.
(MP.8) Exercises: a) Write the constant term.
i) 2x – 5
ii) 3x – 2
iii) − 5x + 4
iv) 3x
b) Evaluate each expression from part a) at x = 0.
c) What do you notice about your answers to parts a) and b)?
Answers:
a) i) −5, ii) −2, iii) 4, iv) 0, v) −1;
b) i) −5, ii) −2, iii) 4, iv) 0, v) −1;
c) the answers to parts a) and b) are the same v) 5x − 1
Variables, coefficients, and constant terms in word problems. Write on the board:
A phone company charges a flat fee of $10 plus $3 per hour of phone conversation.
Have students write the expression for the total cost of talking on the phone for h hours.
($(3h + 10)) ASK: What is the coefficient? (3) What does the coefficient represent? (the hourly
rate) What is the constant term? (10) What does the constant term represent? (the flat fee)
What does the variable represent? (the number of hours)
(MP.2, MP.6) Exercises: A taxi company charges a $5 flat fee and $2 per kilometer.
a) Write an expression for the amount the taxi charges, and say what the variable represents.
b) Write “variable,” “coefficient,” or “constant term.”
i) The __________________ represents the flat fee.
ii) The __________________ represents the cost per kilometer.
iii) The __________________ represents the number of kilometers.
Answers: a) 2x + 5, where x is the distance traveled in kilometers; b) i) constant term,
ii) coefficient, iii) variable
Extensions
(MP.1, MP.7) 1. Investigate: How can you get the value of an expression ax + b at x = 1 from
the coefficient and the constant term?
Answer: Add the coefficient and the constant term. For example, at x = 1, the expression 3x + 4
is 3 + 4 = 7.
(MP.1, MP.6) 2. Write an expression with constant term 0 for the sum of five consecutive even
numbers. Say what the variable represents.
Answer: 5x, where x is the middle number.
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-55
EE7-11
Equivalent Expressions
Pages 94–95
Standards: 7.EE.1
Goals:
Students will learn to write 3(x + 2) as 3x + 6 or a(x + b) as ax + ab.
Prior Knowledge Required:
Can apply the distributive property
Vocabulary: associative property, distributive property, equation, expression, simplify, unknown
Using mass to model expressions. Draw on the board:
weighs 1 kg.
weighs x kg.
Tell students that a circle has mass 1 kg but you don’t know the mass of the triangle. We call its
mass, x, an unknown, because we don’t know what it is. Draw on the board:
Have a volunteer write an expression for the mass of the whole set. (x + 2) Then draw 3 copies
of the set, one on top of each other, and repeat the question. (the mass is 3x + 6 because there
are now 3 triangles (x’s) and 6 circles) ASK: How many times more is the mass than it was
before? (3 times) How do you know? (you just copied the set 3 times) Write on the board:
3(x + 2) = 3x + 6
Exercises: Draw circles and triangles for each expression.
a) x + 3
b) 3x + 1
c) 2x + 5
d) 5x + 3
e) 4x + 4
f) x + 1
g) 2(x + 1)
h) 3(x + 1)
i) 4(x + 1)
j) 5(x + 1)
Selected answers:
a)
, b)
, h)
ASK: Which two quantities are the same? (parts e) and i)) Point out that both have 4 of each
shape, so they weigh the same, no matter what the triangle weighs. That means they are
equivalent expressions.
D-56
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
Exercises:
1.Draw a picture to write an equivalent expression without brackets.
a) 2(x + 3)
b) 3(x + 2)
c) 4(x + 3)
d) 3(x + 5)
e) 5(x + 3)
f) 3(2x + 1)
g) 5(3x + 2)
h) 4(3x + 2)
Answers: a) 2x + 6, b) 3x + 6, c) 4x + 12, d) 3x + 15, e) 5x + 15, f) 6x + 3, g) 15x + 10,
h) 12x + 8
2. Draw a picture to write each of the first five expressions without brackets; then look for a
pattern to do part f).
a) 2(x + 1)
b) 2(x + 2)
c) 2(x + 3)
d) 2(x + 4)
e) 2(x + 5)
f) 2(x + 100)
Bonus: 2(x + 1,000,000,000)
Answers: a) 2x + 2; b) 2x + 4; c) 2x + 6; d) 2x + 8; e) 2x + 10; f) 2x + 200;
Bonus: 2x + 2,000,000,000
SAY: The 2 outside the brackets is telling you to multiply both terms inside the brackets by 2.
Demonstrate how that was done for 2(x + 100) = 2x + 200. SAY: 2x is 2 times x and 200 is 2
times 100.
Exercises: Multiply both terms inside the brackets by 3 to write an equivalent expression.
a) 3(1 + 4)
b) 3(2 + 4)
c) 3(7 + 4)
d) 3(x + 4)
Answers: a) 3 + 12, b) 6 + 12, c) 21 + 12, d) 3x + 12
ASK: Which property did you use? (the distributive property)
Exercises: Use the distributive property to write each expression without brackets:
a) 3(x + 5) = 3x + _____
b) 2(x + 7) = 2x + _____
c) 5(x + 4) = 5x + _____
d) 4(x + 6) = ____________
e) 9(x + 8) = ____________
f) a(x + b) = ____________
Answers: a) 3x + 15, b) 2x + 14, c) 5x + 20, d) 4x + 24, e) 9x + 72, f) ax + ab
When the expression in brackets has coefficient greater than 1. Write on the board:
3(2x + 4) = 6x + 12
SAY: To multiply the whole thing by 3, you multiply the number of triangles by 3 and the number
of circles by 3. There were 2 triangles and 4 circles, so now there are 6 triangles and 12 circles.
Exercises: Use the distributive property to write an equivalent expression.
a) 2(5x + 4)
b) 3(2x + 6)
c) 5(3x + 4)
d) 2(4x + 1)
Answers: a) 10x + 8, b) 6x + 18, c) 15x + 20, d) 8x + 2
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-57
Tell students that the distributive property holds even when you can’t draw a picture. Write on
the board:
3(2x − 4) = 6x − 12
SAY: You can’t draw a picture of 2 triangles and −4 circles, but it still works. Write on the board:
3(2x − 4) = (2x − 4) + (2x − 4) + (2x − 4)
= 2x + 2x + 2x − 4 − 4 − 4
= 6x − 12
Point out that each term inside the brackets is being multiplied by 3: 6x is 3 times 2x and −12 is
3 times −4.
Exercises: Use the distributive property to write an equivalent expression.
a) 2(5x − 4)
b) 3(2x − 6)
c) 5(−3x + 4)
d) 2(−4x − 1)
Answers: a) 10x − 8, b) 6x − 18, c) − 15x + 20, d) − 8x − 2
Write on the board:
7 − 2(3x + 4)
SAY: To simplify this expression, first rewrite the amount in brackets as a sum of two terms
instead of as a multiplication. Write on the board:
= 7 − (6x + 8)
SAY: Both terms are being subtracted. Write on the board:
= 7 − 6x − 8
because −(+) = −
Have a volunteer continue the simplification. You may need to remind students that they can
combine like terms:
= − 6x + 7 − 8
= −6x − 1
(MP.7) Exercises: Simplify.
a) 3x − 2(x + 5)
b) 3x − 2(4x + 3)
c) 3x − 2(4x − 3)
d) 2 − 3(2x − 4)
e) 4x + 1 − (x − 5)
f) − 3x + 2 − 3(4x + 5)
Answers: a) x − 10, b) −5x − 6, c) −5x + 6, d) 14 − 6x, e) 3x + 6, f) − 15x − 13
D-58
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
(MP.1) When the constant term is 0. SAY: When the constant term is 0, you can use the
associative property instead of the distributive property. Write on the board:
3(5x) = (3 × 5)x
= 15x
Exercises:
1. Check that 3(5x) = (3 × 5)x for …
a) x = 2
b) x = 4
Bonus: x = 100
Answers:
a) 3(5(2)) = 3(10) = 30 and (3 × 5)(2) = (15)(2) = 30;
b) 3(5(4)) = 3(20) = 60 and (3 × 5)(4) = (15)(4) = 60;
Bonus: 3(5(100)) = 3(500) = 1,500 and (3 × 5)(100) = (15)(100) = 1,500
2. Multiply.
a) 2(3w)
b) 3(7x)
Answers: a) 6w, b) 21x, c) 40x, d) 18r
c) 5(8x)
d) 9(2r)
Tell students that the associative property works for negative coefficients, too. Write on the
board:
3(−5x) = (3(−5))x
= −15x
Exercises: Multiply.
a) 3(−4x)
b) 2(−8x)
c) 3(−7x)
Answers: a) −12x, b) −16x, c) −21x, d) −16x
d) 4(−4x)
Extensions
(MP.3) 1. Which property is being used? Write “commutative,” “associative,” or “distributive.”
a) 8 + n = n + 8
b) 5(8n) = (5 × 8)n = 40n
c) 3n + 7n = (3 + 7)n = 10n
d) 5n = n × 5
e) 3(n + 7) = 3n + 3(7) = 3n + 21
Which properties are being used?
f) 8 + (n + 5) = 8 + (5 + n) = (8 + 5) + n
g) (5n)(6) = (n × 5) × 6 = n × (5 × 6)
Answers: a) commutative; b) associative; c) distributive; d) commutative; e) distributive;
f) commutative, associative; g) commutative, associative
(MP.1) 2. Write three expressions equivalent to 2x + 10.
Sample answers: 2(x + 5), 10 + 2x, x + 7 + x + 3, 2n + 10
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-59
(MP.1, MP.3) 3. In the magic trick below, the magician can always predict the result of the
sequence of operations performed on any chosen number. Try the trick with students; then
encourage them to figure out how it works. To figure out how it works, students who are
struggling can use blocks to represent the mystery number and counters to represent the
numbers that are added. Give students lots of hints as they manipulate the concrete materials.
Following along with the chart in the “Concrete Materials” column, students see that they always
end up with three circles no matter what number x represents.
The Trick
Pick any number.
The Algebra
x
Add 4.
x+4
Multiply by 2.
2(x + 4)
Rewrite the expression.
2x + 8
Subtract 2.
2x + 6
Divide by 2
x+3
Subtract the mystery number.
x+3−x=3
D-60
The Concrete Materials
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
EE7-12
Using Pictures to Make Equivalent Expressions
Pages 96–98
Standards: 7.EE.1
Goals:
Students will use pictures to make equivalent expressions.
Vocabulary: expand, factor, greatest common factor (GCF)
Writing a sum of two numbers as a multiple of a sum of two numbers that have no
common factor. Write on the board:
20
+
12
5+3
5+3
5+3
5+3
= 4 × (5 + 3)
SAY: Because 20 is 4 × 5 and 12 is 4 × 3, the sum 20 + 12 is 4 × (5 + 3). We could write the
right side of the equation as 4 × 8, but keeping it as 4 × (5 + 3) helps us to keep track of where
the 8 comes from—it is 5 + 3. If we wrote the expression as 4 × 8, we might forget that we got
the 8 from the sum 5 + 3, and sometimes it is helpful to remember. Write on the board:
12 + 18 = 2 × (____ + ____)
ASK: 12 is 2 times what? (6) Write “6” in the first blank. ASK: 18 is 2 times what? (9) Write “9” in
the second blank.
(MP.7) Exercises: Fill in the blanks.
a) 30 + 12 = 6 × (___ + ___)
b) 14 + 8 = 2 × (___ + ___)
c) 15 + 20 = 5 × (___ + ___)
d) 15 + 21 = 3 × (___ + ___)
e) 16 + 28 = 4 × (___ + ___)
f) 20 + 24 = 4 × (___ + ___)
Bonus: 12 + 9 + 24 = 3 × (___ + ___ + ___)
Answers: a) 5 + 2, b) 7 + 4, c) 3 + 4, d) 5 + 7, e) 4 + 7, f) 5 + 6, Bonus: 4 + 3 + 8
Factoring numeric expressions without being given the first factor. Write on the board:
30 + 42 = _____ × ( ____ + ____ )
Have students list the factors of 30 and the factors of 42. Have volunteers tell you the common
factors of 30 and 42 (1, 2, 3, and 6).
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-61
Then write on the board:
30 + 42 = 1 × (30 + 42)
30 + 42 = 2 × (___ + ___)
30 + 42 = 3 × (___ + ___)
30 + 42 = 6 × (___ + ___)
Have volunteers fill in the blanks. (15 + 21, 10 + 14, 5 + 7) Point out that the numbers inside the
brackets get smaller as the number outside the brackets gets bigger. Tell students that you like
to work with small numbers whenever you can because you find them easier to work with than
large numbers. ASK: Which expression uses the smallest numbers overall? (the last one) Point
out that this is the simplest way of rewriting 30 + 42 and it uses the largest factor that 30 and 42
have in common: 6. ASK: What do we call the largest factor that 2 numbers have in common?
(greatest common factor, or GCF) Do the part a) of the exercises below together as a class,
then have students do the rest individually.
Exercises: Find the GCF of the two numbers being added. Then rewrite the sum with the GCF
as a factor.
a) 18 + 42 = ____ × (___ + ___)
b) 15 + 40 = ____ × (___ + ___)
GCF
GCF
c) 21 + 35
d) 24 + 40
e) 22 + 18
f) 25 + 60
g) 15 + 30
h) 8 + 40
Bonus: 14 + 35 + 63
Answers: a) 6 × (3 + 7), b) 5 × (3 + 8), c) 7 × (3 + 5), d) 8 × (3 + 5), e) 2 × (11 + 9),
f) 5 × (5 + 12), g) 15 × (1 + 2), h) 8 × (1 + 5), Bonus: 7 × (2 + 5 + 9)
Factoring expressions with variables. Tell students that they can do the same thing with
variables that they just did with numbers. Write on the board:
8 = 4(_____)
8x = 4(_____)
12 = 4(_____)
ASK: 8 is 4 times what? (2) 8x is 4 times what? (2x) 12 is 4 times what? (3) Write on the board:
8x + 12 = 4(2x + 3)
SAY: We can show this with pictures by using x for a group of shaded dots. x can be any
number of shaded dots, but we’ll draw unshaded dots too. Draw on the board:
xx
xx
xx
xx
8x + 12
xx
xx
xx
xx
=
4(2x + 3)
SAY: In both pictures, we have 8 groups of shaded dots, and 12 unshaded dots.
D-62
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
(MP.7) Exercises: Fill in the blanks.
a) 6x + 4 = 2(____ + _____)
b) 10x + 15 = 5(_____ + _____)
c) 30x + 12 = 6(_____ + _____)
d) 18x + 27 = 9(_____ + _____)
Answers: a) 3x + 2, b) 2x + 3, c) 5x + 2, d) 2x + 3
SAY: When you take out a common factor of the coefficient and the constant term, this is called
factoring the expression.
Exercises: Find the GCF of the coefficient and the constant term. Then factor the expression
with the GCF as one of the factors.
a) 5x + 20
b) 6x + 9
c) 8x + 10
d) 14x + 35
e) 4x + 10
f) 6x − 10
g) 30x − 9
h) 5x + 15
i) 8x − 12
Answers: a) 5(x + 4), b) 3(2x + 3), c) 2(4x + 5), d) 7(2x + 5), e) 2(2x + 5), f) 2(3x − 5),
g) 3(10x − 3), h) 5(x + 3), i) 4(2x − 3)
Write on the board:
−15x + 18
SAY: You can ignore the negative signs for the first step. ASK: What number is a factor of both
15 and 18? (3) Write on the board:
−15x + 18 = 3(_____x + _____)
SAY: For the second step, you need to look at the negative signs. ASK: What number times 3 is
−15? (−5) Write that in the first blank. ASK: What number times 3 is 18? (6) Write that in the
second blank.
Exercises: Factor the expression.
a) −3x − 9
b) −6x + 15
c) −10x + 8
d) −10x − 8
e) −12x − 20
Answers: a) 3(−x − 3), b) 3(−2x + 5), c) 2(−5x + 4), d) 2(−5x − 4), e) 4(−3x − 5)
Expanding an expression. If you have 2(2x + 3), you can write it as 4x + 6 because you are
multiplying both terms in the brackets by 2. That is called expanding 2(2x + 3).
Exercises: Expand the expression.
a) 3(2x + 5)
b) 4(−3x − 4)
c) 2(5x − 3)
d) 8(−2x + 5)
f) 3(−x − 5)
g) 2(−4x + 7)
h) 7(−5x − 2)
e) 5(4x − 3)
Answers: a) 6x + 15, b) −12x − 16, c) 10x − 6, d) −16x + 40, e) 20x − 15, f) −3x − 15,
g) −8x + 14, h) −35x − 14
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-63
Using areas to create equivalent expressions. Draw on the board:
s
3
2
ASK: What is the area of the shaded part? (2s) What is the area of the unshaded part? (6) Write
on the board:
Area of big rectangle = 2s + 6
SAY: The area of any rectangle is the product of the length and width. ASK: What is the length
of the big rectangle? (s + 3) SAY: So the area is also twice s + 3. Write on the board:
Area of big rectangle = 2(s + 3)
SAY: These are two ways to write the area of the big rectangle—multiply its length and width or
add the areas of the two smaller rectangles. So using areas is another way to see whether the
expressions are equivalent.
(MP.2, MP.7) Exercises: Write the area of the whole rectangle in two ways.
s
3
2r
5
a)
b)
4
3
Bonus:
3s
2
4s
s+1
5
Answers: a) 4s + 12 = 4(s + 3), b) 3(2r + 5) = 6r + 15, Bonus:
5(3s + 2 + 4s + s + 1) = 15s + 10 + 20s + 5(s + 1)
Making three equivalent expressions from border areas. Draw on the board:
Tell students that you want to find three equivalent expressions. Write on the board:
4 × _____
D-64
2 × ___ + 2 × ____
6 × ____ + 2 × ______
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
Guide volunteers to fill in the blanks. Point out that the first picture has 4 areas (rectangles) that
are the same. ASK: What is each area? (3) Write that in the blank. Then repeat for the second
picture (this time 2 rectangles have area 5 and 2 have area 1) and for the third picture (6 have
area 1 and 2 have area 3). Write on the board:
4×3=2×5+2×1=6×1+2×3
SAY: All these expressions represent the same area.
(MP.2, MP.7) Exercises: Write three different expressions for the same area.
a)
b)
Answers:
a) 2 × 3 + 2 × 4 and 2 × 1 + 2 × 6 and 6 × 1 + 2 × 4,
b) 2 × 3 + 2 × 5 and 2 × 1 + 2 × 7 and 6 × 1 + 2 × 5
Draw on the board:
1
1
1
1
s
1
1
3
s
1
1+s+1=s+2
3
Have volunteers tell you how many pieces of each area there are in the first picture (6 pieces
with area 1 and 2 pieces with area s). Then have another volunteer write an expression for the
area (2s + 6(1)). Repeat for the other two pictures (areas 2s + 2(3) and 2(s + 2) + 2(1)) Write on
the board:
2s + 6(1) = 2s + 2(3) = 2(s + 2) + 2(1)
SAY: These expressions are all equal no matter what you substitute for s. So they are
equivalent expressions.
(MP.2, MP.7) Exercise: Calculate the area of the shaded part in three different ways to find
three equivalent expressions.
1
s
Answer: 2s + 8(1) = 2s + 2(4) = 2(s + 2) + 2(2)
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-65
Extensions
(MP.2) 1. A square has perimeter 20x + 36. What is the length of each side?
Answer: 5x + 9
2. a) If 12x + 20y = 100, what is 3x + 5y?
Answers: a) 25, b) 34
b) If 25x + 20y = 85, what is 10x + 8y?
(MP.1, MP.7) 3. a) Use the picture to explain why 2 × (1 + 2 + 3 + 4) = 4 × 5.
b) Draw a picture to show that 2 × (1 + 2 + 3 + 4 + 5) = 5 × 6. Check the equation by evaluating
each side.
c) Calculate 1 + 2 + 3 + 4 + 5 + … + 100 by comparing it to a product.
d) Draw a picture to show that 2 × (3 + 4 + 5 + 6) = 4 × 9.
e) What product is 2 × (5 + 6 + 7 + 8) equal to? Draw a picture to show your answer.
f) What product is 2 × (54 + 55 + 56 + 57) equal to? Use the product to evaluate 54 + 55 + 56 + 57
without adding.
g) What product is 2 × (1 + 3 + 5 + 7) equal to?
h) What product is 2 × (1 + 3 + 5 + 7 + 9) equal to?
i) What do the first 100 odd numbers add to?
Answers: a) both expressions are equal to 20, which is the area of the rectangle; b) draw a 5
by 6 rectangle with a similar staircase; c) 2 × (1 + 2 + 3 + … + 100) = 100 × 101 = 10,100, so
1 + 2 + 3 + … + 100 = 5,050;
d)
e) 4 × 13; f) 4 × 111, so 54 + 55 + 56 + 57 = 222; g) 4 × 8; h) 5 × 10; i) 100 × 200 ÷ 2 = 10,000
4. Write an expression for the area by finding the areas of the smaller rectangles.
a)
b)
30
4
30
20
20
7
t
s
(30 + 4)(20 + 7)
(30 + s)(20 + t)
= _____ + _____ + _____ + _____
= _____ + _____ + _____ + _____
Answers: a) 30(20) + 30(7) + 4(20) + 4(7), b) 30(20) + 30t + 20s + st
D-66
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
5. Draw a picture to calculate 23 × 35.
Answer:
20
+3
30
+
5
So 23 × 35 = 20(30) + 20(5) + 3(30) + 3(5) = 600 + 100 + 90 + 15 = 805
Teacher’s Guide for AP Book 7.1 — Unit 3 Expressions and Equations
D-67