Check Your Readiness

1.

Jada is collecting stickers. After getting 15 more stickers, she has 60 stickers in total.

Select all the equations Jada can solve to find $x$ , the number of stickers she started with.

A.

$x + 15 = 60$

B.

$x - 15 = 60$

C.

$x = 60 + 15$

D.

$x = 60 - 15$

E.

$15x = 60$

F.

$x = 60 \boldcdot 15$

G.

$x = \frac{60}{15}$

Answer: A, D

Teaching Notes

Students will solve more advanced equations in this unit, building from the equation types they have worked with in sixth grade.

If most students struggle with this item, plan to revisit it before Activity 2 of Lesson 4. Ask students to draw a tape diagram to represent the situation, and then try matching equations again. You may also choose to revisit the stories in Lesson 2 and ask students to write equations to match the tape diagrams and stories, understanding that the story connects to both an addition and a subtraction equation.

2.

Solve each equation.

$p + 12 = 17$

$\frac 7 3 = q + \frac 2 3$

$90 = 20r$

$\frac 1 3 s = 7$

$15 = 1.5t$

$79 + u = 65$

$6v = \text-9$

Answer:

$p= 5$
$q = \frac 5 3$
$r=\frac 9 2$ (or equivalent)
$s= 21$
$t= 10$
$u=\text-14$
$v=\frac {\text{-}9}{6}$ (or equivalent)

Teaching Notes

Students should have experience solving these types of equations for non-negative rational numbers. The last two parts extend students’ understanding to equations involving negative numbers.

If most students struggle with this item, plan to incorporate practice solving equations with one operation into earlier lessons. You may choose to use the equations from the item for practice. Consider using a Math Talk to give students practice reasoning about equations of the form $x+p=q$ and $px=q$ .

3.

Lin is selling flowers. Each flower costs $3. Lin’s goal is to earn more than $40 selling flowers.

If Lin sells 10 flowers, will she make her goal?
If Lin sells 20 flowers, will she make her goal?
If Lin sells $x$ flowers, write an inequality (using the symbol $<$ or $>$ ) that will be true whenever Lin makes her goal and false whenever she does not.
Graph your inequality on the number line.

Answer:

No. Sample reasoning: She only earns $30 for selling 10 flowers.
Yes. Sample reasoning: She earns $60 for selling 20 flowers, which is more than $40.
Sample response: $x > 13$ , because $40 \div 3$ is between 13 and 14.
The solution is the graph of $x > 13$ , with an open circle at 13 and shading to the right. Alternately, the solution is a set of closed circles at the whole numbers from $x = 14$ and above because Lin cannot sell part of a flower, and therefore all the solutions are whole numbers.

Teaching Notes

The first two parts ask students to test individual values, encouraging the strategy of testing when an inequality is true or false. That key concept will be developed further in this unit to help students solve and graph more complicated inequalities.

Check to see if students recall the “open circle” concept for graphing inequalities. It is unlikely that students will graph the solution as a set of points, but technically the number of flowers must be a whole number.

If most students do well with this item, it may be possible to move more quickly through Activity 1 and Activity 2.

4.

Select all the equations that are true when $x$ is -4.

A.

$\text-8 = 2x$

B.

$\text-12 = x \boldcdot \text-3$

C.

$\text-12 = x+x+x$

D.

$\frac x 4 = \text-1$

E.

$x + 4 = \text-8$

F.

$x^2 = \text-16$

Answer: A, C, D

Teaching Notes

This work with negative number arithmetic previews work that will come up when solving equations in this unit. Students’ general understanding that a number is a solution to an equation when using that value for the variable makes the equation true is crucial. Watch for errors in students’ arithmetic work. Students selecting B or F may need to be reminded about the properties of multiplication by negatives throughout the unit.

If most students struggle with this item, plan to use a Math Talk routine to address students’ needs before this lesson. Note whether the struggle is a result of arithmetic needs or understanding how to determine if a value for the variable makes the equation true.

5.

Which expression is equivalent to $2(3x-4)$ ?

A.

$3x-4$

B.

$5x-6$

C.

$6x-4$

D.

$6x-8$

Answer:

$6x-8$

Teaching Notes

If most students struggle with this item, plan to spend additional time on the Warm-up of Lesson 3. Use area models and tape diagrams to help students recall what they learned about the distributive property in grade 6. If students need additional practice with the distributive property, IM Grade 6, Unit 6, Lessons 9–11 focus extensively on the distributive property.

6.

Next to each equation, write A, B, or neither, to indicate whether it matches Diagram A, Diagram B, or neither diagram.

Tape diagram, two unequal parts labeled 4, 3, total 7. — A

Tape diagram, 4 equal parts each labeled 3, total 12 — B

$7=3+4$
$4-3=7$
$7-4=3$
$4 \boldcdot 3 = 7$
$3+3+3+3=12$
$12=4 \boldcdot 3$
$12 \div 4 = 3$
$3 \boldcdot 3 \boldcdot 3 \boldcdot 3=12$

Answer:

A
neither
A
neither
B
B
B
neither

Teaching Notes

In this unit, students use tape diagrams to represent the structures $px+q=r$ and $p(x+q)=r$ . They will match tape diagrams to situations and use the diagrams to help decide how a situation can be represented algebraically. Note that not all the equations listed are true, which is fine.

If most students struggle with this item, plan to revisit it as part of the Warm-up in Lesson 2. Consider showing students the most popular answers from the item and ask if they agree or disagree with their classmates’ choices. Students have several chances to work with tape diagrams in the first section of this unit.