Check Your Readiness

1.

Lin is collecting coins. After giving away 13 coins, she has 75 coins remaining. Select all the equations Lin can solve to find $x$ , the number of coins she started with.

A.

$x+13=75$

B.

$x-13=75$

C.

$x=75-13$

D.

$13x=75$

E.

$x=75+13$

F.

$x=75 \boldcdot 13$

G.

$x=\frac{75}{13}$

Answer: B, E

Teaching Notes

Students will solve more advanced equations in this unit, building from the equation types students 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 subtraction equation.

2.

Solve each equation.

$\frac{9}{5}=p+\frac{3}{5}$

$\frac{1}{4}q=7$

$r+18=24$

$35=3.5s$

$80=30t$

$98+u=37$

$8v=\text-18$

Answer:

$p=\frac65$
$q=28$
$r=6$
$s=10$
$t=\frac {8}{3}$ (or equivalent)
$u=\text-61$
$v=\text-\frac{18}{8}$ (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.

Noah is selling boxes of greeting cards. Each box costs $4. Noah’s goal is to earn more than $50 selling cards.

If Noah sells 20 boxes of cards, will he make his goal?
If Noah sells 10 boxes of cards, will he make his goal?
If Noah sells $b$ boxes of cards, write an inequality (using the symbol $<$ or $>$ ) that will be true whenever Noah makes his goal and false whenever he does not.
Graph your inequality on the number line.

Answer:

Yes. Sample reasoning: He earns $80 for selling 20 boxes, which is more than $50.
No. Sample reasoning: He only earns $40 for selling 10 boxes.
Sample response: $b>12$ , because $50\div4$ is between 12 and 13.
Sample response: The solution is the graph of $b>12$ with an open circle at 12 and shading to the right. Alternately, the solution is a set of closed circles at the whole numbers from $b = 13$ and greater, because the number of boxes must be a whole number.

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 boxes must be an integer.

4.

Select all the equations that are true when $x$ is -6.

A.

$18=x+x+x$

B.

$4=x+10$

C.

$\text-12=2x$

D.

$9-x=3$

E.

$\frac{x}{2}=\text-3$

F.

$\text-18=x \boldcdot \text-3$

Answer: B, C, E

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 E or F may need to be reminded about the properties of multiplication by negative numbers 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.

Select all expressions that are equivalent to $5n-30$ .

A.

$n+n+n+n+n-30$

B.

$5(n-30)$

C.

$(n-6) \boldcdot 5$

D.

$5n-30n$

E.

$5n+ \text-30$

Answer:

A, C, E

Teaching Notes

This is another problem reminding students of the distributive property. 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.

$17=5+12$
$17-12=5$
$12-5=17$
$4+4+4+4+4=20$
$20=5 \boldcdot 4$
$12 \boldcdot 5=17$
$4 \boldcdot 4 \boldcdot 4 \boldcdot 4 \boldcdot 4=20$
$20 \div 4=5$

Answer:

B
B
neither
A
A
neither
neither
A

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 the unit.