- Subject:
- Algebra
- Material Type:
- Lesson Plan
- Level:
- Middle School
- Grade:
- 7
- Provider:
- Pearson
- Tags:

- License:
- Creative Commons Attribution Non-Commercial
- Language:
- English
- Media Formats:
- Text/HTML

# Gallery Problems Exercise

## Overview

# Gallery Overview

Allow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit's concepts or to assist students who may have fallen behind on work.

# Gallery Descriptions

**Match Inequalities**

Students match inequalities to their solutions.

**Product Between One-Half and One**

Students find a range of values for an inequality situation.

**Inequalities about Numbers**

Students write inequalities to solve problems about the sums of three consecutive numbers.

**School Dance**

Students use equations and an inequality to model the costs and revenues of holding a school dance.

**What Could My Number Be?**

Students use inequalities to identify possibilities for a number given certain conditions.

**Batting Average**

Students use an inequality to find the number of hits needed to get a desired batting average.

# Match Inequalities

# Answers

$10>8+6x$

| $x<\frac{1}{3}$

$6x+10>8$ | $x>-\frac{1}{3}$

$-4x-15>-27$ | $x<3$

$15x+10<5$ | $x<-\frac{1}{3}$

$4x+7>19$ | $x>3$

$-7x+25<46$ | $x>-3$

## Work Time

# Match Inequalities

Match the inequality cards with their corresponding solution cards.

HANDOUT: Matching Inequalities

# Product Between One-Half and One

# Answers

1a. $\frac{1}{2}$

$\begin{array}{c}2a<1\\ 2a\xf72<1\xf72\\ a<\frac{1}{2}\end{array}$

or

$\begin{array}{c}2a<1\\ 2a\cdot \frac{1}{2}<1\cdot \frac{1}{2}\\ a<\frac{1}{2}\end{array}$

1b. $\frac{1}{4}$

$\begin{array}{c}2a>\frac{1}{2}\\ 2a\xf72>\frac{1}{2}\xf72\\ a>\frac{1}{4}\end{array}$

or

$\begin{array}{c}2a>\frac{1}{2}\\ 2a\cdot \frac{1}{2}>\frac{1}{2}\cdot \frac{1}{2}\\ a>\frac{1}{4}\end{array}$

1c. Possible answers: $\frac{1}{3},\text{\hspace{0.17em}}\frac{3}{8},\text{\hspace{0.17em}}\frac{5}{12}$

2a. $-\frac{1}{4}$

$\begin{array}{c}-2b>\frac{1}{2}\\ -2b\xf7-2<\frac{1}{2}\xf7-2\\ b<-\frac{1}{4}\end{array}$

or

$\begin{array}{c}-2b>\frac{1}{2}\\ -2b\cdot -\frac{1}{2}<\frac{1}{2}\cdot -\frac{1}{2}\\ b<-\frac{1}{4}\end{array}$

2b. $-\frac{1}{2}$

$\begin{array}{c}-2b<1\\ -2b\xf7-2>1\xf7-2\\ b>-\frac{1}{2}\end{array}$

or

$\begin{array}{c}-2b<1\\ -2b\cdot -\frac{1}{2}>1\cdot -\frac{1}{2}\\ b>-\frac{1}{2}\end{array}$

2c. Possible answers: $-\frac{1}{3},\text{\hspace{0.17em}}-\frac{3}{8},\text{\hspace{0.17em}}-\frac{5}{12}$

**Work Time**

# Product Between One-Half and One

1. The product of 2 and some number *a* is between $\frac{1}{2}$ and 1.

$2a>\frac{1}{2}$

and $2a<1$

a. What number does *a* have to be less than? Justify your answer mathematically.

b. What number does *a* have to be greater than? Justify your answer mathematically.

c. Write three possible values for *a*.

2. The product of –2 and some number *b* is between $\frac{1}{2}$ and 1.

a. What number does *b* have to be less than? Justify your answer mathematically.

b. What number does *b* have to be greater than? Justify your answer mathematically.

c. Write three possible values for *b*.

# Inequalities about Numbers

# Answers

- 33, 34, 35

$n+(n+1)+(n+2)>100$ - 31, 33, 35

$n+(n+2)+(n+4)<100$ - 30, 32, 34

$n+(n+2)+(n+4)<100$ - –36, –34, –32

$n+(n+2)+(n+4)<-100$

**Work Time**

# Inequalities about Numbers

Write an inequality to solve each problem.

- What are the least three consecutive whole numbers that have a sum greater than 100?
- What are the greatest three consecutive odd numbers that have a sum less than 100?
- What are the greatest three consecutive even numbers that have a sum less than 100?
- What are the greatest three consecutive even integers that have a sum less than –100?

# School Dance

# Answers

1. $d=4\cdot 50-60$

$d=\$140$

2. $d=ab-c$

At least 178 tickets must be sold in order for the class to make a profit of at least $700.

$\begin{array}{c}700\le 4.5b-100\\ 800\le 4.5b\\ 177.\overline{7}\le b\end{array}$

or $b\ge 177.\overline{7}$

Because a fraction of a ticket cannot be sold, the inequality tells us that the minimum number of tickets sold must be the smallest whole number greater than $177.\overline{7}$

**Work Time**

# School Dance

Mr. Washington’s class is holding a school dance to raise money.

- Let
*a*represent the price, in dollars, of each ticket. - Let
*b*represent the number of tickets sold. - Let
*c*represent the cost, in dollars, of renting music equipment. - Let
*d*represent the profit, in dollars, the class made.

- If
*a*= 4,*b*= 50, and*c*= 60, write an equation to determine the profit,*d*.- Solve the equation to determine how much profit the class made.
- Write an algebraic equation to express
*d*in terms of the variables*a*,*b*, and*c*.

- If each ticket costs $4.50 and the cost of renting the music equipment is $100, what is the minimum number of tickets that must be sold for the class to make a profit of at least $700? Explain how to use an algebraic inequality to solve this problem.

# What Could My Number Be?

# Aswers

Possible values: 2, 1, 0

$\begin{array}{c}2n<6\\ n<3\end{array}$Possible values: 54, 55, 56

$\begin{array}{c}8<n\xf76\\ n>48\end{array}$Possible values: −2, 0, 2

$\begin{array}{c}-6n<18\\ n>-3\end{array}$Possible values: −12, −13, −14

$\begin{array}{c}n+7<-4\\ n<-11\end{array}$Possible values: $4\frac{1}{2},\text{}5,\text{}10$

$\begin{array}{c}\frac{n}{2}>2\\ n>4\end{array}$Possible values: −11, −12. −13

$\begin{array}{c}8+n<-2\\ n<-10\end{array}$Possible values: 16, 17, 18

$\begin{array}{c}16>n-3\\ n<19\end{array}$Possible values: 3, 2, 0

$\begin{array}{c}7+3n<18\\ n<3\frac{2}{3}\end{array}$

**Work Time**

# What Could My Number Be?

For each problem, write and solve an inequality that shows possible values for your number. Then give three possible values for the number.

- The product of 2 and my number is less than 6.
- 8 is less than my number divided by 6.
- The product of –6 and my number is less than 18.
- The sum of my number and 7 is less than –4.
- My number divided by 2 is greater than 2.
- The sum of 8 and my number is less than –2.
- 16 is greater than my number minus 3.
- 7 plus the product of 3 and my number is less than 18.

# Batting Average

# Answers

A batting average of 0.286, or $\frac{6}{21}=0.2857$.

At least 10 more hits.

$\begin{array}{c}\frac{(6+x)}{53}>0.300\\ 6+x>15.900\\ x>9.900\end{array}$

**Work Time**

# Batting Average

A baseball player’s batting average is determined by dividing the total number of hits by the total number of official at bats (total number of at bats not including walks). The average is always rounded to the nearest thousandth.

- So far this season, you have been at bat 21 times and have gotten 6 hits. What is your batting average so far?
- Suppose you have 32 more times at bat this season. How many hits will you need in order to reach a batting average of more than 0.300 by the end of the season?