### Examples

*Example:*23% of the junior population are taking an art class this year. What is the ratio of juniors taking an art class to juniors not taking an art class?

*Example: *The ratio of boys to girls in a class is 3:2. What percentage of the students are boys in the class?

### Clarifications

*Clarification 1:*Instruction includes discounts, markups, simple interest, tax, tips, fees, percent increase, percent decrease and percent error.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**7

**Strand:**Algebraic Reasoning

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

- MA.7.NSO.1.2
- MA.7.NSO.2
- MA.7.AR.4.4
- MA.7.AR.4.5
- MA.7.DP.1.3
- MA.7.DP.1.4
- MA.7.DP.1.5
- MA.7.DP.2.2
- MA.7.DP.2.3
- MA.7.DP.2.4

### Terms from the K-12 Glossary

- Percent of Change
- Percent Error
- Rate
- Simple Interest

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 6, students solved mathematical and real-world problems involving percentages, ratios, rates and unit rates. Students then solve multi-step real-world percent problems in grade 7 and solve multi-step linear equations of any context in grade 8.- Instruction includes discounts, markups, simple interest, tax, tips, fees, percent increase, percent decrease and percent error
*(MTR.7.1).*- Markdown/discount is a percentage taken off of an original price. Instruction includes showing the connection between subtracting the calculated discount or taking the difference between 100% and the discount and multiplying that by the original price.
- For example, if there was a 15% discount on an item that costs $15.99, students could take 85% of $15.99 or take 15% of $15.99 and subtract that value from the original price of $15.99.

- Markup showcases adding a charge to the initial price. Markups are often shown in retail situations.
- Simple interest refers to money you can earn by initially investing some money (the principal). The percentage of the principal (interest) is added to the principal making your initial investment grow. The formula, $I$ = $P$$r$$t$, represents $I$=interest, $P$=principal, $r$=rate, and $t$=time. When using simple interest, provide the formula as students should not be expected to memorize this.
- Tax, tips and fees are an additional charge added to the initial price. Students can add the calculated tax, tip or fee to the original price or add 1 to the tax, tip or fee to reach the final cost.
- For example, if there was a 6% sales tax on clothing and a t-shirt costs $7.99. Students can add 100% to the 6% and multiply that value to $7.99 or students can find 6% of the $7.99 and add that to the original value of the t-shirt.
- Percent Increase/Percent Decrease asks students to look for a percentage instead of a dollar amount. Students should discover that they can use the formula below to help become more flexible in their thinking.$\frac{\text{}}{}$
- Percent Error is a way to express the size of the error (or deviation) between two measurements.

- Markdown/discount is a percentage taken off of an original price. Instruction includes showing the connection between subtracting the calculated discount or taking the difference between 100% and the discount and multiplying that by the original price.
- Use bar models to model percent increase and decrease problems.
- For example, if you are finding percentages that are in multiples of 10%, your bar model may look like the model below.To showcase the percent increase, you would add additional boxes into the bar model. If you are showcasing a percent decrease, then you would cross out boxes for the decrease
*(MTR.2.1).*

- For example, if you are finding percentages that are in multiples of 10%, your bar model may look like the model below.
- Use bar models, double number lines, tables or other visual representations to model relationships between percentages and the part and whole amounts
*(**MTR.2.1*).- Double Number Line
- TableInstruction includes the use of patterns when using a table. In the example above, students can use the idea of 100% being 300 and using this knowledge to find other percentages. 10 is of 100, so students can divide by 10. To find 20%, students can multiply their solution from 10% by 2. The pattern can continue to relate common connections between percentages
*(MTR.5.1).*

- Double Number Line
- Reinforce how percentages relate to fractions and decimals. Help students write equivalent ratios to represent problems using reasoning about the relationships between the quantities.
- Instruction includes using proportional relationships and multiplicative reasoning to solve problems.

### Common Misconceptions or Errors

- Students may incorrectly place the decimal point when calculating with percentages. If students have discovered the shortcut of moving the decimal point twice, instruction includes understanding of how a percent relates to fractions and decimals. Refer to MA.7.NSO.1.2 to emphasize equivalent forms.
- Students may forget to change the percent amount into decimal form (divide the percent by 100) when setting up an equation
*(MTR.3.1).* - Students may incorrectly believe all percentages must be between 1 and 100%. To address this misconception, provide examples of percentages below 1% and over 100%.
- Students may incorrectly believe a percent containing a decimal is already in decimal form.
- For example, emphasize that 43.5% is 43.5 out of 100 and dividing by 100 will provide the decimal form.

- In multiple discount problems, students may incorrectly combine the discounts instead of working them sequentially
*(**MTR.5.1*).- For example, 25% off, then 10% off could incorrectly lead to 35% off rather than finding 25% off before calculating the additional 10% off.

- Students may incorrectly invert the part and the whole in the percent problem. To address this misconception, students should use bar models to help visualize and make sense of the problem
*(**MTR.2.1*).

### Strategies to Support Tiered Instruction

- Instruction includes the use of estimation to find the approximate solution before calculating the actual result to help with correct placement of the decimal point and reasonableness of the solution.
- Teacher provides opportunities for students to use a 100 frame to review place value for and the connections to decimal, fractional, and percentage forms.
- Teacher provides support for students in dividing by 100 to change percent into decimal form. Teacher supports by providing calculators, manipulatives and base ten blocks to multiply decimals.
- Instruction includes having students take different percentages of the same amount, such as 40% of 80, 4% of 80, 0.4% of 80, 0.04% of 80 and 400% of 80. Students can be given the flexibility to provide the answer as decimal or fraction and compare.
- Teacher provides support for students when solving multi-discount problems and combining the discounts. Instruction might begin with a single step discount problem in a real-world context.
- For example, teacher can include local sale flyers with products that students are interested in buying. Have students explain how to apply the multi-discounts with a comparison of the difference in costs when combining the discounts incorrectly.

- Teacher provides opportunities for students to reason and think about multiple discount problems by providing prompts.
- For example, “if a pair of jeans are 50% off with an additional 50% off, does that mean the jeans are 100% off, or free?” or “what if the jeans are 75% off with an

- additional 50% off, does that mean the jeans are 125% off and the store now owes
- you money to take them?”
- Teacher provides opportunities for students to comprehend the context or situation by engaging in questions (laminating these questions on a printed card for students to utilize as a resource in and out of the classroom would be helpful).
- What do you know from the problem?
- What is the problem asking you to find?
- Can you create a visual model to help you understand or see patterns in your problem?

- Teacher provides support when solving multi-discount problems, by providing students with a table to keep track of the information in the problem.
- Instruction includes the use of a three-read strategy. Students read the problem three different times, each with a different purpose (laminating these questions on a printed card for students to utilize as a resource in and out of the classroom would be helpful).
- First, read the problem with the purpose of answering the question: What is the problem, context, or story about?
- Second, read the problem with the purpose of answering the question: What are we trying to find out?
- Third, read the problem with the purpose of answering the question: What information is important in the problem?

- Teacher encourages the use of bar models to help visualize and make sense of the problem.
- Instruction includes understanding of how a percent relates to fractions and decimals if students have discovered the shortcut of moving the decimal point twice. Refer to MA.7.NSO.1.2 to emphasize equivalent forms.

### Instructional Tasks

*Instructional Task 1*

**(MTR.4.1,***MTR.7.1*)SurfPro Shop and The Surfer Store both sold surfboards for $350. In February, SurfPro Shop wanted to increase their profits so they increased the prices of their boards by 15%. When this increase failed to bring in more money, they decreased their price again by 10% in November. To beat their competitor who had increased prices, The Surfer Store decided to decrease their price of surfboards by 10% in March. However, when they started to lose money on the new pricing scheme, they increased the price of surfboards in November by 15%.

- Part A. If no other changes were made after November, which store now has the better price for surfboards?
- Part B. What is the difference between their prices?

### Instructional Items

*Instructional Item 1*

A college’s intramural soccer team has 30 players, 60% of which are women. After 22 new players joined the team, the percentage of women was reduced to 50%. How many of the new players are women?

*Instructional Item 2*

Miguel takes out a loan that adds interest each year on the initial amount. What is the interest Miguel will pay on the loan if he borrowed $5,000 at an annual interest rate of 4.5% for 15 years? (Use the formula $I$ = $P$$r$$t$, where $I$ is the interest, $P$ is the principal or initial investment, $r$ is the interest rate per year, and $t$ is the number of years.)

*Instructional Item 3*

Massimo lost his mathematics textbook. The school charges a lost book fee of 70% of the original cost of the book. If Massimo received a notice he owed the school $73.50 for the lost textbook, what was the original cost?

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Educational Game

## Formative Assessments

## Lesson Plans

## Original Student Tutorials

## Perspectives Video: Professional/Enthusiast

## Perspectives Video: Teaching Idea

## Problem-Solving Tasks

## Teaching Ideas

## Tutorial

## STEM Lessons - Model Eliciting Activity

In this activity, students will engage critically with nutritional information and macronutrient content of several fast food meals. This is an MEA that requires students to build on prior knowledge of nutrition and working with percentages.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

In this Model Eliciting Activity, MEA, students will work in cooperative groups to discuss and come up with a procedure to rank the banks from best to worst by estimating the simple interest and total loan amount.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

In this MEA, students will decide how many wolves to introduce into Yellowstone National Park's ecosystem. The number of wolves could influence many factors, from the tourism industry to local farming businesses, as well as the populations of other species in the area. Students must choose to introduce the number of wolves they feel will be most beneficial to the preservation of Yellowstone National Park as determined by the mission statement of Yellowstone and the National Park Service.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

## MFAS Formative Assessments

Students are given gasoline prices from a year ago and today and are asked to calculate the percent change.

Students are asked to calculate the amount of sales tax and total price, given prices of individual items to purchase.

## Original Student Tutorials Mathematics - Grades 6-8

Follow Hailey and Kenna as they estimate tips and sales tax at the mall, restaurants, and the hair salon in this interactive tutorial.

Let's calculate markups and markdowns at the mall and follow Paige and Miriam working in this interactive tutorial.

Calculate simple interest and estimate monthly payments alongside a loan officer named Jordan in this interactive tutorial.

Explore sales tax, fees, and commission by following a customer service representative named Julian in this interactive tutorial.

Learn to solve percent change problems involving percent increases and decreases in in this interactive tutorial.

## Student Resources

## Original Student Tutorials

Follow Hailey and Kenna as they estimate tips and sales tax at the mall, restaurants, and the hair salon in this interactive tutorial.

Type: Original Student Tutorial

Let's calculate markups and markdowns at the mall and follow Paige and Miriam working in this interactive tutorial.

Type: Original Student Tutorial

Calculate simple interest and estimate monthly payments alongside a loan officer named Jordan in this interactive tutorial.

Type: Original Student Tutorial

Explore sales tax, fees, and commission by following a customer service representative named Julian in this interactive tutorial.

Type: Original Student Tutorial

Learn to solve percent change problems involving percent increases and decreases in in this interactive tutorial.

Type: Original Student Tutorial

## Educational Game

In this activity, students are quizzed on their ability to estimate sums, products, and percentages. The student can adjust the difficulty of the problems and how close they have to be to the actual answer. This activity allows students to practice estimating addition, multiplication, or percentages of large numbers. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Educational Game

## Problem-Solving Tasks

This purpose of this task is to help students see two different ways to look at percentages both as a decrease and an increase of an original amount. In addition, students have to turn a verbal description of several operations into mathematical symbols. This requires converting simple percentages to decimals as well as identifying equivalent expressions without variables.

Type: Problem-Solving Task

In this task, students are presented with a real-world problem involving the price of an item on sale. To answer the question, students must represent the problem by defining a variable and related quantities, and then write and solve an equation.

Type: Problem-Solving Task

Students are asked to determine how to distribute prize money among three classes based on the contribution of each class.

Type: Problem-Solving Task

This problem includes a percent increase in one part with a percent decrease in the remaining and asks students to find the overall percent change. The problem may be solved using proportions or by reasoning through the computations or writing a set of equations.

Type: Problem-Solving Task

Students are asked to make comparisons among the Egyptian, Gregorian, and Julian methods of measuring a year.

Type: Problem-Solving Task

5,000 people visited a book fair in the first week. The number of visitors increased by 10% in the second week. How many people visited the book fair in the second week?

Type: Problem-Solving Task

Students are asked to determine which sale option results in the largest percent decrease in cost.

Type: Problem-Solving Task

The sales team at an electronics store sold 48 computers last month. The manager at the store wants to encourage the sales team to sell more computers and is going to give all the sales team members a bonus if the number of computers sold increases by 30% in the next month. How many computers must the sales team sell to receive the bonus? Explain your reasoning.

Type: Problem-Solving Task

After eating at your favorite restaurant, you know that the bill before tax is $52.60 and that the sales tax rate is 8%. You decide to leave a 20% tip for the waiter based on the pre-tax amount. How much should you leave for the waiter? How much will the total bill be, including tax and tip?

Type: Problem-Solving Task

The purpose of this task is for students to calculate the percent increase and relative cost in a real-world context. Inflation, one of the big ideas in economics, is the rise in price of goods and services over time. This is considered in relation to the amount of money you have.

Type: Problem-Solving Task

The purpose of this task is to see how well students students understand and reason with ratios.

Type: Problem-Solving Task

## Tutorial

Learn how to find the full price when you know the discount price in this percent word problem.

Type: Tutorial

## Parent Resources

## Problem-Solving Tasks

This purpose of this task is to help students see two different ways to look at percentages both as a decrease and an increase of an original amount. In addition, students have to turn a verbal description of several operations into mathematical symbols. This requires converting simple percentages to decimals as well as identifying equivalent expressions without variables.

Type: Problem-Solving Task

In this task, students are presented with a real-world problem involving the price of an item on sale. To answer the question, students must represent the problem by defining a variable and related quantities, and then write and solve an equation.

Type: Problem-Solving Task

Students are asked to determine how to distribute prize money among three classes based on the contribution of each class.

Type: Problem-Solving Task

Tom wants to buy some protein bars and magazines for a trip. He has decided to buy three times as many protein bars as magazines. Each protein bar costs $0.70 and each magazine costs $2.50. The sales tax rate on both types of items is 6½%. How many of each item can he buy if he has $20.00 to spend?

Type: Problem-Solving Task

This problem includes a percent increase in one part with a percent decrease in the remaining and asks students to find the overall percent change. The problem may be solved using proportions or by reasoning through the computations or writing a set of equations.

Type: Problem-Solving Task

Students are asked to make comparisons among the Egyptian, Gregorian, and Julian methods of measuring a year.

Type: Problem-Solving Task

5,000 people visited a book fair in the first week. The number of visitors increased by 10% in the second week. How many people visited the book fair in the second week?

Type: Problem-Solving Task

Students are asked to determine which sale option results in the largest percent decrease in cost.

Type: Problem-Solving Task

The sales team at an electronics store sold 48 computers last month. The manager at the store wants to encourage the sales team to sell more computers and is going to give all the sales team members a bonus if the number of computers sold increases by 30% in the next month. How many computers must the sales team sell to receive the bonus? Explain your reasoning.

Type: Problem-Solving Task

After eating at your favorite restaurant, you know that the bill before tax is $52.60 and that the sales tax rate is 8%. You decide to leave a 20% tip for the waiter based on the pre-tax amount. How much should you leave for the waiter? How much will the total bill be, including tax and tip?

Type: Problem-Solving Task

The purpose of this task is for students to calculate the percent increase and relative cost in a real-world context. Inflation, one of the big ideas in economics, is the rise in price of goods and services over time. This is considered in relation to the amount of money you have.

Type: Problem-Solving Task

The purpose of this task is to see how well students students understand and reason with ratios.

Type: Problem-Solving Task