Benchmark Quest: AR and NSO Review Activities

Materials Needed

• Notebook or paper
• Pencil, calculator, and colored pens or highlighters
• Two index cards or small pieces of paper
• Timer
• Optional: digital whiteboard, spreadsheet, or graphing tool
• Practice exercise and answer-key sections below

Lesson Overview

Length: Two class periods, 45 minutes each

Focus Areas:

• AR—Algebraic Reasoning: Simplifying expressions, solving equations, interpreting linear relationships, and applying algebra to real situations.
• NSO—Number Sense and Operations: Working with rational numbers, percentages, scientific notation, exponents, and reasonable estimates.

Learning Objectives

By the end of the two lessons, the learner will be able to:

1. Simplify algebraic expressions by combining like terms and using the distributive property.
2. Solve one- and two-step linear equations and check solutions.
3. Use an equation or table to interpret a real-world linear situation.
4. Compare, calculate with, and estimate using positive and negative rational numbers.
5. Convert between scientific notation and standard form.
6. Apply exponent rules and percentage reasoning in practical situations.
7. Explain the reasoning used to solve a problem, not just provide an answer.

Success Criteria

A successful review shows that the learner can:

• Show organized steps and use correct mathematical symbols.
• Check at least two solutions by substituting an answer back into the original problem.
• Use estimation to decide whether an answer is reasonable.
• Explain mistakes and correct them.
• Earn at least 80% on the final mixed review or identify a clear plan for correcting missed skills.

Day 1: NSO—Number Sense and Operations

Introduction and Hook — 5 Minutes

Ask:

You have $50. You spend 30% on food, lose $8, and then receive a $12 refund. How much money do you have now? What steps would help you avoid a careless mistake?

Have the learner estimate the answer first. Explain that today’s goal is to use number sense to calculate accurately and decide whether an answer makes sense.

Learning Targets — 2 Minutes

Say: “Today I will review rational numbers, scientific notation, exponents, and percentages. I will model the process, we will solve examples together, and then you will complete independent challenges.”

I Do: Teacher or Parent Modeling — 10 Minutes

NSO Example 1: Operations with Rational Numbers

Problem: Evaluate: -6 + 2.5 - (-3)

1. Subtracting a negative is the same as adding: -6 + 2.5 + 3.
2. Combine positive numbers: 2.5 + 3 = 5.5.
3. Combine: -6 + 5.5 = -0.5.

Answer: -0.5

Reasonableness check: The negative 6 is greater in absolute value than the positive amount 5.5, so a small negative answer makes sense.

NSO Example 2: Scientific Notation

Problem: Write 0.000047 in scientific notation.

1. Move the decimal point to create a number between 1 and 10: 4.7.
2. Count the moves: 5 places to the right.
3. Because the original number is less than 1, use a negative exponent.

Answer: 4.7 × 10-5

NSO Example 3: Percent Change

Problem: A jacket costs $80 and is on sale for 25% off. What is the sale price?

1. Find the discount: 0.25 × 80 = 20.
2. Subtract the discount: 80 - 20 = 60.

Answer: The sale price is $60.

Shortcut: Paying 75% means 0.75 × 80 = 60.

We Do: Guided Practice — 10 Minutes

Work through the following prompts together. The learner should explain each step aloud or write it down.

1. Estimate the answer to -19.8 + 7.2 before calculating.
2. Explain why 3.2 × 104 is greater than 3.2 × 103.
3. A $45 meal has an 18% tip. Is a $5 tip reasonable? Calculate the exact tip.

Quick check: Ask, “What sign should the answer have?” and “How can estimation help us catch an error?”

You Do: NSO Practice Exercises — 12 Minutes

Complete the problems independently. Show all work and estimate where appropriate.

1. Evaluate: -12.5 + 4.8 - (-2.5).
2. Evaluate: (-3)2 - 4(2).
3. Write 6,300,000 in scientific notation.
4. Write 8.04 × 10-3 in standard form.
5. A phone originally costs $720 and is discounted by 15%. What is the sale price?
6. A test score increases from 68 to 80. What is the percent increase? Round to the nearest whole percent.
7. A hiking trail is 12.6 miles long. You have completed 7.85 miles. How many miles remain?

NSO Review Game: “Estimate or Calculate?” — 4 Minutes

Read each statement. The learner holds up one card labeled Estimate or Calculate.

• Deciding whether a $19.95 item plus tax will cost more than $25.
• Finding the exact amount of a 17% tip.
• Checking whether 4.1 × 105 is larger than 4.1 × 104.

Closure and Reflection — 2 Minutes

Ask the learner to complete these statements:

• “One strategy that helped me today was…”
• “One mistake I need to watch for is…”
• “I know my answer is reasonable because…”

Day 2: AR—Algebraic Reasoning

Introduction and Hook — 5 Minutes

Present this situation:

A streaming service charges a $9 sign-up fee and $7 per month. Another service charges $15 per month with no sign-up fee. After how many months will the two services cost the same?

Invite the learner to make a prediction before solving. Connect the problem to phone plans, subscriptions, transportation, or other real-life choices.

Learning Targets — 2 Minutes

Say: “Today I will review how to simplify expressions, solve equations, and use algebra to make decisions. I will show my reasoning and check that my answers fit the situation.”

I Do: Teacher or Parent Modeling — 10 Minutes

AR Example 1: Simplifying an Expression

Problem: Simplify: 4(2x - 3) + 5x

1. Distribute 4: 8x - 12 + 5x.
2. Combine like terms: 13x - 12.

Answer: 13x - 12

AR Example 2: Solving an Equation

Problem: Solve: 3x + 7 = 25

1. Subtract 7 from both sides: 3x = 18.
2. Divide both sides by 3: x = 6.
3. Check: 3(6) + 7 = 25, so the solution is correct.

AR Example 3: Comparing Two Linear Situations

Problem: Plan A costs 9 + 7m dollars, and Plan B costs 15m dollars. When do they cost the same?

1. Set the expressions equal: 9 + 7m = 15m.
2. Subtract 7m: 9 = 8m.
3. Divide by 8: m = 1.125.

Interpretation: The costs are equal after 1.125 months. Since billing usually occurs by whole months, compare the plans at Month 1 and Month 2. Plan B becomes less expensive beginning in Month 2.

We Do: Guided Practice — 10 Minutes

Solve the following together. Ask the learner to identify the operation that should be undone first.

1. Simplify: 6x + 4 - 2x + 9.
2. Solve: 5x - 11 = 24.
3. A ride-share service charges $3 plus $2.50 per mile. Write an expression for the cost of m miles and find the cost of a 6-mile ride.

Quick check: Ask, “What does the variable represent?” and “How can we check the answer?”

You Do: AR Practice Exercises — 12 Minutes

Complete the problems independently. Explain the meaning of the variable in every real-world problem.

1. Simplify: 3(2x + 5) - 4x.
2. Simplify: 7a - 2(3a - 4) + 1.
3. Solve: 4x - 9 = 31.
4. Solve: 2(3x + 1) = 20.
5. Solve and check: 5x + 6 = 2x + 24.
6. A gym charges a $25 registration fee plus $18 per month. Write an expression for the cost after m months and find the cost after 8 months.
7. Option A costs 40 + 12m dollars. Option B costs 18m dollars. At how many months are the costs equal? Which option is cheaper after 5 months?

AR Challenge: “Create the Deal” — 4 Minutes

Choose one context: music subscription, gaming membership, transportation, or tutoring.

1. Create two pricing plans using a fixed fee and a monthly fee.
2. Write an expression for each plan.
3. Find when the plans cost the same.
4. State which plan is better for a short-term customer and which is better for a long-term customer.

Closure and Reflection — 2 Minutes

Have the learner explain:

• How simplifying an expression differs from solving an equation.
• Why checking a solution is important.
• How algebra can help someone compare real-world choices.

Final Mixed Assessment — 10 Minutes

Use this assessment at the end of Day 2 or as a separate short quiz. The learner should work independently and show reasoning.

1. NSO: Evaluate -8 + 3.5 - (-4).
2. NSO: Write 0.00092 in scientific notation.
3. NSO: A $120 item is discounted by 20%. Find the sale price.
4. AR: Simplify 5(2x - 1) + 3x.
5. AR: Solve 7x - 4 = 38 and check your answer.
6. AR: A taxi charges $4 plus $2 per mile. Write an equation for a $22 trip and solve for the number of miles.

Answer Key

NSO Practice

1. -5.2
2. 1
3. 6.3 × 106
4. 0.00804
5. $612
6. Approximately 18%
7. 4.75 miles

AR Practice

1. 2x + 15
2. a + 9
3. x = 10
4. x = 3
5. x = 6
6. Expression: 25 + 18m; after 8 months: $169
7. Equal at m = 10 months; after 5 months, Option B costs $90 and Option A costs $100, so Option B is cheaper.

Final Mixed Assessment

1. -0.5
2. 9.2 × 10-4
3. $96
4. 13x - 5
5. x = 6
6. Equation: 4 + 2m = 22; m = 9 miles

Differentiation and Adaptation

Support for Learners Who Need More Scaffolding

• Use a number line for positive and negative numbers.
• Provide a step organizer: distribute, combine, isolate, check.
• Allow the learner to use a calculator after estimating first.
• Color-code like terms and underline operations that must be undone.
• Complete fewer problems while requiring a clear explanation for each one.

Extension for Learners Ready for More Challenge

• Rewrite an equation in multiple equivalent forms.
• Create a situation involving a percent increase followed by a percent decrease.
• Graph the two pricing plans and identify the intersection point.
• Explain why a 20% discount followed by a 20% increase does not return an item to its original price.

Flexible Delivery Options

• Homeschool: Use a parent-child discussion and complete the challenge with household expenses or subscriptions.
• Classroom: Have students work in pairs, compare strategies, and use cards for quick-response checks.
• Digital or training setting: Use a shared document, online poll, calculator, or spreadsheet to model calculations and compare plans.

Feedback and Reflection

After reviewing the answer key, have the learner mark each problem:

• Green: I can solve this independently.
• Yellow: I understand the idea but need more practice.
• Red: I need the skill retaught or modeled again.

End by selecting one yellow or red problem and correcting it using this sentence frame: “My first approach was ____. The error occurred when ____. The correct strategy is ____.”