7th Grade Unit 1: Rationals

Unit Driving Question 

What are rational numbers and why are they useful?

Essential Questions  

1. How do we mathematically represent rational numbers?

2. How do multiplication and division apply to rational numbers?

3. When is it appropriate to use estimation and/or approximation?

4. How important are estimations in real-life situations?

5. How do I make a reasonable estimate?

6. Where are rational numbers found in real-life? 

Big Ideas 

1. Equivalent rational numbers can be represented in multiple ways.
2. Multiplication and division with integers can be estimated and illustrated.
3. Rational numbers are found in the real world. 

   Technology Resources

• 7th Grade Technology Resources

Launch Task 

1 Lesson 

• Notice/Wonder: Use the graphics and examples 1.1, 1.2, and 1.3 in this lesson to let students share what they notice and wonder about the relationship between the numbers. Use the conversation to point out the various number types (decimal, fraction, negative, positive).  Teacher Resource

Big Ideas for Development Lessons 

3-4 Weeks (approximately 1 week per big idea) 

Big Idea 1: Equivalent rational numbers can be represented in multiple ways.

Key Resources  

1. Converting Fractions to Decimals - The purpose of this task is to demonstrate the decimal point and zero comes from exchanging the whole number for equivalent tenths.
2. The Repeater Vs. The Terminator (updated 4/2021)- In this task, students will convert fractions to decimals.
3. Rational Speed Matching Activity - This activity asks students to identify a fraction, decimal or percent from a given representation. Students are asked to work with equivalent representations of rational numbers to demonstrate equivalent relationships.
4. Points on a Number line (Open Up; updated 04/2021)- The purpose of this activity is to reestablish the interpretation of distance on the number line in the context of negative numbers. They then create folded number lines to reason about different representations of opposites.
5. Ordering Rational Numbers (Open Up; updated 04/2021 log in as a teacher to view this resource) - This lesson solidifies what students have learned in the past several lessons about the ordering of rational numbers on the number line. Students practice ordering rational numbers and use precise language to describe the relationships between numbers plotted on a number line.
6. Translating Between Fraction, Decimals and Percents (Math Shell, updated 07/2022) - This lesson is designed to help students convert between fraction, decimals and percents.

Big Idea Probe

1. Comparing Rational Numbers Probe  

Evidence of Understanding 

Students can:

• Identify and classify rational numbers and irrational numbers.
• Identify and communicate the pattern of rational numbers being expressed as a ratio.
• Identify and communicate the pattern of rational numbers being expressed as a terminating or repeating decimal.
• Use bar notation when working with repeating decimals. 
• Accurately express rational numbers in their decimal equivalent form.
• Decide which equivalent representation of a rational number is most useful.
• Graphically illustrate rational numbers on a number line.
• Utilize a number line to explain the order and value of rational numbers.
• Compare rational numbers written in different forms or different representations. 

Big Idea 2: Multiplication and division with integers can be estimated and illustrated.

Key Resources 

1. Multiplying Rational Numbers (Open Up) - This task asks students to model the multiplication of integers using tables to extend patterns.
2. Multiplying Integer Expressions (Open Up)- In this task, students are asked to examine integer expressions to determine if expressions are equivalent.
3. Modeling Multiplication & Division of Integers - This activity is a great way for students to explore and model the rules for multiplying and dividing integers.
4. Multiplying Integers Investigation - In this activity, students explore patterns in multiplication with whole numbers, and apply/extend what they see there to make sense of multiplication involving negative integers.
5. Dividing Rational Numbers (Open Up) - In this lesson, students complete their work extending all four operations to signed numbers by studying division. They use the relationship between multiplication and division to develop rules for dividing signed numbers. In preparation for the next unit, students look at a context, drilling a well, that is modeled by an equation y=kx where k is a negative number.             

 Big Idea Probe 

1. Integer Multiplication and Division 

Evidence of Understanding  

Students can:

• Model multiplication and division of integers using a variety of methods.

• Accurately solve multiplication and division of integer problems with 3 or more signed numbers.

• Use models of multiplication and division of integers to explain integer rules.

• Use estimation to verify their solutions.

• Justify the reasonableness of their solutions using estimation.

• Communicate the reasonableness of their estimation.

• Communicate the rules of signed numbers and how they are utilized in a problem to determine a solution.

Big Idea 3: Rational numbers are found in the real world.

Key Resources 

1. Real-World Problems (Open Up)- Students will solve a variety of real-world problems involving rational numbers with their peers. This will take an entire class period.
2. Integers to Positive Exponents - Students will show the expanded form before solving the problem. This will allow students to better understand why the solution is positive or negative.
3. Exponents (Positive & Negative Bases) - Students will show their understanding of exponents by solving a variety of problems with positive and negative bases.
4. Real-World Problems including Exponents - Students will solve real-world problems using their understanding of exponents.

Big Idea Probe 

1. Rational Numbers Probe

Evidence of Understanding 

Students can:

• Identify a base and exponent.

• Accurately solve positive integer exponents represented in a variety of ways.

• Explain how integer rules can help determine the solution of exponents.

• Generate mathematical representations of rational numbers in a real-world situation.

• Solve real-world integer problems.

• Justify the reasonableness of a solution to a real-world problem utilizing integers.

Unit Closure

1 Week (includes time for probes, re-engagement, and assessment)