Comparing Numbers

Today's Number Talk

For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation. For this Number Talk, I am encouraging students to represent their thinking using an array model.

Task 1: 24 x 13

For the first task, 24 x 13, I was amazed at the students' abilities to represent this problem in multiple ways. Here, a student shows 24x13=(20+2+2) x (10 x 1 x 2). I tried to encourage the use of more precise language as I conferenced with him. 

Task 2: 24 x 26

During the next task, we discussed 24 x 26. Students solved this problem using a wide range of strategies: 

• 24 x 26 Model 1
• 24 x 26 Model 2
• 24 x 26 Model 3
• 24 x 26 Model 4
• 24 x 26 Model 5
• 24 x 26 Model 6

Lesson Set Up:

For this lesson, I wanted to provide students with the opportunity to compare numbers in a real world situation (Math Practice 4: Model with mathematics). So I researched the prices of automobiles, ranging from $15,000 to almost $470,000. Next, I created the Automobile Comparison document. On each page of this document is a set of automobiles, ready to compare! After printing this document, I sorted each set of automobiles into four complexity levels and labeled them accordingly: Level A, Level B, Level C, and Level D. We will use these leveled comparisons during Partner Practice time. 

In addition, I printed a Place Value Chart for each student and placed it inside a sheet protector. I passed this out to students prior to the lesson. 

Lesson Introduction:

To begin, I invited students to join me on the carpet with a white board and a Place Value Chart. I then introduced the lesson's Goal: I can compare multi-digit number using >, =, and < symbols. To provide students with context, I explained: Let's say you're looking to buy a new car today! Students were immediately intrigued! Is it important to be able to compare the prices of cars? Turn and talk with a partner! I then explained the meaning of >, =, and < using the Comparison Symbol Posters (Source). 

Comparing One-Digit Numbers:

I continued: In order to compare the prices of cars, we first need to make sure we understand how to compare simpler numbers using the comparison symbols. Using a set of Colossal Playing Cards (pictured below), I created the following equation on the board: 4 > 3 and asked students to help me determine which symbol would correctly complete the equation. Then we continued to practice using the comparison symbols with other equations that compared one-digit numbers: 4 < 6 and 4 = 4. Students loved using the large playing cards and were always eager to see which card was drawn next! 

Flipping the Equation:

Comparing Multi-Digit Numbers:

To gradually increase the complexity of the task, I used the playing cards to deal two digit numbers and then three digit numbers. I slowly built up to six digit numbers. For each comparison task, I asked for student volunteers to place the correct comparison symbol between the numbers and to read the sentence out loud. I also asked students to use their place value charts to model their thinking: Place Value Chart 165 > 125. Here's the outline of the learning progression:

• 28 < 62 
• 22 < 92
• 165 > 125 
• 922 < 925
• 9725 < 9851
• 59,733 < 59,751

Comparing Car Prices:

Partners & Materials: 

Assigning partners was quick and easy as I already have students desks carefully placed in groups. I passed out a Bag of Money and a set of Comparison Symbols to each pair of students.

Explaining Task: 

Next, I explained the Partner Practice task: Today, you'll be working together with your partner to compare the prices of two cars at a time. There are four levels on the counter: Level A, B, C, and D. Please compare two sets of cars from each level before moving on to the next level. Make sure to represent the cost of each car using your bag of money and each partner needs to use a Place Value Chart to compare the costs of each car. When you are ready to move on to another set of cars, raise your hand so I can check your work!  

Comparing the Cost of Cars: 

During this time, I conferenced with each group of students to check on their understanding. I would often ask questions to encourage more precise language and or to help develop the students' abilities to construct viable arguments (Math Practice 3). 

Here, students use their place value charts and money to explain the cost of the Forester vs Tacoma. Another set of students shared how they analyzed the costs of the Ford Fiesta vs Smart ForTwo. One student comments on the name of the car. It's clear that he is really thinking and taking ownership of his learning!

Next, students compared more costly cars, such as the Bently vs Lamborghini and the Lamborghini vs Rolls Royce. Students were very engaged in the process of comparing such expense vehicles! 

After cleaning up math materials, I provided each student with an exit slip to check student learning: Exit Task. I wanted to make sure all student were able to successfully compare two multi-digit numbers. 

As students finished, I asked them to check their answers with a peer. This was a great opportunity for students to catch their mistakes and construct viable arguments.