Modeling Mixed Number Addition & Subtraction

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SWBAT use models to add and subtract mixed numbers.

Big Idea

Students use digital fraction models to represent the addition and subtraction of mixed numbers.


5 minutes

To prepare students for the lesson today, I start with a journal entry about estimation.  Often times I will have a journal entry connect mathematics to real life situations.  Today, I wanted to keep it simple so that the students would practice the skill of estimating the sums and differences of mixed numbers as a quick review. 

I posted one addition and one subtraction equation on the board for students to estimate.  While students were working, I walked around and prompted those who were a little "stuck" by asking, if someone told you they were 10 and 3/5 years old, what age would you round that to? With a little prompting, students were able estimate each problem.

It is important to take time to share various responses and approaches for estimation of sums and differences because of the various approaches that can be used.

• rounding to the nearest whole

• rounding to a benchmark fraction

• using compatible fractions 

As part of the lesson, students will be working in pairs and also estimating.  It is important to reestablish the expectation that not all estimates will be the same.

"Play Time"

10 minutes

After reviewing the students estimation journal entries, I explain the objective of the day.  

Today we will be use laptops to find the sums and differences of mixed numbers.  You and a partner will use estimation to judge the reasonableness of your answers.  

I quickly demonstrate how to access the fraction tools software and then allow pairs to take a computer, sign in, and play with the fraction tiles. 

Students have 5 minutes to "play with the tools" so they are familiar with how to use them before starting the lesson.

This plan has two benefits:

1. The students are able to familiarize themselves with the different features of the program

2.  This provides you with time to help all students get up and running and address any glitches that they encounter.  

If I don't plan this into the lesson, I find that I get frustrated with the technology, not excited by it.

Guided and Independent Practice (part 1)

20 minutes

The handout for today's independent work is intended to guide student thinking, step-by-step, so that when they interact with the problems given they don't skip/miss steps (see reflection).

I model 1 addition problem and 1 subtraction problem, making sure to use the term REGROUPING.

Next, I have students try a subtraction problem in pairs - then check in with me. This is a critical step, because if I don't catch student errors, misconceptions, etc. now these will then become a part of their practice.

When all students are prepared with the online program we're using - Tools for Math - and have had at least a minute to try it out, I call the students to sit in front of the white board.  This way, they have to move away from their computers and focus on the guided practice. 

I decide to break this lesson into two parts, with modeling addition first. For the initial model, I choose 1 and 7/8 + 1 and 1/4. I provide students with a copy of the handout for the day.  This hand out walks them through the steps.

• Estimate - students turn and talk to estimate this solution, more than one option is recorded.

• Do you have common denominators? Through interactive modeling, students determine that 8 is a good common denominator.  

• Model - Using the tiles, I make 1 whole and 7/8 and then another 1 whole and 2/8.  I then show the students the "exchange tool" so 8/8 can be exchanged for 1 whole.  I compare this regrouping to regrouping 10 ones for 1 ten.  

• Solution- the solution is recorded. 

After one model, students return to their computers and solve 3 more problems from the book.  I am careful to assign problems with a common denominator that can be represented with the tiles we have.

4 and 1/10 + 6 and 1/2

9 and 7/12 +4 and 3/4

5 and 1/8 + 3

8 and 3/4 + 7 and 3/4

Guided and Independent Practice (part 2)

20 minutes

After students have become comfortable with adding mixed numbers using models, I call the group back together.  As students gather, I ask them to share something they learned while using the models. 

I explain that now, they will help me solve an example of subtracting mixed numbers. The handout is the same, so this time I focus only on the content of the modeling.  For this example I solve 7 and 2/3 - 3 and 5/6. 

First, students estimate.  

Next, we look for common denominators and decide on 6ths.  The problem can be rewritten as 7 and 4/6- 3 and 5/6.  

To get started, I put down 7 wholes and 4/6.  Then, I put down 3 and 5/6.  Next, I ask them how I should show subtraction.  Through turn and talk, students decide that I should use the x tool.  I cross out the 3 and 5/6 and end up with the 7 and 4/6 I started with.  Since this doesn't match up with the original estimate, I ask for help.

A student comes to the board to model starting with the whole, 7 and 4/6 and then using the x to take away the 3 wholes.  Stop the student from taking away any of the 6ths because this is a critical moment that you will want to guide the class through.

After taking away 3 wholes, I need to take away 5/6.  But there are only 4/6 here.  BUMMER!  What should I do about this one? 

I encourage students to turn and talk to come up with the solution of trading a whole for 6/6. Again, I connect this with regrouping.  I trade 1 whole out for 6/6 and also cross out the 7, write a 6 and then put 4/6 +6/6 on the board. 

Now, I can take away 5/6.  The final answer can be simplified, so we do that together as well.

Students return to their computers to solve

• 6 - 2 and 4/5

• 6 and 1/3 - 5 and 2/3

• 9 and 1/2 - 6 and 3/4

• 7 and 1/2 - 7/10

Keep in mind that it takes time for students to work with models.  Some students will only get to work on a few examples, this is ok because the point of our work is quality rather than quantity.

Ticket Out

5 minutes

The ticket out is posted 5 minutes before the students wrap up their partner work.  Students know to look for the question prior to the end of their partner work, so that they are prepared for this question.

How is subtracting with mixed numbers similar to spending money?

Students talk about this in their partner groups, then return to their seats to complete the ticket out. 

For this open-ended prompt, I have no specific expectations, I look to see what they come up with and how they justify their response.  I will share some tomorrow to launch the lesson - adding and subtracting mixed numbers (without models).