Adding, Subtracting, and Understanding Decimals

For today's Warm-Up, I will have my students complete three problems that I will use to assess the pre-requisite skills necessary to successfully master the concepts taught in this lesson.

1. 4,000 – 2,379 = (This problem is to be used to determine how well students regroup over zeros)      ANSWER: 1,621
2. 4,783.526 In what place value is the underline value? (This problem is to be used to determine how well students understand place value)    ANSWER: HUNDREDTHS
3. Joshua had a collection of 364 basketball cards he gave his friend 35 cards that were all duplicate. His uncle gave him 42 cards. How many cards does Joshua have now? (This problem is to be used to determine a student's ability to problem solve)  ANSWER: 371

In my opening exercise, my students will be using 10 by 10 grids for the purpose of practicing how to represent decimal quantities in the 10ths place and 100ths place.

In order to show that they understand the concept, the student will have to do the following:

• Write the decimal represented by the models or shade in the model so that it represents the decimal quantity.
• Determine which decimal quantity is the greater quantity of the pair.
• Tell why the quantity is greater.
• Determine how much greater the quantity is.
• Determine which decimal quantities are equivalent.

Once again, like the warm-up, this opening exercise assesses prerequisite knowledge that is necessary to master the concepts and skills surrounding adding and subtracting decimals.

Using the opening exercise, I will facilitate discovery concerning the following:

• Equivalency among decimal quantities
• Ensuring that students understand that 0.5 is equivalent to 0.50 and 0.500... Ultimately any number of zeros can be added to this decimal and it will create an equivalent decimal quantity.
• I will also make sure that they understand why these decimal quantities are equivalent by showing them what 0.5 looks like in comparison to 0.50. I will let them know that ultimately, when going from tenths to hundredths to thousandths... It is like taking a cake and cutting it into 10 equally sized pieces, then taking a cake of the same exact size and cutting it into 100 equally sized pieces and then yet another cake of the exact same size into 1000 equally sized pieces. I will ask the students, "The cake itself did not change in size but, what happened to the size of the pieces?" "If we look at the cake cut into 100 pieces, how many of those pieces will it take to equal one of the pieces of the cake cut into 10 pieces." I will continue to ask questions concerning equivalency between the three equivalent cakes cut into different amounts of pieces.
• Comparing decimal quantities
• Once I know that the students understand equivalency then I can ask them questions involving comparing decimals such as 0.5 to 0.234. Often times, students think that 0.234 would be the greater quantity because it has a greater number of digits. To do away with this misconception, I will show them visually that 0.5 is greater and refer back to what we just discussed concerning equivalency.
• Understanding why decimal quantities must be lined up before you are able to add or subtract them
• I will use the opening exercise to facilitate discussion concerning why the decimal points must be lined up before you can add or subtract decimal quantities. During this discussion, I am looking for students to make connections between size/units. Students need to recognize that tenths can only be added or subtracted with 10ths, hundredths with hundredths, so on and so forth. The same goes for whole number quantities, ones must be added and subtracted with ones, tens with tens, hundreds with hundreds, so on and so forth. Teacher should try to pull a correlation between this and the fact that fractions must have the same denominator before they can be added or subtracted. Just like you have to have the same denominator, you must line up place values so that you are lining up the same size "pieces." I will try to further develop understanding using a scenario of a fair trade. I will ask students, "Would it be fair to you if I traded you 1/100 of a dollar (which is a penny) for your 1/10 of a dollar (a dime)?" Discussion from this question can be used to facilitate discovery as to the importance of lining up decimals and place value before adding or subtracting. As a teacher I make sure to tell students to line up their place values instead of telling them to line up their decimals. This is beneficial when there is no decimal present in quantity being added or subtracted with a decimal quantity.

During the facilitation of this lesson, I will review place value as well as the difference and locations of whole number quantities versus decimal quantities. I will be sure to pose the question “Why is there no one-ths place?” The answer that I am looking for is “because once you reach one you are now in the whole number territory.” I am also looking for students to discern that each place value is either 10 times more or one-tenth of the place value next to it depending upon its orientation and assuming all place values are represented with the same digit.

Ex: 2,222,222.222222

Students should recognize the ones place as a point of reflection.

Helpful Instructional Videos (LearnZillion):

Adding Decimals Using Base Ten Blocks

Subtracting Decimals Using Base Ten Blocks

Decimals in the Real World

To show that they understand what was presented during instruction, I will have my students complete three problems then the class will go over the three problems as a transition into exploration.

     1.  4.679 + 2.34 =

This problem is to be used to check simple calculation skill.

     2.  72 – 0.479 =

This problem is to be used to check student ability to annex zeros and regroup using zeros.

     3.  Kevin needs three wooden boards to repair his porch. The lengths he needs are 4.12

          meters, 2.5 meters, and 2.25 meters. He purchases a board that is 10 meters long and

          cuts the three sections. How much of the board that Kevin purchased will be left?

This problem is to be used to check student ability to problem solve.

During this time, the students will complete the exploration individually but, later they will be given an opportunity to confer and share with a peer.

Problems for Exploration:

The students will solve problems 1 and 2 numerically and then they will explain their method in words and provide a complete answer.

1. A video game costs $39.99 and a new video game controller cost $24.95. How much change will you receive if you pay for both items with a $100?
2. The current balance in Addison’s account is $468.54. If Addison writes a check for $286.29 what would be the new balance of his account?

Students will solve problem 3 by writing a brief essay.

     3.  Jeremy found the answer to the following subtraction problem 74.6 – 9.73. His answer was 65.13 which is incorrect. Write a brief statement to Jeremy explaining his mistake and providing him with the correct answer and method to solve this problem.

At this time, the students will be provided with 7 minutes to confer with their group (my classroom is set up into groups. If you do not have your classroom set up into groups, you may have to provide transition time for them to get into groups or with partners). Before student confer, I will have already provided a written display of expectations during this time of conferring and posted these expectations where all of my students can see the expectations and understand the consequences for not meeting those expectations.

Expectations:

• Compare and debate answers
• Read and critique another student explanations to solving the problems
• Prepare a brief presentation that represents a collaborative answer of all group members

The students will then be given an opportunity to share their discoveries. Each question will be critiqued. A group will be selected to present a problem selected by the teacher. Then, the teacher will allow discussion of that problem to commence. Each question will be allotted a time limit of no more than 4 minutes for presentation and discussion.

After presentations, the students will complete the ticket out the door which should already be in their possession. The ticket out the door will be a single word problem students will answer and give to teacher on the way out. It is necessary to prepare the "tickets" prior to students entering the classroom by cutting out the "tickets" from the provided document.

TOTD: Jeff has two bills in his pocket and no change. He wants to buy a shirt for $12.98 and a pair of shoes for $54.69. What is the least amount each bill could be worth in order for Jeff to be able to purchase the items he wants?