The problems that my students will complete for today's Warm Up are as follows:
Justin buys 12 pencils for $0.56 each. He pays with a $10 bill. How much change will he receive? (ANSWER: $3.28) In this problem, I am looking for 3 things in total. The first thing is, I am looking for my students to show me that they can solve multi-step word problems. I am also looking to make sure that when they subtract $6.72 from 10, that they do exactly that. Often times students will see something like 10 - 6.72 and because they think that 6.72 is the greater number simply because it has more digits, they will switch the problem and have 6.72 as the minuend and 10 as the subtrahend. In this case, they do not line up the place values and therefore, they end up subtracting $0.10 from $6.72 rather than subtracting $6.72 from $10. The last thing that I am looking for my students to do is show that they are capable of regrouping over zeros when subtracting.
A bee hummingbird has a mass of 1.8 grams. How many grams are 6 hummingbirds and a 4-gram nest? (ANSWER: 14.8 grams) Once again, I am looking to see if my students are able to solve a multi-step word problem.
A mile is equal to approximately 1.609 kilometers. How many kilometers is 2.5 miles? (ANSWER: 4.0225 kilometers) In this problem, I am looking to see that my students are able to multiply multi-digit decimal quantities. I am also looking to see that they are able to place the decimal in its proper place when arriving at a solution.
First, I will open up the floor for discussion. The discussion prompts will be:
What is division? What does it mean to divide?
Next, I will give all my students 12 cubes. Then, I will ask the students to divide up the cubes. After that, I will then spend about a minute asking selected students how they decided to divide the cubes. Once I receive a few responses, I will then ask, is there a difference between the commands “divide” and “divide evenly.” The students should respond “yes.” Once receiving this response, I will ask the following questions:
What is the difference? When answering this questions, I will make sure that eventually, my students understand that it is possible to divide unevenly. To demonstrate this fact, I will present a scenario to my students. I will ask them, "How many of you have a younger brother or sister?" I will then ask them how many of you have divided up candy between you and your younger sibling?" Then, I will ask, "How many of you cheated your little brother or sister out of their fair share?" I will tell them, you divided the candy, but you DID NOT divide evenly.
In math when we use the term divide, do we mean divide or divide evenly?
What keywords tell us that we need to divide? (Make a List)
This discussion is to illustrate to students what it means to divide in a mathematical sense. When they divide up the 12 cubes, they should be able to see that they are dividing the cubes in to even groups.
After the discussion of the concept of division, I will segue into instruction by presenting a (deliberately simple) multiplication problem:
3 x 4=12
Of course my students know that 3x4=12 and they understand that if you have 3 groups of 4 items then you have 12 items. I will ask, "Did anyone divide their cubes in this manner?"
Again, I expect a lot of yes's. I'll follow up by asking, "What connections can we make between this multiplication problem and the idea of "divide evenly"? I am looking for students to express the following ideas in their own words:
Division is the inverse of multiplication
If 3 groups of 4 items is equal to 12 items then, 12 items divided into 3 groups will give us 4 items in each group.
I will then transition into multi-digit division by first presenting the examples indicated below.
351 divided by 9
878 divided by 31
6,493 divided by 74
1,460 divided by 365
54,912 divided by 44
With the concept of "divide evenly" on everyone's mind, we will use these examples to apply and to move towards mastery of the standard algorithm for division. In order to begin with a concrete representation, I often use Division modeling strategies to build a deeper conceptual understanding of this algorithm.
At this point, I will have my students solve the following problems as guided practice.
The first 4 problems are simple division problems however, when it comes to the last problem, I want my students to mark keywords and important values by highlighting or underlining. After doing that, then the students may solve the problem.
For independent practice, I will have my students solve the problems presented in the worksheet attached to this section. The students will have to mark the keywords and or phrases that tell them that they need to divide. The students will also have to mark important values before solving each problem.
Selected students will be chosen to present one of the six problem that they completed for independent practice. They will present their solution under the document camera so that the manner in which they solved the problem will be visible to the entire class. I will ask questions concerning how they solve the problem and what they marked as keywords.
You bought a new package of 10 hot wheel cars for $9.97. How much did you pay per car?
This problem will be presented on the digital projector. I will give each student an index card upon which they will solve this problem.