Problem Solving and the Common Core

Activities

Lesson 1 – Concept of a Unit

Objective: Students will learn the importance of a unit and how they need to use that information to help them understand and solve a word problem.

For this lesson, there is an assumption that my students have already practiced with word problems with whole numbers and have developed a beginning understanding of the different problem types. As research suggests, students need to understand the problem before they can solve it. Throughout this activity the student will have to continue to reference to the unit being used in the problem.

Students in this activity will be presented with various scenarios. I want them to use visuals to help them understand and then solve the problems. The activity will start with basic questions that ask students to manipulate between the different units being discussed. For example:

The discussion students have about these problems help in their understanding of a unit. At the end of this lesson I want students to be able to work with partners and answer questions that involve problem solving. The problems I would then present would still be referencing the same scenario.

During this part of the activity I will differentiate my instruction based off the needs of my students. This next scenario can be used for students to work collaboratively or the teacher can pull a small group of students. Throughout these discussions students need to be identifying the unit and how it dictates how to draw the picture. This understanding will help them as they start to learn about the unit fraction in the next lesson.

Lesson 2 – Unit Fraction – Fraction Strips

Objective: Students will develop an understanding of what a fraction represents.

Materials: Pre-cut 15-by-2-inch strips of construction paper (at least 6 different colors), large gallon bags to hold the strips, and scissors

Students will first need to create their own set of fraction strips that they will use and refer back to throughout the unit. Each strip of paper will represent a different denominator. For each strip I will have the students cut apart the wholes into the specified fractional part. This allows them to manipulate the pieces and look at each piece as a unit fraction of the whole. For example students will create the following fraction strips:

Once the fraction strips have been created then I want the students to start manipulating these different amounts. To begin the lesson I would have my students get with a partner. This allows them to both construct a different fraction with the same denominator and then they can combine their pieces to represent amounts greater than one whole. For example, I might ask them to represent two fractions with the same denominators such as 2/6 and 5/6. Once they have been able to represent the amounts correctly I want to show them how they can construct number sentences to illustrate those amounts. In their math journals they can write the following number sentences: 1/6 + 1/6 = 2/6 and 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 5/6. Then I can ask the students to think about what would happen if you put those two fractions together? This question will have students realize that fractions can be larger than one. With their fraction strips the students will be able to first show me the amount using their manipulatives and then we will connect it back to a number sentence. In their journals I want to see the following equation: 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 7/6 or I can also have students write 2/6 + 5/6 = 7/6. At this time I will introduce the idea of an improper fraction and how fractions can have numerators greater than the denominator, since we can take any number of copies of a unit fraction. From this number sentence I can ask, “What is another way to write this number sentence?” At this time I would like to have students discuss how multiplication by a whole number can represent this same amount. For example, 7 × 1/6 = 7/6. I will give the students several different examples that will require them to model a proper fraction and then an improper fraction. With each problem I will have the students write number sentences to represents the fractional amounts they are representing.

Lesson 3 – Unit Fraction – Pattern Blocks

Objective: Students will investigate the relationships between blocks when different blocks are designated a whole.

Materials Needed: Pattern Blocks, 1 tub containing only hexagons, trapezoids, blue rhombi, and triangles.

It will helpful to first review the names of the pattern blocks and if necessary create a chart that can be used as a reference. Once my students become familiar with the names of the shapes I will identify the hexagon as a whole. Using their pattern block have student determine how many different ways they can cover the hexagon, using only one type of block. They should discover there are three ways, using two trapezoids, three blue rhombi, or six triangles.

At this time discuss what a unit fraction represents. In this case my students will discover that the blue rhombus represents one-third, the trapezoid represents one half, and the triangles represent one-sixth. The denominator of the unit fraction is determined by the number of copies needed to make the whole. At this time I will point out, or have the students observe, that larger denominators means smaller unit fractions. Once this has been established it is important to pose the question, “What would happen if I changed the whole”? I have created some questions that they can work on with a partner to help with this understanding. For example:

Then my students can work together as they determine the values of the various pattern blocks. To organize their thinking I would create a graphic organizer like the one below.

As the whole changes there need to be a discussion on how that affects the unit fraction. For example, when the rhombus becomes a whole they will find that the triangle, which was one sixth of the hexagon, is only one-half of the rhombus. To aid in their understanding here are some questions I would consider:

• How does the size of the whole affect the fractional value of the pattern blocks?
• What does the unit fraction tell you about the whole?
• What do you notice about the size of the pieces in the whole?
• What does the numerator represent?
• How can you describe the denominator in a fraction?

Lesson 4 – Representing Fractions using Linear and Area Models

Objective: Students will be able to represent fractions using area models and number lines.

Materials: chart size grid paper laminated, dry erase marker, plain white paper

Starting with the area model I want the students to be able to represent various types of fractions. I would refer back to the fraction strips in the previous lesson since they resemble an area model. This is a model that they can recreate on their own as they continue their study of fractions. The first set of fractions will consist of all general fractions less than or equal to one whole. Students can represent the following fractions: 2/3, 3/5, 4/6, 6/8, and 4/10. I would then continue with students modeling improper fractions using the same types of models. Through this discussion, I hope students will come to the understanding that they will need to draw more than one whole for these fractions. Students can then represent the following fractions: 5/3, 6/5, 10/6, 12/8, and 15/10.

Now students need to understand that a number line is just another way to represent fractions. Using the grid paper they will draw number lines to represent various types of fractions. Time needs to be spent on how you partition a number line so there are equal spaces between each fraction amount. I would pick a certain number of squares and make that the whole. For example, I would have my students draw a number line that measures 30 squares in length. Then with the number they will be able to represent the following denominators: 2, 3, 4, 5, 6, 8, and 10. Introducing the denominators in a specific sequence will show the relationships between the fractions. Here is the sequence in which to introduce the denominators:

After the introduction of these denominators I will have them show improper fractions on the number line. At this time students should be practicing the linear model and representing fractions greater than one whole.

Lesson 5 – Problem Solving with Fractions

Objective: Students will learn how to identify the types of word problems based off the taxonomy of addition and subtraction problems.

As stated earlier in the unit, I will have my students focus their attention on one type of problem. When I first introduce the change increase and decrease problems I will use whole numbers. This helps my students because they already have a background with whole numbers so they will be better able to focus on the content of the problems. Therefore, after sufficient time is spent dealing with whole numbers the teacher can substitute in fractions.

When introducing the problems we will first start with proper fractions and then move to improper fractions and mixed numbers. For each problem my students will draw a model and write a number sentence to explain their thinking. The problems can be used for partner practice or independent. In my appendix I have included word problems that can be used for this lesson. I will pose the questions in triples, with any of the three variables unknown, and I will expect my students to be able to do the same thing.

Students will then start to write their own word problems. They will need to continue to use visual model and equations to prove their thinking. My students will get a chance to share their problem with their classmates who will then in return identify the type of problem. All problems created by students can be collected and inserted into a class word problem book.