Private: Learning Math: Number and Operations
Classroom Case Studies, K-2 Part C: Problems That Illustrate Reasoning About Number and Operations (55 minutes)
As this course comes to a close and you reflect on ways to incorporate your new understanding of number and operations into your teaching, you have both a challenge and an opportunity: to enrich the mathematical conversations you have with your students around number and operations. As you are well aware, some students will readily grasp the ideas being studied, and others will struggle.
In Part C, you’ll look at several problems that are appropriate for students in grades K-2. For each problem, answer these questions:
a. What is the solution to this problem?
b. What is the number and operations content in this problem?
c. What skills do students need to work through this problem?
d. If students are having difficulty, what questions might help them work through this problem?
e. What questions might extend students’ thinking beyond this problem?
You have the following cards:
How many different ways can you put two cards in the squares so that their sum equals 6?
The rectangles are all the same size, and the pieces within each rectangle are all the same size. Which rectangle has the most shaded? Which has the least shaded? How do you know?
If you can trade 1 blue rhombus for 2 green triangles, how would you solve the problems below? Explain your reasoning.
|3.||Is 8 green triangles a fair trade for 4 blue rhombuses? How do you know?|
If you can trade 1 red trapezoid for 3 green triangles, how would you solve the problems below? Explain your reasoning.
|3.||Is 10 green triangles a fair trade for 4 red trapezoids? How do you know?|
Using counters to represent numbers, we can observe the following:
|One is an odd number because one counter has no partner (i.e., it cannot form a pair with another counter).|
|Two is an even number because both counters have a partner (i.e., they form a pair).|
|Three is an odd number because one counter still has no partner (i.e., it does not form a pair with another counter).|
Keeping this in mind, fill in the blanks below:
|1.||An even number plus an even number gives an ________ number. (How did you decide?)|
|2.||An odd number plus an odd number gives an ________ number. (How did you decide?)|
|3.||An even number plus an odd number gives an ________ number. (How did you decide?)|
Consider one of the above problems. What kind of lesson could you generate around this topic?
a. Solution: There are seven different ways:
0 + 6, 1 + 5, 2 + 4, 3 + 3, 4 + 2, 5 + 1, 6 + 0
b. The number and operations content in this problem is addition of whole numbers. There is also an exposure to equations and the symbols involved, such as the equality symbol.c.Students need to be able to recognize the numerals for the numbers 0 through 6 and to be able to add whole numbers with sums to 6. They also need to understand the equality and addition symbols. And they need to use logical reasoning and have the ability to make an organized list.d.For students who are having difficulty, give them six counters and ask them to arrange the counters in the two squares, starting with all six in the right square. Ask them to record this number fact: 0 + 6 = 6. Then ask them to move one counter to the left square and record a new number fact: 1 + 5 = 6. Have them continue until all the counters are in the left square.e.
Choosing a sum other than 6 can extend this problem. A nice extension activity would be to ask students to make a table with all the ways to get each sum from 1 to 10:
Number of Ways
Another extension would be to allow more than two addends. Ask, “How many different ways can you add counting numbers to get 6?” (Notice that 0 is not allowed in this problem.) Solution: There are 31 different ways to get 6 by adding counting numbers. (Remember that 2 + 1 + 2 + 1 is different from 1 + 2 + 2 + 1, etc., so there are six different ways to write these four numbers.)
a. Solution: Rectangle B has the most shaded because it has the greatest number of parts. As the number of parts increases, the size of the parts decreases. Rectangle B is only missing one part out of eight parts. The others have fewer parts. Rectangle D has the least shaded because it is divided into the fewest number of parts, so the missing part is greater than any of the others.
b. The number and operations content is the basic concept of fractions and the inverse relationship between the number of parts and the size of the parts.
c. Students need to be able to make sense of visual representations, extract information from a visual representation, and make observations about what they see.
d. Students who are having difficulty with this problem need to have concrete materials to represent the fractions. They need to be able to compare the size of 1/3 to the size of 1/4 and so on.
e. This problem can be extended by asking students to order all five of the rectangles from least shaded to most shaded. You could also include more fractions and not use the pictures.
|1.||You can trade 2 blue rhombuses for 2 sets of 2, or 4, green triangles.|
|2.||There are 3 sets of 2 green triangles, so you can trade for 3 blue rhombuses.|
|3.||Yes. Eight green triangles is 4 sets of 2, so you can trade for 4 blue rhombuses.|
|1.||There are 3 trapezoids, so there should be 3 sets of 3 triangles.|
|2.||There are 2 sets of 3 triangles, so there should be 2 trapezoids.|
|3.||No. There should be 3 triangles for each trapezoid. Three sets of 3 triangles is 9 triangles, and 4 sets of 3 triangles is 12 triangles. That means that 10 triangles is not a fair trade for 4 trapezoids.|
b. The number and operations content of this problem is the concept of fractions and proportional reasoning. It also involves basic exposure to equations.
c. Students must be able to add numbers to 9, count by twos and threes, and count backward by twos or threes. They need to understand the equality symbol. Students will also be asked to make observations based on visual representations and to use logical thinking.
d. If students are having trouble, give them pattern blocks to use. Have them cover each blue rhombus with 2 green triangles and each red trapezoid with 3 green triangles.
e. One extension of this problem is to use more complex relationships. For example, if 1 hexagon can be traded for 2 red trapezoids, how many green triangles can it be traded for? How many blue rhombuses?
|1.||Even. Every counter has a partner.|
|2.||Even. The two counters without partners can be partners, so every counter has a partner.|
|3.||Odd. The counter without a partner still has no partner.|
b. The number and operations content of this problem is number theory (odd and even numbers) and addition.
c. Students must understand odd and even numbers and be able to add one-digit numbers. Logical thinking and working with word problems are also present.
d. If students are having difficulty, give them counters. Ask them to choose two odd numbers and make as many partners as they can for each number. They will see that one is left over in each number. Ask them what happens when the two numbers are added. Do the same with the other problems.
e. One extension of this problem is to ask students to think about numbers that are multiples of 3. Describe these numbers as numbers you say when you count by threes. What happens when you add two numbers that are multiples of 3?
Answers will vary, but here is one example:
Many students at this level work with pattern blocks. A lesson could be structured around exploring pattern blocks and the relationships that exist among different shapes. Once students gain a familiarity with these, this knowledge can be further extended into writing up those relationships as simple equations. For example, for Problem C3, students could begin by using the triangular pattern blocks to “make” a rhombus before they work on the written equations. This way, students can use manipulatives to reason about and justify their answers. By writing these simple equations, students are not only learning about the equality symbol, but also gaining some early groundwork on fractions, multiples, and addition (for example, one half of a rhombus is a triangle, and three triangles make one trapezoid).
Session 1 What Is a Number System?
Understand the nature of the real number system, the elements and operations that make up the system, and some of the rules that govern the operations. Examine a finite number system that follows some (but not all) of the same rules, and then compare this system to the real number system. Use a number line to classify the numbers we use, and examine how the numbers and operations relate to one another.
Session 2 Number Sets, Infinity, and Zero
Continue examining the number line and the relationships among sets of numbers that make up the real number system. Explore which operations and properties hold true for each of the sets. Consider the magnitude of these infinite sets and discover that infinity comes in more than one size. Examine place value and the significance of zero in a place value system.
Session 3 Place Value
Look at place value systems based on numbers other than 10. Examine the base two numbers and learn uses for base two numbers in computers. Explore exponents and relate them to logarithms. Examine the use of scientific notation to represent numbers with very large or very small magnitude. Interpret whole numbers, common fractions, and decimals in base four.
Session 4 Meanings and Models for Operations
Examine the operations of addition, subtraction, multiplication, and division and their relationships to whole numbers. Work with area models for multiplication and division. Explore the use of two-color chips to model operations with positive and negative numbers.
Session 5 Divisibility Tests and Factors
Explore number theory topics. Analyze Alpha math problems and discuss how they help with the conceptual understanding of operations. Examine various divisibility tests to see how and why they work. Begin examining factors and multiples.
Session 6 Number Theory
Examine visual methods for finding least common multiples and greatest common factors, including Venn diagram models and area models. Explore prime numbers. Learn to locate prime numbers on a number grid and to determine whether very large numbers are prime.
Session 7 Fractions and Decimals
Extend your understanding of fractions and decimals. Examine terminating and non-terminating decimals. Explore ways to predict the number of decimal places in a terminating decimal and the period of a non-terminating decimal. Examine which fractions terminate and which repeat as decimals, and why all rational numbers must fall into one of these categories. Explore methods to convert decimals to fractions and vice versa. Use benchmarks and intuitive methods to order fractions.
Session 8 Rational Numbers and Proportional Reasoning
Begin examining rational numbers. Explore a model for computations with fractions. Analyze proportional reasoning and the difference between absolute and relative thinking. Explore ways to represent proportional relationships and the resulting operations with ratios. Examine how ratios can represent either part-part or part-whole comparisons, depending on how you define the unit, and explore how this affects their behavior in computations.
Session 9 Fractions, Percents, and Ratios
Continue exploring rational numbers, working with an area model for multiplication and division with fractions, and examining operations with decimals. Explore percents and the relationships among representations using fractions, decimals, and percents. Examine benchmarks for understanding percents, especially percents less than 10 and greater than 100. Consider ways to use an elastic model, an area model, and other models to discuss percents. Explore some ratios that occur in nature.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants) who have adapted their new knowledge to their classrooms.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants) who have adapted their new knowledge to their classrooms.