Learning Math: Number and Operations
Classroom Case Studies, 3-5 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 3-5. 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?
John, Sarah, and Mary Beth each threw three beanbags at this target. All the beanbags landed somewhere on the target. Each person’s score was the sum of the numbers in the rings where that person’s three beanbags landed.
1. What is the greatest possible score?
2. What is the lowest possible score?
3. How many different possible scores are there?
4. John and Mary Beth got the same score but didn’t hit the same rings. What was their score?
|1.||Which is greater, 5/6 or 5/8? How do you know?|
|2.||Which is greater, 2/3 or 3/4? How do you know?|
|3.||Which is greater, 5/8 or 2/3? How do you know?|
|4.||Put these fractions in order from smallest to largest, and explain your reasoning:
1/2, 5/6, 2/5, 5/8, 2/3, 9/10, 5/3, 3/4
1. What is the smallest number of blocks you can remove so that there are 3 red blocks for every 4 green blocks left in the box? Which blocks did you remove? How did you figure it out?
2. What is the smallest number of blocks you can remove so that there are 3 red blocks for every 2 green blocks left in the box? Which blocks did you remove? How did you figure it out?
One, 2, and 3 are consecutive integers. If you add 1 + 2 + 3, you get 6, which is 2 • 3.
Two, 3, and 4 are consecutive integers. If you add 2 + 3 + 4, you get 9, which is 3 • 3.
1. What is the sum of 3 + 4 + 5? Does it follow the pattern?
2. What is the sum of 7 + 8 + 9? Does it follow the pattern?
3. What can you tell about the sum of any three consecutive integers?
|1.||The greatest possible score is 8 + 8 + 8, or 24.|
|2.||The lowest possible score is 5 + 5 + 5, or 15.|
|3.||There are 10 different score combinations possible, but since a score of 21 can be gotten two different ways, there are only nine different scores possible:
8 + 8 + 8 = 24
|4.||John and Mary Beth both got 21, because that’s the only score that can be gotten two different ways.|
b. The number and operations content of this problem involves the ability to use the commutative and associative laws to find a sum with three or more addends. Students will also develop flexibility with numbers.
c. Students need to use logical thinking and to be able to recognize that the order in which they add three whole numbers doesn’t affect the sum. They also need to be able to compare sums of numbers in an organized way.
d. If students are struggling, ask them to start by writing out the sums. It is important to have an organized system — for example, add all the sums that start with the 8s (first with three 8s, then two 8s, then one 8) and then repeat for the 7s and 5s.
e. To get students to think beyond this problem, you can turn it around. Make up several scores that players said they got, and have students decide if these scores are possible.
|1.||Five-sixths is greater. You know this because five parts out of six is more than five parts out of eight — the pieces are larger.|
|2.||Three-fourths is greater. Two pies are each missing one piece. The pie with four pieces has less missing, so it is bigger.|
|3.||Two-thirds is greater. If you change the 2/3 to 6/9, then you have two pies each missing three pieces. The pie with nine pieces has less missing, so it is bigger.|
|4.||The order is 2/5, 1/2, 5/8, 2/3, 3/4, 5/6, 9/10, and 5/3. People will explain their reasoning in different ways, but here’s one example:
b. The number and operations content of this problem involves understanding the concept of fractions (the roles of the numerator and denominator; understanding that the larger the denominator is, the smaller the piece is; etc.), how to compare and order fractions, equivalent fractions, and benchmarks.
c. Students will probably want to convert to decimals or get common denominators for all the fractions. The procedures discussed above will help students learn to reason about fractions so that they can avoid tedious computation. Students will also need to use logical thinking.
d. If students are struggling, you can ask the following questions to try to elicit the following answers:
• Why not just get a common denominator for all the fractions and then order them? (Because this is much more time-consuming and does not help to give us an intuitive feeling about the magnitude of fractions.)
• Why not convert all the fractions to decimals and then order the decimals? (Because this is much more time-consuming and loses the fractions — and thus our understanding of the fractions — entirely.)
Essentially, students need to think about the benchmarks. They should be able to look at a fraction and know if that fraction is between 0 and 1/2, between 1/2 and 1, or greater than 1, and which of those benchmarks it’s closer to. In addition, if the denominators are the same, they should know to compare numerators, and vice versa.
e. There are several ways to help students think beyond this problem. The simplest is to change the fractions. You could also increase the number of fractions, use larger numbers, or include some decimals in the list. You could also have students put the numbers on a number line.
|1.||The smallest number of blocks you could remove is 4 — 3 red blocks and 1 green block. To have 3 red blocks for every 4 green blocks, you must have a multiple of 3 + 4, or 7 (e.g., 7, 14, 21) blocks remaining in the box. This could be 3 red and 4 green, 6 red and 8 green, 9 red and 12 green, and so forth. However, since there are only nine of each color in the box, the most you could have is 6 red and 8 green blocks.|
|2.||The smallest number of blocks you could remove is 3 (green blocks). To have 3 red blocks for every 2 green blocks, you must have a multiple of 3 + 2, or 5 (e.g., 5, 10, 15) blocks remaining in the box. This could be 3 red and 2 green, 6 red and 4 green, 9 red and 6 green, and so forth. However, since there are only nine of each color in the box, the most you could have is 9 red and 6 green blocks.|
b. The number and operations content of this problem is ratio and proportional reasoning, patterns, and developing mathematical language (such as understanding what “for every” means).
c. Students need to use logical thinking. They also need to understand the inverse relationship between the number of parts and the size of the parts. Students need to be able to find equivalent fractions and compare fractions. They need to understand that when they’re doing part-part relationships, they must identify the whole. For example, in the block problem, to have 3 red for every 4 green blocks, they need to be aware that the total will be 7 or a multiple of 7.
d. If students are struggling, have them use actual blocks. They can make two piles and then notice the relationships that are evident.
e. To help students think beyond the problem, ask them to solve a more complicated version of it, using nine red blocks, nine blue blocks, and nine green blocks. Tell them that there must be 3 red blocks for every 7 non-red blocks and 3 blue blocks for every 7 non-blue blocks. (There are two possible answers: 6 red, 6 blue, and 8 green; and 3 red, 3 blue, and 4 green.)
|1.||The sum of 3 + 4 + 5 = 12. Yes, because 12 = 4 • 3.|
|2.||The sum of 7 + 8 + 9 = 24. Yes, because 24 = 8 • 3.|
|3.||The sum of any three consecutive integers is three times the middle number.|
b. The number and operations content of this problem is number theory, addition of whole numbers, fluent computation, the idea of average, and patterns.
c. Students need to be able to use logical thinking. They need to be able to divide or multiply by 3, add three numbers, and understand what is meant by “three consecutive integers.”
d. If students are struggling, encourage them to make an organized list in which they list the options one by one. They can also use manipulatives, such as Unifix cubes, to help make the patterns more evident. Once they have the numbers represented by stacks, they can take the top cube off the tallest stack and put it on the smallest to see that they are all the same height.
e. To help students think beyond this problem, ask them to think about five consecutive numbers. (In this case, the sum will be five times the middle number.)
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.