Math Methods 350 Exam 1

Comparison Problems

In multiplicative comparison problems, there are really two different sets or groups, as there were with comparison situations for addition and subtraction

Equal-Group Problems

In multiplicative problems, one number or factor counts how many sets, groups, or parts of equal size are involved. The other factor tells the size of each set, group, or part. The third number in each of these two structures is the whole or product and is the total of all of the parts.

What are the two types of Division Structures?

1.Partition/Fair Sharing 2.Measurement/Repeated Subtraction/Grouping

The Associative Property for Addition

(a+b)+c=a+(b+c) The associative property for addition states that when adding three or more numbers, it does not matter whether the first pair is added first or if you start with any other pair of addends. There is much flexibility in addition, and students can change the order in which they group numbers to work with combinations they know.

Distributive Property

- Models - Connected to Additive Strutures

Area/Row by Column Multi. Structures

-Can be modeled using arrays -Direct correlation between area and product (with the exception of the associated unit) -Model takes the shape of a rectangle

Measurement/Repeated Subtraction/Grouping Division Structure

-Problem situations in which the number of sets is unknown. -The whole is measured off in sets of a given size.

Partition/Fair Sharing Division Structure

-Problems in which the size of the set is unknown. -The whole is distributed among a known number of sets.

Additive Multiplication Structure

-Repeated Addition -Equal sets and number of sets are essential elements. -Multiplicative comparison (can be modeled using Singapore model diagram)

Commutative Property

-The any order property - Representation

Develop the Written Record...There are essentially four steps:

1. Share and record the number of pieces put in each group. 2. Record the number of pieces shared in all. Multiply to find this number. 3. Record the number of pieces remaining. Subtract to find this number. 4. Trade (if necessary) for smaller pieces, and combine with any of the same-sized pieces that are there already. Record the new total number in the next column.

Nineteen addition facts have zero as one of the addends. Though adding 0 is generally easy, some students overgeneralize the idea that answers to addition problems are bigger than the addends. They also may have a harder time when the 0 comes first (e.g., 0 + 8). Use story problems involving zero and use drawings that show two parts with one part empty.

Adding Zero

What are the three structures for multiplication?

Additive Area/Row by Column Combination

Start with a context and give students a drawing of a rectangular garden 47 cm by 6 cm. What is the area of the garden? Let students solve the problem in groups using base-ten materials before discussing it as a class.

Area Model

What distinguishes area (also known as product-of-measures problems) from the others is that the product is literally a different type of unit from the two factors. In a rectangular shape, the product of two lengths (length * width) is an area, usually measured in square units

Area and Array Problems

4X3X2 -> Order doesn't matter

Associative Property

Teachers move from having students draw large rectangles and arrange base‐ten pieces using the ___________________. On the grid paper, students can draw accurate rectangles showing all of the pieces. Do not impose any recording technique on students until they understand how to use the two dimensions of a rectangle to get a product.

Base-Ten Grid Paper

When selecting numbers for multiplicative story problems or activities, there is a tendency to think that large numbers pose a burden to students or that 3 X 4 is somehow easier to understand than 4 X17. An understanding of products or quotients is not affected by the size of numbers as long as the numbers are within your students' grasp.

Choosing Numbers for Problems

Many students prefer to go to a fact that is "close" and then add one more set to this known fact, as shown in Figure 10.14. For example, think of 6 * 7 as 6 sevens. Five sevens is close: That's 35. Add one more seven to get 42. Using set language "5 sevens" is helpful in remembering that one more 7 is needed (not one more 5).

Close Facts

One approach to multi-digit multiplication is called _______________.This approach encourages students to use facts and combinations they already know in order.

Cluster Problems

_____________________provides students with a sense that problems can be solved in different ways and with different starting points. They use facts they already know to slove a problem.

Cluster problems

_____________________(also called Cartesian products) involve counting the number of possible pairings that can be made between two or more sets (things or events). Students often start by using the model shown in where one set is the row (pants) and the other the column (jackets). This visual links to the array and the area representations. It is possible— rarely—to have related division problems for the combinations concept

Combination Problems

Perhaps the most important strategy for students to know is __________________.

Combinations of 10

Order does not matter (product) 5X2=2X5-> Same product different representation

Communitive Property

What are the two Connections to Mathematical Properties?

Commutative Property - The any order property - Repersentation Distributive Property - MODELS - Connected to additive structures

It is not obvious that 3 X 8 is the same as 8 X 3 or (8X3)X2 is the same as (2X8)X3 is the same or that, in general, the order of the numbers makes no difference

Commutative and Associative Properties of Multiplication

There are ten doubles facts from 0 + 0 to 9 + 9, as shown here. Students often know doubles, perhaps because of their rhythmic nature.

Doubles

____________ is a very effective reasoning strategy in helping students learn the difficult facts.

Doubling

The first is to think about it as a "separate" situation. For 14 - 8, the thinking is to first take away 4 to get to 10, then take away 4 more to get the answer of 6. Another way to think about this is as a "comparison," finding the difference" or distance between the two numbers.

Down Under 10

_________ are the absence of reasoning has repeatedly been demonstrated as ineffective. ________ is only appropriate after students know strategies and have moved from phase 2 to phase 3.

Drill

Comparison Models

Comparison situations involve two distinct sets or quantities and the difference between them. The same model can be used whether the difference or one of the two quantities is unknown.

Comparison situations.

Comparison situations involve two distinct sets or quantities and the difference between them. The same model can be used whether the difference or one of the two quantities is unknown.

Students and adults look for ways to manipulate numbers so that the calculations are easy.

Compensation Strategies

When teaching multiplication and division, it is essential to use interesting _______________ instead of more sterile story problems (or "naked numbers"). However, as with addition and subtraction, there is more to think about than simply giving students word problems to solve. Consider the following problem.

Contextual Problems

Contextual Problems

Contextual problems might derive from recent experiences in the classroom; a field trip; a discussion in art, science, or social studies; or from children's literature. Because contextual problems connect to life experiences, they are important for ELLs, too, even though it may seem that the language presents a challenge to them.

Combination Multi. Structures

Counting the number of possible pairings that can be made between two or more sets.

Notice if students recognize the commutative property, there are only 15 facts to learn. These remaining facts can be learned by using foundational facts.

Derived Multiplication Fact Strategies

The reminder is simply disposed. The question relates to the whole number of full packages that could be created.

Dispose

7X6=(5X6) + (2X6)

Distributive Property

________________________ of multiplication refers to the idea that you can split (decompose) either of the two factors in a multiplication problem into two or more parts and then multiply each of the parts by the other factor and add the results.

Distributive Property

Mastery of multiplication facts and connections between multiplication and division are key elements of ______________ mastery. For example, to solve 36, 4, we tend to think, "Four times what is thirty-six?" In fact, because of this, the reasoning facts for division are to (1) think multiplication and then (2) apply a multiplication reasoning fact, as needed. Missing factor stories can assist in making this connection.

Division Facts

Mastery of multiplication facts and connections between multiplication and division are key elements of division fact mastery. For example, to solve 36, 4, we tend to think, "Four times what is thirty-six?" In fact, because of this, the reasoning facts for division are to (1) think multiplication and then (2) apply a multiplication reasoning fact, as needed. Missing factor stories can assist in making this connection.

Division Facts

problems in which the exact quotient needs to be calculated. Typically this is expressed in a decimal or fraction form.

Exact

_______________________ instruction is intended to support student thinking rather than give the students something new to remember.

Explicit Strategy Instruction

the problems requires us to have an extra package.

Extra

Students who have not mastered their addition facts by third grade or their multiplication facts by fourth grade (or beyond) are in need of __________________ that will help them master the facts. More drill is not an intervention!

Fact Remediation

________________ is the problem of dividing a set of resources among several people who have an entitlement to them, such that each person receives his/her due share.

Fair Sharing (Division)

Multiply any # by 1, the answer will be that number... Ex: 1 X 5=5-> One group of 5

Identity Property

A memorization approach does not help students develop strategies that could help them master their facts. Barony (2006) points out these three limitations...

Inefficiency. There are too many facts to memorize. Inappropriate applications. Students misapply the facts and don't check their work. Inflexibility. Students don't learn flexible strategies for finding the sums (or products) and therefore continue to count by ones.

the remainder itself is the appropriate response.

Leftover

All of the basic facts with sums greater between 11 and 20 can be solved by using the ________ strategy. Students use their known facts that equal 10 and then add the rest of the number onto 10

Making 10

Model-based Problems

Many students will use counters, bar diagrams, or number lines (models) to solve story problems. These models are thinking tools that help them understand what is happening in the problem and keep track of the numbers and steps in solving the problem.

This approach moves from presenting concepts of addition and multiplication straight to _________________ of facts, not devoting time to developing strategies

Memorization

As you move students from single‐digit to two‐digit factors, there is a value in exposing them to products involving multiples of 10 and 100. This supports the importance of place value and an emphasis on the number rather than the separate digits.

Multiplication of Multi-digit Numbers

_______________ are also called "doubles-plus-one" or "doubles minus-one" facts and include all combinations where one addend is one more or less than the other.

Near-Doubles

You can look at division with an eye to partial quotients by using a version of a bar diagram model blended with the repeated subtraction approach. Take a look at the problem 1506, 3.

Partial Quotients Using a Visual Mode

Traditionally, if the problem 4)583 was posed, we might hear someone say, "4 goes into 5 one time." Initially, this is quite mysterious to students. How can you just ignore the "83" and keep changing the problem? Preferably, you want students to think of the 583 as 5 hundreds, 8 tens, and 3 ones, not as the independent digits 5, 8, and 3.

Partition or Fair-Share Model

Students break numbers up in a variety of ways that reflect an understanding of place value.

Partitioning Strategies

Arthur Baroody, a mathematics educator who does research on basic facts, describes three phases of learning facts...

Phase 1: Counting strategies: Using object counting (e.g., blocks or fingers) or verbal counting to determine the answer. (Example: 4 + 7 = ___. Student starts with 7 and counts on verbally 8, 9, 10, 11.) Phase 2: Reasoning strategies: Using known information to logically determine an unknown combination. (Example: 4 + 7. Student knows that 3 + 7 is 10, so 4 + 7 is one more, 11.) Phase 3: Mastery: Producing answers efficiently (quickly and accurately). (Example: 4 + 7. Student quickly responds, "It's 11; I just know it.")

How would you use REASONING STRATEGIES to promote fact fluency?

Skip counting and doubles provide foundations for strategies. Strategies can be modeled based on additive multiplication structure.

More often than not in real-world situations, division does not result in a simple whole number. For example, problems with 6 as a divisor will result in a whole number only one time out of six.

Remainders

Another well‐known algorithm is based on repeated subtraction and the measurement model of division.

Repeated Subtraction

Repeated Subtraction (grouping)

Repeated Subtraction (grouping)

The Zero Property

Story problems involving zero and using zeros in the three-addend sums are also good opportunities to help children understand zero as an identity element in addition or subtraction

When students solve simple multiplication story problems before learning about multiplication symbolism, they will most likely begin by writing addition equations to represent what they did. This is your opportunity to introduce the multiplication sign and explain what the two factors mean.

Symbolism for Multiplication and Division

This excellent strategy is not as well known or commonly used in the United States but is consistently used in high-performing countries. It takes advantage of students' knowledge of the combinations that make 10, taking the initial value apart into 10 and ones. This is how it works for 15 - 8

Take from 10

The Commutative Property for Addition

The commutative property (sometimes known as the order property) for addition means you can change the order of the addends and it does not change the answer. Although the commutative property may seem obvious to us (simply reverse the two piles of counters on the part-part-whole mat), it may not be as obvious to children.

Addition and Subtraction problems are separated into what 3 categories?

change problems (join and separate) part-part-whole problems compare problems.

In real contexts, remainders sometimes have what three additional effects on answers?

The remainder is discarded, leaving a smaller whole number answer. The remainder can "force" the answer to the next highest whole number. The answer is rounded to the nearest whole number for an approximate result.

Choosing Numbers for Problems

The structure of the problem will change the difficulty of the task, but you can also vary the difficulty of the problem by the numbers you choose to use. If a student struggles with a problem, use smaller numbers to see if it is the size of the numbers causing the obstacle.

Part-Part-Whole Problems

These are different from change problems in that there is no action of physically joining or separating the two quantities. In these situations, either the missing whole (total unknown), one of the missing parts (one addend unknown), or both parts (two addends unknown) must be found.

As the label implies, in this strategy students use known addition facts to produce the unknown quantity or part of the subtraction (see Figure 10.7). If this important relationship between parts and the whole—between addition and subtraction—can be made, subtraction facts will be much easier for students to learn.

Think-​Addition

Subtraction as Think-Addition

Thinking about subtraction as "think-addition" rather than "take-away" is significant for mastering subtraction facts. Because the tiles for the remaining part or unknown addend are left hidden under the cover, when children do these activities they are encouraged to think: "What goes with the part I see to make the whole?"

The use of an anchor (5 or 10) is a reasoning strategy that builds on students' knowledge of number relationships and is therefore a great way to both reinforce number sense and learn the basic facts.

Using 5 as an Anchor

Multiply any number by zero, the answer will be zero Ex: No groups of 5=nothing

Zero Property

What are the properties of Multiplication?

Zero Property Idenity Property Communitive Property Distributive Property Associative Property

Factors of 0 and, to a lesser extent, 1 often cause conceptual challenges for students. In textbooks, you may find that a lesson on factors of 0 and 1 has students use a calculator to examine a wide range of products involving 0 or 1 (423 * 0, 0 * 28, 1536 * 1, etc.) and look for patterns. The pattern suggests rules for factors of 0 and 1 but not a reason.

Zero and Identity Properties

Subtraction facts prove to be more difficult than ___________.

addition

In ___________ situations, the comparison is an amount or quantity difference between the two groups.

additive

What are the three parts of joining in Change Problems (Join and Separate)?

an initial or start amount a change amount (the part being added or joined) the resulting amount (the total amount after the change takes place).

For the action of joining, there are three quantities involved...

an initial or start amount a change amount (the part being added or joined) and the resulting amount (the total amount after the change takes place).

The ________________ is a model for an equal-group situation. It is shown as a rectangular grouping, with one factor representing the number of rows and the other representing the equal number found in each column

array

As in addition, there is an ________________________ that is fundamental in flexibly solving problems. This property allows that when you multiply three numbers in an expression, you can multiply either the first pair of numbers or the last pair and the product remains the same.

associative property of multiplication

Addition and Subtraction problems are separated into categories...

change problems (join and separate) part-part-whole problems and compare problems.

When either the group size is unknown (How many in each group?) or the number of groups is unknown (How many groups?), then the problem is a ...

division situation

What are the four different classes of multiplicative structures?

equal groups (equal groups and multiplicative comparison, are by far the most prevalent in elementary school) comparison area (CCSS includes arrays with area) combinations

Problems in which the size of the group is unknown are called ...

fair-sharing or partition division problems

When children are exposed to new problems, the ________________________ in generalizing from similar problems on which they have practiced.

familiar characteristics will assist them

Basic facts for addition and multiplication are the number combinations where both addends or both factors are less than 10. Basic facts for subtraction and division are the corresponding combinations. Thus, 15 - 8 = 7 is a subtraction fact because the corresponding addition parts are less than 10. The goal with basic facts is to develop _____________.

fluency

(1s, 2s, 3s,... up through 9s) The basic facts are limited to single-digit factors.

foundational facts

In _____________________ the teacher may not explain a strategy, but carefully set up tasks where students notice number relationships.

guided invention

To help children focus on the commutative property, pair problems that...

have the same addends but in different orders. Using different contexts helps children focus on the significant similarities in the problems. Ex: 2+3 and 3+2

Story problems that promote think-addition are those that sound like addition but have a missing addend which are composed of...

join initial part unknown part unknown

Another approach for recording multi-digit multiplication is known as ________________. Historically this method has been used in a variety of cultures. Here students use a grid with squares split by diagonal lines to organize their thinking along diagonally organized place-value columns.

lattice multiplication

If the number of groups is unknown but the size of the equal group is known, the problems are called...

measurement division or sometimes repeated-subtraction problems

As with addition and subtraction, it is helpful to place multiplication tasks in context. Let students ___________________ in ways that make sense to them.

model the problems

The parts and wholes terminology is useful in making the connection to addition. When the number and size of groups are known, the problem is a ...

multiplication situation.

In _______________, the comparison is based on one group being a particular multiple of the other (multiple copies). With multiplication comparison, there are three possibilities for the unknown: the product, the group size, and the number of groups.

multiplicative situations

The _____________ is a semi-concrete representation of the area model and can be successfully used after students actually experience several constructions of area models with base-ten materials. Starting with a blank rectangle, students can mark off areas (the number of subdivisions depends on the digits in the factors) that

open array

Using a _________________ and focusing on reasoning strategies are just as important, if not more so, for developing mastery of the multiplication and related division facts. As with addition and subtraction facts, start with story problems as you develop reasoning strategies.

problem-based approach

There is also a subtle difference between equal group problems and those that might be termed _____________ ("If there are four apples per child, how many apples would three children have?"). In a __________, students are working with a composed unit (in this case, apples per child)

rate problems

Sometimes equal-group problems have been called repeated-addition problems, as the equal group is being added over and over. And in fact, multiplication is an efficient way to carry out a...

repeated-addition situation

Students who are not yet comfortable decomposing numbers into parts will approach the numbers in the sets as _______ .

single groups

The __________________________ is probably the most challenging of the four algorithms when students have not had plenty of opportunities to explore their own strategies first. It can be meaningfully developed using either a repeated-addition model or an area model.

standard multiplication algorithm

Compare Problems involve...

the comparison of two quantities.

Compare problems involve...

the comparison of two quantities.

Contextual problems

the primary teaching tool that you can use to help children construct a rich understanding of the operations.

With the emphasis on children explaining their ideas and reasoning, lessons should focus on...

two or three problems and the related discussions.

You will need to present a _______________________ as well as recognize which structures produce the greatest challenge for students

variety of problem types (within each structure)