MTE280

Through group problem solving and interactive activities, this course aims to better future educator's ability to break down and teach elementary math concepts in a way where their students understand the actual math processes that are happening as they do them. When doing computations, understanding what is actually happening with the numbers is key to longterm success in mathematics.

By definition, a numeration system is a collection of symbols that represent numbers in a systematic way. - The system that we use today is the Hindu-Arabic system, which consists of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, otherwise known as a base-10 system. - Systems like this are called place-value systems. In these systems, the value of a digit depends on its position in the numeral. For example 43 represents 4 tens and 3 units.

Base-10 blocks provide an excellent way to visually represent a number. Understanding place-values is not a concept that is going to come automatically to students just getting started with mathematics, so utilizing base-10 blocks is very important when it comes to developing math sense.

Base-10 is not the only place-value system out there. Base-5 is one example. Where base-10 contains the digits 0 through 9 and regroups in powers of 10, base-5 contains the digits 0, 1, 2, 3, and 4, and regroups in powers of 5.- There are also place systems with more digits than base-10. Base-12, otherwise known as the duodecimal system, contains twelve digits. Base-12 uses the digits 0 through 9, and then uses T and E to symbolize what we would think of as 11 and 12.

Getting a handle on counting and doing computations in other bases can be confusing. Utilizing base blocks in other bases can be very useful when learning a new base.

The terms carrying and borrowing are commonly used when describing the process of regrouping ten 1s into a ten, ten 10s into a hundred, etc. These problem with these terms is that they don't help students understand the actual math process that is happening when they, for example, "carry the one". Rather, the terms trading and regrouping should be used, as they more accurately describe what the students are doing with the numbers.

The set model and the number line are two very useful visuals for introductory addition.Set ModelThe set model places two groups of objects next to one another in a way that allows the student to count them up and combine the two. For example, 3 + 4 would look like:@ @ @  +  @ @ @ @   Counting up all the symbols gives you 7.Number LineThe number line can be used to visually demonstrate every operation in math, not just addition. It is useful for helping students develop their number sense, and seeing what is happening when you add two numbers.

Commutative Property3 + 4 + 1 = 1 + 3 + 4Associative Property(3 + 1) + 4 = (3 + 4) + 1Closure (a whole number plus another whole number will equal a unique whole number):3 + 4 = 7Identity (a number plus zero will equal itself)7 + 0 = 7

Algorithms are clear methods for solving various operations that work in all cases. Addition has several great algorithms for solving problems.Standard Algorithm: 86 + 44 1  86+44------ 130Partial Sums: 34 + 99 34+99------- 120  (30 + 90)+13  (3 + 9)--------133Expanded Notation: 311 + 201 + 187300 + 200 + 100       10 + 80+     1 + 1 + 7---------------------700 + 90 + 9 = 799

Take Away: Starting with an initial quantity and removing a specified amount. Example: Start with 7 cookies and take 3 away@ @ @ @ X X X      You are left with 4 cookies.Comparison: Compare the size of two quantities and determine how much larger one is than the other. Example: Brad has 9 cookies and Jared has 5. How many more cookies does Brad have?@ @ @ @ @ @ @ @ @ : Brad@ @ @ @ @        : Jared                   The difference is 4.Missing Addend: Determine the quantity that must be added to reach a specific amount. Example: John wants to have A's in all six of his classes. Right now he has A's in three of his classes. How many more A's does he need?3 + ___ = 6        He needs 3 more A's.

Standard Algorithm: 471 - 258   6 11 4 7 1- 2 5 8---------- 2 1  3Expanded Notation: 468 - 291 300  160 400 + 60 + 8-200 + 90 + 1______________       1 7 7Integer Subtraction: 512 - 89 512- 89-----------  -7 (2 - 9) -70 (10 - 80) 500 (500-0)          500 + (-70) + (-7) = 423

Subtraction and addition are inverse operations. A simple addition equation will have a fact family of four. Example: 4 + 5 = 94 + 5 = 95 + 4 = 99 - 5 = 49 - 4 = 5

Repeated Addition: 4 x 54 + 4 + 4 + 4 + 4 = 20Equal Groups: 3 x 4@@ @@ @@@@ @@ @@     3 x 4 = 12Area/Array: 5 x 3     3   @@@   @@@5 @@@   @@@           5 x 3 = 15   @@@

Example: 5 x 25         25   ______________________   | ---------- ---------- *****5 | ---------- ---------- *****   | ---------- ---------- *****              5 x 25 = 10 longs + 25 units   | ---------- ---------- *****              10(10) + 25(1) = 125   | ---------- ---------- *****

Traditional Algorithm: 44 x 7 44     4 x 7 = 28x 7    40 x 7 = 280 ----    280 + 28 = 308308Partial Products: 65 x 4 65x 4------  20+ 240 260Area Model: 44 x 61       40        460   2400     240        2400 + 240 + 40 + 4 = 2684 1      40       4

Partition Model (Sharing): Distributing a given quantity among the specified number of groups. Partition model questions will ask you to determine the amount of each group.Measurement Model (Repeated Subtraction): Take a given quantity and create groups of a specified amount. Repeated subtraction questions will ask you to find the number of groups created.

Multiplication and division are inverse operations. With any given simple multiplication equation, there will be a fact family of four. Example: 6 x 7 = 426 x 7 = 427 x 6 = 4242 / 6 = 742 / 7 = 6

Divisibility defines one value's ability to divide another number without a remainder. For example, 8 is divisible by 4, as 8/4 = 2, whereas 8 is not divisible by 3, as 8/3 = 2 and 2/3.Attached is a hyperlink including the divisibility rules for 2,3,4,5,6,8, and 9, as well as a link to a YouTube video that analysis divisibility further.

Prime numbers are numbers which only factors are 1 and itself. Some examples of prime numbers are 3, 5, 7, 11 and 13.

Composite numbers are whole numbers that have more factors than just 1 and itself. For example, 8 is a composite number, as its factors are 1,2,4 and 8. Each whole number has a prime factorization, which is when it is broken down into its product as primes. The prime factorization for 8 is 2*2*2, written as 2^3.Attached are two different strategies for finding the prime factorization of composite numbers.

Finding the greatest common divisor (GCD) means to find the largest whole number that divides both a and b (two whole numbers). Using colored number rods is an excellent way of introducing the topic of greatest common divisor.Attached is a YouTube video demonstrating the use of color rods, as well as a hyperlink to a method of finding the GCD of larger numbers using the process of prime factorization.

The least common multiple (LCM) of two whole numbers a and b is the smallest non-zero whole number that is both multiples of a and b.Similar to GCD, introducing this topic with colored rods is effective for first time learners. When dealing with larger numbers, prime factorization can also be used to find the least common multiples of multiple whole numbers. Attached is a link demonstrating how to do so.

An integer is a whole number that is either positive, negative, or zero. A negative number is written with a raised "-" in front of it, so negative three would be written as -3.-5, -2, 0, 2, and 5 are all examples of integers.Absolute value describes the distance a numerical value is from zero, and is written as |4| = 4.Absolute value works for both positive and negative values, however a number's absolute value will always be positive as distance cannot be negative.The absolute value of 5 (|5|) and the absolute value of -5 (|-5|) are both 5.

Adding with integers is the next step after mastering addition with whole numbers. As stated before, integers come with the inclusion of negative numbers. Learning how to add a positive and a negative number or two negative numbers can be tricky for students at first.The chip model and number lines are excellent ways to introduce this concept. The chip model does a great job of helping learners understand the "cancelling out" that happens when we add a negative number to a positive one, and number lines are especially useful when covering absolute value.The properties of adding integers are:Closure: a + b is a unique numberCommutative: a+b = b+aAssociative: (a+b)+c = a+(b+c)Identity: 0+a = a = a+0

Subtracting integers is similar to adding in that it can be tricky at first. The chip model and number lines are just as useful when it comes to teaching the subtraction of integers. Attached is a hyperlink to two examples of integers being subtracted, one using a number line and the other using the chip model.Subtracting using "adding the opposite" can also be helpful for students:a - b = a + (-b)7 - (-4) = 7 + 4 = 11

Students will hopefully have a good understanding of the nature of positive and negative integers by the time they begin to learn multiplication, but for those who need a little extra help, the chip model and number lines will do the job. Attached are examples of both models.Multiplying with integers should still be viewed as repeated addition, for example:4 times -2 = (-2) + (-2) + (-2) + (-2) = -8The properties of multiplying with integers are: Closure: a*b equals a unique integerCommutative: a*b = b*aAssociative: a*(b*c) = b*(a*c)Identity: 1*a = a = a*1Distributive: a(b*c) = a*b + a*cZero: a*0 = 0 = 0*a

As with learning all the other operations with integers, chip models and number lines are a great visual representation of dividing integers. Attached are examples of both models in action as well as a video that contains a few extra examples.

A rational number is a number expressed as the quotient/fraction of two integers. Rational numbers can look like a:b, a/b or a divided by b. If we take a/b, the numerator is the top integer, so a, and the denominator is the bottom integer, b in this case.There are several models that can be used to illustrate fractions, including the bar model, number lines and set model. Examples of each are attached.A rational number a/b where 0 ≤ a < b is a proper fraction. Ex: 5/7A rational number where a/b > 1 is an improper fraction. Ex: 7/5

Fundamental law of fractions: If a/b is a fraction and a non-zero number, then a/b = an/bnThe simplication of fractions describes the process of converting a fraction into its simplest form. For example, 3/6 can be simplified to 1/2. Attached are a couple of models that can help students see the equality of a set of fractions like 1/2 and 3/6.

The approach you take to the addition and subtraction of fractions depends on the nature of the rational numbers we use, specifically whether the numbers we have are mixed fractions, and if they share common denominators or not. Addition and subtraction with basic, same denominator fractions can be illustrated using fraction circles, number lines and area models. A few examples are attached.Let's say the problem we are trying to solve is 1/2 + 3/4. In order to solve, we need to find a common denominator between these two fractions. 2 and 4 share the multiple 8. We can convert 1/2 into 4/8, and 3/4 into 6/8. Our new problem is 4/8 + 6/8, and we can add the numerators straight across, giving us 10/8 = 1 + 2/8 = 1 + 1/4.

A mixed fraction is a rational number that shows its quotient and its remainder in the form of a whole number plus a fraction. For example:5/2 can be written as a mixed fraction. 5/2 = 2 1/2

Multiplication with rational numbers can be confusing for students at first, as they are used to the product of two numbers generally increasing in size. Multiplying rational numbers is no different than standard multiplication as it is still repeated addition.For example:3 times 3/4 = 3/4 + 3/4 + 3/4 = 9/4 = 2 1/4The properties of multiplying rational numbers are:Identity: 1 * a/b = a/b = a/b * 1Inverse: a/b * b/a = 1 = b/a * a/bDistributive: a/b * (c/d + e/f) = a/b * c/d + a/b * e/fEquality: if a/b = c/d then a/b * e/f = c/d * e/fZero: a/b * 0 = 0 = 0 * a/bFundamental law of fractions: a/b = a*n / b*n if b does not = 0 and n does not = 0.Attached is a video showing the area model for multiplying rational numbers as well as a hyperlink to an example of how to multiply mixed numbers.

Dividing fractions is no different than dividing integers in that the problem is asking how many groups of one value are in another value. For example, the division problem 7/8 ÷ 3/4 is asking how many groups of 3/4 are there in 7/8. Attached is a hyperlink showing how we can solve this problem using a number line.There are several algorithms students can learn to solve these division problems including:Invert and Multiply:2/3 ÷ 5/7 = 2/3 * 7/5 = 14/15*we only ever invert the rational number that is the divisor.Equal Denominators:a/b ÷ c/b = a/c2/3 ÷ 4/7 = 14/21 ÷ 12/21 = 14 ÷ 12 = 14/12

Ratios describe the comparison of two values: a/b, a:b both compare two values. There are several types of comparisons we can make with ratios. Let's consider a classroom of students:Part to whole: ratio of boys in the class (part) to children in the class (whole)Part to part: ratio of boys in the class (part) to girls in the class (part)Whole to part: ratio of children in the class (whole) to girls in the class (part)Attached are examples of how we can use comparison models to find missing parts in ratio proportion problems.

A proportion is a statement given that two ratios are equal, for example, 2/3 = 4/6. The proportion a/b = c/d can be interpreted as "a is to b as c is to d".Additive Relationship:ex) Andrew and Brad type at the same speed, but Andrew started first. If Andrew has 8 pages types while Brad has 4. By the time Brad has 4 pages typed, how many will Andrew have?Additive relationship: 8 + 6 = 14, Andrew will have 14 pages typed.Multiplicative Relationship:ex) Candice can type 8 pages for every 4 pages that Darren types. If Darren has typed 12 pages, how many pages would Candice have typed?solve for x: 8/4 = x/12, 8/4 = 8*3/4*3 = 24/12x = 24, Candice would have types 24 pages.The ratios d/c are all equal: 1/20, 2/40, 3/60, 4/80In this case, d/c = 1/20, d = 1/20c. In this example, 1/20 is the constant of proportionality.

As students progress through their math education, they are going to be using decimals more often than fractions in their computations. Introducing decimal operations with place value blocks is a good way to start. There are also multiple mental computation strategies that students can learn. Examples of both are attached. Rounding Decimals: If a number is exactly halfway between two values, we round up.a) 6.216 to the nearest hundredth: 6.22b) 6.216 to the nearest tenth: 6.2c) 6.216 to the nearest unit: 6d) 7456 to the nearest thousand: 7000e) 745 to the nearest ten: 750f) 74.56 to the nearest ten: 70

One way to approach the multiplication of decimals is to convert the problem into fractions.Ex: 4.62 * 2.4(4.62)(2.4) = 462/100 * 24/10 = 462/10^2 * 24/10 = 11088/10^3 = 11.088Another useful skill is being able to estimate the placement of a decimal point by looking at the problem. Consider 8.2 * 2.63. We know that 8 * 2 = 16, so we can infer that our answer will be somewhere a little above 16 with the decimal point two values from the left. We can find our answer by multiplying 82 * 263 = 21556, and place the decimal. 8.2 * 2.63 = 21.556Attached is a multiplication algorithm students can use to multiply decimals successfully.

Introducing the division of decimals using a 100-square grid is a good way to help students get an understanding of the process of breaking one decimal into groups of another. For more complex decimal division problems, using the long division algorithm works perfectly.Examples of both are attached.