Why multiplication works as a shortcut to addition

How have most students including I been taught to do this? Is the equal sign a gateway to some sort of magic portal that transforms positives to negatives, goodness to evil, or Rob Ford to a respectable political figure? Of course not. There is no justification for changing signs. Why not do something that DOES make sense?

As a recap, equations are math statements with two sides showing that two expressions are equal. If one side of an equation is altered, the same change must be made to the other side in order for the expressions to remain equal.

If 5 is added to one side of an equation, then 5 must be added to the other side in order for the balance to be maintained. This is the essence of solving equations using the Balanced Method.

Terms are removed from an equation by performing inverse operations i. In other words, we undo whatever is happening in an equation. You get to have your cake and eat it too. Why should we use two different strategies when one will suffice? Whenever students share with me that they have learned to solve equations by moving terms to the other side and switching signs, I ask them why they are allowed to do that.

Simply put, the first step to cross-multiplication is to multiply the numerator of one fraction with the denominator of the other fraction and make the products equal. Similar to moving terms to the other side of an equation, this procedure is also without merit. Well, this is embarrassing. It seems that the Balanced Method provides the Usain Bolt of solutions, while the cross-multiplication solution is more like an injured mule. Clearly, the shortcut is the long-cut here.

You may wonder how the Balanced Method can be used if the variable is in the denominator rather than in the numerator. All you would have to do is take the reciprocal of both sides to fix that issue i. So there you have it. By instead focusing on the universal approach of the Balanced Method, math teachers can promote a sound understanding and a better way to solve equations.

After all…. My role is to support the TDSB with challenging academic streaming from K, paying particular attention to inclusive and culturally responsive mathematics instruction. I view math education through the perspective of equity, inclusion and anti-oppression and its intersection with student identities.

As a powerful tool and vehicle for social change, I see math as student empowerment and ensure they see its learning as a social enterprise that challenges them to think critically and collaboratively. I have worked at the school and system levels to support teachers with inclusive math practices and shape policy to remove streaming as a structure barrier to equity and inclusion. Children who are taught these just as rules to memorize, without some understanding, often garble the rules, confusing them with each other.

Does 1 times a number give 1 or the number? The tiny town image helps establish why 1 times any number gives that number. With only one vertical line, the number of intersections will be the same as the number of horizontal lines. Cards with 0 through 5 slots in them can also be especially useful.

Changing which card is placed on top, or which is vertical and which is horizontal, makes no difference. If one card has a single slot, the number of intersections will match the number of slots in the other card when they are superimposed and the other slots are perpendicular to the single slot.

The slot image makes it particularly clear why multiplication by 0 always gives 0. This lesson gives a nice opportunity to use the words horizontal and vertical in context and to connect their use as directions on maps with East, West, North, and South as directions on the earth.

See horizontal and vertical for common confusions about the ideas these words represent. A student project for second grade: Students use a grid that is laid out like a multiplication table, but does not have a row or column for zero.

Using a backward-L-shaped piece of paper, they select a part of the grid; in the bottom right-hand corner of the selection, they write the number of squares they captured which is the same as the area of the rectangle if each little square represents one square unit of area. Notice that the number at the top nearest the blue boundary gives the width of the green rectangle, the number of columns of squares; the number at the left nearest the blue boundary gives the height of the green rectangle, the number of rows it has.

If we move the boundary straight down one step, we add a new row without changing the number of squares per row. This rectangle happens to have the same width and height, so it is a square. The number in the corner the number of tiny squares inside it is therefore called a square number. See the article on difference of squares for more about this intriguing pattern and another particularly effective way for students to practice facts while developing new and useful mathematical ideas.

Both are ways of describing this rectangle. And even if we decide to reserve one of those notations for and the other notation for , the two would still be equal. That diagonal, yellow in these illustrations, contains the square numbers. Getting rid of the distracting numbers and arrows, we see three regions: the diagonal with square numbers, the green region with other products, and the white region, with the same numbers that are in the green region.

See How many facts to learn? Children who can multiply by 10 and can take half can then use those skills to multiply by 5. At that point, only 15 facts — the ones in green — remain to be memorized. Multiplication Topics: Logic and Proof. Meaning This section gives the meaning of multiplication by presenting some images of it, and some of the kinds of problems it solves. Images of multiplication Given the number of rows and columns in a rectangular array, multiplication tells us how many elements are in the array without making us count them one by one or repeatedly add or skip count the elements in each row or column.

A hint at the relationship with the multiplication algorithm. Arrays and the multiplication table Early in second grade, children can solve and enjoy problems like these.

Intersections as a model for making an organized list When the number of possibilities is greater, as it is with the block-tower problem, children tend to miss combinations, or double-list them, unless they are systematic.

Tables as a model for making an organized list Tables are equally good for representing the combinations and organize the task of listing them. Anchor Building the basic facts First steps When we see the same triples of numbers — 3, 5, 15; 4, 3, 12; 2, 5, 10; 6, 4, 24 — popping up in different contexts, they begin to feel familiar even before any conscious effort to memorize them.

Doubling and halving In first grade, children learn to double [[mental arithmetic mentally] all whole numbers up through Small arrays In Think Math! A common situation where multiplying fractions comes in handy is during cooking. The reciprocal is simply the fraction turned upside down such that the numerator and denominator switch places. A complex fraction is one in which the numerator, denominator, or both are fractions, which can contain variables, constants, or both.

A complex fraction, also called a complex rational expression, is one in which the numerator, denominator, or both are fractions. When dealing with equations that involve complex fractions, it is useful to simplify the complex fraction before solving the equation. From previous sections, we know that dividing by a fraction is the same as multiplying by the reciprocal of that fraction. Therefore, we use the cancellation method to simplify the numbers as much as possible, and then we multiply by the simplified reciprocal of the divisor, or denominator, fraction:.

Start with Step 1 of the combine-divide method above: combine the terms in the numerator. To do so, we multiply the fractions in the denominator together and simplify the result by reducing it to lowest terms:. Recall, again, that dividing by a fraction is the same as multiplying by the reciprocal of that fraction:. Exponentiation is a mathematical operation that represents repeated multiplication. Here, the exponent is 3, and the expression can be read in any of the following ways:.

Some exponents have their own unique pronunciations. Exponentiation is used frequently in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography.

Any nonzero number raised by the exponent 0 is 1. The order of operations is an approach to evaluating expressions that involve multiple arithmetic operations. The order of operations is a way of evaluating expressions that involve more than one arithmetic operation.

These rules tell you how you should simplify or solve an expression or equation in the way that yields the correct output. In order to be able to communicate using mathematical expressions, we must have an agreed-upon order of operations so that each expression is unambiguous. For the above expression, for example, all mathematicians would agree that the correct answer is The order of operations used throughout mathematics, science, technology, and many computer programming languages is as follows:.

These rules means that within a mathematical expression, the operation ranking highest on the list should be performed first. Multiplication and division are of equal precedence tier 3 , as are addition and subtraction tier 4.

This means that multiplication and division operations and similarly addition and subtraction operations can be performed in the order in which they appear in the expression. In this expression, the following operations are taking place: exponentiation, subtraction, multiplication, and addition. Following the order of operations, we simplify the exponent first and then perform the multiplication; next, we perform the subtraction, and then the addition:.

Here we have an expression that involves subtraction, parentheses, multiplication, addition, and exponentiation. Following the order of operations, we simplify the expression within the parentheses first and then simplify the exponent; next, we perform the subtraction and addition operations in the order in which they appear in the expression:. Since multiplication and division are of equal precedence, it may be helpful to think of dividing by a number as multiplying by the reciprocal of that number.

Similarly, as addition and subtraction are of equal precedence, we can think of subtracting a number as the same as adding the negative of that number. In other words, the difference of 3 and 4 equals the sum of positive three and negative four. To illustrate why this is a problem, consider the following:.

This expression correctly simplifies to 9. However, if you were to add together 2 and 3 first, to give 5, and then performed the subtraction, you would get 5 as your final answer, which is incorrect. To avoid this mistake, is best to think of this problem as the sum of positive ten, negative three, and positive two. Or, simply as PEMA, where it is taught that multiplication and division inherently share the same precedence and that addition and subtraction inherently share the same precedence.

This mnemonic makes the equivalence of multiplication and division and of addition and subtraction clear. Privacy Policy. Skip to main content. Numbers and Operations. Search for:. Introduction to Arithmetic Operations. Learning Objectives Calculate the sum, difference, product, and quotient of positive whole numbers.

Key Takeaways Key Points The basic arithmetic operations for real numbers are addition, subtraction, multiplication, and division. The basic arithmetic properties are the commutative, associative, and distributive properties.

Key Terms associative : Referring to a mathematical operation that yields the same result regardless of the grouping of the elements. Learning Objectives Calculate the sum, difference, product, and quotient of negative whole numbers.