## Concepts & Procedures

#### What they're learning

Think of concepts and procedures as the why and how. For example, by the end of sixth grade, students need to know how to divide ¾ by ⅔ (a procedure), but they also need to understand why dividing fractions requires multiplication (a concept).

Sixth graders also learn about ratios, which are a way to compare numbers — or quantities — of things that are part of a whole. For example, imagine you have 12 fish in your fish tank. If 8 of them are angelfish and 4 of them are starfish, that’s a ratio of 8 to 4. It can also be written as 8:4.

Watch these sixth graders explain ratios.

Students need to understand how ratios work because they’ll be called on to use this knowledge practically every day in everything from comparing prices and cooking to figuring out distance and knowing how many miles your car gets with each gallon of gas.

One type of ratio, known as a rate, explains the relationship between a quantity and a period of time. If you’re driving at a rate of 50 miles per hour, that’s a ratio of 50:1. Knowing that rate, students can figure out how long it will take to travel 200 miles. Sixth graders should also begin learning about percentages, which are a ratio expressed as a fraction of 100.

Sixth graders start using a number line that includes negative numbers. Students need to see — and understand — that negative 3 is the same distance from zero as positive 3. Working with negative numbers, students gain a new sense of how to put numbers in order on a number line and compare the value of negative and positive numbers.

Your sixth grader should be starting on the formal road to algebra. Students need to learn how to read, write, and evaluate algebraic expressions and equations in which a letter (also called a variable) stands in for an unknown number.

#### Dividing fractions by fractions

Sixth graders divide fractions by fractions — a more conceptually difficult task than, say, dividing a fraction by a whole number like they did in fifth grade. Dividing a fraction by a fraction can be tricky to visualize, so drawing pictures can be helpful for students when solving problems like 910 divided by 410.

Watch this video from LearnZillion for an explanation of dividing fractions by fractions.

Sample problem 1: Dividing a fraction by a fraction #### Using ratios to solve problems

Knowing how to work with fractions helps students as they start studying ratios, which are a main focus in sixth grade. Sixth graders work on three main skills using ratios: 1) ratio and rate language; 2) recognizing, generating, and graphing pairs of equivalent ratios; and 3) comparing ratios. Ratios express the relative size of two (or more) quantities or numbers. Kids talk about ratios using words such as 3 to 2, or 3 for every 2, or 3 out of every 5, or 3 parts to 2 parts.

Watch how sixth graders calculate equivalent ratios.

Sample problem 2: Using ratios to solve problems #### Understanding rates and unit rates

Ratios have associated rates. For example, the ratio 3 feet for every 2 seconds has the associated rate 32 feet for every 1 second. In sixth grade, students describe rates in terms like for each 1, for each, and per. This is called the unit rate.

Students can calculate the unit rate of a ratio to compare two quantities or find proportional rates (like speed). In the following problem, students must draw on their knowledge of rates to determine who walks faster per 1 second. Why? Because it’s difficult to compare walking rates when they’re in different amounts of time.

Sample problem 3: Calculating and comparing unit rates #### Percents and benchmark percents

Students also learn about percents — sort of a sister to ratios that means per 100. As they become familiar with percents, students will be able to do mental math using common percentages known as benchmark percents, such as 10%, 25%, and 50%. Knowing how to do these calculations in your head comes in handy when you want to leave a 20% tip on a \$36 restaurant bill, or when you want to know how much money you’ll save when a \$35 pair of jeans is 15% off.

Sample problem 4: Percents #### Understanding negative numbers

Before sixth grade, students learned about whole numbers (0, 1, 2, 3, etc.) and fractions greater than 0. In sixth grade, kids step backwards, numerically speaking, to learn about negative numbers. On the surface, it may seem impossible for something to be less than zero, but negative numbers serve a purpose in daily life. If you accidentally take too much money out of the ATM machine, your bank balance could go below zero. If you live in a region prone to droughts or floods, the water level may be described as below or above sea level, with 0 representing sea level. Not enough water, or below sea level, is a negative number. Then there’s the temperature; if you’ve lived through a New England winter you already know what it means when the temperature falls into negative numbers, or below zero. By the end of sixth grade, your child should understand situations that involve negative numbers (like the ones just mentioned). Sixth graders should also be able to place negative numbers on a number line to show the relationship between numbers. (For example, -12 is less than -5 because it is further away from 0; and in a context, -12 degrees is colder than -5 degrees.)

Sample problem 5: Negative numbers #### Algebraic expressions and equations

Since kindergarten, students have been writing numerical expressions, starting with basic addition, such as (2 + 3), and moving into more complex problems, such as (8 x 5) + (2 x 6). In sixth grade, students need to convert these into algebraic expressions by using variables, or letters that represent unknown numbers. For example, subtracting y from 5 would be written as 5 – y.

Sixth graders use variables in word problems. A test question might pose this situation: Daniel spent \$10 on 5 candy bars. Write an equation to determine how much each candy bar cost. The answer is 5x = 10 or 10 ÷ 5 = x, where x is the cost of one candy bar. Here’s another one.

Sample problem 6: Writing and solving equations with variables Sixth graders also use variables with ratios and rates. For example, let’s say you’re driving at a speed of 60 miles an hour. Students may be asked to create an equation showing the distance traveled after 3 hours. It would look like this: 3 x 60 = x.

In the following word problem, students must first identify the patterns for x and y before calculating the relationship between x and y.

Sample problem 7: Working with variables in ratios and rates #### Interpreting inequalities

Sixth graders also work with inequalities, which are expressions that say two things are not equal. Inequalities use the signs > (greater than), < (less than), or ≠ (not equal). Students should be able to interpret inequalities and represent them on a number line.

For example, the inequality x > -2 shows that the variable, x, is greater than -2. Since x is a variable, it includes all numbers greater than -2, such as -1, 0, 1, 2, 3, 4, etc. The inequality x > -2 is shown on the number line below. The arrow means that it goes on forever in that direction, meaning that x can equal any number greater than -2. Sample problem 8: Inequalities #### If your child didn't meet the concepts and procedures standard...

• Your child may not fully understand the concept of fractions, which makes it difficult to understand what it means to divide fractions. (See sample problem 1.)
• Your child may not be sure how to use ratios and rates to solve problems. (See sample problem 2.)
• Your child may not understand how to find a rate or a unit rate from a ratio. (See sample problem 3.)
• Your child may not understand percents or how to use benchmark percentages — such as 10%, 25%, 50% — to quickly and accurately calculate a part of a whole or a percentage. (See sample problem 4.)
• Your child may not understand that numbers can be positive or negative. Many kids have trouble understanding negative numbers and how they are used in the real world. (See sample problem 5.)
• Your child may need practice setting up and solving one-step equations and inequalities that have variables, such as 3 + x = 15, x – 10 = 22, 15x = 45, x ÷ 6 = 5, or x + 7 < 12. (See sample problems 6 and 7.)
• Your child may not know how to find the value of an inequality or how to show that value on a number line. (See sample problem 8.)

The most important thing you can do to help your child with math is to have a positive attitude toward math, and find ways to demonstrate that to your child. Why? Because research shows that a parent’s attitude toward math is contagious; so just by having a good attitude you are helping your child with math. So yes, you are a math whiz — and your child will be one, too.

#### Sprinkle math into everyday activities

• Whenever there’s an item on sale, ask your child to figure out what it will cost after the discount. Pose reverse questions, too. For those \$50 jeans your child wants, how much would they need to go on sale for the total price to drop to \$25?
• When you’re going somewhere, ask your child how long it will take to get there if it’s 15 miles away and you’re driving at 60 miles an hour.
• Ask your child to figure out the ratio of flour to milk in your favorite pancake recipe. Then ask him to double (or halve) the recipe.

#### Talk to your child’s teacher

Start by asking your child’s teacher about your child’s grasp of the why and how. First, does your child understand the why behind the concepts in sixth grade? Does he know why dividing a fraction by a fraction requires multiplication? And why a negative number is to the left of 0 on the number line? Next, does your child understand how to do the problems? If not, where does he struggle? (Is it dividing fractions, writing the equation, filling in ratio tables?) Ask the teacher to help identify the specific concepts and procedures that your child is struggling with most.