**Common Core Math in 6th Grade**

In sixth grade different number and arithmetic concepts come together and are used in interesting ways. Students are going to use their knowledge of multiplication and division to understand problems involving ratios and proportions. They'll increase their skill with fractions to include dividing fractions. And they'll begin to use equations and expressions with variables. Along the way, they'll also fill in the number line with another type of number as they begin to understand and work with negative numbers.

These topics are all highly interrelated. Students will use tables, graphs, number lines, and diagrams to represent a situation with ratios as different approaches to problem solving and to highlight different structure. For example, suppose a juice blend uses 5 cups of grape juice for every 2 cups of peach juice. A student might produce the following table:

Using this they might be able to figure out how many cups of grape juice for 7 cups of peach juice. Graphing these pairs on a coordinate plane would show further structure, such as seeing 2.5 cups of grape juice for every cup of peach juice, and prompt further insights. If students do “cross multiply” to solve ratio problems, it will be a result of a solid understanding of the meaning of ratios.

Geometry in this grade provides some great opportunities. Students will reason about formulas for length, area and volume, and by doing so reinforce their work with equations and expressions, which are new in this grade. At this level, a wide range of applications also opens up.

**Common Core Math in 7th Grade**

Seventh grade math is some of the most useful throughout life. Calculating discounts, taxes, interest, etc. are something all adults need to do regularly. At this point students do more work of recognizing how a percent or proportion comes about and what it means. For example, we can look at a lot of items at a store and ask for each what would be better: a $20 discount or a 20% discount? Letting students figure out that 20% is best for items over $100, and $20 is best for items under $100, from examples (and reason about why) helps them learn about functions later. In fact, one can set this up as a function problem, but reasoning directly perhaps drawing a picture (like the “tape diagrams” borrowed from Singapore) is more intuitive for many.

Learning about negative numbers will also have an emphasis on both context (money owed, temperatures below zero, blocks to the left and right of some landmark) and how previous arithmetic must apply to it. For example, they’ll justify why a negative times a negative must be a positive using area calculations of rectangles with negative numbers (e.g. one side is 10 + -3 =7 feet long). Algebra with linear expressions is also introduced in seventh grade.

As data is a key part of understanding our world now, this will be a focus. Students will look at two quantities or two populations, and try to understand not only how they are related but how certain they can be about the relationship. Students can generate data by for example randomly surveying their peers. They not only make estimates of averages or percentages from the data, but start to understand how far these guesses are likely to be away from the true average or percentage. The related work in probability at this grade is important in its own right, and reinforces fraction arithmetic.

**Common Core Math in 8th Grade**

Extensive work with linear equations (equations whose graph is a line) ties together much of what your student will learn this year. They'll understand them in the context of functions and represent them using tables, graphs, and equations. They'll take data that suggest a linear relationship, find an appropriate line, and make predictions based on the graph or the equation. Geometry will center around lines as well — shifting, stretching or reflecting 2- and 3-dimensional objects using specific lines as a reference. Linear functions will be one basis for understanding more complicated functions such as quadratic and trigonometric functions, and links to the extensive sixth and seventh grade work with proportional relationships. They will also be analyzing angles formed when lines intersect, and finding the distance between two points on a line using the Pythagorean Theorem.