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Math 8 | Unit 1 2 3 4 5 6 7 8 9

Inputs and Outputs

Linear Functions and Rates of Change

Cylinders and Cones

Dimensions and Spheres

This week, your student will be working with functions. A function is a rule that produces a single output for a given input.

Not all rules are functions. For example, here’s a rule: the input is “first letter of the month” and the output is “the month.” If the input is J, what is the output? A function must give a single output, but in this case the output of this rule could be January, June, or July, so the rule is not a function.

Here is an example of a rule that is a function: input a number, square it, then multiply the result by π. Using r for the input and A for the output, we can draw a diagram to represent the function:

We could also represent this function with an equation, A=πr2. We say that the input of the function, r, is the independent variable and the output of the function, A, is the dependent variable. We can choose any value for r, and then the value of A depends on the value of r. We could also represent this function with a table or as a graph. Depending on the question we investigate, different representations have different advantages. You may recognize this rule and know that the area of a circle depends on its radius.

Here is a task to try with your student:

Jada can buy peanuts for $0.20 per ounce and raisins for $0.25 per ounce. She has $12 to spend on peanuts and raisins to make trail mix for her hiking group.

Solution:

This week, your student will be working with graphs of functions. The graph of a function is all the pairs (input, output), plotted in the coordinate plane. By convention, we always put the input first, which means the inputs are represented on the horizontal axis and the outputs on the vertical axis.

For a graph representing a context, it is important to specify the quantities represented on each axis. For example this graph shows Elena’s distance as a function of time. If it is distance from home, then Elena starts at some distance from home (maybe at her friend’s house), moves further away from her home (maybe to a park), stays there a while, and then returns home. If it is distance from school, the story is different.

The story also changes depending on the scale on the axes: is distance measured in miles and time in hours, or is distance measured in meters and time in seconds?

Match each of the following situations with a graph (you can use a graph multiple times). Define possible inputs and outputs, and label the axes.

In each case, the horizontal axis is labeled with the input, and the vertical axis is labeled with the output.

This week your student will be working with volumes of three-dimensional objects. We can determine the volume of a cylinder with radius r and height husing two ideas that we’ve seen before:

Just like a rectangular prism, the volume of a cylinder is the area of the base times the height. For example, let’s say we have a cylinder whose radius is 2 cm and whose height is 5 cm like the one shown here:

The base has an area of π22=4π cm3. Using this, we can calculate the volume to be 20\pi$ cm3 since 4π⋅5=20. If we use 3.14 as an approximation for π, we can say that the volume of the cylinder is approximately 62.8 cm3. Students will also investigate the volume of cones and how their volume is related to the volume of a cylinder with the same radius and height.

This cylinder has a height and radius of 5 cm. Leave your answers in terms of π.

This week, your student will compare the volumes of different objects. Many common objects, from water bottles to buildings to balloons, are similar in shape to rectangular prisms, cylinders, cones, and spheres—or even combinations of these shapes. We can use the volume formulas for these shapes to compare the volume of different types of objects.

For example, let’s say we want to know which has more volume: a cube-shaped box with an edge length of 3 centimeters or a sphere with a radius of 2 centimeters.

The volume of the cube is 27 cubic centimeters since edge3=33=27. The volume of the sphere is about 33.51 cubic centimeters since 43π⋅radius3=43π⋅23≈33.51. Therefore, we can tell that the cube-shaped box holds less than the sphere.

A globe fits tightly inside a cubic box. The box has an edge length of 8 cm.

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.