math resources – Mathberry Lane

28
Apr
2024

Surface Area of Spheres Investigation with Oranges

categories: Geometry

I love a good hands on activity that helps my students remember the math they are learning. A great activity to remember the surface area of a sphere formula is to investigate the surface area of an orange.

Make sure to use oranges instead of clementines as the clementines are not as spherical as oranges. Students trace the outline of an orange onto a piece of paper, keeping their pencil perpendicular to the paper as best they can. Then have students use a compass to copy the same size circle several more times on their paper. Ask students to make a conjecture about how many circles they believe the orange peel will cover once they have peeled the orange.

Once students have their circles and conjecture, it is time to peel the orange. Students peel the orange and then test out their conjectures by covering each circle, one at a time. Students should find that the orange peel is able to cover 4 circles. Then as a class, make the connection between the radius of a circle and the radius of a sphere, like an orange. Ask students to write a formula for the surface area of their orange.

If you would like to grab a free copy of my Surface Area Investigation Activity, here is the link:

Alternate Ways to Facilitate this Activity:

Each student completes the activity on their own(an orange for each student)
Students work in groups of 3 or 4 students (7-10 oranges)
Teacher has student volunteers go to the front of the room to model the activity for the class (1 orange)

01
Aug
2023

Teaching Increasing and Decreasing Functions

categories: Key Features of Functions, Mathberry Lane Resources on TPT

For some reason students tend to have difficulty with this concept. At first glance, it seems rather straightforward. If a graph is going up, then the function is increasing. If the graph is going down, then the function is decreasing. If the graph is a horizontal line, then the function is constant. This is the level of this key feature of functions taught in my algebra 2 class.

In calculus, we take it a step further and look at intervals where the first derivative is positive to see where the function is increasing. And likewise, intervals where the first derivative is negative tells us where the function is decreasing. If the first derivative equals zero, then the function is constant.

Next, are the intervals for increasing and decreasing open or closed? Different textbooks will have different notations. Some will use open intervals while others will use closed. I used to be in the open interval camp, but have switched over to closed. I discuss this with my students about how textbook publishers and even math teachers cannot agree on this topic. With that said, if a function is increasing as it goes to an asymptote or infinity, then an open interval would be used.

Let’s look at the definition for a function increasing or decreasing on an interval.

I start this topic with a what do you notice prompt for the graph shown above. It is an entry point into the lesson where each student can share what they see. We then get out some crayons or colored pencils to shade in different portions of the graph.

Next I ask students to try sketching their own examples of graphs that are increasing, decreasing or constant, followed by a graph that exhibits more than one of these behaviors.

Before we start creating intervals for increasing and decreasing, we first review interval notation. Click HERE if you are interested in my free graphic organizer for interval notation.

Finally, we analyze graphs to identify intervals of increasing, decreasing or constant. After students feel comfortable identifying these intervals, I give them intervals with specified criteria and ask them to create the graph that would meet these criteria.

If you are interested in my lesson, it is available in my TPT shop as part of my Key Features of Functions Unit. I have included 2 versions: closed interval notation and open interval notation. Cheers!

22
Aug
2020

The Importance of Mathematical Modeling

categories: Mathberry Lane Resources on TPT

Many students have a tough time visualizing algebra. One way to help students visualize math is to make a model. Modeling usually starts in the elementary grades.

Mathematical Modeling from numbers to arrays to area models to polynomials

Modeling can start simply by modeling a numerical value with pictures or manipulatives. For example, to model the number five, a student could draw 5 circles, 5 cubes, 5 whatever’s…

As students start learning about larger numbers, they begin to use arrays to model these values. For example, the number 12 might be modeled by an array with 3 rows and 4 columns. This is building the foundation for understanding multiplication. Eventually 3 rows by 4 columns turns into 3×4 =12.

The array modeling of 3×4=12 then leads to area models. A rectangular figure can be labeled with dimensions to represent the side lengths of an area.

You can then use an area model to help students model the distributive property. 5(10+2)

Eventually, we replace a value with x, but we can still use the idea of an area model to help conceptualize the distributive property. Let’s look at 5(x+2).

And finally more complex polynomials can be multiplied such as (x+2)(x+3). I hope that seeing these models helps you understand the importance of the mathematical progression of modeling a basic number like 5, because it helps students conceptualize the abstract later on in algebra.

model multiplying polynomials with algebra tiles

If you are interested in purchasing a resource that has some of these feature, you may want to check out the following resources from my Teachers pay teachers shop.

Mathberry Lane Multiplication Facts — Builds conceptual understanding of multiplication

Mathberry Lane Distributive Property Card Sort — Model the distributive property

Mathberry Lane Multiply Polynomials Task Cards — Also available as Boom Cards or Google Slides Digital Task Cards

Tag: math resources