Blog – Mathberry Lane

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!

30
Jul
2023

Teaching Inverse Functions in Algebra

categories: Teaching Secondary Math

In elementary school, students learn how some mathematical operations are opposites of other operations. Addition is the opposite of subtraction. Multiplication is the opposite of division. As students get older, we introduce the word inverse. Inverse operations undo what has been done and are essential for solving equations. In algebra we expand the idea of inverses to functions. In geometry, the inverse of a conditional statement is when the hypothesis and conclusion are negated. In calculus students learn that integration is the inverse of taking the derivative.

Today we are going to delve into inverses with respect to the algebraic lens of functions, both analytically and with the geometric connection graphically.

Graphical Inverse

A function and its inverses are connected graphically by being the reflection over the line y=x.

When I start teaching inverses of functions, I often will start with points on the coordinate plane and have students switch the x and y coordinates to find the inverse. If you do this with several points and then ask students to graph the line y=x, they often will notice the connection of the reflection over the line y=x on their own. Here is a sample prompt to give your students.

We would discuss the domain and range of the function and its inverse as well as deciding if the inverse is still a function or just a relation.

Next, we look at a linear function, because this is the function that my students are most comfortable with and you do not need to restrict the domain for the inverse to be a function.
I may use a function similar to the graph shown in the first image above, f(x)=2x+4. Students will table the function and then switch the x and y coordinates to graph the inverse. They then would graph the line y=x and really see the connection that the function and its inverse are reflected over the line y=x. It would be good to note whether the inverse of the function is also a function at this point, as it will set up a future discussion when graphing parabolas.

After students start to get a feel for what an inverse looks like graphically, it is time to introduce the algorithm for finding the inverse.

Steps to Find the Inverse of a Function Algebraically

We will go over the steps and an example followed by some practice problems.

Restricted Domains

Finally, we will look at functions that need to have a restricted domain to make the function’s inverse also a function. It should be noted that if the inverse is not a function, then we would classify it simply as a relation.

The parent function for quadratics is a great function to start with when looking at restricted domains.

How do you teach inverse functions? Leave a comment below! If you are interested in my lesson or more teaching resources for inverse functions, visit my TPT store below!

02
Jul
2023

Teaching Interval Notation

categories: Key Features of Functions

Interval notation is a way to describe a set of numbers. In elementary school, students are introduced to number lines and inequalities. Number lines are a wonderful visual tool for students to make sense of numbers and to process whether a value is greater than or less than another value. Eventually, these ideas morph into a solution set when they reach pre-algebra. For example, x is greater than 3 has infinitely many solutions. Its solution set could be displayed on a number line with an open ended ray.

Students first learn to write this analytically as an inequality, x > 3. We can write this same interval of values with interval notation such as (3,∞). In algebra, we start to study intervals of numbers such as the domain and range. Instead of using an inequality to represent an interval of real numbers, interval notation is often used instead. When a set of numbers does not include the endpoint, as shown above, a parenthesis is used to indicate that the interval approaches that number, but does not include it in the interval.

Suppose we would like to include the endpoint of an interval. Let’s look at the interval when x is less than or equal to 2.

As an inequality we would write this as x ≤ 2. However as an interval of values, we would use a square bracket to show that the endpoint 2 is included in the interval such as (-∞, 2]. Note that one can never actually reach infinity, so infinity will always have a parenthesis and not a square bracket.

Sometimes an interval of numbers has a starting and ending point on the number line. This is sometimes referred to as an “and” compound inequality.

As an inequality we would write -3<x<1. And then for interval notation it would look like (-3,1).

We can make a union of intervals when the rays go in opposite directions. This would be an “or” compound inequality.

The set of all real numbers can also be written with interval notation.

A common mistake I students make when using interval notation is to write the larger value first and then the smaller value. So, make sure that when you introduce interval notation to your students that you remind them it looks like (lower bound, upper bound).

You can grab my free interval notation graphic organizer here. I also have a full version Interval Notation Lesson available to purchase in my TPT store that is part of my Key Features Unit.

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

01
Sep
2019

How to Study for a High School Math Test

categories: Teaching Tips

Have you ever heard a student ask how to study for a math test? Or maybe, have they told you that they did not study because they don’t know how to study for a math assessment? Some students still do not know how to study in high school and college is only a few years away. The time to teach them how to study is now.

Teaching your students to study can be tough love. One fall I polled my students to learn about their studying techniques. I got responses such as:

Study with a friend
Watch videos on Khan Academy
Look over your notes
Re-do some homework problems
Do the review the teacher provides
I don’t study

I told my students that I would not be providing them with a practice test type of review. Some of them were not very happy with me as this is how some of them had been trained to study for a math test since middle school. I asked how many of them were planning on going to college. Most students indicated that they had a desire to attend college. I informed them that their college professors most likely were not going to give them a practice test before their exam. It was time to learn how to study for a math test. High school is a safe place to try out a new technique. If you fail, you can learn from your mistakes, and then reassess(at least at my school). I do not know if this is the case in many colleges.

Goal as a teacher:

My goal was for each student to develop strategies for how to independently prepare for a math assessment.

How I achieved this goal…

This goal is achieved over a series of assessments. Just like students practice newly learned skills, they also have to practice newly learned study strategies and make it become a habit. #1 below goes with assessment #1, #2 goes with assessment #3, etc. Students can use these strategies in other courses besides math as well.

#1 Teach students how to deep dive a lesson working as a team

Students work collaboratively in study groups so that the task is not so daunting
Each group is assigned a lesson from the unit of study which they will present to the class at the end of the period
Students are given a set amount of time to deep dive into their assigned lesson; students are given guidelines(see end of post if you would like a template) for the deep dive which include learning targets or competencies
Students present their lesson overview to the class
- Posters
- Over-sized white boards
- Designated area on chalkboard or whiteboard
- Teacher sets up a google slides doc and students add their presentation to the document which becomes a study guide shared with the class

#2 Assign students to deep dive each lesson independently

After students have practiced the lesson deep dive in teams, for the next assessment, ask them to try it on their own.
Give them 10-15 minutes of class time to work on a specified lesson
Have students trade papers and review a peer’s work and provide feedback
Assign remaining lessons for students to reflect upon for next class
Collect students’ reflections and provide them with feedback

#3 Repeat the process at least once more with collecting student reflections and providing feedback

You should see growth and more detailed reflections the second time students independently complete this activity. Teacher feedback is essential for students to see what might be missing from their reflections or praise what was awesome!

You can continue this process in class as needed or let students be self-directed in their studying. I remind students that they should be reviewing each lesson and processing what they have learned. This can become a great activity for early finishers – start on your unit reflection. If you would like a free template for student instructions for how to deep dive a lesson, you can download the ppt file HERE. I had this printed as a booklet for the first time we did the activity independently. After that, I just printed the cover sheet and let students create their reflections as they saw fit.

06
Aug
2019

Back to School 2019

categories: Giveaway

It’s that time of year. My daughter can’t wait to meet her third grade teacher and my son, going into 7th grade, is soaking up every last bit of summer before we all go back to school in a few weeks.

Don’t miss out on the TPT BTS Sale! Save up to 25% in my Mathberry Lane TPT Store with code: BTS19

To kick things off, I am giving away a $10 Teachers pay Teachers gift card! Head on over to my facebook page to enter! Have a wonderful school year!

21
Aug
2018

Geometry Constructions Digital Unit now available!

categories: Mathberry Lane Resources on TPT

I hope that you had a wonderful summer as we gear up for back to school! I have joined the site wide one day sale for Back to School for today, August 21, 2018. Please use code: BTSBONUS18 at checkout. I have recently finished my latest digital unit for geometry, the constructions unit. It is includes 5 lessons and assessments. These lessons do include paper and pencil labs for students to traditionally experience constructions. I have also included an assessment and a constructions book with a rubric so that it may be used as an alternative assessment. It is now ready for purchase. I have finished the first 3 lessons of the digital parallel lines unit, which includes proofs with angles and lines. I hope to finish up that unit soon! See my store for details.

If you would like to enter the giveaway for the $10 TPT gift card, please visit my facebook page for details. I will randomly select a winner at 9:00 pm EST tonight!

I hope you have a fabulous school year with your students!

~Mathberry Lane

26
Jul
2018

Geometry Constructions Unit Compatible with Google Slides

categories: Mathberry Lane Resources on TPT

My latest digital lesson compatible with google slide is now available:

Geometry Constructions Bundle

Take a peek to see more in this video!

Lesson 2.1 Lab*: Compass & Straightedge
Lesson 2.2 Lab*: Construct Parallel & Perpendicular Lines
Lesson 2.3 Lab*: Construct the Circumcenter and Incenter
Lesson 2.4 Lab*: Construct Square & Hexagon
Lesson 2.5 Lab*: Construction Applications
Constructions Mini Book with rubric
Assessments with rubrics

Happy Trails!
Mathberry Lane

12
Feb
2018

TPT Valentine’s Day Sale – Multiplication Card Sort & Win a $10 Gift Card

categories: Giveaway, Mathberry Lane Resources on TPT

We made it halfway through the school year! My latest resource focuses on multiplication facts. It is a card sort with the following different types of cards:

1) Multiplication problem
2) Array of multiplication problem
3) Answer to multiplication problem
4) Area model of multiplication problem
5) Grouping model of multiplication problem.

Save 25% with code: XOXO February 14-15th. Visit my Mathberry Lane facebook page to enter for a chance to win a $10 TPT Gift Card! Winner will be chosen at 9:00 pm on February 14, 2018.

21
Jan
2018

Multiplication Flash Cards – Fluency

categories: Mathberry Lane Resources on TPT

My dad is seventy years old. School was much different when he was a kid. He walked to school in morning and home in the afternoon. He might even have told me that it was uphill both ways! He was not the most diligent student growing up and finds it ironic that he married a school teacher. Being a teacher seems to run in the family.

Anyways, my dad and I were having a chat one day about his schooling. He said that the most useful thing he learned in school was his times tables. He still uses them all the time. Take a minute to think about when you last used a multiplication fact.

Multiplication Flash Cards by Mathberry Lane

I remember being a third-grader and my teacher would have us play the game around the world to practice our times tables. I also remember 2 minute tests. We had to write down each problem that we got wrong ten times on the back of the paper for the next day. I did not like doing this, but I knew all of my multiplication facts by the end of the year.

As a child, I did not fully understand the meaning multiplication. I knew that it was a math operation. It was not introduced as repeated addition 2+2+2 turns into 3 groups of 2 –>> 3×2=6. I hear a lot of parents complaining about “the new math.” However, my own children understand the processes of operations and why they work.

Can you explain the process of borrowing when subtracting? –Not the algorithm, but the reason why we borrow one from the next place value over. I was curious and asked my ten-year-old son. Without batting an eye, he said you take one from the hundreds and make it ten groups of ten. Then you move one group of ten to the next column over to add to that place value and then you can subtract.

I love the newer mathematics for elementary students and the deeper understanding of mathematical concepts they are learning. Fluency is still important after the basic understanding of the topic is learned. Just as students would need to practice words for a language class, they also need to practice their times tables for math class. Flash cards are a great tool for practicing fluency. Keep reading to find where you can download your free set!

I find that there are many high school students who do not know their multiplication tables. This is a crucial prerequisite skill for factoring polynomials as well as many other tasks. Some might argue that a student can just use a calculator to multiply. However, when the skill needed is to find the factors of a number, a calculator will only take a student so far. Without knowledge of times tables, trial and error may be an approach that a student might use. This is a fine approach, but very time consuming.

In an effort to help teachers, tutors, and parents assist students to learn their times tables, I have created these multiplication flash cards with a progress tracker. Please use this resource to help this generation of students become fluent with their times tables.

Visit my M athberry Lane TPT Store to purchase this resource!

Multiplication Flash Cards