Functions are perhaps the most prevalent and important topic covered in secondary math, aside from maybe 1 variable linear equations. The concept of a mathematical function is challenging for many students. This post provides details about a meaning making approach to introducing functions.
The introduction is presented on a Google Jamboard, to allow for movement in the pairing of inputs and outputs. It starts with analogies pairing of items using a gumball machine and a Coke machine and proceeds incrementally towards the various representations. The functions are contrasted with examples of relationships that are not functions.
Slides of the Jamboard
Slides1 and 2 present the gumball and Coke machines. Students can move the items to see how a quarter can result in 2 different color gumballs while the Coke button results in only 1 output.
In slides 3 and 4, the use of an hourly wage introduces input and output with quantities. Slide 4 shows two different pay amounts for the same number of hours worked. This taps into prior knowledge.
The sequencing progresses through
Each includes an example and a non-example.
The last slide provides a sorting activity.
Access to Jamboard
Here is the link. To access the Jamboard, you need to make a copy.
This post provides details about an artifact that has a manipulative and visual representation of tax rate and discount rate. These contexts are used as an introduction to percent change. The manipulatives are presented on a Google Jamboard.
The price or original price is presented as dollar bill. The bill is cut into proportional pieces to show the increase or decrease amount, visually, as a part of the original amount. The pieces can be moved around the Jamboard and replaced by other denominations.
The slides are presented in the slide show below. They are arranged in the following order. the slides show the different positions of the manipulatives, e.g., how the $20 bill is cut into discount and sales price.
Slide 1: 5% Tax Rate for $20 price – compute the total to pay
Slide 2: 20% off discount for $20 original price – compute the sales price
This post provides access to and details about a Google Jamboard with scaffolded background to support multi-digit subtraction with regrouping.
The Jamboard has images of basic base 10 blocks. The background provides side by side tables for numbers and for blocks. Additional blocks are set aside for regrouping. Here is a FB Reel and a YouTube video showing how to use this artifact.
Access to the Jamboard
Here is a link. You need to make a copy to access it
This post provides a conceptual approach to understanding perimeter and area.
Students are prompted to build an rectangular animal pen for some farm animals. The number of fences represents the perimeter. The number of squared segments of grass inside the pen represents the area.
Graphing linear functions may be the most important topic in Algebra 1. While proportional reasoning is a prelude to functions, this is the first formally identified function presented to them. The graphing leads to slope and intercepts, beyond the entry point for graphical representations to functions. This post presents an activity that can serve as the entry point for linear functions.
The activity is presented on a Google Jamboard, which provides manipulatives. It begins with a relevant context for students, money and being paid for a job. This allows them to engage the function using money. Before using numbers, they engage the work context through images. They are presented the table and then graph representations of the function before getting into the equation. Here is a FB Reel showing the movement of the images.
There are 3 categories of slides. Here is a description of each.
Table and graph clocks for hours and dollar bill for the money.
They graph the whole hours first, then fractional hours (1/2 and 1/4) to see that there are points “squeezed in between each other. This leads to the idea of infinite number of points. In turn, this leads to the idea of the line are a visual means to present all the points. The points can be presented as solutions. Hence, the graph presents all the solutions for the function.
Table and graph with numbers on sticky notes that can be moved from the table to ordered pairs to positions on the coordinate plane.
The equation, with sticky notes to show numbers substituted in for the variables and then moved to ordered pairs with parentheses.
This post presents a Google Jamboard manipulative activity to help scaffold the act of subtraction which helps unpack the concept of subtraction.
The Jamboard can be individualized with Google Images. The can allow for context. In this example, maybe the context is there are 7 players and 4 have an injury or on COVID protocol and have to sit out.
The artifact also incorporates scaffolding, color coding, and manipulatives. Subtraction is an operation, which invokes a verb. The kinesthetic aspect of the manipulatives helps to unpack the concept of subtraction.
Write the problem, using color.
Circle the starting amount in the row as the same color as the initial number.
Populate the row with the images of interest to the student.
Base 10 blocks are a go to representation for place value. They are also easy to implement for addition or subtraction with place. With a group model, they are useful for multiplication and division. It is harder to model multiplication of multi-digit numbers with regrouping. This post presents a Google Jamboard with base 10 blocks on a scaffolded chart to provide such a model.
Fractions are challenging. Multiplying fractions is really challenging! This post presents a Google Jamboard to introduce students to the concept of multiplication of fractions.
The artifact is chunked to incrementally move from multiplication of whole numbers to whole number and fraction to multiplication of fractions. The representation of multiplication as number of objects in a group times number of groups is the structure used throughout. Cookies on a plate is the context used to draw upon prior knowledge and make the idea more concrete.
This serves as an introduction. Each chunk can be followed by practice before moving on to the subsequent chunk.
The Jamboard starts with a representation of multiplication as groups of objects, first with the number of objects in a group and the number of groups. This is presented first as cookies per person to connect to prior knowledge. Then presented per plate as the plate is subsequently used to model the fractions.
First, whole number times a fraction is presented. This allows for a connection to prior knowledge and introduces fractions in this representation. There are still 6 cookies per group, but now there is only 1/2 a group.
The students can move the cookies onto the plate to see the group of objects. Then they can cut the group in half.
To help make sense of the fractions used in the multiplication of two fractions, the fractional parts of the cookies are presented first.
For multiplication of fractions, the process is the same. There is 1/4 of a cookie in each group, then there is 1/2 a group. As was done previously, 1/2 the group is removed. Conceptually, you can explain to the students that they have 1/4 of a cookie and they split it with a friend.