## 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.

### Overview

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
• function machines
• equations
• tables
• graphs
• Each includes an example and a non-example.
• The last slide provides a sorting activity.

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.

### Overview

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.

### Slides

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
• Slide 3: generic tax rate
• Slide 4: generic discount rate

Here is a link to the Jamboard. You must make a copy to access it.

## Base 10 Blocks on Jamboard for Subtraction

### The Jamboard

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.

Here is a link. You need to make a copy to access it

## The Product and Quotient Rules are challenging largely because there are few ways to present them in an accessible format. This post outlines an approach using manipulatives on a Google Jamboard.

### Jamboard

The Jamboard activity involves using moveable variables as manipulatives to provide a hands on approach to learning the rules. The first slide presents the idea of exponents as repeated multiplication. The Product and Quotient Rules use the manipulatives approach to unpack the underlying concepts of the rules. Note: there are set problems followed by blank templates. Here is a FB Reel and a YouTube video showing how this works.

### Accessing Jamboard

Make a copy to access the Jamboard

## Visuals and manipulatives allow for a multi-sensory approach to presenting math topics. Google Jamboard makes implementation of both relatively easy and is effective.

### Current Jamboard

Here is an image of Jamboard used to guide multiplication by a 2-digit factor.

### Creation of the Jamboard

Below is a step by step visual guide on how I created a version of the Jamboard seen above. Here is a Facebook Reel version with music only. Here is a YouTube version with music only. Here is a PDF for the slides I show and a YouTube version of me talking about the sldes.

## Operations on integers and integers in general is challenging for many students. Negative numbers are abstract. Whole numbers and fractions can be represented with images. The activity presented draws upon student prior knowledge of thumbs up and down in a vote to make negative more accessible.

The following images are from a Jamboard. Here are accompanying videos on FB Reels and Youtube showing how this works. There are 3 sets of images (or chunks) loosely following a CRA appoach and all referring to the same two situations.

• Prior Knowlege drawing upon a classroom setting (concrete)
• Transition using thumbs (representational)
• Introduction of adding integers, using thumbs (more abstract but still supported by

### Prior Knowledge

The activity starts with a couple of classroom votes using thumbs up and down.

This is followed by scaffolding to focus on how the voting works through a comparison of the quantity of thumbs up vs down.

### Transition

This section introduces thumbs as counters for integers, which is a common instructional strategy (yellow for positive and red for negative). The scaffolding is the same.

Finally, references to thumbs is replaced with the integers values. The thumbs tokens are maintained to allow for continued concrete representation.

### Accessing Jamboard

To access the Jamboard you must make a copy.

## Intro to Ratios using Jamboard

Below are images from a Google Jamboard for a hands on introduction to ratios. (See image at the bottom for how to make a copy in order to use it.) The images are from Clever Cat Creations and provide a visual representation. The moveable items engage the students kinesthetically. It also helps unpack the concept of ratio as a comparison of two quantities as the students count out the quantities and represent them as numbers in a ratio. The scaffolding guides the process.

First, students move the terms to make a connection between the statement and the ratio.

Then the objects are counted and moved.

Then the ratio is written.

The quantities can be flipped to show an alternative ratio.

There is a blank to create your own and another with shapes.

You have to make a copy in order to move the pieces.

## Solving Proportions – Scaffolded and Jamboard

Below are images from a Jamboard and a handout that scaffold cross multiplying to solve a proportion. (See image at bottom to make a copy of the Jamboard.) This is an entry point, with a focus on how to write the ensuing equation. Solving would be a prerequisite skill so it is not addressed (but obviously would follow). This allows for less task demand placed on the students and for more time spent on the new steps.

The arrows and shading scaffold the cross multiplication step. Students move the terms from the proportion to the equation. This allows for kinesthetic engagement and helps students see how the equations are formed. The scaffolding for the equation guides students to writing the equation, which I have found a challenging step for some students. The equation is written first as factors to reinforce the idea of multiplication, then the students simplify for the second equation.

The handout draws upon the Jamboard and uses the same scaffolding. The template is blank to allow for use with other handouts. The students can copy problems from another handout and follow the scaffold to get to the equation. The steps and equation can be transferred over to the handout.

Make a copy of the Jamboard to use it.