## 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

## Fractions are challenging. Multiplying fractions is really challenging! This post presents a Google Jamboard to introduce students to the concept of multiplication of fractions.

### Overview

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.

### Prior Knowledge

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.

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### 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.

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

## Word problems are challenging. In middle school and high school, word problems for proportional relationships and linear functions are particularly challenging. This post shows an visualization approach to unpacking the unit rate component of such word problems.

### Visualizing Unit Rate

Below is an excerpt from a handout used to introduce such word problems. It starts with the unit rate and unit quantity provided. This is more accessible for students and allows for a visual representation. This allows students to “see” the multiplication and make a connection to the word problem.

This is followed by situations in which the unit quantity is unknown. Students still draw a visual, but use the 3 dot symbol for etc. to indicate the unknown quantity.

The same problems are used but students now write the symbolic representation. First they write the expression with both factors provided, then with the unknown. The “?” emphasizes there is an unknown quantity before writing the variable.

### The Handout

Here is a link to the handout. It is in WORD format to allow you to enter your own problems.

## 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.

## The concept of fractions as some number of equal parts begins in 1st grade per the Common Core (image below). There are students who struggle with the idea of equal parts and this could undermine student work in subsequent topics. The activity cited in this post is designed to develop the concept of equal parts.

### Jamboard with Sharing Slides

The following images are from a Jamboard used as an introduction to equal parts activity (see photo at the end for access). The activity is chunked to incrementally present more of the ideas underlying equal parts. The use of the Jamboard can be viewed in a FB Reel and on YouTube.

First, the idea of equal is addressed by presenting a situation in which two students are sharing candy. Partitioning out pieces alludes to the set notation of fractions.

The idea of sharing equal amounts transition to sharing a single candy that can be broken into parts. The candy bar image is actually two images of parts. The a non equal sharing is used to unpack equal parts. This is continued for a circular shape and a triangular-ish shape.

### Jamboard with Mathy Slides

There are additional slides to do more “mathy” work with equal parts. First, the students are asked to choose the shape that was cut into equal part (rectangle, circle, triangle). Then the students partition the shapes but with a dotted line as scaffolding.

Each shape can be connected to the food images from above. For example, the student may intuitively understand that a pizza is cut from the crust to the tip. I use pizza fractions to unpack the need for common denominators, which reinforces the significance of the concept of equal equal parts cited previously.

### Handout

Here is an image of an accompanying worksheet. It draws upon the images from the Jamboard and follows the same sequence.

### Accessing the Jamboard

The image below shows how to make a copy of the Jamboard in order to use it.

## Slope is one of the most important topics in algebra but perhaps one of the most challenging topics. One issue is that students often think of the formula as opposed to the idea of slope. The images shown below are for a handout that unpacks the idea or concept of slope.

### Handout

The images show sections of a handout. Slope is a measurement of a graphed line. It measures steepness and also indicates direction. The handout starts by drawing on prior knowledge of measurements for newborns.

Prior knowledge of steepness and up and downhill are invoked.

How to measure steepness is introduced through a focus on stairwells.

The measurement of the length and height of steps leads into the rise and run of a hill. I use “right” and “up” or “down” which focus on direction. The terms “rise” and “run” can be addressed subsequent to the introduction. This part of the handout is scaffolded to reduce task demand and focus on the concept.

Finally, the students are provided a line and apply the ratio procedure. Note: the slopes are not the same as in the prior section.

### Jamboard

An accompanying Jamboard will eventually be shared in this space.