Introduction to Adding Integers

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.

thumbs from Educlips on TPT

Adding Integers

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.

Equal Parts of Fractions – Intro

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.

CCSS Coherence Map

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.

Intro to Slope of a Line

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.

MathBootCamps

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.

Speeding Up and Slowing Down on a Graph looking at Velocity and Acceleration

We all understand speed intuitively. Velocity and acceleration are more mathy and harder for many students to comprehend, especially on a graph. Even harder is the idea of when an object is slowing down or speeding up. In this post I attempt to unpack both.

Velocity is speed with a direction. Negative in this case does not indicate a lower value but simply which way an object is traveling (think of backing up). Both cars below are traveling at equivalent speeds.

Velocity can be graphed (the red curve below). Here is information found in the graph.

  • Where the graph is above the x-axis (positive) the car is traveling to the right. Below is negative which indicates the car is traveling to the left.
  • The 2 points on the x-axis indicate 0 velocity meaning the car stops (no speed).
  • The blue lines are tangent lines with slope equal to acceleration (the derivative of velocity). If the line is going down from left to right, the acceleration is negative. Up, positive.velocity graph with tangents

Below is an example of using prior knowledge and visuals to help make sense of speeding up and slowing down.

As stated previously, the points on the x-axis indicate 0 velocity – think STOP sign. When the car is above the x-axis, it has positive velocity. Below, negative. When the car is headed down, the acceleration is negative. Up, positive.

Now for speeding up or slowing down.

  • Think of a bank account balance.
    • If you have a positive balance and a transaction that is negative, Your balance is moving towards 0.
    • Suppose you have a negative balance in your bank account and you have a positive transaction. Your balance is moving towards 0.
  • As the car moves towards a stop sign it will slow down. The speed is getting closer to 0.
    • Car at top left has positive velocity and negative acceleration so the change is going opposite of the direction. The car is moving towards 0, like the bank balance analogy.
    • Car at top left has positive velocity and negative acceleration so the change is going opposite of the direction. The car is moving towards 0.
  • As the car moves away from a stop sign it speeds up. The speed is moving away from 0.
    • Positive bank balance, positive transaction means balance is moving away from 0.
    • Negative bank balance, negative transaction means balance is moving away from 0.
speeding up and slowing down

Intro to Factoring Out GCF – Scaffolded and Jamboard

The post shows a lesson to introduce factoring out a GCF. The handout is chunked to incrementally add task demands. A Jamboard is used as a complement to flesh out the concept of a factoring by allowing engaging the student kinesthetically, with scaffolding as guidance. (See image at bottom for how to make a copy to access it.)

The sequence starts with a review of factoring and distributive property. The boxes are a preview of the scaffolding for the full factoring to be presented in subsequent chunks.

The Jamboard slides aligned with the first chunk has students pull apart the variables with exponents to help them understand the meaning of the exponents during factoring. For example, there are two Xs overlaid on the x squared. Similarly, there are two 5s, one overlaying the other to allow them to be distributed.

The next chunk in the lesson is factoring out with the terms written with the multiplication symbol written in between. The problems are aligned with the distributive problems to help with the concept of “reverse distribution.” The same problems are presented again in box form as the boxes will be used to support the full factoring. Using the same problems helps the students understand how the boxes work. The Jamboard reinforces the distribution by allowing the movement of the 5s out of the binomial, including with the box method.

The last chunk combines the factoring of individual terms and the factoring out. This is conducted with the box methods for support, and then the boxes are faded. The scaffolding starts by writing the expression with parentheses because that step is easy to overlook when explaining it to students and can be confusing for students.

Here is what the final version looks like.

Note: the handout has blank templates for copying and pasting if you want to create your own handout.

Make a copy of the Jamboard in order to access it.

Intro to Multiplication – A Sequence of Lessons

Below are photos from multiple lessons to introduce multiplication. They are combined into a single document. I use a task analysis approach to first develop conceptual understanding of multiplication as repeated addition. This is followed by skip counting and then using skip counting to multiply. The lessons are not necessary completed in a single day.

Lesson 1 focus is to unpack repeated addition vs simple addition to build on prior knowledge.

Lesson 2 focus is to unpack arrays by identifying rows and columns which are the factors in a multiplication problem. It builds on the previous lesson with repeated addition of groups that are then converted into arrays of items and then into arrays of circles and squares.

Lesson 3 transitions from repeated addition to skip counting (with a future focus of multiplication by skip counting vs fact memory).

Lesson 4 combines skip counting and the rows and columns of arrays into a multiplication sentence.

Lesson 5 uses skip counting to multiply, first with arrays and groups, then as multiplication problems. Here is the link to a post about the Grumpy Cat Jamboard cited in the document.

The nature of the task analysis approach is a sequence of topics building towards the objective of multiplying single digit numbers. Mastery of each of the steps or lessons can be recorded as progress towards mastery of the overall objective. Below is an excerpt from a Google Sheet that is used to record such progress. This can be shared with the team, including parents.

Introduction to Scatter Plots with Google Sheets

This post outlines an activity to introduce linear functions (or scatter plots). The students are tasked with shopping for a used car – a specific make and model. They go to Carmax.com to find mileage and price for 10 cars for sale. They have to find a make and model that has at least 10 cars and can change the search radius to include all locations of Carmax as necessary.

They enter the data for each car into a table on a Google Doc. They are not to include the “k” or the “$” or “,” for price. This allows easier transfer of data. I do not use 0s for the mileage as the slope is more meaningful per thousand miles. For example, -$104 per thousand miles vs $.104 per mile.

Before they graph, you can provide them a common set of data to guide them through a trial run. This way you can show them your graph of the data to allow them to verify that they did it correctly. The data sets shown below are linked at the bottom of this post. (This can be useful for introducing systems of equations as Mustangs typically have a higher intercept and a steeper slope, which allows for a cluster of dots from both in an intersection.)

They copy and paste the mileage and price into a Google Sheet and attempt to graph. You can provide a link to a YouTube video on graphing a scatter plot to free you up to help individuals. The title of graphs should have the variable(s) and the individuals under study. A subtitle can be included to show when data was collected or a data set was accessed. The variables should include units.

A next step would piggy back off of this activity with a Jamboard addressing mileage and price to help students interpret scatter plots.

Here are links to items used for this activity.

Rounding with Scaffolded Handout

This post shows an approach to teach rounding to the nearest 10 and is presented on a handout. Multiple strategies are used. This includes scaffolding, chunking with supports incrementally faded, and a visual.

Overview

A visual of the hand releasing the balloon draws upon prior knowledge and provides a mnemonic for when to round up. The scaffolding provides support to focus on fewer parts of the problem. The supports are faded to incrementally increase task demand placed on the student.

Handout

The first page has maximum scaffolding with the student focusing only on which tens to round to. In turn, an a different element is the focus (e.g., writing the 0s then rounding) and then the scaffolding is increasingly faded.

Here is a link to a WORD document so you can change numbers AND add digits in the hundreds. Here it is in PDF for easier access.

Unpacking Graphs Using a Rollercoaster

I introduce key characteristics with parabolas and use the analogy of a rollercoaster. Riding once (and never a again) the Superman rollercoaster at Six Flags New England got me thinking about this. At one point the roller coaster hits ground level (a zero) and then goes underground (negative y values).

Handout

Here is a handout I use for the introduction. Here are images showing how I use the handouts. The table helps students visualize the x-values and y-values when looking at the graph. The rollercoaster provides extra context for the various characteristics, e.g., increasing means the rollercoaster is going up. Note the scaffolding by adding context clues for each characteristics.

I start with max height of the rollercoaster. I highlight the actual graph first, then the y-values in the table. Then point out we are looking for the x-values for what we highlighted.

The issue of highlighting the vertex for the increasing and decreasing values would be addressed when writing the interval. The idea of the rollercoaster at the tip top provides context to develop the concept.

Similarly, I start with the zeroes. Again, highlight the graph, then the y-values, then the x-values.

The issue of highlighting the zeroes for the positive and negative values would be addressed when writing the interval. The idea of the rollercoaster at the ground level provides context to develop the concept.

I has been effective to have students highlight the parts of the respective axis when discussing the domain and range (not discussed yet). A common challenge is understanding that the x-values continue to the right or left when it appears they simply go down. To address this, I use a very wide parabola to show more lateral movement.

Jamboard

Here is a Jamboard. You have a make a copy to access it (see photo at bottom). Here is a FB Reel and a YouTube video showing how it works.

Intro Linear & Proportional Relationship

This post provides a handout to introduce linear functions and proportional relationships. Multiple instructional strategies are implemented. There is context that draws upon prior knowledge, scaffolding, and visual representations. Graph and table representations of the functions are used as an entry point, without the use of equations.

Overview

There is a difference between proportional relationships and linear functions, which is addressed in another post. Proportional relationships are a subset of linear functions. They can be used as an entry point by citing it as prior knowledge and then showing how they have a constant rate of change.

Proportional Relationship

The students complete the table with the images then the table with the variables. This leverages the context to help them make sense of the table and graph. You can follow up by asking them the total for 0. This allows you to highlight the intercept.

Linear Functions

Linear functions are introduced in similar fashion, including with 0 toppings for the intercept.

Handout

Here is a link to the handout.