0 and Negative Exponents – Scaffolded

This post presents a scaffolded and meaning making approach to exponents that are 0 or negatives.

Handout

The slide show below presents all 4 pages.

  • The handout starts with an initiation to preview the prerequisites for what is presented in the lesson. It also introduces a chart that will be used for discovery.
  • Page 2 presents a discovery activity of following a pattern of dividing by 2 down to the 0 exponent. The concept of exponents is presented as the number of occurrences of the base. This leads to the idea of a 0 exponent indicating the base is no longer present, but there is still 1.
  • Similarly, on the 3rd page the pattern of dividing continues into negative exponents to show the resulting fractions. The negative exponents are then presented as reciprocals.
  • For terms with multiple factors (e.g., 5x vs just x) the students are presented steps to write the factors separately. This unpacks the reason why the negative exponent acts only on one of the factors (unless both are grouped with parentheses).

Access to Handouts

Here is a link to the student handout, and a link to the teacher handout.

Intro to Concept of Functions

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.

Access to Jamboard

Here is the link. To access the Jamboard, you need to make a copy.

Prerequisite Skills and Current Content

The effort to provide intervention to fill in gaps is challenging for different reasons. One reason is the effort to balance support for current content while filling in gaps. This post shows an example of how to fill in gaps while working through the current topic.

Overview

Various rubrics used to assess teacher instruction includes an effort to build on or connect to prior knowledge. If the student has gaps with prior knowledge, the lesson becomes less accessible for students with the gaps. Previously, I addressed how to support both current content and fill in gaps. The idea is to systematically fill in gaps by addressing prerequisite skills as they arise in new lessons.

Example

The handout out below shows an example of how this can play out. The first page is used as a do now for the content presented on page 2. If you are teaching a student how to solve 1 step equations and are moving into integers, page 1 is a a means of supporting the new content while filling in possible gaps. The first image shows the student will need to evaluate -13 – 3 as part of the solving in the lesson. This can be addressed in the do now, as shown in the 2nd image, page on the right. (Notice all the problems on page 1 are steps to solve on page 2 problems.) This is useful for students with special needs and for differentiation.

Handout from mathworksheets4kids

Intro to Percent Change on Jamboard

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

Access to Jamboard

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

Base 10 Blocks on Jamboard for Subtraction

This post provides access to and details about a Google Jamboard with scaffolded background to support multi-digit subtraction with regrouping.

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.

Access to the Jamboard

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

Introduction to Perimeter and Area

This post provides a conceptual approach to understanding perimeter and area.

Overview

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.

Google Jamboard

The slides are presented below. This video shows how the manipulatives work.

Access to Jamboard.

Here is a link to the Jamboard. Need to make a copy to use it.

Intro to Graphing Linear Functions using Jamboard

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.

Overview

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.

The Slides

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.

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

Complete the Square for Vertex Form

This post provides a handout that guides students through the various steps for completing the square to transform an equation into vertex form. Students are guided through each step in isolation.

Overview

Students are presented each step in a separate chunk of the lesson. Then the steps are chained together, with scaffolding that is faded. This is a different approach than presented in a previous post. The chunks, examples, and scaffolding help make students more independent in completing the work. This frees up the teacher to provide more 1 on 1 support.

Chunks of the Lesson

The initiation addresses prerequisite skills: factoring, perfect squares, fractions, and doubles. In lieu of having students divide by 2, I focus on identifying fractions that add to the linear coefficient as you will see in the second page.

Desmos Activity to See Completing the Square

To introduce completing the square, I recommend a visual activity like this one from Desmos.

The students identify the constant that results in a perfect square. They do so by identifying doubles that result in the linear coefficient (e.g., 6 = 3 + 3). The examples help guide them through this process. This section could be presented after a hands on activity on

Students are then tasked with factoring perfect squares in isolation, including those with fractions. The doubles are modeled for whole numbers first, generalized to fractions.

At this point, the students have identified the constant to complete the square and then factored expressions. The next sections have students complete the square and then factor in equations. Note that the equations are structured as a step after the students would have subtracted the original constant, leaving the quadratic and linear terms on the right.

The last section chains all the steps together, first with scaffolding then without. Additional practice would be generated with other handouts that have problems in isolation.

Access to the Handout

Here is a link to the handout.

Subtraction within 10 – Jamboard

This post presents a Google Jamboard manipulative activity to help scaffold the act of subtraction which helps unpack the concept of subtraction.

Overview

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.

Steps

  1. Write the problem, using color.
  2. Circle the starting amount in the row as the same color as the initial number.
  3. Populate the row with the images of interest to the student.
  4. Physically take away the identified amount.
  5. Write the remaining amount as the answer.

Access to Jamboard

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

Base 10 Chart for Multiplication

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.

Overview

The structure aligns with the group representation of multiplication. The # of items in each group is presented first as this aligns with unit rate and slope problems.

Steps

The steps are listed in each photo in the gallery below. Here is a Youtube and FB Reel video showing the steps.

Access to Jamboard

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

%d bloggers like this: