Tag: concrete

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Monthly Budget – Introduction

Helping students understand and implement a monthly budget is challenging, especially for students with disabilities that make it harder for students to conceptualize abstract ideas. I previously posted about a full budget activity. This post shows a means of scaffolding the concept of partitioning money in a budget context. The idea is to keep it simple for now and build from there.

Set Up

A parent of a student I support came up with the following idea. We start with a couple major budget items (rent, groceries, utilities) and the temporary idea that the remaining money is discretionary (not the word we use with the student). Money is printed (legal if the printed bills are small enough and only 1 sided) in lieu of fake money that does not look like the bills they would see.

The activity is guided by slides on a Google Slides presentation (link at bottom of this post). Note: the activity can be rerun as needed and the Google Slides slides can be copied, pasted, and information removed. This allows you to keep a record of each trial with this activity.

Job and Pay

The student can either search for a job on a site like Indeed.com or an ad for a job can be provided. The hourly rate is established and the student is prompted to compute the total pay on Google Calculator to allow a screenshot to be produced.

The student then uses the chart to provide a visual and scaffolding to compute the total pay for a month. I go with 4 weeks of 5 work days each, with no taxes to keep it simple.

The student counts out the money, first by grouping hundreds together to get a $1,000. Then the total is moved next to the envelopes.

The pay is entered into a bank balance table to provide practice with the format of a check register. This helps provide structure and having the money counted out on the table allows the student to see a concrete representation of the bank balance table. (Note: I slide the money to the left to allow space to move the money to the envelopes as the student pays bills.)

Paying Bills

The first bill is rent. The student is prompted to search for an apartment on a website like Apartment.com, take a screenshot, and paste into a slide.

The student then pays the bill by counting out the money and sliding the money towards the envelope.

The student then enters the rent into the bank balance. I then point to the money pile on the right in image above and refer to it as rent. I then point to the rent entry into the bank balance. Similarly, I point to the pile on the left, refer to it as the balance and count it out, then point to the new balance in the table. This provides a concrete representation for the bank balance.

I found a website that provides average bill amounts for our state. The student clicks on the link, takes a screen shot of the average costs, and pastes it into the slide.

We focused only on heat and electricity. The student identifies both amounts (I round to the nearest 5 to keep it simple) and then pays both by moving the money over.

Both bills are entered into the bank balance. I then point to the two piles of money used to pay the bills, point to the entries into the table below, point to the pile of remaining money, and point to the entry into the balance in the table below.

Finally, the student makes a shopping list of food items for all 3 meals for the week. To make it easy, we can assume the the same meal each day. The student is provided a lot of leeway in what he or she chooses and what amounts. The amounts they choose may not be enough for a week. That can be addressed in grocery shopping activities conducted in isolation.

The student shops online for the items and takes a screenshot of the cart.

The student completes the table to determine the total cost for a month.

The student moves the money over and then this total cost is entered into the bank balance. The same comparisons between money piles and cost and balance are presented. Then the remaining money is free to use for whatever the student wants. At this point, you can have the student go shopping for clothes or whatever.

The Google Slides File

Here is the Link to the Google Slides file. You can make a copy to access it.

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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.

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Fraction Multiplication with Cookies

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.

f

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.

Access to Jamboard

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

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Exponent Rules with Jamboard

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

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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.

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Combining Like Terms – From Concrete to Symbolic Form

Here is a Jamboard (see photo at very bottom for how to make a copy to edit) that presents like terms as visual manipulatives and then eases into the symbolic form – the “mathy” stuff. The following shows each slide as is and how it looks after completion.

Start with two groups of common items, pull them together because they are alike, and compute the total. This slide is designed to introduce the students to how the manipulatives work for this Jamboard and to introduce them to the concept of like terms.

This slide introduces actual like terms. It is the “mathy” representation of the previous slide and is the most basic form of simplifying algebraic expressions. (The phrase, simplifying algebraic expressions, can be introduced later to allow the focus to remain on “like terms.”)

This slide introduces different types of terms at a conceptual level and how they are rearranged into like groups. It also introduces the use of a binary operation between the two groups of terms (vs adding everything together as the students are wont to do). This slide also establishes color coding by like terms, which is useful for when the work shifts from concrete form (manipulatives) to abstract form (written symbols).

This is the mathy version of grouping by like terms. This slide is crucial as it presents the Associate Property with the binary operation symbols acting for a moment as unary operations. In other words, the addition symbol follows the 2b, the 4t, and the 1b before latching on to another term. It is easier to address this while all operations are addition. Present the original problem as an algebraic expression and not individual pieces “3t plus 2b plus 4t plus 1b”. NOTE: I keep the coefficient of 1 to reduce task demand – one less thing to think about. I address the implied coefficient of 1 after students have had ample exposure to like terms.

The use of 1 dollar bills is intended to introduce constants. I fluctuate between whether to write the terms for the bills with or without units (4 dollars vs 4).

At this point, I suggest giving students independent practice with expressions that have addition only before moving on to the following slide.

This slides shows how I introduce subtraction and negative terms. The image shows a woman eating a taco, hence it cancels one of the tacos.

I present the eating 1 taco image as a negative, with the terms separated as opposed to being an expression. This allows the students to see the “-” symbol as a unary operations (negative) and then as a binary operations (subtraction). In other words, they see the symbol “attached” to the term. It is a prelude to the use of the Associative Property with subtraction.

The new image is of a person eating a burrito. The slide introduces the concept of the negative term in the Associate Property.

**The students are now presented with the what is likely the most challenging aspect combining like terms, which is the “-” fluctuating between being a unary and binary operation. The original problem is again presented as an algebraic expression, “2b minus 2t plus 4t minus 1b”. The minus 2t is converted into negative 2t while the minus 1b is converted temporarily converted into negative 1b as it is moved, but then is converted back into minus 1b. This is the opportunity to unpack this situation. You redo the problem with the 4t minus 2t and negative 1b + 2b (and add the plus symbol) as part of the discussion. You can also duplicate this slide and revise into different problem.**

The next two slides have the color and the “?” sticky note faded. They can duplicate an existing sticky note to record the final answer (or you can add in the “?”).

Finally, students are presented static algebraic expressions. I return to color coding as this is a support they can take with them to handouts. Eventually, the color is faded on the handouts, but they can still use shapes to indicate the different types of terms.

Copy the Jamboard and to edit and use it.

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Group Model for Proportional Relationships and Slope Application

If you find this helpful, maybe consider making a small donation to help build an accessible playscape for children with disabilities.

In my experience, many students struggle with translating word problems modeled by a proportional relationship or a linear function into a algebraic expression or equation. (Image below.)

Here is a link to a Jamboard that can be used to introduce the algebraic expression to model the unit rate. (See image below on how to copy and edit.)

Here is how you can use this to introduce modeling the word problem.

  • Start with the unit rate concept. In this case there is $45 “in” every hour. This is modeled in slide 1 (top 2 photos).
  • The next 2 photos show slide 2 in which the student duplicates the $45 image and fills 2 hours, with $45 “in” each. They complete the multiplication expression by multiplying by 2.
  • This is followed by the same steps for 3 hours (photo bottom left) and sequentially to 6 hours.
  • In the last slide there are no hours shown because the # of hours is unknown. This leads to using “X” to represent the unknown NUMBER of hours (I don’t let students get away with x=hours) and finally the algebraic expression (bottom photo).

You can make a copy and edit it.

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Grumpy Cat Multiplication

If you find this helpful, maybe consider making a small donation to help build an accessible playscape for children with disabilities.

Here is a CRA approach to multiplication that can be individualized. I created this one for a student who loves cats.

Here is a link to the Jamboard. You make a copy by clicking on the 3 dots at the top right, then you can manipulate the items.

Here is how I use this Jamboard.

  • You have a slide with a problem (5×2) and the appropriate grouping (2 in this case).
  • Go to slide 1 and choose the appropriate number of items in the group (5 in this case).
  • Copy (use CTRL C) and paste the appropriate # of grouped items into the groups (boxes).
  • Click on the =? to enter the answer.
  • The slides have groups up to 6.
  • You can have students personalize by choosing their own image. They can paste the image repeatedly to create the grouped items then snip the grouped items as a single image.
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Online Personalized Consumer Math Board Game

The game is played on a Jamboard. There are moveable game pieces on the left (Lego figures chosen to mirror the players – no hair is me), along with movable bills. There is a white rectangle partially covering the cashier’s money. It is a moveable rectangle I use to reveal the money when the cashier pays out money to a player. The money is subsequently covered again. When money is paid, the appropriate number of bills are moved to the cashier’s counter. Change can be computed and given. (technical note: you can click on an object on Jamboard to change its order, e.g., click on the bills to move them back, behind the rectangle.)

The game is a version of the Allowance Game, which is appears to be a version of Monopoly. The goal is to simulate budgeting and real life spending situations in an interactive and gamified way. The spaces can be revised to cater to the interests and reality of the players. The activities are all ones that I have used in isolation with students I help. The game can be played online and with multiple players who need to learn consumer math topics. (When you share the Jamboard with others, you can make them editors which allows them to move pieces.)

Players start with money in a bank account (center of the board) and then roll a virtual die and move accordingly. If a player does not have enough money for a spending activity, the activity cannot be completed. For some experiences, they are limited in what they can spend, e.g., buying a birthday present. (For rent, I will use an IOU until the player makes enough money – obviously a lot to possibly unpack in this scenario.) Spaces have an activity that falls into one of three categories:

  • earn money at a job by rolling dice for the number of hours worked
  • have a static money experience, e.g., get $20 from birthday or spend $12 on tooth brush
  • have a dynamic money experience, e.g., spend money on Amazon or attend a baseball game and the player goes to a related website (for example, a player buys a Red Sox ticket and a YouTube video of highlights of a game is shown – maybe 2 minutes)

A couple notes: I left the START space empty and am thinking I will move the find a job activity to that space. As of this posting I did not have students find a job yet and simply opened a link to a Target store site for employment and showed them a job ad. I think I will start the game with each player finding a job and rolling the die to earn money at the start.

Below are the steps I use to create and revise the game. If you have any suggestions, please post in the comment section.

Here is a link to a master copy of the game on a Google Doc. It can be copied and then revised. I store my game pieces on here as well.

Here is a link to a master copy of a WORD document I use to position the board with a cashier to create the image shown on the Jamboard.

The image created can be uploaded to Jamboard using the Set background function.

Here is a link to the dice rolling site I use. Each player can open it or I will roll for everyone.

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Unit Cost and Actual Shopping

I previously shared that grocery shopping has a lot of tasks that are overlooked. One is working with unit costs. There are two math tasks related to unit cost, interpreting what a unit cost is and computing the total cost for buying multiple items.

When I take a student into a grocery store to work on unit costs, where is what I do. I start with a pack of items (photo below) and ask the student to compute the cost for 1 item, in this case, “what is the price for the pack of chew sticks, how many in a pack, how much for 1 chewstick?” Then I prompt the student to compute the total if he buys 3 packs. This allows the student to differentiate between the two tasks.

The cost per items is easier to grasp and then is followed with the same prompts for a jar of sauce (below). I have the student compare unit costs for the large vs the small jars and ask, “do you want to pay $4.99 per ounce or $5.99 per ounce.” This language is more accessible than “which is a better deal?” You can work towards that language eventually.

These tasks can be previewed at school with a mock grocery store. The price labels can be created on the computer.

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