Tag Archives: scaffolding

Introduction to Solving Equations

I introduce solving equations by building off of the visual presentation used to introduce equations. The two photos below show an example of handouts I use. Below these two photos I offer an explanation of how I use these handouts.

intro to solving equations

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First I develop an understanding of a balanced equation vis-a-vis an unbalance equation using the seesaw representation.intro to equations balanced vs unbalanced

I then explain that the same number of guys must be removed from both sides to keep the seesaw balanced.

intro to solving equations balance and unbalanced

I then apply the subtraction shown above to show how the box (containing an unknown number of guys) is isolated. I explain that the isolated box represents a solution and that getting the box by itself is called solving.

intro to solving equations adding

I use a scaffolded handout to flesh out the “mathy” steps. This would be followed by a regular worksheet.

solving 1 step equations add scaffolded

I extend the solving method using division when there are multiple boxes. I introduce the division by explaining how dividing a Snickers bar results in 2 equal parts. When the boxes are divided I explain both boxes have the same number of guys.intro to solving equations multiplying

The students are then provided a scaffolded handout followed by a regular worksheet.solving 1 step equations multiply scaffolded

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One More or One Less Scaffolding

more-less-screen-for-hundreds-table

The photo above shows a screen for a hundreds table was shared by one of my students in a Math for Children graduate course. She found it on Pinterest for use a class presentation. I love this idea and came up with some revisions I think can make it more effective. It seems to me that this screen may be too busy with 4 different numbers showing. Additionally, the view of the other numbers outside the screen could be distracting.

Below are a couple revisions I would suggest.

one-less-hundreds-table-screen one-more-hundreds-table-screen

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Making Slope Less Complicated

slope-graph-real-life-application

Slope is the rate of change associated with a line. This is a challenging topic especially when presented in the context of a real life application like the one shown in the photo. The graphed function has different sections each with a respective slope.

One aspect of slope problems that is challenging is the different contexts of the numbers:

  • The yellow numbers represent time
  • The orange numbers represent altitude
  • The pink numbers represent the slopes of the lines (the one on the far right is missing a negative)

Before having students find or compute slope I present the problem as shown in the photo above and discuss the meaning of the different numbers. What I find is that students get the different numbers confused and teachers often overlook this challenge. This approach is part of a task analysis approach in which the math topic is broken into smaller, manageable parts for the student to consume. Once the different types of numbers are established for the students we can focus on actually computing and interpreting the slope.

This instructional strategy is useful for all grade levels and all math topics.

 

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Fraction Word Problems Scaffolded

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Above is a handout used to scaffold student work with fraction word problems (more on this below).

Here is a type of word problem I recently encountered in working with a student with special needs: There are 60 students. 3/5 of the students are girls. How many students are girls? The student struggled with this problem in regards to the concept of fractions and in determining a fractional amount of a total.

To address the concept of fractions I used the handout seen in the photo above. (The handout can be printed in color to show the actual colors of the birds – see this handout.) A pink highlighter (red is too dark) is used to help the student connect the actual red birds with the number of redbirds used in writing the fraction.

To work out the 60 students problem the following approach is used to develop conceptual understanding – see photo below. (Note: for students in upper grades who struggle teachers often turn first to showing how to solve by multiplying. Students who are working at below grade level typically need the conceptual piece to be addressed first.)

  1. Break the all the items into groups based on the denominator, in this case it is groups of 5.
  2. Mark the fractional amount in each group, in this case mark 3 of the 5 circles.
  3. Find the total number of circles marked in.

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To scaffold this approach I use a task analysis approach and break the strategy into steps.  First I use a handout focusing students on circling a fractional amount in each group – see photo below.

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The next step is to count the number of items circled (or marked).

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The next step is to use a situation where the items to circle are not colored (the circles in the student word problem are not colored but are hand drawn figures). In the photo below Students are tasked with circling and counting.

 

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Then students take the next step is to answer the question to find the total number.

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Finally, students are tasked with creating their own drawings before circling and counting.

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Targeting Gaps in a Math Topic

percent vs monetary value

A key to intervention for math is to drill down into a topic to see which step is causing a student problems. This is a big reason why ongoing progress monitoring is vital to intervention.

In this case a student in a previous session had occasionally added the percent to the dollar amounts – the step that was problematic. He conceptually wasn’t thinking about the meaning of the values but just added or subtracted numbers he saw. In response the next session focused on helping the student discern between the percent rate and the monetary values.

In the photo above is the work of the student as review of the previous session. This was followed by highlighting the dollar amounts in green and the percent amount in yellow. It was emphasized that the yellow was not used in the calculation in the bottom row.

This was followed by the task seen in the photo below. The focus is strictly on the one step that was problematic. This was followed by work on IXL.com (2nd photo below) with the student writing in values on the handout shown on the  bottom photo to help the student focus on this tep

percent vs monetary value 2

percent vs monetary value 3

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Assessment of a Math Objective

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List all the steps for the objective. Use this table (above) as a pretest to identify gaps.

Provide instruction on the gaps. In the photo below I used color coding to show what to multiply and scaffolding to align the digits in ONES and TENS place. NOTE: I provide the problems with some steps already completed to focus on the steps for which gaps were identified.

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After providing instruction on the steps with gaps data is collected on mastery of these isolated steps. NOTE: The problems are identical in nature to the gaps and the problems used in instruction. (Link to the data sheets – WORD so you can revise.)

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CEC 2014 Presentation on CCSS Math Support

highlight constant

This link is for a drop box that contains the handouts for this presentation. Please email me with follow up questions ctspedmathdude at gmail.com.

 

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Counting Out Total Value of Coins

The chart shown in the photo below was created and used by my former co-teacher and I to teach students in a high school life skills program how to count out the total value for coins (dimes, nickels and pennies). Here is how we implemented it.

  • The students are given a pile of coins, set next to this chart.
  • Students start with dimes (identifying dimes as the coin to start with is a prerequisite step that can be taught in isolation if necessary)
  • They line up the dimes in the dimes column as shown below.
  • They count out the total value of dimes (and can look at the number under the last dime)
  • Then the students identify the nickels as the next coin to use.
  • The place the first nickel in the nickels column, starting at the row below the dimes (you can use a highlighter to highlight the last dime row to scaffold where the student places the nickels)
  • Have the student count off 5 and place the next nickel (on the dimes column) etc.
  • Then follow the same steps to transition to pennies.
  • Have the student identify the total value by looking under the last penny.

coin chart

The idea is to fade the use of the chart and have the student count out the value without the chart. This is more possible if the task demand is incremented with pennies only, then dimes only, the pennies and dimes etc. Here is a link to handouts for those.

 

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Interpreting Slope Intercept

Slope is one of the most important topics covered in high school algebra yet it is one of the least understood concepts. I have two observations about this. First, slope is often introduced with the formula and not as a rate of change. Second, students intuitively understand slope as rate of change conceptually when presented in a relevant, real life context. The challenge is compounded when slope is presented with the y-intercept.slope intercept scaffolding

In the photo I present slope and y-intercept in a context students can understand (money is their most intuitive prior knowledge). The highlighting makes it easier for them to see the context, specifically the variables. I have the students work on this handout and I circulate and ask questions.

Here’s a typical exchange – working through problems 11, 12:

  • Me: “Look at the table, what’s changing?”
  • Student: “the cost”
  • Me: “How much is it changing?”
  • Student: “20”
  • Me: “20 what?”
  • Student: “20 cost”
  • Me: “What are you counting when you talk about cost?”
  • Student: “money…dollars”
  • Me: “So the price is going up 20 what?”
  • Student: “Dollars”
  • Me: Show me this on the yellow” (student knows from before  to write +$20)
  • Me: “What else is changing?”
  • Student: “People”
  • Me: “By how much”
  • Student: “1 people…person”
  • Me: “write that on the green”
  • Me: “Now do this same thing on the graph. Where do you start?” (they put their pencil on the y-intercept
  • Me: “What do you do next?” (they typically know to move over and up)
  • Me: “Use green to highlight the over” (they highlight)
  • Me: “How much did you go over?”
  • Student: “1…1 person”
  • Me: “Now what?” (Student goes up.)
  • Me: “Highlight that in yellow.” (They highlight.)
  • Me: “How much did it go up?”
  • Student: “2…20…20 dollars”
  • Me: “What is a rate?” (I make them look at their notes until they say something about divide or fraction or point to a rate)
  • Me: “So what is the rate of change?”
  • Student: “$20 and 1 person”
  • Me: “Look at the problem at the top. What is the 20?”
  • Student: “$20 per person.”

I point out that you can find this rate or slope in the equation, the table and in the graph.

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Intro to Slope as Rate of Change

Slope may be the most challenging concept to teach in algebra yet it is one of the most important concepts. I use the following sequence to introduce slope: rate of change, rise over run, rise over run as a rate of change. The first photo is a map of Manhattan with directions on counting city blocks. This builds on prior knowledge to introduce rise and run.

Manhattan map for rise and run

The photo below builds on the map and transitions students into coordinate planes. They are introduced to rise over run and positive and negative as indicators of the direction of a line.

rise over run on grid

The photo below combines rise and run with rate of change. The hourly wage is prior knowledge they can much more easily comprehend. A major issue is getting students to include units and to understand what units are. This would have been addressed in the previous unit on rates and proportions.

rise over run as rate of change

This is the handout used with a train activity in which I use battery operated trains and time them as they travel 200″. I project a stop watch on the screen as the train moves. The kids pick up right away that Percy is “slow.” As Percy is traveling I ask them how they know it is slow and get answers like “it takes a long time.” This is a concrete representation which they can draw upon as they work with the graph and calculations.

train activity

The photo below shows a scaffolded version of a Smarter Balance (Common Core assessment) test question. The original question simply shows the graph and asks for average rate of change from 0 to 20 years. Even with the scaffolding many problem areas appear: units vs variable (student wrote “value” as opposed to $), including $ with the 1000, finding unit rate, and even identifying the part of the graph at 0 years.

rate of change Smarter Balance problem scaffolded

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