Tag Archives: scaffolded handout

Sales Price Entry Point

A pseudo- concrete representation of a sales price problem is shown below. This is what I use as an entry point for teaching these problems.

  • The entire shape represents the total price of $80. This is 100%, which in student language is “the whole thing.”

  • The discount rate is 25%. Cut with scissors to lop off the 25% which also lops off $20, which is the actual discount. Explain to the student that this 25% is part of the “whole thing.”

  • What remains is 75% or $60. This is the “new price” which is called the sales price.

IMG_20180526_165232.jpg

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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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Elapsed Time Scaffolded

Elapsed time scaffolded

The photo above shows a scaffolded handout to break down elapsed time for a student. The problem is divided into 3 parts: time from 10:50 to 11:00, time from 3:15, time from 11:00 to 3:00 (see photo below). This is based on how we may compute elapsed time by focusing on minutes then on hours. Notice the 3 clocks (in photo above) with no hands which allows the student to engage the clocks by having to determine and show how many minutes passed, e.g. 10:50 to 11:00.

The final answer would be 4 hours and 25 minutes.

Elapsed Time Scaffolded and completed

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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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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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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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Scaffolding Multi-step Math Problems

sidewalk problem

This is the figure from Mrs. Olsen’s Sidewalk Problem from the CT CAPT test 2010 (released to public). The problem has four major steps: divide the figure into common shapes, use Pythagorean Theorem to find height of the resulting triangles, find area of these shapes and compute total cost for  pouring the asphalt for the sidewalk. Simply finding an entry point into this problem is a major challenge as is keeping track of the multiple steps.

Below are photos of the handouts I use to break the problem down into parts – scaffolding. Eventually the students have to learn to find an entry point and navigate the steps on their own. They learn to do this incrementally with the teacher shaping the problem-solving skills.

sidewalk problem scaffolded

 

side walk problem PT

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Percent Discount Scaffolded and Concrete

percent discount scaffolded

I continue to be surprised at how much of a challenge computing percent discount is for students. It’s prior knowledge. If you ask them to explain what a discount is in their own words you’ll get a response like “it makes something cost less.” The students may even have mastery of computing a percent of a number. I believe that in part this is a working memory issue – extra step to process is a little too much.

The photo shows a scaffolded handout (links below) I created to help with the conceptual development of the steps for computing the new, discounted price along with the actual mathematical steps. In another post I showed the use of ten-dollar bills to conceptualize percents. This handout builds on that activity and the scaffolding makes it easier for the students to access the concept. The students have an item that costs $250 and is on sale for 60% off  (allows for nice “round” numbers). The students cut out the discount and have the new price in their hand. When a student has trouble with the mathematical steps at the bottom of the page I review the act of cutting to help with concrete understanding.

On a follow-up pop quiz the vast majority of students were able to compute the new discounted price.

money handout (print in black and white per legal restrictions)

table handout

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Combining Like Tacos (scaffolded handout)

tacos for combining like terms

Here is a snippet of a scaffolded handout I use to teach students how to combine like terms. The tacos bring in prior knowledge and makes the concept more concrete. The scaffolding guides the students as to where and what to write. The full handouts are available by following the links below. I’ve included my webpage address. I don’t care about getting credit but please leave the address on there if you share these handouts. If anyone has follow up questions they know where to find me.

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