Matches for: “meaning” …

Multiplication with Integers: a Meaning Making Approach

I previously tackled the difference between the ” – ” symbol used to represent a negative and a subtraction problem. This post gets into multliplication through the context of buying a Wendy’s frosty.

One comment to preface what is presented here. Negative numbers are abstract and challenging for many students. Multiplying by negatives is even more so. The approach presented here for multiplying gets a little complex, which is inherent to the topic. In other words, this is an involved process as “there is no royal road to geometry.” What I present here is a path I develop over time, as seen in the sequence below.

First, review of a couple building blocks. Multiplication can be represented at groups of items. 2 x 3 can be represented as 2 groups of 3 $1bills, e.g., you bought 2 Frosties for $3 each.

Negative in terms of money can mean you owe money. Hence, -3 means you owe $3, e.g., you order a frosty and owe $3.

If you change your mind and cancel the frosty, the $3 you owe is cancelled and you get your money back. +$3.

If you order 2 frosties, you owe $3 and another $3, which is -3 + -3. (You owe $6 or -6.)

Cancel those two frosties and you get your money back. -(-6) is now +6

2 x -3 means you means you have 2 negative 3s – repeated addition, or -3 + -3.

If you ordered the 2 frosties and owe $6 then cancel, you get your $6 back or +6.

In mathy terms, cancelling the 2 x -3 is written as -(2 x -3) or -2 x -3.

The – for the -2 can be held out front to focus on 2 orders of frosties or 2 x -3. That was covered previously. Then the extra negative cancels that order so you get your money back. And so you have multiplying two negatives!

All images were generated on this jamboard.

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Making Proportions Meaningful (and Therefore Accessible)

A student reported to our schools math lab where I reside. He had a handout on proportions shown in the photo below and stated that he didn’t know what to do.

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I find that in the vast majority of situations like this the student lacks the conceptual understanding of the topic. As is typically the case, I started my sessions with the student by focusing on something he more intuitively understood. Teens know money, phones, games, music and food.

In this case I started by showing him a photo on my phone shrunk then enlarged the photo and talked about how I could double the size of the photo. We talk about what doubling means then I show him a handout with the photo in two sizes (below).

I explained that the small photo was 3×2 inches and that I wanted to enlarge it. The bottom of the big photo is 6″ but I needed to figure out the height (vertical length) which is marked with an X.

I had him figure out the height (4). Then I explained that proportional means the shape is the same but bigger or smaller. In this case both the side and bottom were multiplied by 2. Then I showed him the “mathy” way of doing the problems. This progressed towards the handout he brought into math lab. By the end he was doing the proportions independently.

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Introduction to Equations – (Meaning Making)

This is a meaning making approach to introducing equations. I will walk through the parts shown in the photo in the space below this photo. (A revised edition of this handout will be used in a video on this topic.)

intro to equations

First I explain the difference between an expression (no =) and an equation (has =). An equation is two expressions set equal to each other (21 is an expression).

intro to equations definition equation

I then develop the idea of a balanced equation and will refer to both sides of the see saw as a prelude to both sides of the equation. I also focus on the same number of people on both sides as necessary for balance.intro to equations balanced vs unbalanced

At this point I am ready to talk about an unknown. Here is the explanation I use with the photo shown below.

  • I start with the seesaw at the top. The box has some guys in it but we don’t know how many.
  • We do notice the seesaw is balanced so both sides are equal.
  • This means there must be 2 guys in the box.
  • I follow by prompting the students to figure out how many guys are in the box(es) in the bottom two seesaws.
  • Finally, I explain that the number of guys in the box is the solution because it makes the seesaw balanced.

intro to equations definition solution

There are multiple instructional strategies in play.

  • Connection to student prior knowledge – they intuitively understand a seesaw. This lays the foundation for the parts of an equation and the concept of equality.
  • Visual representation that can be recalled while discussing the symbolic representation, e.g. x + 1 = 3
  • Meaning making which allows for more effective storage and recall of information.
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Fractions! Meaning Making for Comparing Fractions

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Fractions is one of the most challenging math topics. Many high school and college students struggle to some degree with fractions.  The Common Core of State Standards (CCSS), despite all the criticism, includes components to address the conceptual understanding of fractions. Below is a photo showing a 4th grade Common Core standard regarding fractions along with an objective for a class lesson I taught at an elementary school in my district. I subsequently presented on this at the national CEC conference in 2014. Notice the bold font at the bottom, ¨justify…using a visual fraction model.¨ The photo above shows an example of a model I used in class.

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The photo below shows a handout I used in the lesson. The first activity involved having students create a Lego representation of given fractions. These would eventually lead to the photo at the top with students comparing fractions using Legos. The students were to create the Lego model, draw a picture version of the model then show my co-teacher or I so we could sign off to indicate the student had created the Lego model.

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The Lego model is the concrete representation in CRA. In this lesson I subsequently had students use fractions trips (on a handout) and then number lines – see photos below.

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FRACTIONS! Meaning Making for Adding Fractions

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Fractions is one of the most challenging topics in math. Here’s an approach to help introduce fractions.

I show the photo above, explain to a student that he and I both paid for the pizza. We are going to finish eating the pizza and I get the slice on the left. I ask “is this fair?” This leads into a discussion about the size of the slices and what 1/2 and 1/4 mean. The pizza on the left was originally cut into 2 slices so the SIZE of the slices is halves. The SIZE of the slices in the one on the right is fourths. I have 1 slice left and it is a half so my pizza is 1 half or 1/2. He has 1 slice left and it is a fourth so his pizza is 1/4.  The bottom number is the size and the top number is the # of slices.

We cannot count the number of slices because they are not the same size. So we need to change my pizza.  So I slice my pizza and now I have 2 slices and they are cut into fourths. So now I have 2/4.  Note: I don’t show the actual multiplication to show how I got the 2 and 4.  I am sticking with the visual approach to develop meaning before showing the “mathy” approach.

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Now that I have slices that are all the same size, I can now count the # of slices. “1, 2, 3…3 slices and they are cut in fourths.”

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A Meaning Making Approach to Word Problems

Here is a typical story or word problem.

Dakota helped her father bake cookies. They baked 9 sugar cookies and 3 chocolate chip cookies. How many cookies did they bake total?

When solving word problem the focus is often on following steps, e.g. read the problem and identify important information. There is also a focus on identifying key words, e.g. “total.” The problem with both is they rely on rote memorization. How do we identify “important” information? Focusing on the word such as total does not address the concept of total but is more of a signaled command like “sit.” Students see “total” and they know they are supposed to add. The problem is they often don’t understand why.

The entry point to word problems should be a focus on the underlying concepts. For example, present the word problem with cutouts of the actual cookies and physically demonstrate “total” by pulling all the cookies together. Similarly, you can have cutouts of the tadpoles and demonstrate the concept of how many are left.

word problems focus on concept first concept first approach

Words are symbolic representations of ideas. Same with math symbols (below). Addressing the concepts, vocabulary and the process with this approach is a concrete-representational-approach (CRA). The equations below would not be addressed until the conceptual understanding was developed. When word problems presented do not include the term “total” the student can process the context as opposed to being reliant on the signal.

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Making Discount Meaningful

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Educators teaching math typically start with the “mathy” stuff first. For example, for finding the sales price teachers may start with showing students the steps to calculate (photo below).

I start with the concept, either with a pictorial representation or actual objects to represent the underlying concept. In the photo above, I show an object (related to the student’s interest – this student is into weight training) on sale. The $50 circled in yellow represent the original price. I explain the concept of being on sale and discount and show that 20% is $10 to take away (marked out). This leaves $40 (in green) which is the sales price. This allows for conceptual understanding before showing him the “mathy” way of doing the problem.

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Analogies: Making Math Meaningful

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Math is an esoteric subject for most people. Good instruction makes information meaningful. One method for making information meaningful is to connect new information to prior experience.

In this situation the new information involves determining whether shapes are similar (see photo below). One example of student prior experience with this topic would be shrinking people down. In the photo above I use Mini Me and Dr. Evil and their respective (and fabricated) weights and shoe sizes as measures that will eventually give way to measures of sides of a polygon (below). When working on the problem below the students can be prompted by recalling the analogy of Mini Me and Dr. Evil.

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Intro to Constant of Proportionality Using Hourly Pay

An effective instructional strategy is to make the new math topic meaningful. A fellow Facebook group member asked about teaching the topic constant of proportionality. My suggestion is to use hourly wages as an introduction.

I created a handout that starts with students finding a job with an hourly pay stated and then completing a time sheet.

This is followed by unpacking the relationship between hours and pay.

This establishes a context and a situation that many if not most students may find interesting and to connect to the math topic. This handout is intended as an introduction and not the formal unpacking of the term.

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