Tag Archives: word problems

CCSS MATH Coherence Map – 3rd Grade Groups of Objects to HS Slope

A hidden treasure is the Common Core of State Standards Math Coherence Map. It is an interactive flow chart that shows connections between the various standards.

  • If you are teaching math, you can see the connections between what you are teaching, what was taught previously, and how you are preparing students for their future math education.
  • If you are a special education teacher, you can see the progression of prerequisite skills. If you write IEP objectives for grade level standards, you can address the prerequisite standards and progress made through these prerequisites can show progress towards mastery of the IEP objectives.

In this post, I show the progression from 1st grade standard on the = sign and 2nd grade standard of repeated addition all the way to interpreting slope in high school math.

After clicking “Get Started” you will narrow your focus to the grade, the cluster, and then the math domain.

The flow chart shows connections between a selected standard and others, including prerequisites. In this case 8.F.B.4 – 8th grade content that is a prerequisite for the high school math work. Click on the 8.F.B.4 standard and it pops up (below right).

In turn, the 8th grade standard is connected to a 7th grade prerequisite regarding ratios and proportions.

Notice that the 7th grade standard, 7.RP.A.2, appears to be a dead end (bottom left). I picked up the path by clicking on Grade seen at the top left of the screenshot and made my back to that standard and the connections to prerequisites appeared. (Same happened in 3rd grade shown further down in this post.)

The path continues from ratio and proportions in 7th grade to unit rate in 6th grade, multiplication word problems and multiplication in elementary school.

I want to emphasize that students are working on unit rate and slope problems in ELEMENTARY SCHOOL! 3.OA.A.1 below addresses groups of objects model for multiplication and 4.OA.A.2 addresses word problems involving multiplication.

I was recently working with a student entering middle school on multiplication word problems. To unpack the word problems and the concept of multiplication in context, I had her draw (photo below) groups and groups of objects to help her identify the unit rate (although we don’t use that term yet). This work will impact her math education through the high school math and even into college (slope has been a common gap for the college students in the math courses I taught at various colleges and universities).

This approach I used with the student could be used for high school students, especially those with special needs.

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Multiplication Problems as Intro to Unit Rates and Slope (follow up)

I previously related elementary school word problems with math topics in secondary schools. The photo below shows a method to help elementary school students unpack the multiplication problem, to help middle school students identify the unit rate, and to help algebra students identify slope (you can focus on simple problems like this as an entry point to the linear function type problems).

In advance of this method, a review is conducted on the representation of multiplication using the groups of items model (below). By drawing a picture for the two parts of the problem that have a number, the students are guided to break the problem into parts and then to unpack the parts. The “5 boxes of candies” is represented by squares (or circles if you prefer) with no items inside. The “each box holds 6 candies” is represented by a single square with 6 items (dots) inside.

In turn, the drawing of the group of items leads to the multiplication statement, “6 candies x number of boxes.” I prompt students to include the items with the number as sometimes they will write this statement as “6 x number of candies”. I point out that 6 and candies go together. As seen in the previous blog post, the next step in this problem would be to replace “number of boxes” with the quantity given and then compute.

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Slippery Slope – 3rd Grade Multiplication Word Problems to Slope in Algebra

Top left is a scaffolding I use to help students learn to solve math problems using multiplication (3rd grade). The situations are typically rate problems (e.g., 5 pumpkins per plant or $2 per slice of pizza) although the term “rate” is not used yet. The same concept of rate plays out in high school with slope of a line, applied to real life situations (top right).

These types of problems start in 3rd grade (below, top left), play out in 6th grade (below, top right), into 8th grade (below, middle), and into high school algebra and statistics (below, bottom). I referenced this connection previously regarding word problems and dominoes. This highlights how crucial it is that strategically selected gaps in a student’s math education be addressed in context of long range planning.

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Evaluating the Effectiveness of Math Objectives

iep-math-objective-improve

The math objectives present in the photos on this post were written for former students of mine. These types of objectives are ineffective and ubiquitous. When I have sat in IEP meetings the majority of the time I am the only person who is capable of evaluating IEP math objectives. This post provides some guidance for others to evaluate these objectives.

In the photo above the objective has 3 major flaws.

  • “Using tools” is ambiguous. A 4 year old can use a calculator even if he does not know what he is doing.
  • “Problem solving skills” is a broad term that needs to be defined.
  • “Improve” can mean a student increases a success rate from 0% to 1%. That speaks for itself.

iep-math-objective-create-visual-aids

In the objective above there are 2 major problems.

  • “Multi-step word problems” is very broad. If a student shows she can solve a problem that requires addition and then subtraction but no multiplication or division is this mastery?
  • Often the accommodations are built into the objective and therefore the assessment. I have repeatedly had educators tell me this is what the student needs. That is valid if there is no intention for the student to do the work independently but often that point is overlooked.

iep-math-objective-demonstrate-understanding

The objective above is similar to the previous example. The examples in the objective include “solving” and “graphing.” Is the student supposed to demonstrate mastery in all the different types of algebra concepts? Or, if he can solve equations is the objective mastered?

How can caregivers evaluate these objectives? 

The language of an effective objective can be used, almost verbatim, as  problem. For example

  • Objective: Billy will add fractions with like denominators.
  • Problem: Add the following fractions (with like denominators).

Have the person writing the objectives provide an example problem that can be used to assess mastery of the objective. If the problem includes additional information or language beyond what is written in the objective then the objective is ineffective. For example:

  • In objective 1 above (the first one) the objective is to use tools to improve problem solving skills.
  • Below is a possible problem (from LearnZillion.com) that could be used. I would ask the author of the objective to explain what problem solving skills should be demonstrated and to explain what constitutes improvement. Neither of these terms is explicitly stated in this problem. It is very likely that valid responses to these questions is not possible and hence the objective needs to be revised.

story-problem-2-step

Solve the word problem above using a calculator.

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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.

word problems focus on concept first traditional approach

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