Category Archives: Uncategorized

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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1 and 2 Step Word Problems – Important First Dominoes PART 2

In PART 1 of Word Problems I went over my approach to teaching 1 and 2 step word problems involving addition and subtraction. In this post I cover multiplication, which is exponentially harder (pun intended – lay people see photo below).

As seen in PART 1, I color code the different parts of the problem: blue for the multiplication or division (rate), yellow for stand alone numbers, green for addition or subtraction, and orange for the unknown quantity.

To identify the multiplication or division parts, I focus on situations that involve groups of items, e.g. two cupcakes in every package or $5 in every lawn (for every lawn) as opposed to key words (as explained in PART 1). The students focus not on a single term such as “each” but on the situation. I use the groups of items as the structure for the equation, e.g., 5 x # lawns. The additional step in a two-step word problem can be connected within this structure, e.g., 12-9 in the top photo below.

Before working on the actual word problem handouts, I present the problems with a Google Jamboard to help flesh out the concept of multiplication as groups of items. Here is a link to the scaffolded handout.

After the Jamboard, I will use a scaffolded handout to help them unpack the structure. This is a scaffolded handout I use for 1 step multiplication word problems and the additional step, and show the additional step off to the side. This would be followed by problems on a typical worksheet as shown in excerpts above.

The problem below is a division problem. For division problems, I like to continue the focus on groups of items, in this case groups of wings. The difference is the number of items in a group is not given. This is a prompt for students to divide (which is how they will compute unit rate in the future). The division provides the main structure of the problem and the additional step can be attached, as is the case with 34 + 11 shown below. This way division is built on their prior knowledge of how to do word problems and they learn one additional step.

1 and 2 Step Word Problems – Important First Dominoes PART 1

Several special education teachers responded in a poll indicating that the most difficult math topic to teach in elementary school is solving multi-step word problems. This happens to be a topic that is massively important and the first of several dominoes that will fall all the way through high school and beyond. One and two-step word problems are cited in the Common Core domain of Operations and Algebraic Thinking (images below) and the CCSS Coherence Map shows how these two standards lead to future algebraic thinking and algebra topics.

There are two aspects of word problems in elementary school that are incredibly important building blocks in terms of math education. First, these problems establish math as a language used to represent real life situations. Second, the multiplication word problems develop the student understanding of rates, which is a major topic in middle school math and in algebra of all levels.

Before I get into what I call a conceptual approach to word problems, which I recommend, I will share that I am not in favor of the key word approach (image below). The major flaw, as I see it, involves how the brain stores or memorizes information. The key word approach is based on rote memorization. For many of the students with special needs, this is exactly what they do NOT need, more taxes on their working memory.

Here is the approach I use, with a focus on addition and subtraction first (followed by a forthcoming PART 2 blog post on multiplication and division). The handout used is from Math-aids.com.

  • I train students to highlight the quantities cited, along with any verbs. In the top photo is a legend for the elements of the word problem I highlight.
  • The yellow is used for quantities given in isolation. For example, Jason found 7 seashells but this was not presented after an preceding number.
  • In contrast, Fred found 6 seashells, which was in addition to what Jason found. Hence, Fred added to the number found and the green highlighting indicates this. Also note that the “+” is highlighted to indicate the adding on context.
  • Orange is used to indicate the quantity that is unknown. This helps focus their attention on the number they are looking for and is an ancestor to the eventually use of a variable.
  • The blue will be addressed in the PART 2 blog post. (Note: I do not use the term rate but wrote it for emphasis for the blog posts.)

The two-step addition and subtraction problems follow the same structure and involve minimal additional processing.

Additional notes about the process I follow.

I use a chunking approach in which I present the students several problems and have them practice 1 step at a time.

  1. highlight just the unknowns (orange)
  2. highlight the given values (yellow)
  3. highlight the additional values (green)
  4. then have new problems where they highlight all 3
  5. then I would have them write the equation for the the previously highlighted problems
  6. finally, they would attempt all the steps on a 3rd set of problems

There are additional types of problems such as Billy and 5 more tokens than Joey. If Joey has 8 topics, how many does Billy have? I would address these after the students show fluency with the process and the concept of using an equation to model a word problem. They do not follow the same type of structure I present above.

Our students may need help developing a conceptual understanding of addition and subtraction as well as the concepts underlying word problems. In my work with students I often find this to be a major obstacle in student progress with word problems. Hammering out conceptual understanding is likely to be a highly effective investment with a long range effect. It is not as easy to implement as the keyword strategy but we get what we pay for.

IEP and the General Ed Curriculum – Keep Your Eye on the Ball

In the movie Caddy Shack, there is a scene in which a putt to win a contest is resting on the lip of the hole. Meanwhile, Bill Murray is the groundskeeper who is attempting to rid the course of a gopher. To do so, he sets off explosives in various gopher holes. As everyone is looking around at the explosions (photo on the left), the contest referee ignores the commotion and keeps his on the ball (photo on the right).

For many of the students I have helped, the ball is the general curriculum. (For many students, like my son, the general curriculum is not appropriate. For readers in this situation, I recommended this post as the ball becomes the math necessary for non-academic setting.) Connecticut’s State Education Resource Center (SERC) has a rubric to evaluate IEPs (see excerpts below). In the rubric is an explicit and extensive focus on the general ed curriculum.

Here are a couple points about the IEP and the general ed curriculum.

In the first image above, highlighted on the left, is an indication that focusing on standardized testing in isolation to determine current levels of performance is problematic. A key element in my assessment and support for students is use of curriculum based assessment that is aligned with the Common Core. This allows me to gauge student achievement and ability on the actual curriculum and on the various standards at different grade levels as opposed to assigning a single grade level for a student’s math ability.

When evaluating IEP objectives for math, keep in mind the indicator shown in the bottom photo above (highlighted), “IEP goals and objectives are driven by the age-appropriate grade-level general education curriculum.” Given this, the focus of IEP objectives should not be determined by gaps or weaknesses in isolation but in respect to the general ed curriculum through an assessment similar to the one cited above.

I take this a step further and write IEP objectives to align with the upcoming math curriculum to focus the IEP on the entire course vs a handful of math topics. In turn, this would focus the programming and services associated with the objectives on the general curriculum, i.e., we would be keeping our eye on the ball!

Prerequisites for Algebra – An Illusion

The orange circle on the right looks bigger, but in fact both are the same size. The deception is based on the additional sensory input.

Similarly, the prerequisites for taking algebra are often considered to be basic skills. This is largely an illusion. I routinely encounter students who are referred to me for help as they have been caught in an infinite loop of working on basic math such as number operations (adding, subtracting, multiplication, and division) before moving on to algebra, with limited progress. I am not suggesting basic math skills are not important but am focused on the context of prerequisites needed to engage algebra. Many of the students I have helped who were in this situation. We worked to quickly move them into algebra where they were successful.

One student worked on half a year of 4th grade math during her 7th grade year. During the spring of that 7th grade year and the subsequent summer, I worked with her on algebraic thinking and algebra topics. She successfully completed algebra 1 during her 8th grade year.

The Common Core of State Standards (CCSS) for Math maps out the prerequisites as seen in the CCSS math domains (below). Throughout elementary school, Operations and Algebraic Thinking topics are covered. The Algebraic Thinking standards establish for the students a foundation for algebra taught in middle and high school. A focus of algebra is to model or represent patterns or relationships in real life situations using equations, tables, and graphs. These include quantities modeled by variables.

Below is a break down of this foundation in elementary school. If you are supporting a student in middle or high school who is taking algebra and has major gaps in his or her math education, look to these standards for the essential prerequisite skills.

First Grade: represent situations in word problems by adding or subtracting, and introduce equations (and equal sign).

Second Grade: Represent, solve word problems, introduce multiplication as groups of objects.

Third Grade: represent, solve word problems, explain patterns

Fourth Grade: Solve word problems, generate and analyze patterns

Fifth Grade: Write expressions (equations are 2 expressions with an = in between), analyze patterns and relationships

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Addressing Multiplication as a Gap

There is a delineated sequence for teaching multiplication over the years, including repeated addition, set modeling, arrays, single digit etc (below). It exists to build conceptual understanding of the multiplication facts that are at some point memorized by many students. When I work with students who are a more than a year behind in the sequence for multiplication, I find that programming for these students to help them catch up sometimes involves shortcuts such as a reliance on rehearsal or resorting to use of the multiplication table in isolation. I am not against use of the table or narrowing the focus, but am promoting a more comprehensive approach.

Here is a sequence, on a Jamboard, I used for a recent student who was struggling for a long time with multiplication (explanation of each step shown below images). The student was interested in Minecraft so I used Minecraft items such as stone bricks and a wagon. I would spend as much time on each step, as necessary.

  • Count out the total number of stone bricks. This allows an assessment of how the student counts: by 3s or individually. If individually, I would prompt the student to count by 3s.
  • Add 3 + 3
  • Show a short video on the wagon (this adds interest and gives the students a bit of a break)
  • Present the bricks in 2 groups of 3, in context of 2 wagons with 3 bricks each.
  • Present the same problem as a multiplication problem but with the image for one of the factors in lieu of two numbers.
  • Use the multiplication table to skip count.
  • Present additional multiplication problems for independent attempts. The student completed both problems independently, without the table. For him this was a major success.

The follow up to this would be to assess his ability to do higher groups of 3s and groups of other numbers. For some students, I work on mastery of individual numbers before moving on. This builds confidence and allows for fluency in the process of skip counting out to the appropriate number. NOTE: I don’t worry about rote memorization of the facts but of fluency in the process of skip counting out the answers.

For students who are older, I sometimes recommend that the student be presented problems with visuals but then use a calculator to compute. This can develop conceptual understanding and also address the working memory and other related issues that undermine learning math facts.

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Q&A Regarding Math and Online Learning

I will be fielding questions about math and online learning in real time. As a follow up, I will respond to questions through this blog post. If you did not catch the Instagram session and have questions, you can post them here through a comment. I will post replies on this post.

Below is a list of links to resources, e.g. online handouts, activities, which align with the discussion.

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Online Academic Practice – Especially for School Closures!

I recommend 3 sites for online academic practice in multiple content areas.

IXL.com can be purchased for an individual or a family with access to multiple content areas. You can try 10 problems for free. It allows students to work on specific skills. Math grades covered range from kindergarten to precalculus!

Here is a link to handouts to help you access Khan Academy which covers a wide range of math grade levels and other content areas, and the ACCUPLACER practice app to prepare for college placement tests.

 

Base Ten Using Popsicle Sticks

I use the following method as a entry point for double digit numbers.

The photo below shows 2 packs of Popsicle sticks counted as 10 each, followed by single sticks counted as 1 each. The student counts on from 20, with the use of the scaffolded handout (photo at bottom). The handout focuses only on counting on from 20 and shows a photo of 2 of the bundles of sticks. Similar handouts involve counting on from 10 or from 30 etc.

By engaging in the actual counting, the student learns the 10s by doing. This would be followed by counting on from each 10 without the handout.

The use of Popsicle sticks is useful for 2 reasons. First, a bundle of items like shown below is more concrete than the rods for Base 10 blocks. Second, pulling packs of sticks apart of bundling 10 sticks together is an act that is concrete for students and ties into their prior knowledge regarding the grouping of objects (e,g. pack of gum).

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