Tag Archives: multiplication

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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Hack for Multiplication (and division) Facts

A common method to learn multiplication facts is through skip counting. In turn, this is a means of learning division facts (see next paragraph). The challenge for many students is they struggle to learn the skip count routines or cannot engage brute force memorization effectively (e.g., have a working memory deficit).

https://www.homeschoolcreations.net/skip-counting-charts-from-2-through-15-printable/

The challenge with multiplication by skip counting is keeping track of two sets of numbers while memorizing the order of the skip counting. That is another example of the rubbing belly and patting head phenomena in math where one extra task demand undermines the process.

A hack I use to scaffold this process to reduce the task demand during the learning process is to provide rows from a multiplication chart (below) for the facts of focus (3s and 4s in this example). The same approach can be used for division facts, e.g., in the image below right I have the student choose the row of the divisor (3) and then skip count to until reaching the dividend (12). The idea is the student has less task demands while learning the process and seeing the number pattern. This allows for more repetitions or rehearsal.

For students more severely impacted by a disability or who simply struggle with the patting head and rubbing belly of skip counting, the appropriate times table row can be provided for each problem to allow the student to circle (below). This allows for a hands on approach with even less task demand. You could also laminate the rows to make them reusable in lieu of several consumable ones requiring more paper. I like the consumable as I use that for data collection.

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Handouts for Multiplication and Unit Rate Word Problems

Here are excerpts from two handouts I use to help students understand how to write multiplication and rate word problems as math expressions. The image, below at top, shows a problem from the first handout I present. The students draw a single group represented by the rate expression (for elementary school word problems the term rate is not used). The image at the bottom is the same problem with scaffolding to write the multiplication problem. I find that students working on rates and slope in middle school, high school, and even in college struggle with this topic. I use this approach as part of a review of prerequisite skills before getting into rate and slope.

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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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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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Multiplying and Carrying a Tens Digit

Carrying the TENS digit in a multiplication problem is a sticking point for many students. To address this, I use a task analysis approach to zero in on the step of identifying the product for the ONES as a prelude to carrying.

In the example below, 5 and 4 are in the ONES place and the product is 20. The task analysis steps involved:

  • compute the product
  • identify the digits in the product
  • identify the digit in the ONES
  • identify the digit in the TENS
  • Understand that the TENS digit must be carried to the TENS column

By creating a place holder for the product and scaffolding it to differentiate between the TENS and the ONES, the student can visualize the product. This reduces the demand placed on working memory. Once mastery with the place holder is demonstrated, it can be faded (and used as necessary as part of corrective feedback).

NOTE: I started this mini-lesson for a student with ADHD by having him warm up with problems without carrying. Also, extra line below the 60 and 20 are used for multiplying by 2 digit numbers (next in the sequence).

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Superhero Math

I was recruited to help a middle school student who is having a very rough time at this time in his life. It was shared with me that he likes Marvel superheroes and he is struggling with counting money and multiplication. Below are some ideas I presented for a test run and photos of the items I ordered for these suggested activities.

  • For multiplication

    • Put the heroes (or villains) in groups of 2 and have him count out 4 groups and compute. Use different groups and number per group. (IGNORE the numbers on the cards)

    • Get a group of 10 villain cards. Pretend heroes have to travel in groups of 2 and ask how many groups to get 10 heroes to fight the 10 villains. (IGNORE the numbers on the cards). Variations of this.

    • After gets the idea of groupings, focus on the number on the cards and show him two 5s and have him compute. Variations of this.

    • Play a game where he draws two cards and has to multiply the cards (start with very low numbers or maybe show him a 2 card and he has to pick another card to multiply by 2.

marvel playing cards

  • For Money

    • Tell him he earn money to buy these figures, one at a time – a monetary version of a token economy. Have him rank them by his favorite to least favorite and come up with a price for each with his favorite figures costing more. Start with the least favorite and make the price such that with a little practice he could count out the coins to pay for it. Maybe 17 cents with dimes and pennies. He has to count out the money correctly and independently to actually buy the item.

Avengers Figurines

  • Other options

    • If he needs work with addition you can play WAR in which 3 cards are played and each person adds to find the total. For subtraction do the same with 2 cards.

    • You can play subtraction in which one person has superheroes and the other has villains. In order for a villain to win a villain card has to be higher than a hero card by 3 or more.

You can write an 11, 12 and 13 on the J, Q, K cards respectively. All the games can be presented though Direct Instruction – I do, we do, you do. The You do can be used as daily progress monitoring. If he needs prompting this can be recorded. This can be used for your progress reports. Attached is a data sheet I use for activities.

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Modified Multiplication Table – Area Model Included

Modified Multiplication Table Using Area Model

The idea is that the student will have to count squares. By doing so the student is more engaged (or less passive) in determining the product and has to engage the visual representation.

Here is a link to the document.

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