{"id":3331,"date":"2026-04-29T10:02:06","date_gmt":"2026-04-29T17:02:06","guid":{"rendered":"https:\/\/theothermath.com\/?p=3331"},"modified":"2026-04-29T11:56:25","modified_gmt":"2026-04-29T18:56:25","slug":"tapes-diagrams-versus-algebra","status":"publish","type":"post","link":"https:\/\/theothermath.com\/index.php\/2026\/04\/29\/tapes-diagrams-versus-algebra\/","title":{"rendered":"Tapes diagrams versus Algebra"},"content":{"rendered":"

Take a moment to solve this problem using any method you wish\u2026<\/span><\/p>\n

 <\/p>\n

A farm has a total of 480 llamas and alpacas.<\/span><\/strong><\/em><\/p>\n

\\(\\frac{4}{5}\\) of the number of llamas at the farm are equal to \\(\\frac{4}{7}\\) of the number of alpacas.<\/span><\/strong><\/em><\/p>\n

How many more alpacas than llamas are there?<\/span><\/strong><\/em><\/p>\n

 <\/p>\n

I’ll wait.<\/span><\/p>\n

.<\/span><\/p>\n

.<\/span><\/p>\n

.<\/span><\/p>\n

.<\/span><\/p>\n

.<\/span><\/p>\n

Can I move on?<\/span><\/p>\n

It is likely you tried some sort of algebra.<\/span><\/p>\n

If so, it might have looked a little like this\u2026<\/span><\/p>\n

\"\"<\/a><\/p>\n

Gross.<\/span><\/p>\n

Using tape diagrams, it would look like this\u2026<\/span><\/p>\n

\"\"\"\"<\/a><\/p>\n

Take a moment to look deeply at the tape diagram above.<\/span><\/p>\n

What do you notice? What do you wonder?<\/span><\/p>\n

.<\/span><\/p>\n

.<\/span><\/p>\n

.<\/span><\/p>\n

Using tape diagrams, this problem requires no mathematics beyond an understanding of 4th grade division and 4th grade fractions. The SAME problem can be solved using fancy algebra or simple arithmetic.<\/span><\/p>\n

Tape diagrams are visual representations that allow students to see the underlying mathematical truths of the abstract mathematics we teach. Let’s look at the “simple” fraction concept of converting an improper fraction into a mixed number, for example.<\/span><\/p>\n

\"\"\"\"<\/a><\/p>\n

A traditional approach to this concept would have the teacher explaining that the fraction 8\/3 also means to divide 8 by 3. Why? Don’t ask\u2026just do it. Then the teacher goes through some sort of song and dance explaining why the top number goes into the “house” and the bottom number goes outside.<\/span><\/p>\n

The cowboy rides the horse. Of course, the cowboy gets to go into the house\u2026not the horse.<\/span><\/em><\/p>\n

The top dog gets the house.<\/span><\/em><\/p>\n

What other lame mnemonics have you heard?<\/span><\/p>\n

Then the teacher continues the fraction conversion by relying on the 4th grader’s shaky understanding of long division. What do we do with the remainder? We just swoop it up to the top and bring over the three as the denominator. Why? Don’t ask\u2026just do it.<\/span><\/p>\n

In contrast, we use a tape diagram to represent 8 thirds. We use brackets or circles to show that 3 thirds equals 1 whole. We have enough thirds left over to show another 1 whole. We have 2 thirds left over. If at any time a students asks WHY, the teacher can produce a more meaningful response than “Don’t ask\u2026just do it.”<\/span><\/p>\n

Why do we draw 8 boxes? Because we have 8 thirds and each unit represents a third.<\/span><\/em><\/p>\n

Why do we put a bracket around 3 thirds? Because 3 thirds equals 1 whole.<\/span><\/em><\/p>\n

So what is the lesson of this short blog post? Tape diagrams are visual representations that make math make sense. Tape diagrams increase access to rigorous mathematics for ALL students.<\/span><\/strong><\/p>\n

 <\/p>\n

But here is the dilemma:<\/span><\/h3>\n

Virtually no teachers in the classroom today learned mathematics through regular use of the tape diagram model. And yet, we expect teachers to develop an understanding of tape diagrams on their own, so they can then integrate them into their daily lessons for their students.<\/span><\/p>\n

This is a remarkably unfair and unreasonable expectation to place on teachers. Instead, site leaders and district leaders need to acknowledge the elephant in the room that for many, many teachers we are expecting them to teach mathematics that they themselves do not yet understand.<\/span><\/p>\n

How do we fix this? Provide mathematics content coaches for all teachers.<\/span><\/h3>\n

There is lots of research connecting a teacher’s math content knowledge, their mathematical knowledge for teaching, and their strong use of student discourse and math vocabulary to an increase in student mathematical achievement.<\/span><\/p>\n

Math content knowledge:<\/span><\/strong><\/p>\n

This refers to the ability to get the answer in any fashion. Can the teacher convert 8\/3 into a mixed number? Yes? Check.<\/span><\/p>\n

While most teachers know most of the math they teach, it is a common experience for our mathematics instructional coaches to be pulled aside by a teacher who sheepishly asks us to explain how to do some sort of mathematics they about to teach in the coming days.<\/span><\/p>\n

What to do:<\/em> Leaders need to create time in the teachers’ busy schedules for them to do math together. Perhaps a grade level team can print an FIAB from the CAASPP website and complete it in its entirety. Teachers can discuss the answers and the variety of ways in which each problem can be solved. Math content coaches can provide additional support\u2026especially with tape diagrams\u2026to help the grade level grow as mathematicians.<\/span><\/p>\n

 <\/p>\n

Math knowledge for teaching:<\/span><\/strong><\/p>\n

I have written extensively\u00a0about it here<\/a><\/span>, but MKT is the teacher\u2019s ability to<\/span><\/p>\n