From Arithmetic to Algebra: Variables, Word Problems, Fractions and the Rules

CONTENTS OF CURRICULUM UNIT 17.05.04

  1. Unit Guide
  1. Introduction
  2. Rationale
  3. Background
  4. Content and Objectives
  5. Teaching Strategies
  6. Classroom Activities
  7. Example Problems
  8. Resources
  9. Appendix – State Standards
  10. Biography for Teachers
  11. Endnotes

Math World Problems and the Diné Hooghan

Jolene Rose Smith

Published September 2017

Tools for this Unit:

Background

My goal in writing this curriculum unit is to embed our culture and language in math word problems, and to strengthen our students’ confidence and determination to solve math word problems. During the course of the “From Arithmetic to Algebra” seminar, I have created a math curriculum unit with word problems that will be relevant to our Diné children. Most of our students know the basic use and purpose of the two hooghans, and how they are used in our culture. However, their knowledge of these cultural features is not very detailed or specific. Our students need to know the culture and language better and also need to develop solid math skills to let them continue learning math. Therefore, this unit will use problems about hooghans to help students learn how to use effective math strategies while learning to read and solve math word problems and eventually become problem solvers.

The word problems will be designed to both deepen my students’ ability to do mathematics and to strengthen their knowledge of Diné culture and language. The problems will require students to learn a substantial amount of new math vocabulary. A math word wall will be created to assist special education and ELL students in learning the new vocabulary. In addition to vocabulary, students need to know and learn how to comprehend math word problems and solve problems.

There are numerous ways and methods, publishers, administrators and teachers teach students how to solve math word problems. The method Roger Howe explained and demonstrated during his seminars when solving common math word problem, is based on the taxonomies of one-step addition/subtraction problems, and one step multiplication/division problems in Tables 1 and on 2 of The Appendix to the Common Core State Standards in Mathematics (CCSSM). Refer to corestandards.org/Math/ Content/mathematics-glossary/Table-1/1 The taxonomy is not intended to be taught to students, but is for teachers to use as a frame for writing various math problems.

Below is a table showing example math word problem models for common addition and subtraction problems. Then, the next table is the common multiplication and division examples. I have written problems connecting to the hooghan’s activities like construction, ceremony preparation, and events surrounding the hooghan. The Diné words in the math problems are in italic in the front of the English word. I put the Diné words first because our children need to read and say their language, even so in mathematics.  In a sense, “they will become trilingual,” because students will need to read and know the math problems in Diné and English and also to know math vocabulary word. This will also be true for Dine teachers and for non-Native teachers. They will also need background knowledge to understand the specific events within the word problems. For my students, this will be supplied in a separate social studies unit.

Furthermore, these problems will vary throughout the school year. I know as a teacher on the Dine Nation, our children need constant practice and review of how to read and solve math problems throughout the school year. The math word problems will become more advanced and difficult as students’ progress into the school year. This is when the strategies and classroom activities come into play.

TYPE / UNKNOWN

Result Unknown

Change Unknown

Start Unknown

add to (also known as change plus)

·  Four (chidí) trucks were hauling logs for building a hooghan.

·  Then 12 more (chidí) trucks joined them.

·  How many (chidí) trucks in all were hauling logs for the hooghan?

·  Four (chidí) trucks were hauling logs for building a hooghan.

·  Some more (chidí) trucks joined them to haul logs.

·  Then were 16 (chidí) trucks.  

·  How many (chidí) trucks joined the first four?

·  Some (chidí) trucks were hauling logs for a hooghan.

·  Twelve more (chidí) trucks joined them to haul logs.

·  Then there were 16 (chidí) trucks hauling logs.

·  How many (chidí) trucks were hauling logs at first?

TYPE / UNKNOWN

Result Unknown

Change Unknown

Start Unknown

take from (also known as change minus)

·  50 buckets of sand clay were brought to make the base of the hooghan.  

·  47 were used.

·  How many buckets of sand clay are left over?

·  50 buckets of sand clay were brought to make the base of the hooghan. 

·  3 were left over.

·  How many buckets of sand clay were used?

·  Some buckets of sand clay were brought to make the base of the hooghan. 

·  47 were used.

·  Now 3 are left.

·  How many buckets were brought for making the base?

Total Unknown

Addend Unknown

Both Addends Unknown

Put together/take apart (also known as part-part-whole)

·  7 men and 9 women are in the hooghan.

·  How many adults are in the hooghan?

·  16 adults enter the hooghan before the songs begin.

·  7 are men and the rest are women.

·  How many are women?

Not commonly used for word problems

Difference Unknown

Bigger Unknown

Smaller Unknown

Compare (plus fewer/minus greater)

How many more?

How many fewer?

·  The Tótsoníí family made 25 frybread.

·  The Ashįįhí family made13 frybread.

·  How many more frybread did the Tótsoníí family make?

·  The Ashįįhí family made 13 frybread.

·  The Tótsoníí family made 25 frybread.

·   How many fewer frybread did the Ashįįhí family make than the Tótsoníí family?

·  The Tótsoníí family made 12 more fry bread than the Ashįįhí family.

·  The Ashįįhí family made 13.

·  How many fry bread did the Tótsoníí family make?

·  The Ashįįhí family made 12 fewer frybread than the Tótsoníí family.

·  The Tótsoníí family made 25.

·  How many frybread did the Ashįįhí family make?

·  The Tótsoníí family made 25 fry bread.

·  That was 12 more fry bread than the Ashįįhí family made.

·  How many fry bread did the Ashįįhí family make?

·  The Ashįįhí family made 12 fewer frybread than the Tótsoníí family.

·  The Tótsoníí family made 25.

·  How many frybread did the Ashįįhí family make?

Unknown Product

Group Size Unknown (“How many in each group?” Division)

Number of Groups Unknown (“How many groups?” Division)

equal groups

measurement

·  There are 12 (ts’aa’) baskets with four popcorn balls in each (ts’aa’) basket.

·  How many popcorn balls in all?  

·  You need ten lengths of logs; each log is 8 feet long.

·  What is the total length of all these logs?

·  If 48 popcorn balls are shared equally in twelve (ts’aa’) baskets, then how much popcorn will be in each basket?

·  You have 80 feet of logs and you will cut them into 10 parts.

·  How long will each part be?

·  If 48 popcorn balls are to be packed 4 into a (ts’aa’) basket, then how many (ts’aa’) baskets are needed?

·  You have 80 feet of trunks to cut into logs, and you will cut the logs into 8 foot lengths.

·  How many lengths of logs will you get?

arrays

area

·  There are 6 rows of (łįį’) horses with 8 in each row.

·  How many (łįį’) horses are there?

·  What is the area of a 20 foot by 12 feet rectangular (chaa’oh) shadehouse?

·  If 48 (łįį’) horses are held in 6 equal rows, how many (łįį’) horses will be in each row?

·  The (chaa’oh) shadehouse has the shape of a rectangle, and has an area of 240 feet.

·  If one side is 12 feet long, how long is a side next to it?

·  If 48 (łįį’) horses are arranged into equal rows of 8.

·  How many rows of (łįį’) horses will there be?

·  The (chaa’oh) shadehouse has the shape of a rectangle, and has an area of 240 feet.

·  If one side is 20 feet long, how long it a side next to it?

compare

measurement

·  A flee market retailer charges $6 for (tsint’eesh) ash in a tin can.

·  (Ch’ilawééh) tea in a tin can costs 5 times as much as the (tsint’eesh) ash.

·  How much does a can of the (ch’ilawééh) tea cost?

·  During her Kinaalda, a puberty ceremony, Jolene is required to run east every morning. Each day, Jolene needs to run a longer distance from the hooghan.

·  On the first day, she ran 2 miles.

·  One the last day of her (kinaalda) ceremony she ran 4 times longer.

·  How many miles did she run on the last day?

·  (Ch’ilawééh) tea in a tin can costs $30 and it is 5 times as much as a can of the (tsint’eesh) ash.

·  How much does the (tsint’eesh) ash cost?

·  For each day of her Kinaalda, Jolene needs to run a longer distance from the hooghan.

·  One the last day of her (kinaalda) ceremony she ran 8 miles. That is 4 times than she ran on the first day.

·  How many miles did she run the first day?

·  (Ch’ilawééh) tea in a tin can costs $30 and (tsint’eesh) ash cost $6.

·  How many times as much does a can of (ch’ilawééh) tea cost than a can of (tsint’eesh) ash?

·  For each day of her Kinaalda, Jolene needs to run a longer distance from the hooghan.

·  On the last day of her (kinaalda) ceremony she ran 8 miles.

·  On the first day, she ran 2 miles.

·  How many times longer did she run on the last day?

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